Technical Field

The present disclosure relates to self-calibration of an antenna array system.

Background

An Advanced Antenna System (AAS) includes multiple antenna elements in a matrix, or antenna array. Each antenna element is connected to a radio front-end of a transceiver. A phase tuner and gain stage is connected between the radio front-end and the AAS to enable phase and gain tuning for, e.g., analog beamforming. Specifically, phase and gain adjustments for the AAS can be made to direct a beam of the AAS in a desired direction. For example, phase and gain adjustments can be made to direct a transmission towards a desired receiver.

One issue that arises with an AAS and, in particular, with an analog beamforming transceiver including an AAS, is that there are variations in gain and phase between different antenna elements due to, e.g., manufacturing tolerances for the various components in the associated transmit or receive path, temperature gradients, etc. These variations result in undesired errors in analog beamforming. Therefore, there is a need for systems and methods for calibrating the gain and phase of the antenna elements in an AAS.

Conventional self-calibration techniques used by an analog beamforming transceiver to perform self-calibration of the gain and phase of each antenna element in the AAS are time consuming and need many measurements. Further, even after calibration, there is some residual error due to coupling between antenna elements. This residual error is particularly problematic for self- calibration because the coupling between antenna elements is asymmetrical (i.e., the coupling between one pair of antenna elements may be substantially different than the coupling between a different pair of antenna elements). For a digital beamforming AAS, there is direct access to individual antenna elements and, as such, the analog beamforming transceiver can measure or calculate this error for each individual antenna element and then use the measurement or calculated values to compensate for this error. In contrast, for an analog beamforming AAS, this error is difficult to distinguish as all antenna elements are merged with splitter/combiner networks. One way of compensating for this error in an analog beamforming AAS is to store a coupling matrix in a Look Up Table (LUT), but generating the coupling matrix requires a very large number of measurements. Additionally, overall coupling will vary as the antenna structure may change with temperature variation and power amplifier loading.

Thus, when it comes to self-calibration of an analog beamforming AAS utilizing coupling, there is a need for systems and methods for efficient and accurate calibration that compensates for errors due to asymmetrical coupling properties between antenna elements in the AAS. Summary

Systems and methods relating to self-calibration of an antenna array of a transceiver are disclosed. In some embodiments, a method of operation of a transceiver to perform self-calibration for transmit (Tx) antenna elements and receive (Rx) antenna elements in an antenna array comprises performing gain measurements (G _MTx . _Rx . ) and phase measurements for pairs of Tx and

Rx antenna elements in the antenna array. The gain measurement {G _MTx . _Rx . ) is a measured gain at an output of a receiver coupled to a j-th Rx antenna element during reception of a signal transmitted by a transmitter coupled to an i-th Tx antenna element. The method further comprises processing the gain

measurements (G _MTx . _Rx . ) and the phase measurements based on combinations of Tx and Rx antenna elements having symmetrical coupling properties to obtain gain and phase calibration values for the plurality of Tx antenna elements and the plurality of Rx antenna elements in the antenna array. The method further comprises applying the gain and phase calibration values at the transceiver. In this manner, self-calibration can be performed at the transceiver dynamically with only a small amount of hardware modification, if any, and with only a small amount of processing.

In some embodiments, processing the gain measurements {G _MTx . _Rx .) and the phase measurements comprises selecting a pair of Tx antenna

elements (Tx, and Tx _k where i≠ k), determining a first pair of Rx antenna elements (Rx _n and Rx _m ) having symmetrical coupling properties with respect to the pair of Tx antenna elements (Tx, and Tx _k ), and computing at least one relative gain value and at least one relative phase value for the Tx antenna element Tx, relative to the Tx antenna element Txk based on at least two of: the gain measurement {G _MTx . _RXn ) and the phase measurement {φ _Μ τχβχ _η ) f° ^r the pair of Tx and Rx antenna elements (Tx, and Rx _n ); the gain measurement {G _MTx . _RXm ) and the phase measurement f° ^r the pair of Tx and Rx antenna elements (Tx, and Rx _m ); the gain measurement (G _MTXkRXn ) and the phase measurement (0 _M T¾RX„) f° ^r the pair of Tx and Rx antenna elements (Tx _k and Rx _n ); and the gain measurement {G _MTXkRXm ) and the phase measurement (0 T¾Rx _m ) f° ^r the pair of Tx and Rx antenna elements (Tx _k and Rx _m )- The method further comprises repeating the steps of determining and computing for at least one second pair of Rx antenna elements (Rx _n and Rx _m ) having symmetrical coupling properties with respect to the pair of Tx antenna elements (Tx, and Tx _k ) and averaging the relative gain values and the relative phase values for the Tx antenna element Tx, relative to the Tx antenna element Tx _k across the first pair of Rx antenna elements and the at least one second pair of Rx antenna elements to thereby provide an average relative gain value and an average relative phase value for the Tx antenna element Tx, relative to the Tx antenna element Tx _k . The method further comprises repeating the steps of selecting, determining, computing, repeating, and averaging for at least one second pair of Tx antenna elements (Tx, and Tx _k ) and normalizing the average relative gain values and the average relative phase values relative to a defined reference Tx antenna element to thereby provide gain and phase calibration values for the plurality of Tx antenna elements. In some embodiments, a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _n is approximately the same as a mutual coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _n , a mutual coupling between the Tx antenna element Tx, and the Rx 5 antenna element Rx _m is approximately the same as a mutual coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _m , and the at least one relative gain value and the at least one relative phase value for the Tx antenna element Tx, relative to the Tx antenna element Tx _k comprises a relative gain value defined as:

r _Λ _ { MTXiRXn - G _M Tx _k Rx _n ) + (G _MTx . _RXm ^_ ¾T¾f!x _m )

I ^U &U _Tx . _TXk L) - and a relative phase value defined as:

ί _Λ _ ((pMTX _j RXn - 0 T¾Rx _n ) + ^MTX _j Rx _m ~ 0 T¾Rx _m )

^ΨΤ ^χ ιΓ ^χ ^ ) — 2 "

In some embodiments, a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _n is approximately the same as a 5 mutual coupling between the Tx antenna element Tx _k and the Rx antenna

element Rx _m , a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _m is approximately the same as a mutual coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _n , and the at least one relative gain value and the at least one relative phase value for the Tx 0 antenna element Tx, relative to the Tx antenna element Tx _k comprises a relative gain value defined as:

_Γ _ {G _MTx . _RXn - G _MTXkRXm ) + {G _MTx . _RXm - G _MTXkRXn )

< ^J TXiTx _k ^J — 2

and a relative phase value defined as:

5 In some embodiments, averaging the relative gain values and the relative phase values for the Tx antenna element Tx, relative to the Tx antenna element Tx _k across the first pair of Rx antenna elements and the at least one second pair of Rx antenna elements comprises performing a weighted average of the relative gain values and the relative phase values for the Tx antenna element Tx, relative to the Tx antenna element Txk across the first pair of Rx antenna elements and the at least one second pair of Rx antenna elements.

In some embodiments, processing the gain measurements {G _MTx . _Rx .) and the phase measurements comprises selecting a pair of Rx antenna elements (Rx _n and Rx _m , where n≠ m), determining a first pair of Tx antenna elements (Tx, and Tx _k ) having symmetrical coupling properties with respect to the pair of Rx antenna elements (Rx _n and Rx _m ); and computing at least one relative gain value and at least one relative phase value for the Rx antenna element Rx _n relative to the Rx antenna element Rx _m based on at least two of: the gain measurement (G _MTx . _RXn ) and the phase measurement (φ _Μ τ _Χί ιι _Χη ) for the pair of Tx and Rx antenna elements (Tx, and Rx _n ); the gain measurement {G _MTx . _RXm ) and the phase measurement {<p _M T _Xi R _Xm ) f° ^r the pair of Tx and Rx antenna elements (Tx, and Rx _m ); the gain measurement {G _MTXkRXn ) and the phase measurement f° ^r the pair of Tx and Rx antenna elements (Tx _k and Rx _n ); and the gain measurement {G _MTXkRXm ) and the phase measurement

(0 Tx _fe Rx _m ) f° ^r the pair of Tx and Rx antenna elements (Tx _k and Rx _m )- The method further comprises repeating the steps of determining and computing for at least one second pair of Tx antenna elements (Tx, and Tx _k ) having

symmetrical coupling properties with respect to the pair of Rx antenna elements (Rx _n and Rx _m ) and averaging the relative gain values and the relative phase values for the Rx antenna element Rx _n relative to the Rx antenna element Rx _m across the first pair of Tx antenna elements and the at least one second pair of Tx antenna elements to thereby provide an average relative gain value and an average relative phase value for the Rx antenna element Rx _n relative to the Rx antenna element Rx _m . The method further comprises repeating the steps of selecting, determining, computing, repeating, and averaging for at least one second pair of Rx antenna elements and normalizing the average relative gain values and the average relative phase values relative to a defined reference Rx antenna element to thereby provide gain and phase calibration values for the plurality of Rx antenna elements.

In some embodiments, a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _n is approximately the same as a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _m , a mutual coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _n is approximately the same as a mutual coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _m , and the at least one relative gain value and the at least one relative phase value for the Rx antenna element Rx _n relative to the Rx antenna element Rx _m comprises a relative gain defined as:

AG (3 — ^ ^MTx i ^RXn ~ ^MTXiRXm) + _^MTx _k Rx _n ~ ^MTx^Xm) and a relative phase value defined as:

, _Λ ((pMTXiRXn - (pMTXiRXm) + ((pMTXkRXn - (pMTXkRXm) q>Rx _n Rx _m {3) = 2 ■ In some embodiments, a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _n is approximately the same as a mutual coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _m , a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _m is approximately the same as a mutual coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _n , and the at least one relative gain value and the at least one relative phase value for the Rx antenna element Rx _n relative to the Rx antenna element Rx _m comprises a relative gain value defined as:

J^Q (ή _, _ (.^MTxiRx _n ~ ^MTXkRXm) + _^MTxiRx _m ~ ^MTx _k RXn) and a relative phase value defined as:

, _Λ (<t>MTXiRx _n - 0 T¾Rx _m ) + (<t>MTXiRx _m ~ 0 T¾Rx _n )

In some embodiments, averaging the relative gain values and the relative phase values for the Rx antenna element Rx _n relative to the Rx antenna element Rxm across the first pair of Tx antenna elements and the at least one second pair of Tx antenna elements comprises performing a weighted average of the relative gain values and the relative phase values for the Rx antenna element Rx _n relative to the Rx antenna element Rx _m across the first pair of Tx antenna elements and the at least one second pair of Tx antenna elements.

In some embodiments, performing the gain measurements {G _MTx . _Rx .) and the phase measurements for the pairs of Tx and Rx antenna elements in the antenna array comprises, for each pair of Tx and Rx antenna elements (Tx,, Rx _j ), transmitting a pseudo random IQ signal of a desired bandwidth while the transmit antenna element Tx, and the Rx antenna element Rx _j are active, cross -correlating the transmitted pseudo random IQ signal and a received IQ signal received via the Rx antenna element Rx _j , and computing the gain measurement (G _MTx . _Rx .) and the phase measurement for the pairs of

Tx and Rx antenna elements (Tx,, Rx _j ) based on results of the cross-correlating.

Embodiments of a transceiver are also disclosed. In some embodiments, a transceiver enabled to perform self-calibration for a plurality of Tx antenna elements and a plurality of Rx antenna elements in an antenna array comprises the antenna array comprising the plurality of Tx antenna elements and the plurality of Rx antenna elements, gain and phase adjustment circuitry, one or more transmitters and one or more receivers coupled to the plurality of Tx antenna elements and the plurality of Rx antenna elements, and a baseband processing system. The baseband processing system is adapted to perform gain measurements (G _MTx . _Rx .) and phase measurements for pairs of Tx and

Rx antenna elements in the antenna array. The gain measurement {G _MTx . _Rx .) is a measured gain at an output of a receiver coupled to a j-th Rx antenna element during reception of a signal transmitted by a transmitter coupled to an i-th Tx antenna element. The baseband processing system is further adapted to process the gain measurements {G _MTx . _Rx .) and the phase measurements

based on combinations of Tx and Rx antenna elements having

symmetrical coupling properties to obtain gain and phase calibration values for the plurality of Tx antenna elements and the plurality of Rx antenna elements in the antenna array and apply the gain and phase calibration values at the transceiver via the gain and phase adjustment circuitry.

In some embodiments, in order to process the gain measurements, (GMTxiRxj ) and the phase measurements the baseband processing system is further operable to select a pair of Tx antenna elements (Tx, and Tx _k , where i≠ k), determine a first pair of Rx antenna elements (Rx _n and Rx _m ) having symmetrical coupling properties with respect to the pair of Tx antenna elements (Tx, and Tx _k ), and compute at least one relative gain value and at least one relative phase value for the Tx antenna element Tx, relative to the Tx antenna element Tx _k based on at least two of: the gain measurement {G _MTx . _RXn ) and the phase measurement {φ _Μ τχιΐ{χ _η ) f° ^r the pair of Tx and Rx antenna elements (Tx, and Rx _n ); the gain measurement {G _MTx . _RXm ) and the phase measurement

(0 _MTx . _RXm ) for the pair of Tx and Rx antenna elements (Tx, and Rx _m ); the gain measurement (G _MTXkRXn ) and the phase measurement (0 T¾RX„) for the pair of Tx and Rx antenna elements (Tx _k and Rx _n ); and the gain measurement

(G _MTXkRXjn ) and the phase measurement (0 Tx _fe Rx _m ) for the pair of Tx and Rx antenna elements (Tx _k and Rx _m )- The baseband processing system is further adapted to repeat the steps of determining and computing for at least one second pair of Rx antenna elements (Rx _n and Rx _m ) having symmetrical coupling properties with respect to the pair of Tx antenna elements (Tx, and Tx _k ) and average the relative gain values and the relative phase values for the Tx antenna element Tx, relative to the Tx antenna element Tx _k across the first pair of Rx antenna elements and the at least one second pair of Rx antenna elements to thereby provide an average relative gain value and an average relative phase value for the Tx antenna element Tx, relative to the Tx antenna element Tx _k . The baseband processing system is further adapted to repeat the steps of selecting, determining, computing, repeating, and averaging for at least one second pair of Tx antenna elements and normalize the average relative gain values and the average relative phase values relative to a defined reference Tx antenna element to thereby provide gain and phase calibration values for the plurality of Tx antenna elements.

In some embodiments, a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _n is approximately the same as a mutual coupling between the Tx antenna element Txk and the Rx antenna element Rx _n , a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _m is approximately the same as a mutual coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _m , and the at least one relative gain value and the at least one relative phase value for the Tx antenna element Tx, relative to the Tx antenna element Tx _k comprises a relative gain value defined as:

and a relative phase value defined as:

In some embodiments, a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _n is approximately the same as a mutual coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _m , a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _m is approximately the same as a mutual coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _n , and the at least one relative gain value and the at least one relative phase value for the Tx antenna element Tx, relative to the Tx antenna element Tx _k comprises a relative gain value defined as:

and a relative phase value defined as:

, _Λ (<t>MTXiRx _n - 0 T¾Rx _m ) + (<t>MTXiRx _m ~ 0 T¾Rx _n )

In some embodiments, in order to average the relative gain values and the relative phase values for the Tx antenna element Tx, relative to the Tx antenna element Tx _k across the first pair of Rx antenna elements and the at least one second pair of Rx antenna elements, the baseband processing system is further operable to perform a weighted average of the relative gain values and the relative phase values for the Tx antenna element Tx, relative to the Tx antenna element Txk across the first pair of Rx antenna elements and the at least one second pair of Rx antenna elements.

In some embodiments, in order to process the gain measurements (GMTxiRxj ) and the phase measurements the baseband processing system is further operable to: select a pair of Rx antenna elements (Rx _n and Rx _m , where n≠ m), determine a first pair of Tx antenna elements (Tx, and Tx _k ) having symmetrical coupling properties with respect to the pair of Rx antenna elements, Rx _n and Rx _m , and compute at least one relative gain value and at least one relative phase value for the Rx antenna element Rx _n relative to the Rx antenna element Rx _m based on at least two of: the gain measurement {G _MTx . _RXn ) and the phase measurement (0 _M Tx _j Rx _n ) f° ^r the pair of Tx and Rx antenna elements (Tx, and Rx _n ); the gain measurement {G _MTx . _RXm ) and the phase measurement

(0 _MTx . _RXm ) for the pair of Tx and Rx antenna elements (Tx, and Rx _m ); the gain measurement (G _MTXkRXn ) and the phase measurement (0 T¾RX„) for the pair of Tx and Rx antenna elements (Tx _k and Rx _n ); and the gain measurement {G _M Tx _k Rx _m ) and the phase measurement (0M7¾Rx _m ) for the pair of Tx and Rx antenna elements (Tx _k and Rx _m )- The baseband processing system is further operable to repeat the steps of determining and computing for at least one second pair of Tx antenna elements (Tx, and Tx _k ) having symmetrical coupling properties with respect to the pair of Rx antenna elements (Rx _n and Rx _m ) and average the relative gain values and the relative phase values for the Rx antenna element Rx _n relative to the Rx antenna element Rx _m across the first pair of Tx antenna elements and the at least one second pair of Tx antenna elements to thereby provide an average relative gain value and an average relative phase value for the Rx antenna element Rx _n relative to the Rx antenna element Rx _m . The baseband processing system is further operable to repeat the steps of selecting, determining, computing, repeating, and averaging for at least one second pair of Rx antenna elements and normalize the average relative gain values and the average relative phase values relative to a defined reference Rx antenna element to thereby provide gain and phase calibration values for the plurality of Rx antenna elements.

In some embodiments, a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _n is approximately the same as a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _m , a mutual coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _n is approximately the same as a mutual coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _m , and the at least one relative gain value and the at least one relative phase value for the Rx antenna element Rx _n relative to the Rx antenna element Rx _m comprises a relative gain defined as:

and a relative phase value defined as:

, . _ {(pMTXiRXn ~ 0 TX _j Rx _m ) + (0 T¾Rx _n ^— 0 T¾Rx _m ) V J 2 "

In some embodiments, a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _n is approximately the same as a mutual coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _m , a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _m is approximately the same as a mutual coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _n , and the at least one relative gain value and the at least one relative phase value for the Rx antenna element Rx _n relative to the Rx antenna element Rx _m comprises a relative gain value defined as:

J^Q (ή _, _ (.^MTxiRx _n ~ ^MTxi^RXm) + _^MTxiRx _m ~ ^MTx _k RXn) and a relative phase value defined as:

In some embodiments, in order to average the relative gain values and the relative phase values for the Rx antenna element Rx _n relative to the Rx antenna element Rx _m across the first pair of Tx antenna elements and the at least one second pair of Tx antenna elements, the baseband processing system is further operable to perform a weighted average of the relative gain values and the relative phase values for the Rx antenna element Rx _n relative to the Rx antenna element Rx _m across the first pair of Tx antenna elements and the at least one second pair of Tx antenna elements.

In some embodiments, in order to perform the gain measurements,

(G _MTx . _Rx .) and the phase measurements for the pairs of Tx and Rx antenna elements in the antenna array, the baseband processing system is further operable to, for each pair of Tx and Rx antenna elements Tx,, Rx _j :

transmit a pseudo random IQ signal of a desired bandwidth while the Tx antenna element Tx, and the Rx antenna element Rx _j are active; cross-correlate the transmitted pseudo random IQ signal and a received IQ signal received via the Rx antenna element Rx,; and compute the gain measurement {G _MTx . _Rx .) and the phase measurement f° ^r the pairs of Tx and Rx antenna elements Tx,,

Rx _j based on results of the cross-correlating.

Embodiments related to remote post-processing of measurements made at a transceiver in order to provide self-calibration are also disclosed. In some embodiments, a method of operation of a processing system to provide self- calibration for a plurality of Tx antenna elements and a plurality of Rx antenna elements in an antenna array of a remote transceiver comprises obtaining, from the transceiver, gain measurements {G _MTx . _Rx .) and phase measurements

( pMTxiRxj ) f° ^r P ^a i ^rs °f Tx and Rx antenna elements in the antenna array of the transceiver. Gain measurement G _MTx . _Rx . is a measured gain at an output of a receiver coupled to a j-th Rx antenna element during reception of a signal transmitted by a transmitter coupled to an i-th Tx antenna element. The method further comprises processing the gain measurements (G _MTx . _Rx ) and the phase measurements based on combinations of Tx and Rx antenna elements

having symmetrical coupling properties to obtain gain and phase calibration values for the plurality of Tx antenna elements and the plurality of Rx antenna elements in the antenna array. The method further comprises providing the gain and phase calibration values to the transceiver.

In some embodiments, processing the gain measurements {G _MTx . _Rx .) and the phase measurements comprises selecting a pair of Tx antenna elements (Tx, and Tx _k , where i≠ k), determining a first pair of Rx antenna elements (Rx _n and Rx _m ) having symmetrical coupling properties with respect to the pair of Tx antenna elements (Tx, and Tx _k ), and computing at least one relative gain value and at least one relative phase value for the Tx antenna element Tx, relative to the Tx antenna element Tx _k based on at least two of: the gain measurement (G _MTx . _RXn ) and the phase measurement (φ _Μ τ _Χί ιι _Χη ) for the pair of Tx and Rx antenna elements Tx, and Rx _n ; the gain measurement {G _MTx . _RXm ) and the phase measurement f° ^r the pair of Tx and Rx antenna elements Tx, and Rx _m ; the gain measurement {G _MTXkRXn ) and the phase measurement (0 _M T¾RX„) f° ^r the pair of Tx and Rx antenna elements Tx _k and Rx _n ; and the gain measurement {G _MTXkRXm ) and the phase measurement

(0 T¾Rx _m ) f° ^r the pair of Tx and Rx antenna elements Tx _k and Rx _m . Processing the gain measurements (G _MTx . _Rx .) and the phase measurements

further comprises repeating the steps of determining and computing for at least one second pair of Rx antenna elements (Rx _n and Rx _m ) having symmetrical coupling properties with respect to the pair of Tx antenna elements (Tx, and Tx _k ). Processing the gain measurements {G _MTx . _Rx .) and the phase measurements further comprises averaging the relative gain values and the relative

phase values for the Tx antenna element Tx, relative to the Tx antenna element Tx _k across the first pair of Rx antenna elements and the at least one second pair of Rx antenna elements to thereby provide an average relative gain value and an average relative phase value for the Tx antenna element Tx, relative to the Tx antenna element Tx _k ; repeating the steps of selecting, determining, computing, repeating, and averaging for at least one second pair of Tx antenna elements; and normalizing the average relative gain values and the average relative phase values relative to a defined reference Tx antenna element to thereby provide gain and phase calibration values for the plurality of Tx antenna elements.

In some embodiments, a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _n is approximately the same as a mutual coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _n , a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _m is approximately the same as a mutual coupling between the Tx antenna element Txk and the Rx antenna element Rx _m , and the at least one relative gain value and the at least one relative phase value for the Tx antenna element Tx, relative to the Tx antenna element Tx _k comprises a relative gain value defined as:

and a relative phase value defined as:

In some embodiments, a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _n is approximately the same as a mutual coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _m , a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _m is approximately the same as a mutual coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _n , and the at least one relative gain value and the at least one relative phase value for the Tx antenna element Tx, relative to the Tx antenna element Tx _k comprises a relative gain value defined as:

Λ Γ _ ( MTX _j RXn ^~ G _M Tx _k RXm) + (G _MTx . _RXm - G _MTXkRXn )

^TXiTXk K ) — 2

and a relative phase value defined as:

In some embodiments, averaging the relative gain values and the relative phase values for the Tx antenna element Tx, relative to the Tx antenna element Tx _k across the first pair of Rx antenna elements and the at least one second pair of Rx antenna elements comprises performing a weighted average of the relative gain values and the relative phase values for the Tx antenna element Tx, relative to the Tx antenna element Tx _k across the first pair of Rx antenna elements and the at least one second pair of Rx antenna elements.

In some embodiments, processing the gain measurements {G _MTx . _Rx .) and the phase measurements comprises selecting a pair of Rx antenna elements (Rx _n and Rx _m , where n≠ m), determining a first pair of Tx antenna elements (Tx, and Tx _k ) having symmetrical coupling properties with respect to the pair of Rx antenna elements (Rx _n and Rx _m ), and computing at least one relative gain value and at least one relative phase value for the Rx antenna element Rx _n relative to the Rx antenna element Rx _m based on at least two of: the gain measurement (G _MTx . _RXn ) and the phase measurement (φ _Μ τχβχ _η ) for the pair of Tx and Rx antenna elements Tx, and Rx _n ; the gain measurement (G _MTx . _RXm ) and the phase measurement for the pair of Tx and Rx antenna elements Tx, and Rx _m ; the gain measurement {G _MTXhRXn ) and the phase measurement (0 T¾Rx _n ) for the pair of Tx and Rx antenna elements Txk and Rx _n ; and the gain measurement (G _MTXkRXjn ) and the phase measurement (0 T¾Rx _m ) for the pair of Tx and Rx antenna elements Txk and Rx _m . Processing the gain measurements (G _MTx . _Rx .) and the phase measurements further comprises repeating

the steps of determining and computing for at least one second pair of Tx antenna elements (Tx, and Tx _k ) having symmetrical coupling properties with respect to the pair of Rx antenna elements (Rx _n and Rx _m ) and averaging the relative gain values and the relative phase values for the Rx antenna element Rx _n relative to the Rx antenna element Rx _m across the first pair of Tx antenna elements and the at least one second pair of Tx antenna elements to thereby provide an average relative gain value and an average relative phase value for the Rx antenna element Rx _n relative to the Rx antenna element Rx _m . Processing the gain measurements (G _MTx . _Rx .) and the phase measurements

further comprises repeating the steps of selecting, determining, computing, repeating, and averaging for at least one second pair of Rx antenna elements and normalizing the average relative gain values and the average relative phase values relative to a defined reference Rx antenna element to thereby provide gain and phase calibration values for the plurality of Rx antenna elements.

In some embodiments, a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _n is approximately the same as a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _m , a mutual coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _n is approximately the same as a mutual coupling between the Tx antenna element Txk and the Rx antenna element Rx _m , and the at least one relative gain value and the at least one relative phase value for the Rx antenna element Rx _n relative to the Rx antenna element Rx _m comprises a relative gain defined as:

AG (3 — ^ ^MTx i ^RXn ~ ^MTXiRXm) + _^MTx _k Rx _n ~ ^MTx^Xm) and a relative phase value defined as:

, _Λ

Δφ _{ΚΧηΚΧηι} (3) =

In some embodiments, a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _n is approximately the same as a mutual coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _m , a mutual coupling between the Tx antenna element Tx, and the Rx antenna element Rx _m is approximately the same as a mutual coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _n , and the at least one relative gain value and the at least one relative phase value for the Rx antenna element Rx _n relative to the Rx antenna element Rx _m comprises a relative gain value defined as: ft ^' MTxiRx- ^' MTxiRx- MTx _k Rx-

phase values for the Rx antenna element Rx _n relative to the Rx antenna element Rxm across the first pair of Tx antenna elements and the at least one second pair of Tx antenna elements comprises performing a weighted average of the relative gain values and the relative phase values for the Rx antenna element Rx _n relative to the Rx antenna element Rx _m across the first pair of Tx antenna elements and the at least one second pair of Tx antenna elements.

Embodiments of a processing system enabled to perform self-calibration for a plurality of Tx antenna elements and a plurality of Rx antenna elements in an antenna array of a remote transceiver are also disclosed. In some

embodiments, the processing system comprises a communication interface, at least one processor, and memory comprising instructions executable by the at least one processor whereby the processing system is operable to perform the method of operation of the processing system according to any one of the embodiments disclosed herein.

Those skilled in the art will appreciate the scope of the present disclosure and realize additional aspects thereof after reading the following detailed description of the embodiments in association with the accompanying drawing figures.

Brief Description of the Drawings

The accompanying drawing figures incorporated in and forming a part of this specification illustrate several aspects of the disclosure, and together with the description serve to explain the principles of the disclosure.

Figure 1 illustrates an example embodiment of a transceiver (e.g., an analog beamforming transceiver) that provides self-calibration for an antenna array of the transceiver according to embodiments of the present disclosure; Figures 2A through 2C illustrate examples of a measurement arrangement that may be used when providing self-calibration according to some embodiments of the present disclosure;

Figure 3 is a flow chart that illustrates the operation of a transceiver (e.g., an analog beamforming transceiver) to perform self-calibration of an antenna array according to some embodiments of the present disclosure;

Figure 4 is a flow chart that illustrates the operation of a transceiver (e.g., an analog beamforming transceiver) to perform gain and phase

measurements for different pairs of transmit (Tx) and receive (Rx) antenna elements according to some embodiments of the present disclosure;

Figures 5A and 5B are flow charts that illustrate the operation of a transceiver (e.g., an analog beamforming transceiver) to post-process gain and phase measurements for pairs of Tx antenna elements and pairs of Rx antenna elements having symmetrical coupling properties to obtain gain and phase calibration values for Tx and Rx calibration, respectively, according to some embodiments of the present disclosure;

Figures 6A and 6B illustrate one example of multiple different Rx pairs having symmetrical coupling properties with respect to a particular Tx antenna element pair that can be used for Tx calibration in accordance with the process of Figure 5A;

Figures 7A and 7B are flow charts that illustrate the operation of a transceiver (e.g., an analog beamforming transceiver) to post-process gain and phase measurements for pairs of Tx antenna elements and pairs of Rx antenna elements having symmetrical coupling properties to obtain gain and phase calibration values for Tx and Rx calibration, respectively, according to some other embodiments of the present disclosure;

Figure 8 illustrates the operation of the transceiver to configure the gain and phase adjustment circuitry according to some embodiments of the present disclosure; Figure 9 illustrates a transceiver (e.g., an analog beamforming transceiver) that provides self-calibration for an antenna array of the transceiver according to some other embodiments of the present disclosure;

Figure 10 illustrates a system in which post-processing is performed at a remote processing system according to some other embodiments of the present disclosure; and

Figure 1 1 is a flow chart that illustrates the operation of the remote processing system of Figure 10 according to some embodiments of the present disclosure.

Detailed Description

The embodiments set forth below represent information to enable those skilled in the art to practice the embodiments and illustrate the best mode of practicing the embodiments. Upon reading the following description in light of the accompanying drawing figures, those skilled in the art will understand the concepts of the disclosure and will recognize applications of these concepts not particularly addressed herein. It should be understood that these concepts and applications fall within the scope of the disclosure and the accompanying claims.

Figure 1 illustrates an example embodiment of a transceiver 10 that provides self-calibration for an antenna array of the transceiver 10 according to embodiments of the present disclosure. In some preferred embodiments, the transceiver 10 is an analog beamforming transceiver and, as such, the transceiver 10 is sometimes referred to herein as an analog beamforming transceiver 10. However, it should be appreciated that, in some other

embodiments, the transceiver 10 may, e.g., be partly digital. The analog beamforming transceiver 10 may be, for example, a radio access node in a cellular communications network (e.g., a Third Generation Partnership Project (3GPP) Long Term Evolution (LTE) or Fifth Generation (5G) New Radio (NR) network), an access point in a local wireless network (e.g., an access point in a WiFi network), a wireless communication device (e.g., a User Equipment device (UE) in a 3GPP LTE or Third Generation (3G) NR network), or the like. As illustrated, the analog beamforming transceiver 10 includes a baseband processing system 12, one or more transmitters 14 and one or more receivers 16 coupled to the baseband processing system 12, gain and phase adjustment circuitry 18, and an antenna array that, in this example, is implemented as one or more Advanced Antenna Systems (AASs) 20. The baseband processing system 12 is implemented in hardware or a combination of hardware and software. For example, the baseband processing system 12 may include one or more processors (e.g., Central Processing Units (CPUs), Digital Signal Processors (DSPs), Application Specific Integrated Circuits (ASICs), Field Programmable Gate Arrays (FPGAs), and/or the like). In some embodiments, at least some of the functionality of the baseband processing system 12 described herein is implemented in software that is executed by the processor(s).

The antenna array implemented by the one or more AASs 20 includes multiple antenna elements and, in some implementations, many antenna elements (e.g., tens or hundreds of antenna elements). When an antenna element is used for transmission from the transmitter(s) 14, the antenna element is referred to herein as a Transmit (Tx) antenna element. Likewise when an antenna element is used for reception via the receiver(s) 16, the antenna element is referred to herein as a Receive (Rx) antenna element. In some implementations, a single antenna element may operate as both a Tx antenna element and a Rx antenna element. Due to various parameters (e.g.,

manufacturing tolerances of the various components in the transmit and receive paths, temperature, etc.), there may be variations in gain and/or phase between different Tx antenna elements and/or variations in gain and/or phase between different Rx antenna elements. In particular, the gain and phase of a first transmit path from the output of the baseband processing system 12 to a first Tx antenna element may differ from that of a second transmit path from the output of the baseband processing system 12 to a second Tx antenna element. Likewise, the gain and phase of a first receive path from a first Rx antenna element to the input of the baseband processing system 12 may differ from that of a second receive path from a second Rx antenna element to the input of the baseband processing system 12.

The baseband processing system 12 includes a self-calibration subsystem 22 that operates to provide self-calibration at the analog beamforming transceiver 10 for the variations in gain and/or phase for different Tx and/or Rx antenna elements. The self-calibration subsystem 22 is implemented in hardware or a combination of hardware and software. For example, the self-calibration subsystem 22 may include one or more processors (e.g., CPUs, DSPs, ASICs, FPGAs, and/or the like). In some embodiments, at least some of the functionality of the self-calibration subsystem 22 described herein is implemented in software that is executed by the processor(s). In this example, the self-calibration subsystem 22 includes a signal generator 24, a measurement function 26, and a post-processing function 28, the operation of which is described below in detail. Note that, in some embodiments, some or all of the components used for self- calibration can be dedicated for that purpose (i.e., not used for transmission or reception of normal uplink and downlink signals).

Coupling between the antenna elements is generally asymmetrical.

Specifically, for a particular Tx antenna element, the coupling between the Tx antenna element and some Rx antenna elements is symmetrical whereas the coupling between the Tx antenna element and some other Rx antenna elements is asymmetrical. Likewise, for a particular Rx antenna element, the coupling between the Rx antenna element and some Tx antenna elements is symmetrical whereas the coupling between the Rx antenna element and some other Tx antenna elements is asymmetrical. As discussed below, embodiments of the present disclosure provide self-calibration at the analog beamforming transceiver 10 based on measurements between pairs of Tx antenna elements and pairs of Rx antenna elements having symmetrical coupling properties with respect to one another.

The measurement arrangement for obtaining the measurements used for the self-calibration process may vary. As illustrated in Figure 2A, in some example embodiments, measurements can be performed between individual Tx antenna elements and individual Rx antenna elements between two AASs 20 in a same unit with the same or different polarization. Figure 2B illustrates an example in which measurements can be performed between individual Tx antenna elements and individual Rx antenna elements on different Radio

Frequency Integrated Circuits (RFICs) of a same AAS 20. However, in this case, the polarization should be different because of analog beamforming setup where a Tx antenna element and a Rx antenna element of the same polarization are using the same channel in Time Domain Duplexing (TDD). Figure 2C illustrates an example in which any Tx antenna element and Rx antenna element can be selected for measurement if there is scope to operate the Tx antenna element in Horizontal polarization and the Rx antenna element in Vertical polarization and vice versa. This is one preferred measurement scenario as the Tx antenna element and the Rx antenna element are in proximity and differences in coupling values will be minimal with better symmetry. However, note that Figures 2A through 2C are only examples.

Figure 3 is a flow chart that illustrates the operation of the analog beamforming transceiver 10, and in particular the self-calibration subsystem 22 of the analog beamforming transceiver 10, to perform self-calibration of the antenna array formed by the one or more AASs 20 according to some

embodiments of the present disclosure. In operation, the analog beamforming transceiver 10, and in particular the measurement function 26 of the self- calibration subsystem 22, performs a self-calibration procedure (step 100). In particular, the analog beamforming transceiver 10 performs pair-by-pair gain and phase measurements for all pairs of Tx and Rx antenna elements in the antenna array formed by the one or more AASs 20 and stores the resulting

measurements (step 100A). In other words, for each pair of Tx and Rx antenna elements Tx, and Rx _j where I = 1...Ν _Τ χ and j = 1...N _RX and Ν _Τ χ is the number of Tx antenna elements in the antenna array and N _RX is the number of Rx antenna elements in the antenna array, the analog beamforming transceiver 10 performs a gain measurement (G _MTx . _Rx .) and a phase measurement he details of how the gain and phase measurements G _MTx . _Rx . ^are

performed is provided below with respect to Figure 4. However, in general, an IQ signal (e.g., a pseudo-random IQ signal having a desired bandwidth for the measurement) is generated by the signal generator 24 and provided to the transmitter(s) 14 for transmission via the Tx antenna element Tx,. The

measurement function 26 measures the gain and phase of a received signal that is received via the receiver(s) 16 and the Rx antenna element Rx _j during transmission of the IQ signal from the Tx antenna element Tx,. The measured gain and phase are the gain and phase measurement values G _MTx . _Rx . and

</>MTxiRxj - This process is performed for each pair of Tx and Rx antenna elements.

The analog beamforming transceiver 10, and in particular the postprocessing function 28 of the self-calibration subsystem 22, performs postprocessing of the gain and phase measurements G _MTx . _Rx . and for all i

and j to compute gain and phase calibration values for the Tx and Rx antenna elements of the antenna array (step 100B). The details of step 100B are provided below with respect to Figures 5A and 5B and Figures 7A and 7B. In general, the post-processing function 28 computes the gain and phase

calibration values based on the gain and phase measurements G _MTx . _Rx . and

(p _MTXiRXj and combinations of Tx and Rx antenna elements having symmetrical coupling properties. The combinations of Tx and Rx antenna elements having symmetrical coupling properties are, e.g., predefined or predetermined based on, e.g., a known layout of the antenna elements in the antenna array or computed by the analog beamforming transceiver 10 based on, e.g., a known layout of the antenna elements in the antenna array.

The analog beamforming transceiver 10 applies the computed gain and phase calibration values at the analog beamforming transceiver 10 to thereby compensate for gain and phase variations between the different Tx antenna elements (referred to herein as Tx calibration) and between the different Rx antenna elements (referred to herein as Rx calibration) (step 102). The gain and phase calibration values are applied by the gain and phase adjustment circuitry 18.

Figure 4 illustrates step 100A of Figure 3 in more detail according to some embodiments of the present disclosure. As illustrated, the self-calibration subsystem 22 initiates measurement for self-calibration (step 200). The self- calibration subsystem 22 sets a frequency channel for the measurements (step 202). The signal generator 24 generates a pseudo-random IQ signal of a desired bandwidth for the measurements (step 204). The measurement function 26 selects the AAS(s) 20 for calibration (step 206), selects a TX polarization (step 208), and selects an RX polarization (step 210). For the first iteration, the measurement function 26 initializes a Rx antenna element counter / ^' and a Rx antenna element counter j to, in this example, a value of 0. The measurement function 26 increments (i.e., counts) the Rx antenna element counter j and then determines whether the Rx antenna element counter j is less than or equal to the total number of Rx antenna elements in the antenna array (step 212). If not, the measurement process ends (not shown). If the Rx antenna element counter j is less than or equal to the total number of Rx antenna elements in the antenna array, the measurement function 26 selects and enables the y ^' -th Rx antenna element Rx _j (step 214). For the first iteration, j = 1. The Rx antenna element counter j is incremented in subsequent iterations. The measurement function 26 increments (i.e., counts) the Tx antenna element counter / ^' and then determines whether the Tx antenna element counter / ^' is less than or equal to the total number of Tx antenna elements in the antenna array (step 216). If the Tx antenna element counter / ^' is less than or equal to the total number of Tx antenna elements in the antenna array, the measurement function 26 selects and enables the /-th Tx antenna element Tx, (step 218). For the first iteration, / ^' = 1 . The Tx antenna element counter / ^' is incremented in subsequent iterations.

The measurement function 26 causes the baseband processing system 12 to send the IQ signal generated by the signal generator 24 in step 204 to the transmitter(s) 14 for transmission via the enabled Tx antenna element Tx, and reception via the enabled Rx antenna element Rx _j (step 220). The resulting received signal received by the baseband processing system 12 via the enabled Rx antenna element Rx _j is measured by the measurement function 26. More specifically, the measurement function 26 cross-correlates the transmitted IQ signal and the received IQ signal (step 222) and divides (or subtracts in dB scale) the transmitted IQ signal by the cross-correlated received IQ signal (step 224). In other words, the relative phase measurement can be obtained from the cross- correlation (e.g., by looking for the peak in the cross-correlation to determine the relative delay, or phase, between the Tx IQ signal and the Rx IQ signal. The relative gain measurement can be obtained by division in normal scale or by subtraction in dB scale. The relative phase and relative gain for this

measurement are stored as a relative phase measurement (p _MTXiRXj and a relative gain measurement for G _MTx . _Rx . for the Tx antenna element Tx, relative to the Rx antenna element Rx _j (step 226).

The measurement function 26 then disables the Tx antenna element Tx, (step 228), and the process returns to step 216 such that gain and phase measurements are performed for all Tx antenna elements relative to the Rx antenna element Rx _j . Once gain and phase measurements are performed for all Tx antenna elements relative to the Rx antenna element Rx _j , the measurement function 26 disables the Rx antenna element Rxj (not shown), and the process returns to step 212 to be repeated for the next Rx antenna element. Once phase and gain measurements have been performed (and stored) for all Tx, Rx antenna element pairs, the measurement process ends.

Figures 5A and 5B illustrate step 100B of Figure 3 in more detail according to some embodiments of the present disclosure. In particular, Figure 5A illustrates the operation of the post-processing function 28 to compute gain and phase calibration values for Tx calibration, and Figure 5B illustrates the operation of the post-processing function 28 to compute gain and phase calibration for Rx calibration. In some embodiments, both Tx and Rx calibration are performed. However, in some other embodiments, only Tx calibration or only Rx calibration may be performed. First, as shown in Figure 5A, the post-processing function 28 selects a pair of Tx antenna elements, Tx, and Txk where i≠ k (step 300). The postprocessing function 28 determines a pair of Rx antenna elements, Rx _n and Rx _m where n≠ m, having symmetrical coupling properties with respect to the selected pair of Tx antenna elements Tx, and Txk (step 302). In some embodiments, pairs of Rx antenna elements that have symmetrical coupling properties with respect to the pair of Tx antenna elements Tx, and Tx _k are known (e.g., predefined or predetermined based on, e.g., the layout of the Tx and Rx antenna elements in the antenna array and, e.g., stored in a Look Up Table (LUT), computed based on, e.g., the layout of the Tx and Rx antenna elements in the antenna array, or the like). For Tx calibration, the pair of Rx antenna elements Rx _n and Rx _m that have symmetrical coupling properties with respect to the pair of Tx antenna elements Tx, and Tx _k satisfy either or both of the following two symmetrical coupling scenarios:

1. CTxiRxn CTx _k Rxn and CTxiRx _m CTx _k Rx _m

2. CTxiRx _n CTx _k Rx _m and CTxiRx _m CTx _k Rx _n

where CTxiRx _n is the coupling between Tx antenna element Tx, and Rx antenna element Rx _n , CTxiRx _m is the coupling between Tx antenna element Tx, and Rx antenna element Rx _m , CTx _k Rx _n is the coupling between Tx antenna element Tx _k and Rx antenna element Rx _n , and CTx _k Rx _m is the coupling between Tx antenna element Tx _k and Rx antenna element Rx _m . Further, in this context, the symbol "- " means "approximately equal to" or "approaches." Thus, coupling scenario (1 ) is where: (a) the coupling between Tx antenna element Tx, and Rx antenna element Rx _n is approximately equal to the coupling between Tx antenna element Tx _k and Rx antenna element Rx _n and the coupling between Tx antenna element Tx, and Rx antenna element Rx _m is approximately equal to the coupling between Tx antenna element Tx _k and Rx antenna element Rx _m . Similarly, coupling scenario (2) is where: (a) the coupling between Tx antenna element Tx, and Rx antenna element Rx _n is approximately equal to the coupling between Tx antenna element Tx _k and Rx antenna element Rx _m and the coupling between Tx antenna element Tx, and Rx antenna element Rx _m is approximately equal to the coupling between Tx antenna element Txk and Rx antenna element Rx _n .

The post-processing function 28 computes a relative gain value(s) and a relative phase value (s) between the pair of Tx antenna elements Tx, and Tx _k for the applicable symmetrical coupling scenario(s) based on the gain and phase measurements obtained in step 100A (step 304). Specifically, for the pair of Tx antenna elements Tx, and Tx _k and the pair of Rx antenna elements Rx _n and Rx _m having symmetrical coupling properties, the gain measurements previously obtained are G _MTx . _RXn , G _MTx . _RXm , G _MTXkRXn , and G _MTXkRXm , and the phase measurements previously obtained are φ _{ΜΤΧίΚΧη} , and

The gain measurements may be defined as follows:

GuTxiRx-n ⁼ G _Tx . + G _CTx . _RXn + G _RXn (1 a)

GMTxiRx _m ⁼ G _TXi + G _CTx . _RXm + G _RXm (1 b)

GMTx _k Rx _n ⁼ ^Tx _k + G _CTXkRXn + G _RXn (1 c)

GMTx _k Rx _m ⁼ ^Tx _k + G _CTXkRXm + G _RXm (1 d) where G _Tx . is a gain of the i-th transmit branch (i.e., a gain for the path from an output of the baseband processing system 12 to the Tx antenna element Tx,, G _TXk is a gain of the k-th transmit branch (i.e., a gain for the path from an output of the baseband processing system 12 to the Tx antenna element Tx _k , G _RXn is a gain of the n-th receive branch (i.e., a gain for the path from the Rx antenna element Rx _n to an input of the baseband processing system 12, G _RXm is a gain of the m-th receive branch (i.e., a gain for the path from the Rx antenna element Rx _m to an input of the baseband processing system 12, G _CTx . _RXn is a gain resulting from coupling between the Tx antenna element Tx, and the Rx antenna element Rx _n , G _CTx . _RXm is a gain resulting from coupling between the Tx antenna element Tx, and the Rx antenna element Rx _m , G _CTXkRXn is a gain resulting from coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _n , and G _CTXkRXm is a gain resulting from coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _m . Note that for all of the equations provided herein, gain is represented in decibels (dB) (i.e., G _d B = 20 log-io(GuNEAR)- As one of skill in the art will appreciate upon reading this disclosure, the equations can easily be rewritten in terms of linear gain.

Based on symmetrical coupling scenario (1 ) for Tx calibration given 5 above, a first relative gain value AG _Tx . _TXk (l) for the gain of Tx, relative to the gain of Tx _k can be computed as:

Λ Λ — ( ^G MTx _i Rx„-GMr¾Rx„) + (GMrx _i Rx _m -GMr¾Rx _m )

& _Tx . _TXk L) — ( )

By substituting Equations (1 a) through (1 d) above into Equation (2), it can be seen that:

where e _{CG Tx} (l) is an error term defined as:

Note that, due to the symmetrical coupling properties, the error term e _{CG Tx} (l) is small. In other words, the term (fi _CTx . _RXn - G _CTXkRXji ) will be small since CTxiRx _n 5 - CTx _k Rx _n , and the term {G _CTx . _RXm - G _CTXkRXm ) will be small since CTxiRx _m - CTx _k Rxm- However, there is still some error. As discussed below, this error is mitigated by averaging the computed relative gain values across multiple different pairs of Rx antenna elements having symmetrical coupling properties with respect to the same pair of Tx antenna elements.

0 Based on symmetrical coupling scenario (2) for Tx calibration given

above, a second relative gain value AG _Tx . _TXk (2) for the gain of Tx, relative to the gain of Tx _k can be computed as:

_{Λ /} ~ — { ^G MTx _i Rx _n -GMTx _k Rx _m ) + (GMTx _i Rx _m -GMTx _k Rx _n ) . .

^TXiTx^) - l ^¾ j

By substituting Equations (1 a) through (1 d) above into Equation (5), it can be 5 seen that:

(½Ti _i S¾ -½T¾te _m ) + (½ft _j Ki _ln - GMT¾S¾) _ (

- {. ^U TXi - ^U Tx _k ) + ^e CG,Tx ^ (Ό) where e _{CG Tx} 2 is an error term defined as: _{f Λ} _ (GcTXjRxn -GcTXkRxm) (CcTXjRxm ~ ^G CTx _k Rx _n ) . . eCG x ^L ) — ₂ + 2 ^{■ i} I

Note that, due to the symmetrical coupling properties, the error term e _{CG Tx} 2) is small. In other words, the term (G _CTx . _RXn - G _CTXkRXm ) will be small since CTxiRx _n - CTx _k Rx _m , and the term {G _CTx . _RXm - G _CTXkRXn ) will be small since CTxiRx _m - CTx _k Rx _n . However, there is still some error. Again, as discussed below, this error is mitigated by averaging the computed relative gain values across multiple different pairs of Rx antenna elements having symmetrical coupling properties with respect to the same pair of Tx antenna elements.

In a similar manner, the phase measurements may be defined as follows:

ΜΎΧί Χη ⁼ Φτχι + ΦθΤχιϋχ _η + Φ χ _η (8a)

0 TX _j Rx _m ⁼ Φτχ _ί + 0CTX _j Rx _m + 0Rx _m (8b)

0 T¾Rx _n ⁼ 0Tx _fe + + Φ χ _η (8c)

0 T¾Rx _m ⁼ ΤΛ¾ + 0CTx _fe Rx _m + 0Rx _m (8d) where φ _Τχ . is a phase of the i-th transmit branch (i.e., a phase for the path from an output of the baseband processing system 12 to the Tx antenna element Tx,, φ _ΤΧ]ι is a phase of the k-th transmit branch (i.e., a phase for the path from an output of the baseband processing system 12 to the Tx antenna element Tx _k , φ _ΚΧη is a phase of the n-th receive branch (i.e., a phase for the path from the Rx antenna element Rx _n to an input of the baseband processing system 12, φ _ΚΧηι is a phase of the m-th receive branch (i.e., a phase for the path from the Rx antenna element Rx _m to an input of the baseband processing system 12,

is a phase resulting from coupling between the Tx antenna element Tx, and the Rx antenna element Rx _n , is a phase resulting from coupling between the Tx antenna element Tx, and the Rx antenna element Rx _m , 0 _C T¾R _X „ is a phase resulting from coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _n , and is a phase resulting from coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _m . Based on symmetrical coupling scenario (1 ) for Tx calibration given above, a first relative phase value Δφ _Τχ . _ΤΧ]( (1) for the phase of Tx, relative to the phase of Tx _k can be computed as:

Note that in the term Δφ _Τχ . _ΤΧ]( (1) and similar terms described herein, the parenthetical "(x)" is used to refer to coupling scenario "x" (e.g., Δφ _ΤΛ .. _τ¾ (1) refers to a relative phase value computed for coupling scenario (1 ) whereas refers to a relative phase value computed for coupling scenario (2)). By substituting Equations (8a) through (8d) above into Equation (9), it can be seen that:

(0Mrx _i Rx _n -0Mr _¾ R¾) + (0Mrx _i Rx _m -0Mr _¾ Rx _m ) _ ,

2 - VPTXi - + ^β Οφ,Τχ ¹ ) ^{I U} where β _€φιΤχ 1) is an error term defined as:

Note that, due to the symmetrical coupling properties, the error term ε _{φ Tx} (l) is small. In other words, the term {4 ⁾ _C TxiRx _n ^~ 0C7¾RX„) will ^De small since CTxiRx _n CTx _k Rxn, and the term (0 _CTXiRXm -≠ _C Tx _k Rx _m ) will be small since CTxiRx _m CTx _k Rx _m - However, there is still some error. As discussed below, this error is mitigated by averaging the computed relative phase values across multiple different pairs of Rx antenna elements having symmetrical coupling properties with respect to the same pair of Tx antenna elements.

Based on symmetrical coupling scenario (2) for Tx calibration given above, a second relative phase value Δφ _ΤΛ .. _τ¾ (2) for the phase of Tx, relative to the phase of Tx _k can be computed as:

By substituting Equations (8a) through (8d) above into Equation (12), it can be seen that:

(0Mr _i R _n -0Mr _fc R _m ) + (0Mr _i R _m -0Mr _fc R¾) _ (

2 - where e _{C(j} Tx} 2) is an error term defiend as:

, _Λ _ (φθΤχ - φθΤχ (φθΤχ -φθΤχ

eC<p,Tx V ) — 2 2 ' ^ '

Note that, due to the symmetrical coupling properties, the error term β _{€φ Τχ} 2) is small. In other words, the term {<pcTxiRx _n ~ 0c ₇ ¾Rx _m ) will ^De small since CTxiRx _n - CTxkRxm, and the term (^ _C T _Xi Rx _m ~ 0cT¾Rx _n ) will ^De small since CTxiRx _m - CTx _k Rx _n . However, there is still some error. Again, as discussed below, this error is mitigated by averaging the computed relative phase values across multiple different pairs of Rx antenna elements having symmetrical coupling properties with respect to the same pair of Tx antenna elements.

The post-processing function 28 determines whether there are more Rx pairs that have symmetrical coupling properties with respect to the selected pair of Tx antenna elements Tx, and Tx _k (step 306). Preferably, there are two or more different Rx pairs that have symmetrical coupling properties with respect to the same pair of Tx antenna elements Tx, and Tx _k such that, for each pair of Rx antenna elements, separate gain and phase values are computed in accordance with Equations (2), (5), (9), and (12) above. In this manner, multiple relative gain values AG _Tx . _TXk (l, v) and AG _Tx . _TXk (2, i) and multiple relative phase values A<p _TXiTXk l, p) and Αφ _Τχ . _ΤΧ]( (2, p) are determined for the same pair of Tx antenna elements Tx, and Tx _k , where "p" is an index for the Rx pair such that p = 1 for the first Rx pair. Thus, for example, AG _Tx . _T¾ (l, p) is the relative gain value

G _Tx . _TXk (l) for the p-th Rx pair.

The post-processing function 28 averages the relative gain values

AG _Tx . _TXk (l, p) and AG _Tx . _TXk 2, p) for the pair of Tx antenna elements Tx, and Tx _k across all pairs of Rx antenna elements Rx _n , Rx _m having symmetrical coupling properties with respect to the pair of Tx antenna elements Tx, and Tx _k to provide an average relative gain value AG _Tx . _TXh for the pair of Tx antenna elements Tx, and Tx _k . Similarly, the post-processing function 28 averages the relative phase values Δφ _ΤΛ; . _τ¾ (1, ρ) and Δφ _ΤΛ; . _τ¾ (2, ρ) for the pair of Tx antenna elements Tx, and Tx _k across all pairs of Rx antenna elements Rx _n , Rx _m having symmetrical coupling properties with respect to the pair of Tx antenna elements Tx, and Tx _k to provide an average relative phase value Δφ _Τχ . _ΤΧ]( for the pair of Tx antenna elements Tx, and Tx _k (step 308). The average values are stored.

In some embodiments, the average values are computed as linear average values. However, in some other embodiments, the average values are computed as weighted average values. As one example implementation of weighted averaging, the larger the (average) distance of Rx elements from Tx elements (in Tx calibration), the smaller the weight. In other words, the weight (W1 ) applied to the measurement for a particular pair of Tx antenna elements Tx, and Txk is inversely proportional to the distance (e.g., average distance) of the corresponding Rx antenna elements Rx _n and Rx _m from the Tx antenna elements Tx, and Tx _k . As another example implementation, the closer the Rx elements Rx _n and Rxm to the center of the AAS, the higher the weighting factor. In other words, the weight (W2) applied to the measurement for a particular pair of Tx antenna elements Tx, and Tx _k is inversely proportional to the distance (e.g., average distance) of the corresponding Rx antenna elements Rx _n and Rx _m from the center of the AAS. In some embodiments, the weight applied to the measurement for a particular pair of Tx antenna elements Tx, and Tx _k is equal to W1 *W2.

In some other embodiments, a statistical distribution analysis may be used in lieu of averaging. More specifically, in some embodiments, a statistical distribution analysis is performed on the gain and phase values for the pairs of Tx antenna elements such that extreme results are filtered out and the remaining values are averaged (linear or weighted average). As an example, the extreme results may be gain values that are more than a defined threshold amount from the mean of all of the gain values and phase values that are more than a defined threshold amount from the mean of all of the phase values.

The post-processing function 28 determines whether there are more TX antenna element pairs to be processed (step 310). If so, the process returns to step 300 and is repeated for the next Tx antenna element pair. Once all of the Tx antenna element pairs have been processed, the post-processing function 28 normalizes the average relative gain values AG _Tx . _TXk and the average relative phase value Αφ _Τχ . _ΤΧ]ι with respect to a single Tx antenna element, which is referred to as a reference Tx antenna element (step 312). The normalized gain values are the gain calibration values G _Tx ., and the normalized phase values are the phase calibration values φ _Τχ ..

It is important to note that, in the example above, it is assumed that all Rx pairs for all Tx pairs satisfy both symmetrical coupling scenario (1 ) and symmetrical coupling scenario (2) and that both AG _Tx . _TXk (V) and AG _Tx . _TXk (2) and Αφ _Τχ . _ΤΧ]( (1) and Αφ _Τχ . _ΤΧ]( (2) are computed for each Rx pair. However, the present disclosure is not limited thereto. In some alternative embodiments, for a particular Tx pair Tx, and Tx _k , each Rx pair having symmetrical coupling properties with respect to the Tx pair Tx, and Tx _k may satisfy only symmetrical coupling scenario (1 ) in which case only AG _Tx . _TXk (l) and Δφ _ΤΛ .. _τ¾ (1) are computed for that Rx pair, satisfy only symmetrical coupling scenario (2) in which case only AG _Tx . _TXk 2) and Αφ _Τχ . _{ΤΧ ι} {1) are computed for that Rx pair, or satisfy both symmetrical coupling scenarios (1 ) and (2) in which both AG _Tx . _TXk (l ^~ ) and Δφτχ _; ¾(1) and G _Tx . _TXk 2) and Δφ _Τχ . _τ¾ (2) are computed for that Rx pair. In some other alternative embodiments, the post-processing function 28 may consider only symmetrical coupling scenario (1 ) in which case only AG _Tx . _TXk (l ^~ ) and Δφ _ΤΛ .. _τ¾ (1) are computed. In some other alternative embodiments, the post-processing function 28 may consider only symmetrical coupling scenario (2) in which case only AG _Tx . _TXk (2) and Αφ _Τχ . _ΤΧ]( (2) are computed.

Figure 5B illustrates the operation of the post-processing function 28 to compute gain and phase calibration for Rx calibration according to some embodiments of the present disclosure. First, the post-processing function 28 selects a pair of Rx antenna elements, Rx _n and Rx _m where n≠ m (step 400). The post-processing function 28 determines a pair of Tx antenna elements, Tx, and Tx _k where i≠ k, having symmetrical coupling properties with respect to the selected pair of Rx antenna elements Rx _n and Rx _m (step 402). In some embodiments, pairs of Tx antenna elements that have symmetrical coupling properties with respect to the pair of Rx antenna elements Rx _n and Rx _m are known (e.g., predefined or predetermined based on, e.g., the layout of the Tx and Rx antenna elements in the antenna array and, e.g., stored in a LUT, computed based on, e.g., the layout of the Tx and Rx antenna elements in the antenna array, or the like). For Rx calibration, the pair of Tx antenna elements Tx, and Txk that have symmetrical coupling properties with respect to the pair of Rx antenna elements Rx _n and Rx _m satisfy either or both of the following two symmetrical coupling scenarios:

3. CTxiRx _n CTxiRx _m and CTx _k Rx _n CTx _k Rx _m

4. CTxiRxn ^ CTx _k Rx _m and CTxiRxm ^ CTx _k Rxn

where CTxiRx _n is the coupling between Tx antenna element Tx, and Rx antenna element Rx _n , CTxiRx _m is the coupling between Tx antenna element Tx, and Rx antenna element Rx _m , CTx _k Rx _n is the coupling between Tx antenna element Tx _k and Rx antenna element Rx _n , and CTx _k Rx _m is the coupling between Tx antenna element Tx _k and Rx antenna element Rx _m . Further, in this context, the symbol "- " means "approximately equal to" or "approaches." Thus, coupling scenario (3) is where: (a) the coupling between Tx antenna element Tx, and Rx antenna element Rx _n is approximately equal to the coupling between Tx antenna element Tx, and Rx antenna element Rx _m and the coupling between Tx antenna element Tx _k and Rx antenna element Rx _n is approximately equal to the coupling between Tx antenna element Tx _k and Rx antenna element Rx _m . Similarly, coupling scenario (4) is where: (a) the coupling between Tx antenna element Tx, and Rx antenna element Rx _n is approximately equal to the coupling between Tx antenna element Tx _k and Rx antenna element Rx _m and the coupling between Tx antenna element Tx, and Rx antenna element Rx _m is approximately equal to the coupling between Tx antenna element Tx _k and Rx antenna element Rx _n . Note that coupling scenario (4) is the same as coupling scenario (2) above.

The post-processing function 28 computes a relative gain value(s) and a relative phase value(s) between the pair of Rx antenna elements Rx _n and Rx _m for the applicable symmetrical coupling scenario(s) based on the gain and phase measurements obtained in step 100A (step 404). Specifically, for the pair of Rx antenna elements Rx _n and Rx _m and the pair of Tx antenna elements Tx, and Tx _k having symmetrical coupling properties, the gain measurements previously obtained are G _MTx . _RXn , G _MTx . _RXm , G _MTXkRXn , and G _MTXkRXm , and the phase measurements previously obtained are <t> _MTXI R _XN , and

The gain measurements may be defined as follows:

GMTxiRxn ⁼ G _Tx . + G _CTx . _RXn + G _RXn (15a) GMTxiRx _m ⁼ G _TXi + G _CTx . _RXm + G _RXm (15b) GMTx _k Rx _n ⁼ ^Tx _k + G _CTXkRXn + G _RXn (15c) G _MTXkRX†n = G _TXk + G _CTXkRX†n + G _RX†n (15d) where G _Tx . is a gain of the i-th transmit branch (i.e., a gain for the path from an output of the baseband processing system 12 to the Tx antenna element Tx,, G _TXk is a gain of the k-th transmit branch (i.e., a gain for the path from an output of the baseband processing system 12 to the Tx antenna element Tx _k , G _RXn is a gain of the n-th receive branch (i.e., a gain for the path from the Rx antenna element Rx _n to an input of the baseband processing system 12, G _RXm is a gain of the m-th receive branch (i.e., a gain for the path from the Rx antenna element Rx _m to an input of the baseband processing system 12, G _CTx . _RXn is a gain resulting from coupling between the Tx antenna element Tx, and the Rx antenna element Rx _n , G _CTx . _RXm is a gain resulting from coupling between the Tx antenna element Tx, and the Rx antenna element Rx _m , G _CTXkRXn is a gain resulting from coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _n , and G _CTXkRXm is a gain resulting from coupling between the Tx antenna element Txk and the Rx antenna element Rx _m . Note that for all of the equations provided herein, gain is represented in dB. As one of skill in the art will appreciate upon reading this disclosure, the equations can easily be rewritten in terms of linear gain.

Based on symmetrical coupling scenario (3) for Rx calibration given above, a first relative gain value AG _RXnRXm (3) for the gain of Rx _n relative to the gain of Rx _m can be computed as: AG (3) — ( ^GMTx i ^RXn ~ ^GMTx i ^{RXm +} ( ^GMTx k ^Rx n ~ ^{GM Tx} k ^Rx rn)

By substituting Equations (15a) through (15d) above into Equation (16), it can be seen that: - RXn - ^U Rx _m ) + ^e CG,Rx -3j I ¹ ' )

5 where e _{CG Rx} 3) is an error term defined as:

Note that, due to the symmetrical coupling properties, the error term e _{CG Rx} 3) is small. In other words, the term (G _CTx . _RXn - G _CTx . _RXm ) will be small since CTxiRx _n - CTxjRxm, and the term {G _CTXkRXn - G _CTXkRXm ) will be small since CTx _k Rx _n ->

10 CTx _k Rxm- However, there is still some error. As discussed below, this error is mitigated by averaging the computed relative gain values across multiple different pairs of Tx antenna elements having symmetrical coupling properties with respect to the same pair of Rx antenna elements.

Based on symmetrical coupling scenario (4) for Rx calibration given

15 above, a second relative gain value &G _RXnRXm (4) for the gain of Rx _n relative to the gain of Rx _m can be computed as:

By substituting Equations (15a) through (15d) above into Equation (19), it can be seen that:

n (½Ti _i s¾ - ½T¾te _m )+(½ft _j Ki _m - ½T¾s¾) _ (

^ ^U - \ Rx _n - Rx _m ) + ^e CG,Rx V±) l^ ^U j where e _{CG Rx} (4) is an error term defiend as:

Note that, due to the symmetrical coupling properties, the error term e _{CG Rx} (4) is small. In other words, the term (G _CTx . _RXn - G _CTXkRXm ) will be small since CTxiRx _n 25 - CTx _k Rxm, and the term {G _CTx . _RXm - G _CTXhRXn ) will be small since CTxiRx _m - CTxkRxn. However, there is still some error. Again, as discussed below, this error is mitigated by averaging the computed relative gain values across multiple different pairs of Tx antenna elements having symmetrical coupling properties with respect to the same pair of Rx antenna elements.

In a similar manner, the phase measurements may be defined as follows:

0 TX _j Rx _n ⁼ Φτχι + 0CTX _j Rx _n + Φ χ _η (22a)

0 TX _j Rx _m ⁼ Φτχ _ί + 0CTX _j Rx _m + 0Rx _m (22b)

0 T¾Rx _n ⁼ 0Tx _fe + ΦθΤχ + Φ χ _η (22c)

Φ Τ ⁼ Φτχ _ΐι + 0C7¾Rx _m + 0Rx _m (22d) where φ _Τχ . is a phase of the i-th transmit branch (i.e., a phase for the path from an output of the baseband processing system 12 to the Tx antenna element Tx,, φ _ΤΧ]ί is a phase of the k-th transmit branch (i.e., a phase for the path from an output of the baseband processing system 12 to the Tx antenna element Tx _k , φ _ΚΧη is a phase of the n-th receive branch (i.e., a phase for the path from the Rx antenna element Rx _n to an input of the baseband processing system 12, φ _ΚΧηι is a phase of the m-th receive branch (i.e., a phase for the path from the Rx antenna element Rx _m to an input of the baseband processing system 12,

is ^a phase resulting from coupling between the Tx antenna element Tx, and the Rx antenna element Rx _n , is a phase resulting from coupling between the Tx antenna element Tx, and the Rx antenna element Rx _m , 0 _CT¾RXn is a phase resulting from coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _n , and is a phase resulting from coupling between the Tx antenna element Tx _k and the Rx antenna element Rx _m .

Based on symmetrical coupling scenario (3) for Rx calibration given above, a first relative phase value Δφ _{ΚΧηΚΧηι} (3) for the phase of Rx _n relative to the phase of Rx _m can be computed as:

Αψ (3) - ( ^ΦΜΤχ ί ^ΚΧη ~ ^ΦΜΤχ ί ^Κχ ™ ^φΜΤχ ^ ^{χη Κχ} ι ^η ) ^23)

By substituting Equations (22a) through (22d) above into Equation (23), it can be seen that:

(0Mr _i R _n -0Mr _i R _m ) + (0Mr _fc R _n -0Mr _¾ R _m ) _ , _ (1~) (2Λ where β _{αφ Κχ} (3) is an error term defined as:

, _Λ _ (φθΤχ -φθΤχ >CTx _k Rx _n -0CT¾Rx _m ) . eC0,Rx W - 2 2 ' ^ '

Note that, due to the symmetrical coupling properties, the error term β _{€φ Κχ} 3) is small. In other words, the term will ^De small since CTxiRx _n - CTxjRxm, and the term (p _C Tx _k Rx _n ^_ 0cT¾Rx _m ) will ^De small since CTx _k Rx _n - CTx _k Rx _m - However, there is still some error. As discussed below, this error is mitigated by averaging the computed relative phase values across multiple different pairs of Tx antenna elements having symmetrical coupling properties with respect to the same pair of Rx antenna elements.

Based on symmetrical coupling scenario (4) for Rx calibration given above, a second relative phase value A0 _RXnRXm (4) for the phase of Rx _n relative to the phase of Rx _m can be computed as:

( ) _ {ΦΜΤχ _ι χ _η -φΜΤχ _1ί χ _τη ) + {φΜΤχ _ι χ _ηι -φΜΤχ _1ί χ _η )

By substituting Equations (22a) through (22d) above into Equation (26), it can be seen that:

(0Mrx _i Rx _n -0Mr _¾ Rx _m ) + (0Mrx _i Rx _m -0Mr _¾ R¾) _ ,

2 PRXn - VRXm)

where β _{φ Κχ} (4) is an error term defiend as:

,_ _Q .

Note that, due to the symmetrical coupling properties, the error term β _{φ Κχ} (4) is small. In other words, the term - 0cT¾Rx _m ) will be small since CTxiRx _n - CTxkRxm, and the term ((pcrxiRxm ~ <l>cTx _k Rx _n ) will be small since CTxiRx _m - CTx _k Rx _n . However, there is still some error. Again, as discussed below, this error is mitigated by averaging the computed relative phase values across multiple different pairs of Tx antenna elements having symmetrical coupling properties with respect to the same pair of Rx antenna elements.

The post-processing function 28 determines whether there are more Tx pairs that have symmetrical coupling properties with respect to the selected pair of Rx antenna elements Rx _n and Rx _m (step 406). Preferably, there are two or more different Tx pairs that have symmetrical coupling properties with respect to the same pair of Rx antenna elements Rx _n and Rx _m such that, for each pair of Tx antenna elements, separate gain and phase values are computed in accordance with Equations (16), (19), (23), and (26) above. In this manner, multiple relative gain values G _RXnRXm (3, p and G _RXnRXm (4, p and multiple relative phase values Αφ _{ΚΧηΚΧηι} (3, τρ) and A0 _RXnRXm (4, p) for the same pair of Rx antenna elements Rx _n and Rxm, where "p" is an index for the Tx pair such that p = 1 for the first Tx pair, p = 2 for the second Tx pair, and so on. Thus, for example, AG _RXnRXm (3, p) is the relative gain value &G _RXnRXm (3) for the p-th Tx pair.

The post-processing function 28 averages the relative gain values

AG _RXnRXm (3, Tp) and AG _RXnRXm (4, p) for the pair of Rx antenna elements Rx _n and Rx _m across all pairs of Tx antenna elements Tx,, Tx _k having symmetrical coupling properties with respect to the pair of Rx antenna elements Rx _n and Rx _m to provide an average relative gain value &G _RXnRXm for the pair of Rx antenna elements Rx _n and Rx _m . Similarly, the post-processing function 28 averages the relative phase values A0 _RXnRXm (3, p) and A0 _RXnRXm (4, p) for the pair of Rx antenna elements Rx _n and Rx _m across all pairs of Tx antenna elements Tx,, Tx _k having symmetrical coupling properties with respect to the pair of Rx antenna elements Rx _n and Rx _m to provide an average relative phase value Δφ _{ΚΧηΚΧηι} for the pair of Rx antenna elements Rx _n and Rx _m (step 408). The average values are stored.

In some embodiments, the average values are computed as linear average values. However, in some other embodiments, the average values are computed as weighted average values. As one example implementation of weighted averaging, the larger the (average) distance of Rx elements from Tx elements, the smaller the weight. In other words, the weight (W1 ) applied to the measurement for a particular pair of Rx antenna elements Rx _n and Rx _m is inversely proportional to the distance (e.g., average distance) of the

corresponding Tx antenna elements Tx, and Tx _k from the Rx antenna elements Rx _n and Rx _m . As another example implementation, the closer the Rx elements Rx _n and Rx _m to the center of the AAS, the higher the weighting factor. In other words, the weight (W2) applied to the measurement for a particular pair of Rx antenna elements Rx _n and Rx _m is inversely proportional to the distance (e.g., average distance) of the Rx antenna elements Rx _n and Rx _m from the center of the AAS. In some embodiments, the weight applied to the measurement for a particular pair of Rx antenna elements Rx _n and Rx _m is equal to W1 *W2.

In some other embodiments, a statistical distribution analysis may be used in lieu of averaging. More specifically, in some embodiments, a statistical distribution analysis is performed on the gain and phase values for the pairs of Rx antenna elements such that extreme results are filtered out and the remaining values are averaged (linear or weighted average). As an example, the extreme results may be gain values that are more than a defined threshold amount from the mean of all of the gain values and phase values that are more than a defined threshold amount from the mean of all of the phase values.

The post-processing function 28 determines whether there are more Rx antenna element pairs to be processed (step 410). If so, the processes returns to step 400 and is repeated for the next Rx antenna pair. Once all of the Rx antenna element pairs have been processed, the post-processing function 28 normalizes the average relative gain values &G _RXnRXm and the average relative phase value Δφ _{ΚΧηΚΧηι} with respect to a single Rx antenna element, which is referred to as a reference Rx antenna element (step 412). The normalized gain values are the gain calibration values G _RXn , and the normalized phase values are the phase calibration values φ _ΚΧη .

It is important to note that, in the example above, it is assumed that all Tx pairs for all Rx pairs satisfy both symmetrical coupling scenario (3) and symmetrical coupling scenario (4) and that both &G _RXnRXm (3) and A0 _RXnRXm (3) and &G _RXnRXm (4) and A0 _RXnRXm (4) are computed for each Tx pair. However, the present disclosure is not limited thereto. In some alternative embodiments, for a particular Rx pair Rx _n and Rx _m , each Tx pair having symmetrical coupling properties with respect to the Rx pair Rx _n and Rx _m may satisfy only symmetrical coupling scenario (3) in which case only &G _RXnRXm (3) and A0 _RXnRXm (3) are computed for that Tx pair, satisfy only symmetrical coupling scenario (4) in which case only AG _RXnRXm (4) and A0 _RXnRXm (4) are computed for that Tx pair, or satisfy both symmetrical coupling scenario (3) and (4) in which case both &G _RXnRXm (3) and A0 _RXnRXm (3) and G _RXnRXm (4 and A0 _RXnR¾m (4) are computed for that Tx pair. In some other alternative embodiments, the post-processing function 28 may consider only symmetrical coupling scenario (3) in which case only &G _RXnRXm (3) and Δφ _{ΚΧηΚΧηι} (3) are computed. In some other alternative embodiments, the post-processing function 28 may consider only symmetrical coupling scenario (4) in which case only AG _RXnRXm (4) and A0 _RXnR¾m (4) are computed.

Note that in addition to or as an alternative to computing and storing the phase calibration values φ _Τχ . and the gain calibration values G _Tx . for Tx calibration and the phase calibration values φ _ΚΧη and the gain calibration values G _RXn for Rx calibration, the self-calibration subsystem 22 may compute and store a coupling matrix. The coupling matrix stores the phase coupling value Φατχ _{ί Χ]} and gain G _CTx . _Rx . for each pair of Tx and Rx antenna elements Tx, and Rxj.

Specifically, the values φ _Τχ . and G _Tx . are known from Tx calibration and the values φ _Κχ . and G _Rx are known from Rx calibration as performed above. Then, the coupling values φατχ _Χ] and G _CTx . _Rx . can be computed as:

GcTXiRxj ⁼ ^MTXiRxj ^~ G _Tx . — G _Rx .

These coupling values can be stored and thereafter used by the transceiver 10 for, e.g., self-calibration.

Figures 6A and 6B illustrate one example of multiple different Rx pairs having symmetrical coupling properties with respect to a particular Tx antenna element pair that can be used for Tx calibration in accordance with the process of Figure 5A. In a similar manner, multiple different Tx pairs having symmetrical coupling properties with respect to a particular Rx antenna element pair that can be used for Rx calibration in accordance with the process of Figure 5B can be determined.

While any suitable process can be used to determine pairs of antenna elements having symmetrical coupling properties, one example is as follows. For Tx calibration, assume a square array of antenna elements where all antenna elements are the same distance from their neighboring antenna elements in the x and y direction. Let Tx, be at a known position a on the x-axis and b on the y- axis (referred to as position (ax,by)). Let Tx _k be at a known position c on the x- axis and d on the y-axis (referred to as position (cx,dy)). If Rx _n is at a known position p on the x-axis and q on the y-axis (referred to as position (px,qy)), then the position for Rx _m (referred to as rx, sy) to provide symmetrical coupling properties can be computed as:

rx = cx + ax— px,

sy = dy + by - qy

where rx is the position of Rx _m on the x-axis and sy is the position of Rx _m on the y-axis and all values of x and y should be within the range of possible values considering the square array of antenna elements. Additional Rx pairs for the same Tx pair can be computed by first selecting a new position for Rx _n and then computing the position for Rx _m .

Note that the example above considers distance between a pair of Tx antenna elements and a pair of Rx antenna elements when determining a pair of Rx antenna elements having symmetrical coupling properties with respect to a pair of Tx antenna elements (and vice versa). However, in some embodiments, other factors may additionally or alternatively be considered. For example, for co-polarized coupling, the coupling is stronger in one direction than the other and the coupling in the diagonal direction is low. As another example, for cross- polarization, the coupling in the diagonal direction is stronger than for co- polarized coupling.

Figures 7A and 7B are example implementations of the processes of Figures 5A and 5B, respectively. As illustrated in Figure 7A, for Tx calibration, the self-calibration subsystem 22 initiates measurement for self-calibration, e.g., selects a subarray of the antenna array (step 500). The self-calibration subsystem 22 gets a frequency channel, AAS(s) 20, and TX polarization to calibrate (steps 502 through 506). For the first iteration, the post-processing function 28 initializes a first Tx antenna element counter / ^' and a second Tx antenna element counter k to, in this example, a value of 0. The post-processing function 28 increments (i.e., counts) the first Tx antenna element counter / ^' and then determines whether the first Tx antenna element counter / ^' is less than or equal to the total number of Tx antenna elements in the antenna array (step 508). If not, the process proceeds to step 528, as discussed below. If the first Tx antenna element counter / ^' is less than or equal to the total number of Tx antenna elements in the antenna array, the post-processing function 28 gets the i-th Tx antenna element Tx, and reads the gain measurements G _MTX . _RX . and the phase measurements for the i-th Tx antenna element Tx, for all Rx antenna elements Rx _j (for j 1 ... N _RX where N _Rx is the number of Rx antenna elements) (steps 510 and 512).

The post-processing function 28 increments (i.e., counts) the second Tx antenna element counter k and then determines whether the second Tx antenna element counter k is less than or equal to the total number of Tx antenna elements in the antenna array (step 514). If not, the process returns to step 508 where the first Tx antenna element counter / ^' is incremented and the process is repeated. If the second Tx antenna element counter k is less than or equal to the total number of Tx antenna elements in the antenna array, the post-processing function 28 gets the k-th Tx antenna element Tx _k and reads the gain

measurements G _MTXKRX . and the phase measurements <p _M Tx _k Rx _} for the k-th Tx antenna element Tx _k for all Rx antenna elements Rx _j (for j 1 ... N _Rx where N _Rx is the number of Rx antenna elements) (steps 516 and 518).

The post-processing function 28 then finds, or determines, one or more pairs of Rx antenna elements Rx _n and Rx _m , but preferably multiple pairs of Rx antenna elements Rx _n and Rx _m , having symmetrical coupling properties with respect to the pair of Tx antenna elements Tx, and Tx _k , as discussed above (step 520). As discussed above, for Tx calibration, the pairs(s) of Rx antenna elements Rx _n and Rx _m having symmetrical coupling properties are those Rx antenna pairs for which one or both of the following coupling scenarios are satisfied:

• CTxjRxn - CTxkRxn and CTxiRx _m - CTx _k Rx _m (referred to as

coupling scenario (1 ) above)

• CTxjRxn - CTxkRxm and CTxiRx _m - CTx _k Rx _n (referred to as

coupling scenario (2) above).

As discussed above, for each determined pair of Rx antenna elements Rx _n and Rxm, the post-processing function 28 computes:

· relative gain value AG _Tx . _TXk (l) in accordance with Equation (2) above and/or relative gain value AG _Tx . _TXh (2) in accordance with Equation (5) above, and

• relative phase value Δφ _ΤΛ .. _τ¾ (1) in accordance with Equation (9) above and/or relative phase value Δφ _Τχ . _ΤΧ]( (2) in accordance with Equation (12) above,

assuming both gain and phase Tx calibration (step 522).

As also discussed above, the post-processing function 28 performs averaging of the relative gain values and the relative phase values for the pair of Tx antenna elements Tx, and Tx _k across all of the determined pairs of Rx antenna elements Rx _n and Rx _m (step 524). By doing so, the post-processing function 28 computes an average relative gain AG _Tx . _TXh for the gain of Tx antenna element Tx, relative to the gain of the Tx antenna element Tx _k and an average relative phase value Δφ _Τχ . _{ΤΧ ι} for phase of the Tx antenna element Tx, relative to the Tx antenna element Tx _k . The post-processing function 28 saves the average relative gain AG _Tx . _TXh and the average relative phase value Δφ _Τχ . _{ΤΧ ι} for the Tx antenna element Tx, relative to the gain of the Tx antenna element Tx _k (step 526).

At this point, the process returns to step 514 where the second Tx antenna element counter k is incremented and the process is repeated. In this manner, average relative gain AG _Tx . _TXk and average relative phase Δφ _Τχ . _{ΤΧ ι} values are computed for the Tx antenna element Tx, relative to all other Tx antenna elements (i.e., for all Txk for k≠ i). Once this is done, the process returns to step 508 and is repeated where the first Tx antenna element counter / ^' is incremented and the process is repeated. In this manner, average relative gain AG _Tx . _TXk and average relative phase Δφ _Τχ . _ΤΧ]( values are computed for all Tx antenna element combinations.

Once average relative gain AG _Tx . _TXk and average relative phase Δφ _Τχ . _ΤΧ]( values are computed for all Tx antenna element combinations, the postprocessing function 28 normalizes the average relative gain AG _Tx . _TXk and the average relative phase Δφ _Τχ . _ΤΧ]( values relative to a single reference Tx antenna element (step 528). Normalization results in the gain calibration values G _Tx . and the phase calibration values φ _Τχ . for each Tx antenna element Tx, (for all ί ε l ... N _Tx ).

As illustrated in Figure 7B, for Rx calibration, the self-calibration

subsystem 22 initiates measurement for self-calibration (step 600). The self- calibration subsystem 22 gets a frequency channel, AAS(s) 20, and TX

polarization to calibrate (steps 602 through 606). For the first iteration, the postprocessing function 28 initializes a first Rx antenna element counter n and a second Rx antenna element counter m to, in this example, a value of 0. The post-processing function 28 increments (i.e., counts) the first Rx antenna element counter n and then determines whether the first Rx antenna element counter n is less than or equal to the total number of Rx antenna elements in the antenna array (step 608). If not, the process proceeds to step 628, as discussed below. If the first Rx antenna element counter n is less than or equal to the total number of Rx antenna elements in the antenna array, the post-processing function 28 gets the n-th Rx antenna element Rx _n and reads the gain measurements

G _MTx . _RXn and the phase measurements for the n-th Rx antenna element Rx _n for all Tx antenna elements Tx, (for ί ε 1 ... N _Tx where N _Tx is the number of Tx antenna elements) (steps 610 and 612). The post-processing function 28 increments (i.e., counts) the second Rx antenna element counter m and then determines whether the second Rx antenna element counter m is less than or equal to the total number of Rx antenna elements in the antenna array (step 614). If not, the process returns to step 608 where the first Rx antenna element counter n is incremented and the process is repeated. If the second Rx antenna element counter m is less than or equal to the total number of Rx antenna elements in the antenna array, the postprocessing function 28 gets the m-th Rx antenna element Rx _m and reads the gain measurements G _MTx . _RXm and the phase measurements <p _M TxiRx _m f° ^r the m-th Rx antenna element Rx _m for all Tx antenna elements Tx, (for ί ε 1 ... N _Tx where N _Tx is the number of Tx antenna elements) (steps 616 and 618).

The post-processing function 28 then finds, or determines, one or more pairs of Tx antenna elements Tx, and Tx _k , but preferably multiple pairs of Tx antenna elements Tx, and Txk, having symmetrical coupling properties with respect to the pair of Rx antenna elements Rx _n and Rx _m , as discussed above (step 620). As discussed above, for Rx calibration, the pairs(s) of Tx antenna elements Tx, and Tx _k having symmetrical coupling properties are those Tx antenna pairs for which one or both of the following coupling scenarios are satisfied:

· CTxjRxn - CTxjRxm and CTx _k Rx _n - CTx _k Rx _m (referred to as

coupling scenario (3) above)

• CTxjRxn - CTxkRxm and CTxiRx _m - CTx _k Rx _n (referred to as

coupling scenario (4) above).

As discussed above, for each determined pair of Tx antenna elements Tx, and Tx _k , the post-processing function 28 computes:

• relative gain value &G _RXnRXm (3) in accordance with Equation (16) above and/or relative gain value &G _RXnRXm (4) in accordance with Equation (19) above, and • relative phase value A0 _RXnRXm (3) in accordance with Equation (23) above and/or relative phase value A0 _RXnRXm (4) in accordance with Equation (26) above,

assuming both gain and phase Rx calibration (step 622).

As also discussed above, the post-processing function 28 performs averaging of the relative gain values and the relative phase values for the Rx antenna elements Rx _n and Rx _m across all of the determined pairs of Tx antenna element Tx, and Tx _k (step 624). By doing so, the post-processing function 28 computes an average relative gain G _RXnRXm for the gain of Rx antenna element Rx _n relative to the gain of the Rx antenna element Rx _m and an average relative phase value ^ _RXnRXm for phase of the Rx antenna element Rx _n relative to the Rx antenna element Rx _m . The post-processing function 28 saves the average relative gain AG _RXnRXm and the average relative phase value (p _RXnRXm for the Rx antenna element Rx _n relative to the gain of the Rx antenna element Rx _m (step 626).

At this point, the process returns to step 614 where the second Rx antenna element counter m is incremented and the process is repeated. In this manner, average relative gain G _RXnRXm and average relative phase (p _RXnRXm values are computed for the Rx antenna element Rx _n relative to all other Rx antenna elements (i.e., for all Rx _m for m≠ n). Once this is done, the process returns to step 608 and is repeated where the first Rx antenna element counter n is incremented and the process is repeated. In this manner, average relative gain AG _RXnRXm and average relative phase k(p _RXnRXm values are computed for all Rx antenna element combinations.

Once average relative gain AG _RXnRXm and average relative phase

^Rx _n R _Xm values are computed for all Rx antenna element combinations, the post-processing function 28 normalizes the average relative gain &G _RXnRXm and the average relative phase Δφ _{ΚΧηΚΧηι} values relative to a single reference Rx antenna element (step 628). Normalization results in the gain calibration values G _RXn and the phase calibration values <fi _RXn for each Rx antenna element Rx _n (for ali n e l ... N _Rx ).

Figure 8 illustrates the operation of the transceiver 10, and in particular the self-calibration subsystem 22, to configure the gain and phase adjustment circuitry 18 according to some embodiments of the present disclosure. More specifically, upon computing the gain and phase calibration values as described above, the self-calibration subsystem 22 configures the gain and phase adjustment circuitry 18 to provide the corresponding gain and phase

adjustments. In some embodiments, the self-calibration subsystem 22 configures the gain and phase adjustment circuitry 18 by configuring registers with values that provide the computed gain and phase calibration adjustments. However, in order to do this, the computed gain and phase values must be mapped to the corresponding register values. In this regard, Figure 8 illustrates a process by which the transceiver 10, and in particular the self-calibration subsystem 22, maps the computed gain and phase calibration values to the corresponding register values and configures the gain and phase adjustment circuitry 18 with those register values.

More specifically, the self-calibration subsystem 22 gets, or obtains, the frequency channel for which calibration is desired (step 700) and determines whether a mapping of the computed gain and phase calibration values for the frequency channel to corresponding register values is available, e.g., in a LUT, in this example (step 702). If available, the self-calibration subsystem 22 gets the register values corresponding to the computed gain and phase calibration values for the Tx and Rx antenna elements from the LUT (step 704). The self- calibration subsystem 22 then configures the gain and phase adjustment circuitry 18 by, in this example, setting respective registers to the register values obtained from the LUT (step 706).

Returning to step 702, if a mapping of the computed gain and phase calibration values for the desired frequency channel to register values is not available, the self-calibration subsystem 22 performs a procedure to determine the mapping between the computed gain and phase calibration values and the corresponding register values. More specifically, in step 708, the self-calibration subsystem 22 selects any Tx element Txj, selects any Rx element Rx _m , and measures <l> _MTx . _Rx X) = Φτ _Χ] ( +≠cT _Xj Rx _m +≠Rx _m - The self-calibration subsystem 22 changes the phase of Tx gradually to get new measurements by changing the phase register value (when there is no prior knowledge of register delta phase relation)

4>MTXjRx _m (nQ = 4>TXj(nQ + (PcTXjRXm +≠Rx _m -

The self-calibration subsystem 22 compares </> _M TxjRx _m (O ^~ 0 Tx,Rx _m ( ⁿ O to <l>Txj,REF> which is the desired phase calibration value for Txj. When

<p _MT xjRx _m (.O - (pMTxjRx _m nQ == 4*Txj,REF> the self-calibration subsystem 22 stops changing the phase register value, gets the phase register value, and stores the phase register value (step 710).

In a similar manner, in step 708, the self-calibration subsystem 22 measures G _MTx . _Rx X) = G _Tx .{l) + G _CTXjRXm + G _RXm . The self-calibration subsystem 22 changes the gain of Tx gradually to get new measurements by changing the gain resistor value (when there is no prior knowledge of resistor delta gain relation)

^MTxjRx _m ^0 ⁼ G _Tx .(ni) + G _CTx . _RXm + G _RXm .

The self-calibration subsystem 22 compares G _MTx . _RXm (l - G _MTx . _RXm (ni to

G _{Tx REF} , which is the computed gain adjustment value for Tx _j . When

uMTXjRx _m (i) - ^u MTXjRx _m (ni) == G _{Tx REF} , the self-calibration subsystem 22 stops changing the gain register value, gets the gain register value, and stores the gain register value (step 710). The process of steps 708 and 710 is repeated to compute the gain and phase register values for each Tx and each Rx antenna element.

The self-calibration subsystem 22 then configures the gain and phase adjustment circuitry 18 by, in this example, setting respective register values to the values determined in steps 710 and 712 (step 708). Figure 9 illustrates the analog beamforming transceiver 10 according to some other embodiments of the present disclosure. In this embodiment, the analog beamforming transceiver 10 includes a number of modules 30 that operate to provide self-calibration according to any one of the embodiments described herein. In this particular example, the modules 30 include a signal generating module 30-1 that operates to provide the functionality of the signal generator 24 as described herein, a measuring module 30-2 that operates to provide the functionality of the measurement function 26 as described herein, and a post-processing module 30-3 that operates to provide the functionality of the post-processing function 28 as described herein. Each of the modules 30 is implemented in software.

In the embodiments above, the post-processing of gain and phase measurements is performed locally at the analog beamforming transceiver 10. However, in some alternative embodiments, the post-processing of the measurements is performed remotely by some other processing system. In this regard, Figure 10 illustrates a system including the transceiver 10 and a remote processing system 32 according to some other embodiments of the present disclosure. The transceiver 10 is the same as that described above other than the post-processing function 28. In this embodiment, the post-processing function 28 is implemented at the remote processing system 32. The remote processing system 32 is implemented in a combination of hardware and software. For example, the remote processing system 32 may include one or more processors (e.g., CPUs, DSPs, ASICs, FPGAs, and/or the like) and memory storing software executed by the processor(s) whereby the remote processing system 32 operates to provide the functionality of the post-processing function 28 as described herein. The remote processing system 32 includes a communication interface (e.g., a wired or wireless network interface) that communicatively couples to the transceiver 10. As one example, the transceiver 10 may be part of a radio access node (e.g., a base station) in a cellular communications system, and the remote processing system 32 may be, e.g., another network node such as, e.g., a core network node in a core network of the cellular communications system.

Figure 1 1 is a flow chart that illustrates the operation of the remote processing system 32 and, in particular, the post-processing function 28 of Figure 10 according to some embodiments of the present disclosure. In general, when implemented at the remote processing system 32, the post-processing function 28 operates in the same manner as described above other than having to obtain the gain and phase measurements from the transceiver 10 via a remote connection and returning the gain and phase calibration values to the transceiver via the remote connection.

More specifically, as illustrated in Figure 1 1 , the remote processing system 32, and in particular the post-processing function 28 implemented at the remote processing system 32, obtains pair-by-pair gain and phase

measurements for all pairs of Tx and Rx antenna elements in the antenna array formed by the one or more AASs 20 at the transceiver 10 and stores the resulting measurements (step 800A). In other words, for each pair of Tx and Rx antenna elements Tx, and Rx _j where I = 1 ...Ν _Τ χ and j = 1 ...N _RX and Ν _Τ χ is the number of Tx antenna elements in the antenna array and N _RX is the number of Rx antenna elements in the antenna array, the post-processing function 28 obtains a gain measurement (G _MTx . _Rx .) and a phase measurement from the

transceiver 10. As discussed above, these gain and phase measurements are performed by the measurement function 26 at the transceiver 10. The details of how the gain and phase measurements G _MTx . _Rx . and are performed by the measurement function 26 are provided above with respect to Figure 4.

The post-processing function 28 performs post-processing of the gain and phase measurements G _MTx . _Rx . and for all i and j to compute gain and phase calibration values for the Tx and Rx antenna elements of the antenna array (step 800B), as described above. The details of step 800B are provided above with respect to Figures 5A and 5B and Figures 7A and 7B. In general, the post-processing function 28 computes the gain and phase calibration values based on the gain and phase measurements G _MTx . _Rx and _Μ τ _Χί χ, and combinations of Tx and Rx antenna elements having symmetrical coupling properties. The combinations of Tx and Rx antenna elements having

symmetrical coupling properties are, e.g., predefined or predetermined based on, e.g., a known layout of the antenna elements in the antenna array or computed by the post-processing function 28 based on, e.g., a known layout of the antenna elements in the antenna array.

The post-processing function 28 provides the computed gain and phase calibration values to the transceiver 10 (step 802), where the computed gain and phase calibration values are applied at the transceiver 10 as described above. As discussed above, in addition or alternatively, the post-processing function 28 may compute a coupling matrix and provide the coupling matrix to the transceiver 10.

The following acronyms are used throughout this disclosure.

• 3G Third Generation

• 3GPP Third Generation Partnership Project

• 5G Fifth Generation

• AAS Advanced Antenna System

• ASIC Application Specific Integrated Circuit

• CPU Central Processing Unit

• dB Decibel

• DSP Digital Signal Processor

• FPGA Field Programmable Gate Array

• LTE Long Term Evolution

• LUT Look Up Table

• NR New Radio

• RFIC Radio Frequency Integrated Circuit

• Rx Receive

• TDD Time Division Duplexing

• Tx Transmit • UE User Equipment

Those skilled in the art will recognize improvements and modifications to the embodiments of the present disclosure. All such improvements and modifications are considered within the scope of the concepts disclosed herein.