Method and Apparatus for Channel Estimation in a MIMO-OCDM System

Field

The present application is directed towards a channel estimation method for acquiring channel state information (CSI) for multiple-input multiple-output (MIMO) systems.

Background

Multiple-input multiple-output (MIMO) is an antenna technology which multiplies the capacity of a wireless system by elegantly exploiting the multipath transmission of a radio link. MIMO has been already proved as the one of the most successful techniques in broadband communications, such as, WiFi, 5G and beyond. In MIMO system, channel estimation is the most crucial functional block required to track the multi-antenna transmission channel to ensure the reliable recovery of the high-speed signals. By incorporating other advanced techniques with MIMO, high-speed communications can be realized for not only wireless systems but also other types of systems, such as power-line and fiber- optic systems. For example, the combination of MIMO and orthogonal frequency- division multiplexing (OFDM) has been proven the most successful solution in the 4G and recent 5G mobile networks as the air interface to support high-speed, low-latency wireless access to broadband services.

Recently, orthogonal chirp-division multiplexing (OCDM) has been proposed as an advanced modulation technique, and been both theoretically and experimentally demonstrated as a promising solution for high-speed communications. In contrast to the OFDM that modulates a stream of high-speed data onto a large number of narrowband subcarriers parallel in frequency, OCDM instead multiplexes a large group of linearly frequency-modulated (LFM) waveforms, termed as chirps, for high-speed data modulation.

OCDM is in essence a chirp spread spectrum (CSS) technique that achieves the Nyquist signalling rate, just as the OFDM to frequency-division multiplexing (FDM) signal. Compared to the traditional CSS techniques, which are notorious for their poor spectral efficiency because of the non-orthogonal chirped waveforms, the chirps in OCDM signal are mutually orthogonal, and thus attain the maximum spectral efficiency in terms of the Nyquist signalling rate. On the other hand, by virtue of the spread-spectrum feature inheriting from CSS, OCDM shows superior resilience in combating the detrimental effects in the communication systems, and outperforms other waveform modulation techniques, such as, OFDM. As a result, one can expect that the MIMO-OCDM combination can offer a more appealing physical layer solution for future broadband systems, such as the beyond 5G and 6G mobile networks, and wireless local area networks (WLAN), providing higher data rate and better reliability.

In MIMO-based systems, channel estimation is of crucial importance to guarantee the reliable recovery of the high-speed MIMO signals. Therefore, the present disclosure is focused on channel estimation and, in particular, pilot-based channel estimation schemes.

Blind channel estimation schemes do not need pilot symbols for training. However, without the use of pilot symbols, it takes a longer time for the system to converge on an accurate estimate of channel state information (CSI). In addition, systems using blind channel estimation schemes are susceptible to channel impairments, especially in MIMO scenarios.

The CSI estimation algorithms proposed for the MIMO-OFDM systems, can be adapted for the MIMO-OCDM systems thanks to the compatibility of OCDM and OFDM. However, there exist some drawbacks in the traditional CSI algorithms for the MIMO-based systems. These drawbacks are transposed to OCDM systems when traditional CSI estimation algorithms are used with OCDM systems. For example, the dimensions of pilot matrices should be no less than the number of transmit antennas to reconstruct the full rank of the transfer matrices of the MIMO system. The pilots should be transmitted over multiple OFDM/OCDM symbols. As a result, there needs to be redundancy in pilot allocation, or complicated algorithms need to be used to recover the CSL

To put it differently, pilot symbols should not overlap in either the time domain or the frequency domain. This is essential so that pilot symbols from different transmit (Tx) antennas can be distinguished at the receiver. In other words, when one Tx antenna is transmitting a pilot symbol, silent pilots that contain no symbol information should be transmitted by the other Tx antennas in either the time domain or the frequency domain. However, silent pilots reduce the spectral efficiency and the estimation accuracy of the estimated CSI is limited.

CN1980201 discloses performing channel estimation for a MIMO-OFDM signal, in that the transmit antennas transit pilot signals in the form of linear frequency modulation (chirp) signals. Herein, the chirped pilots are set on even number OFDM subcarriers in the frequency domain, and zeros are set on the odd number OFDM subcarriers, there creating a time domain shifted copy. However, as a result, in the frequency domain, the odd number OFDM subcarriers cannot estimate any information, which means it loses half of the information of the channel when performing channel estimation. In addition, in the time domain, as a result of the zeros on the odd number OFDM subcarriers, the shift in time domain has to be fixed to A//2/Vr with certain limitations as number of antenna increase. Equivalently, the loss of half of the information in the frequency domain, means half of the time-domain information is lost. This makes the pilot recovering and channel estimation at the receiver becomes quite different and complicated. Although, CN1980201 avoids the silent symbol in traditional MIMO-OFDM channel estimation algorithms, it only partially guarantees the orthogonality on the even number of frequency subcarriers in the frequency domain, and also loses half of the pilot for channel estimation, which equivalently reduces the bandwidth utilization for pilot by half.

The document titled “Channel estimation for MIMO space time coded OTFS under doubly selective channels” by Bomfin Roberto et al. discloses a MIMO transmission scheme in which data is transmitted using OTFS, but each data frame is bracketed by a “unique word”. The unique word may be regarded as a chirped pilot signal. However, for the complexity involving matrix operation, said method is not practical in real system. Also, the channel estimation disclosed in said document is only for CDD systems, which is equivalent to a single-input multiple output systems rather than Ml MO-based systems.

Another document titled as “A Robust Baseband transceiver design for doubly dispersive channels” by Bomfin Roberto et al. discloses three different concepts for robust link-level performance under doubly-dispersive wireless channels, namely, i) channel estimation, ii) cyclic prefix (CP)-free transmission, and iii) waveform design. A unique word-based channel estimation is employed, where the channel related errors are decoupled into channel estimation error (CEE) and Doppler error (DE). A polyphase sequence, called Zadoff-Chu sequence, is used, just as in conventional 4G/5G systems, and the channel estimation is done in the frequency domain. However, the Zadoff-Chu sequence does not incorporate the property and advantages of the Fresnel transform, due to that there is no pulse compression. Also, in this way, the frequency domain average problem still exist.

It is an object therefore to provide an improved channel estimation method for acquiring CSI at the receiver for MIMO systems to overcome at least one of the above-mentioned problems.

Summary

The present disclosure is directed towards methods, transmitters, receivers and computer readable storage mediums, the features of which are set out in the appended claims. In particular, the present disclosure is directed towards a method of estimating channel state information (CSI) of a MIMO channel. The method comprises: transmitting pilot symbols from a plurality of transmission antennas, wherein: a number of pilot symbols are transmitted using the chirped signals based on Fresnel transform for generating OCDM signals at a plurality of receiver antennas, the number of pilot symbols overlap in time-domain, and a minimum of single symbol interval is sufficient to estimate the transfer matrix of the MIMO system. Preferably no silent pilot symbols are transmitted. The present application is directed towards solving the above problems through using chirped pilot symbols are used for channel acquisition in Ml MO-based systems. In particular, the present disclosure is directed towards providing improved channel estimation methods and systems for MIMO-based systems which improve the spectral efficiency of channel estimation and, in addition, improve the estimation accuracy thereby providing better performance.

The channel estimation methods and systems of the present invention employ the orthogonal chirps based on the Fresnel transform, and choose several chirps without the limitation of the cyclic shift of A//2/Vr. The cyclic shift limitation is independent of the number of TX antennas, /Vr. Rather, the channel estimation method of the present invention is extremely flexible. Moreover, the chirps are orthogonal in both time and frequency domain. In other words, in the frequency domain, there is no zeros setting on even or odd frequency subcarriers and no interpolation is needed. In the time domain, all the received pilot signals may be used for pilot recovery and channel estimation without discarding any useful information, which means no bandwidth utilization reduction at all. The transmitter of the present invention choses the pilot signal, and stores in memory and cyclic shift. The receiver of the present invention transform the received pilot by a Fresnel transform per antenna and extract the MIMO channel state information, thereby facilitating simpler pilot generation, pilot recovery and channel estimation in contrast to prior art systems that requires separate IFFTs at the transmitter and receivers.

Preferably, the chirped signals used to transmit the number of pilot symbols are linear frequency modulated signals that are orthogonal to each other based on the Fresnel transform.

The method optionally also comprises selecting a pilot symbol of each transmission antenna of the plurality of transmission antennas, whereby the pilot symbols are distinguishable upon reception with pulse compressed operation, and wherein the pilot symbols when are overlapped in both time and frequency domains.

Preferably, each chirped signal used to transmit a pilot symbol is a cyclic shift of another chirped signal used to transmit another pilot symbol. More preferably, the method also comprises performing a cyclic shift on a chirped signal, wherein performing a cyclic shift comprises performing either a discrete Fresnel transform (DFnT) or an inverse discrete Fresnel transform (IDFnT).

Preferably, the number of pilot symbols are transmitted simultaneously from all the plurality transmit antennas in a single time interval over the full bandwidth of the channel.

The present application also discloses a method of estimating channel state information (CSI) of a MIMO channel comprising receiving a number of pilot symbols that were transmitted using chirped signals from the plurality of transmission antennas, wherein the number of pilot symbols overlap in time- domain, and no silent pilot symbols are received.

Preferably, the method also comprises performing an inverse cyclic shift on the received chirped signals. More preferably, the inverse cyclic shift is performed using either a discrete Fresnel transform (DFnT) or an inverse discrete Fresnel transform (IDFnT).

The present application is also directed towards a transmitter for estimating channel state information (CSI) of a MIMO channel comprising: a plurality of transmission antennas configured to transmitting pilot symbols, wherein: the transmitter is configured to encode the pilot symbols into chirped signals; and the transmitter is configured to transmit pilot symbols such that they overlap in time- domain, and no silent pilot symbols are transmitted.

Preferably, the transmitter is configured to perform a cyclic shift on a chirped signal, wherein performing a cyclic shift comprises choosing suitable vectors from either a discrete Fresnel transform (DFnT) or an inverse discrete Fresnel transform (IDFnT) matrix.

A receiver is also provided. The receiver is for estimating channel state information (CSI) of a MIMO channel and comprises: a plurality of receivers configured to receive a number of pilot symbols that were transmitted using chirped signals from a plurality of transmission antennas, wherein the number of pilot symbols overlap in time-domain, and no silent pilot symbols are received.

Preferably, the receiver is configured to perform a pulse compression on a plurality of overlapped chirped signals, which includes performing either a discrete Fresnel transform (DFnT) or an inverse discrete Fresnel transform (IDFnT).

A computer readable storage medium is also provided. The computer readable storage medium comprises instructions which, when executed by a processor operatively couple to a plurality of antenna, cause the processor to perform at least one of the methods from this summary of the invention described above.

Further, a mobile computing device comprising either one of the transmitters describe above, one of the receivers described above, or both.

Brief Description of the Drawings

The invention will be more clearly understood from the following description of an embodiment thereof, given by way of example only, with reference to the accompanying drawings, in which:-

Figure 1 is a block diagram of a preferred spatial multiplexing (SM) MIMO system;

Figure 2 illustrates a set of chirped pilots;

Figure 3a shows a one-shot observation of received pilot signals;

Figure 3b shows recovered CFR functions of the MIMO system;

Figure 4 shows the theoretical MSE Vs SNR performance for different numbers of TX antennas; Figure 5 is a pair of 2D plot diagrams illustrating the performance of a channel estimator;

Figure 6 illustrates the performance of the proposed estimator against the window width;

Figure 7 shows a performance comparison of an estimator in accordance with the present disclosure and a prior art SAW estimator; and

Figure 8 shows MSE performance as a function of the received SNR for both the estimator of the present disclosure and an LS estimator with SAW algorithm.

Detailed Description of the Drawings

As noted above, the present application is directed towards solving the above problems through using chirped pilot symbols based on Fresnel transform for channel acquisition in Ml MO-based systems. Preferably, linear-frequency modulated signals that are orthogonal to each other are used to transmit the pilot symbols. The method exploits the time-frequency properties of the chirped symbols. According to the present disclosure, the orthogonal chirped pilot symbols are transmitted simultaneously from the transmit antennas (preferably all the transmit antennas) in a single time interval. Thus, the chirped pilot symbols from different transmit antennas overlap in both time and frequency domains. In other words, all the chirped pilot symbols occupy a single time interval over the entire bandwidth. By carefully choosing and assigning the orthogonal chirped pilot symbols for different transmit antennas, the pilot symbols are distinguishable at the receiver. Thus, the transfer matrix of the MIMO channel can be estimated accordingly. As a result, according to the present disclosure the CSI of a MIMO channel is acquired within a single symbol interval. Thus, the spectral efficiency of the system is significantly improved. Furthermore, the CSI estimation accuracy is also improved due to the unique time-frequency property of the chirped pilots.

According to the present disclosure, a set of chirped pilot symbols is chosen from the Fresnel basis and assigned to the transmit antennas for channel estimation, without inserting any silent symbols. At the receiver, although the received pilot symbols from various transmit antennas superpose in time and occupy the entire bandwidth, they are distinguishable thanks to the orthogonality of the chirps and the convolution preservation property of the Fresnel transform.

As a result, the transfer matrices of the MIMO system can be easily estimated and used for signal recovery after pulse compression. This CSI estimation technique avoids the problems related to the bandwidth wastage caused by silent symbols through providing better spectral efficiency (SE). In addition, this CSI estimation technique also improves the estimation accuracy by providing a CSI estimation better tailored than other techniques to the unique features of the chirped pilots.

The rest of this disclosure is organized as follows: in Section I, the system model of the MIMO-OCDM system is disclosed. A channel estimation algorithm according to the present disclosure is introduced in Section II, and the performance of the algorithm is analyzed in comparison with the least-square (LS) algorithm in Section III. Section IV provides the simulation results to study the performance of the proposed algorithm and validate its advantage.

For the avoidance of doubt, normal italic letters are used to denote variables, and boldface lowercase and boldface capital letters for vectors and matrices, respectively. The superscript, (·)*, is complex conjugate operator, and (·) ^T , (·) ^H , and denotes respectively: the transpose, Hermitian transpose, and pseudo- inverse of a matrix. Tr{·} and are respectively the trace and Frobenius norm of a matrix. is the circular convolution. 8 {·} is the expectation or ensemble average operator. is the discrete Fourier transform (DFT), and the discrete Fresnel transform (DFnT).

I. SYSTEM MODEL OF MIMO-OCDM

Figure 1 shows a preferred spatial multiplexing (SM) MIMO system 1000 with M _T transmit (TX) antennas 1 101 and M _R receive (RX) antennas 1201 , where M _R ≥ M _T > 1 . Those skilled in the art will of course recognise that, the channel estimation method and systems of the present application can be readily adapted for other types of MIMO systems. For example, space-time block code (STBC) MIMO, for which M _R < M _T , could also be used.

At the transmitter 1 100, the bit stream is serial-to-parallel (S/P) converted, grouped in blocks, and mapped to symbols for modulation. The symbols are then processed by the MIMO encoder 1 102 and divided into M _T streams 1 103 for OCDM waveform modulation. In comparison to OFDM that generates the time- domain signal by the discrete Fourier transform (DFT), OCDM modulation generates the signal instead using a discrete Fresnel transform (DFnT). This transform can be efficiently realized using fast Fresnel transform (FFnT) algorithms. FFnT algorithms essentially have the same arithmetic complexity as the fast Fourier transform (FFT). Given that each OCDM symbol consists of N chirps, the discrete-time signal on the q-th TX antenna, s _q (n) for q=1,...,M _T , and n = 0,1 - 1 , is generated by an inverse DFnT (IDFnT), as where is the DFnT operator, is the inverse DFnT operator (IDFnT), and x _q (k) is the symbol modulating the -th chirp on the q-th TX antenna, for k = 0,...,N -1.

Guard intervals (GIs) are inserted between each OCDM symbol to avoid inter- symbol interference ( IS I) from adjacent symbols caused by the spread of differing delays which are typical of multipath systems such as MIMO. Based on the property of DFnT, the Gl should be in the form of a cyclic prefix (CP) to maintain circular-convolution. In practice, the length of the CP should be sufficiently larger than the maximum delay spread (i.e. the difference in propagation times between the fastest propagation path and the slowest propagation path) of the channel. The baseband signals are then parallel-to-serial (P/S) converted and up- converted for transmission.

The transmitted signals will go through the wireless MIMO channel with attenuation, reflection, and scattering, and arrive at the receiver 1200. In order to simplify the design of the system, it is assumed that the MIMO channel is quasistatic, i.e., the state of the channel remains unchanged within one frame and varies from frame to frame. At the output of the MIMO system, the received signals are down-converted back to the baseband and sampled to digital domain by analog-to-digital converters (ADCs). After synchronization, the OCDM signals are S/P converted with the GIs removed, and grouped into blocks. The received signal on the p-th RX antenna, for p = 1 ,...,M _R , is the superposition of the transmitted signals, where h _p , _q (n) is the channel impulse response (CIR) function of the path from the q-th TX antenna to the p-th RX antenna, v _p (n) is the additive noise on the p-th RX antenna, and © is the circular convolution operator. It should be noted in this equation that the circular convolution results from the effect of CP, which converts the linear convolution to circular convolution.

The single-tap frequency-domain equalization (FDE) can be adapted to efficiently recover the MIMO-OCDM signals. The received signal on the p-th RX antenna is transformed by a DFT to the frequency domain, as and is the channel frequency response (CFR) function from the q-th TX antenna to the p-th RX antenna, wΩ( _P )(n) is the frequency domain additive noise on the p- th RX antenna, and T*(m) is a phase coefficient, which is in fact the eigenvalue of DFnT with respect to (w.r.t.) DFT.

The third equation in (3) is arrived by using the Eigen-decomposition identity of the DFnT, i.e., where Γ* (m) is the m-th eigenvalue of IDFnT w.r.t DFT.

To facilitate the representation, we formulate the system in matrix form. Stacking the received signal in (3) w.r.t. p, the received signal vector in the m-th frequency bin is Where is the received frequency-domain signal vector, and j _s the DFT of transmitted symbol vector in the m-th frequency bin with its q-th element defined as is the channel transfer matrix, and j _s the frequency-domain noise vector.

Based on Eq. (5), once the CFR matrix, A(m), is estimated by some channel estimation method, the effect of the channel imposed on the received signal can be compensated. The phase Γ* (m) can be easily rotated back as it is a known scalar. For example, if a linear equalizer is adopted, the equalized signal is where =(m) is the M _T xM _R equalization matrix on the m-th frequency bin. If zero- forcing (ZF) criterion is adopted, and if MMSE criterion is adopted, where i.e. p is the signal-to-noise ratio (SNR).

After channel equalization, the transmitted symbols on each layer can be readily recovered with another IDFT. Taking the ZF equalizer for example, the symbols on the q-th TX antenna for decision are: where is the q-th entry of the equalized noise vector in the frequency domain. It should be noted that although the ZF equalizer can completely compensate the channel, the noise in the vicinity of frequency nulls will be severely enhanced. The MMSE equalization can effectively alleviate the noise enhancement problem as the MMSE coefficients approach to the matched filter in low SNR regime and the ZF equalizer for high SNR.

II. CHANNEL ESTIMATION ALGORITHM BASED ON ORTHOGONAL CHIRPED PILOTS

In this section, to introduce the proposed channel estimation algorithm, we define a family of orthogonal chirps as where are in fact the column vectors of an N xN IDFnT matrix. The n-th element of is defined as for k = 0, and the rest are the cyclic shift of , as

Ψ _k (n) = Ψ ₀ (n - k) (13)

If we carefully choose a subset of A _Ψ of size-M _T as the pilot signals for channel estimation, the pilots can be simultaneously transmitted over the M _T TX antennas within a single OCDM symbol period. The receiver is able to recover the CSI of the MIMO system from the received pilot signals by utilizing the pulse- compression property of

Suppose that the pilot signal assigned to the q-th TX antenna is where 0 < D _q ≤ N - 1 is the index of the chirp on the q-th TX antenna, the transmitted pilot signals are

Substituting equation (14) into equation (2) yields the received pilot signals, and performing DFnT on the received pilot signal on the p-th RX antenna, we have The second equation is obtained by exploiting the convolution-preservation property of the DFnT that the Fresnel transform of a convolution of two signals equals the Fresnel transform of either one convolving with the other.

Inspecting equation (15), the received pilot on the p-th RX antenna is the superposition of the CIR from all the TX antennas to the pth RX antenna shifted by {D _q }. In the above equations, as well as following discussion, considering the cyclic convolution, the domain of the sequences and functions, , and h _p , _q (n), etc., is a cyclic group of order N. That is, when imposing the sequence domain n with shift operation, one has

(n + D _q ) = (n + D _q mod N.

As the spread of real-world channels is time-limited, the CIR functions h _p,q may still be recoverable if {D _q } are carefully designed. Given that the maximum delay spread of the channel is L _CIR , which is in practice smaller than the length of CP, i.e., L _CIR < L _CP . One can easily prove that for any D _q1 ,D _q2 , where 1 ≤ q1,q2 ≤ M _T and , if

|D _q1 - D _q2 | > L _CI (16) the CIR functions can be recovered without any inter-antenna interference. With the condition in (16), the CIR function from the q-th TX antenna to the p-th RX antenna can be estimated using merely shift operations, as by properly confining n. For simplicity, we consider that {D _q } are in order, i.e., 0 < D1 < ... < D _MT ≤ N, without loss of generality. Even if {D _q } are not in order, we can always find a mapping f _Q : q→ q’, so that {D _q } are in order. Thereby, the estimated

CIR functions are where D _q is the domain of h _p , _q defined as

The P x Q CIR functions can be formulated based on equation (18) to characterize the state of the MIMO channel for signal recovery. For the choice of the chirped pilots, the most intuitive way is to uniformly choose the chirped pilots with

Here a 4 x 3 MIMO system is taken as an example. Figure 2 illustrates a set of chirped pilots 200a, 200b, 200c respectively for the (a) 1 ^st , (b) 2nd, and (c) 3rd TX antennas in a MIMO-OCDM systems with M _T = 3 TX antennas. It can be seen that the chirped pilots 200a, 200b, 200c are cyclic shift of each another. After MIMO transmission, the received pilot signals will be the superposition of the transmitted pilots as indicated in (2). Pulse compression are achieved by performing a DFnT on the received pilot, given in (15). Fig. 3a provides a one- shot observation of the received pilot signals. It can be seen that the received pilots after pulse compression are the superposition of the CIR functions from different TX antennas without interfering each other. Thus, the CIR functions can be readily retrieved based on (17) for signal recovery and channel equalization.

When FDE is adopted, the CFR functions can be obtained by DFTs, as where wΩ´( _p , _q ) (m) is the frequency-domain noise defined as for n ∈ D _p . Thus, the transfer matrix on the m-th frequency bin is formed with its (p,q )-th element to be in (21 ).

In Fig. 3b, the CFR functions of the MIMO system are recovered for illustration.

III. ANALYSIS

In this section, the performance of a channel estimation algorithm in accordance with the present disclosure is compared with the least-square (LS) channel estimator of traditional MIMO-OFDM systems. A discussion about the practical implementation for MIMO-OCDM systems is also provided. Further, an effective noise suppression algorithm is introduced. In order to simplify signal processing, it is assumed that the MIMO channel is linear. Preferably the MIMO channel is also assumed to be quasi-static. More preferably the MIMO channel is also assumed to be stochastic. The additive noises are independent and identically distributed circularly symmetric Gaussian with zero means and variance No.

A. Performance Analysis of an Estimator in accordance with the present disclosure

In this subsection, we first consider the case that the chirped pilots are transmitted using one OCDM symbol. Based on (21 ), the MSE of the proposed estimation is given by where js the noise matrix with its (p,q )-th element to be w _Ω´ (m ) as defined in equation (22). The detailed derivation is provided in Appendix A. Moreover, the chirped pilots can be transmitted over multiple symbols to improve the estimation accuracy. For a fair comparison, we assume the same overhead as the LS estimator in Section IV-A, i.e., using M _LS symbols. The performance of the proposed estimator can be further improved by a factor of M _S , as

Thus, we can see that the MSE of the proposed estimator is times that of the LS estimator. That is, with the same overhead, the performance of the proposed estimator is improved by a factor of Mr to the LS estimator.

Figure 4 shows the theoretical MSE Vs SNR performance for different numbers of TX antennas, M _T for both an estimator in accordance with the present disclosure and an LS estimator. Both have the same overhead using M _LS = 4 pilot symbols. It can be seen that the MSE of both estimators are proportional to the received noise power. However, in terms of the number of TX antennas, M _T , the MSE of the estimator of the present disclosure is linearly proportional to M _T , while that of the LS estimator is proportional to , as indicated in equations (26) and (28), respectively. In other words, the estimator of the present disclosure degrades slower than an LS estimator. Further, figure 4 also shows that the estimator of the present application performs better than the LS estimator as the number of TX antennas increases.

B. Discussions on Practical Implementation

To further improve estimation accuracy, smoothing/averaging algorithms are usually applied to suppress the noise. For example, sliding average window (SAW) is an effective algorithm adopted in the 4G/5G systems to improve the estimation accuracy [39]. Once the CFRs are estimated through the pilots, as shown in equation (25), adjacent pilots are averaged in the frequency domain to suppress the noise. However, the window width should be carefully designed. In the 4G/5G systems, as the SAW algorithm will generally introduce distortion on the estimated CSL In practice the average window width should be chosen no greater than 19 as a balance between the noise suppression and the estimation deviation.

In a further improvement over the prior art a more efficient and unbiased smoothing algorithm dedicated for the proposed estimator is provided. This improvement is achieved by exploiting the pulse-compression property of chirped waveforms based on the Fresnel transform. The new smoothing algorithm is based on the realisation that the received pilot signal after pulse compression is equivalent to the sum of the pilots from different TX antennas, as shown in equation (15). Thus, the noise exceeding the length of CIR can be removed in addition to the pilots from unwanted TX antennas. Utilizing the finite impulse response (FIR) of the channel, a more accurate estimation can be obtained by filtering out the excessive noise. To achieve this, a window function П _G (n) is defined. This function can be a rectangular function of width L _G . Other types of window function can also be adopted to remove the excessive noise, such as e.g. a raised cosine function. The windowed CIR functions are accordingly obtained as

If we take the rectangular function as the window for example, i.e., П _G (n) = 1 for and 0 otherwise, the MSE of the proposed algorithm is

The detailed derivation is in Appendix B.

Comparing equations (29) and (27), the MSE’s of both estimators consist of two sub-terms; one is the PDP term and the other is noise term. However, in the proposed algorithm, the noise term is instead proportional to Lg, and smaller the window width smaller the noise. This is because the proposed algorithm rejects the excessive noise beyond the window. Thus, we term it as noise- rejection window (NRW). In the NRW algorithm, the PDP term is the sum of the tail of the PDP function outside the window. Considering the FIR feature of real- world channels, that |σ (n)| ² = 0 for n ≥ L _CIR , the estimation of the NRW is unbiased. The PDP term in equation (27) vanishes as long as the window is wider than the maximum delay spread of the channel.

Although both algorithms can effectively suppress the noise effects, they behave differently. The SAW algorithm suppresses the noise by smoothing the CFR function, and the estimation deviates from the actual system. As a result, there is always a trade-off between the noise and deviation yielding a sub-optimal performance.

In contrast, the NRW algorithm removes excessive noise directly removed the estimation after pulse compression. The estimation is unbiased if the window is wider than the maximum delay spread of the channel, i.e., , and can optimally converge to the system under estimation. Even if the window width is reasonably smaller than L _CIR , there won’t be severe distortion because the tails of the CIR functions are usually much smaller than its main path. For example, for a channel with exponential PDP, which is applicable for most practical channels, the tail beyond 3 times of the root mean square (rms) delay spread is less than 5 percent of the total energy.

IV. RESULTS

In this section, numerical results are provided to evaluate the performance of a channel estimation technique in accordance with the present disclosure. The OCDM system has a bandwidth of 20 MHz with N = 2048 chirps for modulation. The length of Gl is 256. The analysis is based on a multipath fading MIMO channel with an exponential PDP. This is a typical channel model in practical systems. The PDP function for such channels is defined as where TO is the rms delay spread of the channel. Substituting equation (28) into equation (27), the analytical MSE can be further given by as derived in (34) in Appendix B. The optimal window width and the minimum MSE are given in equations (36) and (37), respectively.

Figure 5 is a pair of 2D plot diagrams illustrating the performance of a channel estimator using NRW algorithm. For the purposes of illustration, the channel is part of a MIMO-OCDM system with three TX antennas and 4 RX antennas. In figure 5, MSE is shown as a function of the received SNR and the width of noise rejection window, with ro = (a) 0.4 μs and (b) 0.8 μs, respectively. Along the SNR axis, the MSE is a monotonic function, and it decreases along with the SNR axis. The MSE is a convex function with respect to the window width, Lg. For a fixed SNR, there exists an optimal Lg for providing the minimum MSE. The optimal window width is given as a function of SNR (see equation (36) in Appendix B). For a given SNR, a larger Lg allows more noise passing through but has much less deviation. When the window width is much larger than the delay spread of the system, , the remainder noise passing through the window determines the MSE performance. The MSE goes larger as Lg increases because the noise term is linearly proportional to the window width.

When the window width is comparable to the delay spread, the MSE of the proposed estimator is the interplay of the noise and estimation deviation. However, when the widow width is relatively smaller than the delay spread, the deviation of the estimation will dominate the performance degradation. Especially, the MSE degrades dramatically as Lg decreases. Similar trend can be observed in Fig. 5a and 5b. The difference between them is that a larger delay spread results in worse performance, and the optimal window width is scaled by a factor of ~ τ ₀ .

Figure 6 illustrates the performance of the proposed estimator against the window width. In particular, figure 6 shows the CIR and noise terms are depicted in the dashed and the dotted lines, respectively. As can be seen, the noise term is inversely proportional to the window width, Lg, as indicated in equation (29). For example, in the case of τ ₀ = 0.4 s, the MSE curve is well fitted with the corresponding noise curve for Lg > 128 (16ro), and increases with . The deviation of the CIR term in this case is less than -58 dB for , which is negligible. As the window becomes narrower, the MSE degrades dramatically, for values of Lg < 64 (8ro). In this case, the deviation is greater than -24 dB, and dominates the degradation on MSE. Similar trend can be observed in the cases of τ ₀ = 0.2 and 0.8 μs. The slope of the CIR curves is much steeper curve of the noise term discussed with reference to Figure 5. This is because the deviation increases exponentially as becomes narrower. It also implies that an unbiased estimator is usually preferred in practice.

Figure 7 shows a performance comparison of an estimator in accordance with the present disclosure and a prior art SAW estimator. In particular, figure 7 shows that MSEs of both the proposed and the SAW algorithms. Both algorithms effectively suppress the noise term as shown by the overlap of the dotted lines overlap for both algorithms. However, the MSE performances of the estimators is significantly different due to the deviations of their estimations.

In the SAW algorithm, the averaging operation is performed in the frequency domain, and the CIR function is equivalent to being weighted by a Dirichlet sine function. The SAW estimation always deviates from the MIMO system under estimation, i.e., L _S > 1 , as indicated by the dashed lines in Fig. 7. In fact, if there is no delay spread, namely in the condition of h _p,q (n) = δ(n), the SAW algorithm is also unbiased. However, in practice this condition cannot realistically be satisfied in real wireless systems.

In contrast, in the proposed estimator, there is negligible deviation until the window size is comparable to delay spread of the channel. Strictly speaking, the proposed estimator is unbiased as long as

Figure 8 shows MSE performance as a function of the received SNR for both the estimator of the present disclosure and an LS estimator with SAW algorithm. Both systems equipped 3 TX antennas and 4 RX antennas. The MSE of both estimators without any smooth/average algorithms (circled lines) has no error floor, and the estimator of the present disclosure exhibits better sensitivity to the distortion than the SAW algorithm. Figure 8a shows MSE performance as a function of the received SNR for τ ₀ = 0.4 μs and figure 8b shows MSE performance as a function of the received SNR for τ ₀ = 0.8 μs.

With reference to figure 8a, the system of the present disclosure requires 5 dB less SNR to achieve the same MSE than the LS estimator if there are no smoothing algorithms applied. For the estimator of the present disclosure, the MSE performance is improved when the NRW algorithm is applied. The MSE reduces as the noise-rejection window width becomes smaller in the low SNR region. However, in the high SNR region although there is no obvious degradation for Lg = 96, error floor occurs for Lg ≤ 64. This is because in the low SNR region, noise dominates the performance and smaller the window width, less the noise effect. As the SNR increases, the noise becomes small, and the distortion due to deviation after NRW begins to dominate for a small window width, e.g., for τ ₀ = 0.4 μs.

For the LS estimator with SAW algorithm, as the averaging window increases, the MSE of the estimation decreases, especially in the low SNR region. However, error floors occur as long as SAW is applied. For example, in the case of Ls = 5, slight degradation can be observed for SNR > 35. In the case of Ls = 9, obvious degradation can be observed for SNR > 25, and an error floor at MSE = 1 x10 ^-3 exists. In contrast, even if the NRW algorithm has an error floor for a small Lg, the error floor is much lower than the SAW algorithm.

With reference to figure 8b, similar trends as those shown in figure 8a can be observed, but the overall performance is worse as a result of a larger delay spread. The error floor of LS estimator becomes much obvious for a larger τ ₀ . For example, the proposed algorithm with has no degradation, and the performance improves linearly along with the SNR. On the other hand, a slight degradation can be observed in the SAW algorithm even for a small L _S = 3. As Ls increases to 7, although the performance outperforms the proposed estimator with for SNR < 15 dB, the performance degrades with an error floor at MSE = 5 x 10 ³ . In particular, error floor occurs in the proposed estimator only for with a BER = 3 x 10 ^-4 .

The embodiments in the invention described with reference to the drawings comprise a computer apparatus and/or processes performed in a computer apparatus. However, the invention also extends to computer programs, particularly computer programs stored on or in a carrier adapted to bring the invention into practice. The program may be in the form of source code, object code, or a code intermediate source and object code, such as in partially compiled form or in any other form suitable for use in the implementation of the method according to the invention. The carrier may comprise a storage medium such as ROM, e.g. a memory stick or hard disk. The carrier may be an electrical or optical signal which may be transmitted via an electrical or an optical cable or by radio or other means.

In the specification the terms "comprise, comprises, comprised and comprising" or any variation thereof and the terms include, includes, included and including" or any variation thereof are considered to be totally interchangeable and they should all be afforded the widest possible interpretation and vice versa.

The invention is not limited to the embodiments hereinbefore described but may be varied in both construction and detail.

APPENDIX A - PROOF OF THE MSE OF THE PROPOSED ESTIMATOR

From the third equation in (23), the noise term is where for n = 0,1 ,...,D _q+1 . Substituting back into equation (23), the MSE can be given as

It is possible to show that the noise terms are independent over different transmit antennas.

APPENDIX B - MSE OF THE WINDOWING ALGORITHM

Except to the noise term, there is another term due to the deviation of the channel estimation.

If we consider a channel with exponential decaying channel model, Eq. (33) can be further given as

In addition, we can derive the minimum MSE with respect to the gate size as to get the optimum window width and the minimum MSE