REFERENCE TO RELATED APPLICATIONS

[001] This application is a continuation of U.S. Nonprovisional Application

No. 1 3/844,543, fi led March 1 5, 201 3, entitled "System and Method for Broadband Doppler Compensation," which claims the benefit of U.S. Provisional Application No. 61/731 ,406 entitled "Broadband Doppler Compensation" filed November 29, 2012, both of which are incorporated by reference in their entirety.

FEDERALLY SPONSORED RESEARCH OR DEVELOPMENT

[002] The United States government may hold license and/or other rights in this invention as a result of financial support provided by governmental agencies in the development of aspects of the invention. The claimed invention described herein was supported by a grant from the Department of the Navy, Office of Naval Research, under grants ONR MURI N00014-07-1 -0738 and ONR N00014-07-1 - 031 1 .

BACKGROUND

[003] Unless otherwise indicated herein, the materials described in this section are not prior art to the claims in this application and are not admitted to be prior art by inclusion in this section.

[004] Doppler Effects and other sources of discrepancy between the time- scale of emission and the time-scale of reception play a significant role i n many communication and other signal transmission systems. In Broadband communication cases, a received signal is typical ly distorted because of Doppler effects, which induce a change in the relative time-scale between the signal emission process and the signal reception process. As such, a significant chal lenge in underwater acoustic (UWA) communications is the proper compensation for time-scale differences between the emission signal at the transm itter and the received signal at the receiver. These time-scale differences occur due to motion of either the source or receiver, changes i n the motion or velocity of the medi um of propagation, or even due to electronic means, such as asynchrony i n the timi ng references used i n the source and receiver.

[005] Further, due to its large delay spread and rapid time variation, a UWA chan nel is particularly chal lengi ng for high-data-rate digital communications. The transmitted signal bandwidth can be also a substantial fraction of its center frequency maki ng the common narrowband assumptions i nval id. Broadband transmissions as i n UWA communications can experience high ly time-varyi ng Doppler. However, conventional approaches to Doppler compensation often assume a constant velocity difference between the source and receiver, or equivalently, a constant time-scale factor between the source and receiver processes. Conventional approaches estimate an average or bul k Doppler factor di rectly from the received waveform. For compensation, the received baseband signal is then resampled and phase corrected based on this factor and a phase- locked loop is employed to remove any residual Doppler. Th is approach is prone to i nstabi l ities, si nce an estimation error i n the Doppler Effect may be i ncreasi ngly ampl ified with time.

[006] Therefore, there is a need for a system and method that address and overcome the above discussed disadvantages and l imitations.

SUMMARY

[007] Disclosed herein are an improved method and system for broadband

Doppler compensation in underwater acoustic communications.

[008] In one aspect, an embodiment of a Doppler compensation system includes a transmitter unit for transmitting a signal, wherein the transmitted signal being associated with an emission time-scale, a receivi ng unit for receivi ng a signal, wherein the received signal is associated with a receive time-scale that is not equivalent to the emission time-scale, and a Doppler compensating unit configured to estimate an inverse temporal distortion function, wherein the Doppler compensating unit implements the inverse temporal distortion function to estimate the transmitted signal .

[009] In another aspect, a Doppler compensation system includes a transmitting unit for transmitting a signal, wherein the transmitted signal being associated with an emission time-scale; a receiving unit for receiving a signal, wherein the received signal is associated with a receive time-scale that is not equivalent to the emission time-scale, and a Doppler compensating unit configured to estimate an inverse temporal distortion function, wherein the inverse temporal distortion function is estimated on a sample-by-sample basis that may be at a faster rate than that of the data symbol rate, and wherein the Doppler compensating unit implements the inverse temporal distortion function to estimate the transmitted signal.

[0010] In yet another aspect, an apparatus for Doppler compensation includes a Doppler compensating unit configured to estimate an i nverse temporal distortion function, wherein the Doppler compensating unit implements the inverse temporal distortion function to estimate data symbols of a signal transmitted by a transmitting unit, and an equalizer for removing signal dispersion and multipath effects. [0011] In yet another aspect, a Doppler compensation method includes transmitting a signal, wherein the transmitted signal includes a sequence of data symbols transmitted based on an emission time-scale, receiving a signal, wherein the received signal is associated with a receive time-scale that is different from the emission time-scale, and estimating sample times using a Doppler compensating unit for the received signal so that the received signal and the transmitted signal share the same time scale.

[0012] These as well as other aspects, advantages, and alternatives will become apparent to those of ordinary skill in the art by reading the following detailed description, with reference where appropriate to the accompanying drawings. Further, it should be understood that the disclosure provided in this summary section and elsewhere in this document is intended to discuss the embodiments by way of example on ly and not by way of limitation.

BRIEF DESCRIPTION OF THE DRAWINGS

[0013] Figure 1A and 1 B are block diagrams i llustrating components of a transmitting unit (1 A) and of a receiving unit (1 B);

[0014] Figure 2 is a flow chart i llustrating an exemplary embodiment of a method for broadband Doppler compensation in an underwater acoustic communication;

[0015] Figure 3 is a block diagram i ll ustrating an exemplary embodiment of a broadband Doppler compensator;

[0016] Figure 4 is a block diagram i l lustrating an exemplary embodiment of an acoustic system that incl udes an equalizer;

[0017] Figure 5 is a block diagram illustrating an exemplary embodiment of an

SISO equalizer;

[0018] Figure 6 is a flow chart illustrating an exemplary embodiment of a method for broadband Doppler compensation in an underwater acoustic communication that utilizes a SISO equalizer;

[0019] Figures 7 - 12 are graphs that illustrate experimental set-ups and results of a broadband Doppler compensation approach; and

[0020] Figure 13 is a schematic diagram illustrating a conceptual partial view of an example computer program product.

DETAILED DESCRIPTION

[0021] In the fol lowing detai led description, reference is made to the accompanying figures, which form a part hereof. In the figures, similar symbols typical ly identify simi lar components, unless context dictates otherwise. The i llustrative embodiments described in the detai led description, figures, and claims are not meant to be l imiti ng. Other embodiments may be uti lized, and other changes may be made, without departing from the spirit or scope of the subject matter presented herein. It wil l be readi ly understood that the aspects of the present disclosure, as general ly described herein, and i l lustrated i n the figures, can be arranged, substituted, combined, separated, and designed in a wide variety of different configurations, al l of which are explicitly contemplated herein.

Overview

[0022] Wireless communication is an essential component of many underwater operations, such as those involving underwater vehicles. However, the conductivity of salt water prevents electromagnetic waves from penetrati ng deep into the ocean, thus preventing the use of radio based communication technology in the ocean envi ronment. As a result, engineers have made use of acoustics in an attempt to meet underwater communication needs. There are several acoustic modems on the market that provide a transparent data link and can reach data rates of the order of 10 Kbits/second, but when communication signals have multiple interactions with scatterers, such as the surface or the ocean bottom, harsh multi- path arises. Under these conditions, existing modems perform poorly and on ly achieve data rates of the order of 100 bits/second. These effects are typical ly most severe in horizontal, long-range communication.

[0023] In addition to multi-path, both the communication platforms and the communications medium itself may have significant motion. Unl ike in mobi le radio systems, these influences cannot be neglected in the design of acoustic communication systems. "Stationary" acoustic systems may move at speeds of several meters per second due to the forces induced by currents, tides, and waves. Autonomous underwater vehicles (AUVs) move at comparable speeds and submarines can move even faster. This leads to Mach numbers on the order of 1E-3 and higher. In comparison, a velocity of 400 km/h in radio channels translates to a Mach number of only 3.7E-7, i.e., several orders of magnitude smaller. Motion always manifests as time-varying temporal scaling of the received waveform. This temporal distortion is a time-scale distortion, in that the transmitted and received signals are effectively defined over different, possibly time-varying time-scales. In radio channels, this is typically negligible, while in acoustic communications, it can be catastrophic if not compensated dynamically. The research community has explored adapting the most sophisticated techniques known from radio communication such as Space-Time coding, Turbo Equalization, Orthogonal Frequency Division Multiplexing (OFDM), and Low Density Parity Check (LDPC) coding but only data rates in the range of 6 bits/s to 48 Kbit/s have been realized. One fatal flaw of many of these works is that the channel model is borrowed from the radio communication community and only slightly modified, if at all, and hence does not properly respect the physics of the underlying acoustic system. A popular assumption is that the Doppler is constant (i.e., a fixed time-scaling between transmit and receive signals) over the time of a data block and the remaining channel effect is linear and near time-invariant, but in reality the Doppler can be highly time-varying and different wave propagation paths can experience different Doppler (i.e., different time-scale distortions along each path). The UWA channel remains one of the most difficult communication channels and our understanding of it is still in its infancy.

[0024] At the receiver in a radio communication link, equalizers are employed in order to remove the inter-symbol interference introduced by channel dispersion and multi-path propagation, but the Doppler Effect cannot be compensated for in this manner. One approach to Doppler Effect compensation is configured to obtain an estimate of the average Doppler Effect factor. For compensation, the received baseband signal is then resampled and phase corrected based on this factor and a phase-locked loop is employed to remove any residual Doppler. This approach, however, only works if the motion induced temporal scaling of the received waveform is quasi-time-invariant. In the present disclosure, a different approach is introduced in which time-varying Doppler is explicitly modeled, tracked and compensated throughout a block transmission in an underwater acoustic communication system. This method has been field-tested successfully on data from multiple underwater acoustic experiments at sea. Performance examples are shown from the MACE 10 experiment. To illustrate the robustness of this approach to high rates of Doppler variation, as well as the ability of this approach to track source - receiver position and velocity, a variety of simulations are also provided.

[0025] Accordingly, a novel Doppler Effect compensation method in information bearing signals is provided that recursively tracks an optimal resampling and phase drift correction of a received waveform. Further, the recursive tracking accurately tracks the propagation path distance and propagation path velocity. Moreover, results are provided from extensive performance evaluations of a sample- by-sample, recursive resampling technique, in which time-varying Doppler is explicitly modeled and tracked throughout a block transmission in a UWA communication system. Performance examples are described hereafter in the results of an underwater acoustic communications experiment, referred to as the MACE experiment. To illustrate the robustness of this approach to high rates of Doppler variation, as well as the ability of this approach to track source-receiver and velocity, a variety of simulations are also provided.

[0026] Referring to Figures 1A and 1 B, an AUW communications system 100 includes an acoustic transmitter (transmitting unit) 102 configured to transmit an acoustic signal, and an acoustic receiver (receiving unit) 104. The position of transmitter 102 relative to receiver 104 can vary with time thereby defining a relative motion, which can also vary with time. In accordance with one embodiment, transmitter 102 incl udes a data generator 106, a modulator 108, a low-pass filter 1 10, an encoder 1 12, an interleaver 1 14, a mapper 1 1 6, a processi ng unit (processor) 1 1 8, and a memory unit 120. Receiver 104 includes a demodulator 122, a low-pass fi lter 124, a Doppler compensator 1 26, a demapper 128, a deinterleaver 1 30, a decoder 1 32, a processing unit (processor) 1 34, and a memory unit 1 36. Doppler compensator 126 is configured to include a sampler 1 38 and a phase compensator 140.

[0027] In one exemplary embodiment, transmitter 102 is configured to send a sequence "s" of symbols "s„" from a finite set of signal constellation points A c C over a frequency selective channel that also experiences a time-varying Doppler Effect. The sequence 5 is mapped to a waveform s(t): R→ C s(t) = ^ Si p(t — IT) Equation 1 by use of a basic pulse function p(t) time shifted by multiples of a symbol period T. This signal is then modulated to a passband at carrier frequency fc and transmitted over the channel to yield:

Sp _B (t) = 2&{s(t) e ^{2Ti f} C ⁽ } Equation 2

The time-dispersive effects of the channel are assumed to be linear and quasi- stationary and can hence be modeled by a linear time invariant system with some kernel function h _PB (t) as follows: f(t) = Sp _B (t) * hp _B (t) Equation 3

[0028] If one of transmitter 102 and receiver 104 is in motion or both of them are in motion, receiver 104 can observe a time warped version of f(t) in additive noise. In order to defi ne this effect more rigorously, a position vector x _tx (t) of transmitter 102 and a position vector x _rx (t) of receiver 1 04 are i ntroduced. Since signal s _PB (t) needs to travel some distance to reach receiver 1 04, there wi l l typical ly be a (time-varyi ng) delay i n the received signal f(t) relative to transmitted signal s _PB (t). I n particular, a pulse observed by receiver 1 04 at time t has actual ly been em itted by transmitter 1 02 at some earl ier time r(t) that is a sol ution to the fol lowi ng implicit equation: Equation 4 where c is a wave speed. Note that under a realistic assumption, because both transmitter 1 02 and receiver 1 04 move at a speed less than c, τ(ΐ) is a strictly increasing continuous function and is referred to as a temporal distortion function. The signal r _PB (t) that is observed at receiver 1 04 can be represented as fol lows: r _PB (f) = r (T (t)) + v _PB (t) Equation 5 where the process v _PB (t) is an additive white Gaussian noise. Equation 5 i l lustrates that the signal observed at receiver 1 04 has a different time-scale than the signal emitted by transmitter 1 02, and one can convert time values from one scale to time values on the other scale using a temporal distortion function r(t). For the application of underwater acoustic commun ications, a scale distortion between transmitted and received waveforms is indeed a time-scale distortion, and is typical ly caused by motion of the source and receiver as wel l as motion of the propagation medium or scatterers in the path of propagation. Potential other sources of scale distortion between signals at the transmitter and receiver incl ude spatial differences, as in data storage or other spatial ly distributed data. For example, in a magnetic storage application transmitter 1 02 writes data symbols onto a medium that is later read by receiver 1 04. If the velocity of the read equipment over the recording medium used to recover the signal from the medium is different from the velocity over the medi um when it was written to the medium, as might occur due to disk drive or tape processing equipment differences, then the signals as read from the storage medium (receive signal) will exhibit similar time-scale distortion as caused by the Doppler effects described above. Similar distortion could arise from spatial aberrations in the medium, such as stretching or contraction of the medium from heat, stress, or other external influences.

[0029] In one embodiment, a time-varying Doppler Effect is expressed as fol lows: d(t) = 1 - τ ^' (ή Equation 6 when the relative motion between transmitter 1 02 and receiver 104 along the line of signal propagation has constant velocity, d(t) = d and the Doppler effect simply scales the time-axis of the signal f(t) by 1— d.

[0030] Further, a convolution of p(t) with h(t), which represents an equivalent base-band channel impulse response, yields:

%{t) = h(t) *p(t) Equation 7

Also a convolution of h(t) with s(t) yields: r (t) = h(t) * s(t) =∑i si h(t - IT) Equation 8

The received signal r _PB (t) is then expressed as follows: r _PB (t) = 2 R{f« ) e ² ^ ⁰ } + v _PB (t) Equation 9

Once captured by receiver 104, the signal r _PB (t) is demodulated by f _c and low-pass filtered to yield: r(t) = ^ ^{2Mfc (T (t t)} r(x (t)) + v(t) Equation 1 0 where v(¾)denotes the demodulated and filtered noise process. Doppler Compensation

[0031] In one embodiment, a novel approach to temporal distortion compensation estimates the inverse temporal distortion function τ ² (ΐ) at t = 0, T, 2T, ... and then resamples the received signal at those time values in order to remove any temporal distortion from it. A particular temporal distortion compensation case involves no multi-path effects and no signal attenuation. Equation (10) then reads: r(t) = (j ^{27cfc(T(t t)} S(T (t)) + v(t) Equation 11

Moreover, if a value of τ ² (ηΤ) was known, then the sequence s„ could be recovered in additive noise. In practice, a precise value of τ ² (ηΤ) is unknown, but its approximation T„/„-f ² can be obtained. A synchronization between transmitter 102 and receiver 104 can readily yield a value of τ _η / _η . ² for n = 0. In One embodiment, a recursive algorithm is structured to obtain τ„ ₊ ι/„ ^~2 from τ„/„.{ ² for any n. In accordance with this recursive algorithm, a state τ _η+1 / _η ^~2 is introduced and represents a time derivative of τ ³ (t) at t=nT, and its estimate is denoted as r„ ₊ ;/„ ^_J '·

[0032] In one exemplary embodiment, the above-introduced algorithm is configured to evaluate r(t) at ⁷ as follows: r(T _n/n -i') = e ^{i2nfc (nT+ ¾"} ™ ^~r> (S(nT+ ε _η )) + ν(τ _η/η -ι ^~ ') Equation 12 where the error ε„ nT. The signal r(t) is available at the receiver in some form. For band limited r(t) this may include equally spaced samples of r(t) at a period of Ts seconds, i.e. r[k]=r(kTs), where, typically, Ts < T/2. The signal may also be available in other forms. In either case, since the desired sample of r(t) is typically not one of the samples available in rfkj, then computing can be accomplished either through resampling the stored signal or computing this value as a function of the samples available in r[k], through interpolation, estimation, or other signal processing methods. [0033] Further, in this exemplary embodiment, the recursive algorithm is configured to obtain an estimate s _n of s _n by removi ng the phase drift 2nf _c (nT- τ _η / _η .Γ ^ν ) s _n = _η/η -ί') e ^{i2nfc (n T} -™-' ^} Equation 1 3

§ _n = _e ⁱ² *fc ^£ n _s(nT + _Sn) + e ^{i2Kfc (nT~} ™- ^r> Equation 14

Based on Equations 1 2 and 1 3, one can note that the smaller the magnitude of the error ε _η and the additive noise are, the better is the estimate s„. The error s„ is typically small enough to lead to the fol lowing expression or the sequence s _n . s(nT + S _n ) ~ ¾ Eq uation 1 5

However, in a phase term the error S„ is amplified by by 2nfc Further, in one embodiment, an argument, which can be expressed as shown in Equation 1 6, can be used as a measurement of error S„:

Ω„= arg(s„ Sn ) ~ 2nf _c S„ Equation 16

Moreover, the recursive algorithm is configured to compute the approximation τ _{η +} ι/ _η ^~ ' from Tn/n-f ¹ based on the measurement Ω„ as follows:

Tn+i/n-' = Tn/n-r ¹ - μ Ω _η Equation 1 7

Tn+l/n ¹ = Tn/n-f ¹ + Tn+J/n^ ¹ T Equation 18 where μ is a step size of the recursive algorithm.

[0034] In UWA communication scenarios, the transmitted signal may also be subject to dispersion and multi-path effects. In this case, a computation in Equation 14 can be used to obtain an estimate of r(nT), which is then processed by an equalizing unit to obtain s„. This estimate of r(nT), can then be processed by an equalizer to obtain s„. In one embodiment, the equalizer can be as shown in Figure 5, where the Doppler compensator is coupled to a linear or decision feedback equalizer for the channel. In this embodiment, the transmitted data symbols are estimated as a linear function of these Doppler compensated signal values using one of a number of channel equalization methods. One embodiment includes a minimum mean-square error channel equalizer, where the channel weights, w _j „ in Figure 5, are updated using a recursive algorithm, such as the RLS, Kalman, or LMS algorithm, using knowledge of the transmitted symbol alphabet and the output of the channel equalizer. If the data symbols have been protected by an error- correction code, then the channel equalizer can also be coupled to a decoder for the error correction code. An output of the decoder can also be used in the equalizer either as inputs or as part of the update of the weights of the equalizer. This is commonly done in Turbo Equalization.

[0035] Note that the range between transmitter 102 and receiver 104 at time

T ^~ '(t) can be estimated from the value of the approximation to Τηη- ¹ using

Equation 4:

I \x _t (τ(τ'(ηΤ)) - Equation 19

~ C {-nT - Tn/n-i ¹ ) Equation 20

When several hydrophone elements are used at receiver 104, such a range estimate can be computed for each one and multilateration can be used to determine the relative position of transmitter 102.

[0036] Now referring to Figure 2, a flow diagram shows an exemplary method

200, initiated at Step 201, for broadband Doppler compensation in underwater acoustic communications. At step 202, transmitter 102 is configured to transmit a signal that includes a sequence of data symbols transmitted based on an emission time-scale. At Step 204, receiver 1 04 is configured to receive a signal associated with receive ti me-scale that is different from the em ission time-scale. Subsequently, a Doppler compensati ng unit is configured to estimate sample times for the received signal so that the transm itted signal and the received signal share the same time scale, at Step 206.

[0037] Now referring to Figure 4, an exemplary embodiment of a commun ication system 400 i l lustrates a process of estimating a set of symbols that were subject to an underwater acoustic transmission . As shown, an encoder 402 is configured to encode a set of data "a" into a set of data "d" that is transformed a set of data "e" by an interleaver 404. The set of data "e" is then mapped by a mapper 406 into a quadrature phase-shift keying (QPSK) arrangement, which may include one bit per symbol or two bits per symbol for example, to generate a sequence S that is transmitted by transmitter 1 02 th rough a communication channel 408 on a signal carrier having a frequency fo. The transmitted sequence S, which may be subject to Doppler Effects and noise during the transmission, is received as a signal r(t) by receiver 1 04. In order to derive an estimate of the transmitted sequence S, the received signal r(t) is processed by an equalizer 41 0, wh ich may include a demodulator, a low-pass fi lter, a non-uniform sampler, and a phase compensator, as discussed above. In another exemplary embodiment, equalizer 41 0 may be a Soft-In-Soft-Out (SISO) equal izer, as shown in Figure 5. The estimated sequence s is then de-mapped using a demapper 41 2, de- interleaved using a deinterleaver 41 4, and then decoded using a decoder 41 6 to generate sequence a. When equalizer 41 0 is a SISO linear equal izer, sequence ά is provided back to equalizer 41 0, as shown in Figure 4, to further minimize an error in the estimation process of the transmitted sequence S.

[0038] Now referri ng to Figure 6, a flow chart 600 shows an exemplary method 602 for an UWA temporal distortion compensation usi ng a SISO equal izer. Fol lowi ng the i nitiation of the method 602 at Step 601 , receiver 1 04 is configured to synchronize positions Tx and Rx of transm itter 1 02 and receiver 1 04, respectively, at Step 602. At Step 604, equalizer 310 and temporal distortion compensator 126 are initialized, and a non-uniform sampling time defined by recursive evaluation of the inverse retarded time function is updated, at Step 606, as follows: Ti+i/i ¹ = Ti/i-f ¹ + βη+ι/η T _s . Then, at Step 608, the phase correction, determined as discussed above, is updated as follows: Wi+ = W -i ¹ + 2f _c T(l- β„/„-ι). Subsequently, at Step 610, a temporal distortion compensation is implemented as follows: Yi ₌ e ^lw "'-' γτι _/ΐ - _ΐ ^'1 ). Then at Step 612, receiver 104 is configured to determine whether / is a multiple of n. In the affirmative, a SISO filter is used to generate an estimate s„, at Step 614. Otherwise, the process goes back to Step 606 to update the non-uniform sampling time. Following the generation of estimate s„, receiver 104 is configured to determine a phase lag or lead using the above-derived phase measurement Ω„= arg(s„ s„ ^* ) _f at Step 616. Based on the determination of the phase lag lead, receiver 104 is configured to update a jump between successive estimated symbols of the transmitted sequence, at Step 618, as follows: β _η +ι/ _η = βη/η-ι - μΩ„. Then, at Step 620, receiver 104 is configured to determine whether the estimated symbol s _n is the last symbol. In the affirmative, this symbol estimation process is terminated, at Step 622. Otherwise, index n is augmented by one digit, at Step 624, and the process is repeated for this new index n+1.

Experimental Results

[0039] A performance of one embodiment of the provided temporal distortion compensating compensator is evaluated using real and synthetic data sets. The real data set stems from a Mobile Acoustic Communications Experiment (MACE) conducted at a location (site) having a depth of about 100 m. As shown in Figure 7, a mobile V-fin with an array of transducers attached was towed along a "race track" course approximately 3.8 km long and 600m wide. A maximum tow speed was 3kt. (1.5 m/s) and a tow depth varied between 30 and 60 m. A receiver 12 channel hydrophone array was moored at a depth of 50 m. A range between the transmitter and the receiver array was between 2.7 and 7.2 km. [0040] For a Single Input Multiple Output (SIMO) transmission with one transducer and 12 hydrophones, a rate 1/2, (1 31 , 1 71 ) RSC code and puncturing was used to obtain an effective rate of 2/3. Blocks of 1 9800 bits were generated, interleaved, and mapped to 1 6-QAM symbols. A carrier frequency was about 13 kHz. The receiver sampling rate was 39.0625 k samples/second. Data was transmitted at a symbol rate of 1 9.531 3 k symbols/second. A 1 0% overhead for training symbols for the equalizer achieved a data rate of 23.438 Kbit/s. A square-root raised cosine filter with a roll-off factor 0.2 was used in both the transmitter and the receiver.

[0041] At receiver 1 04, an LMS direct-adaptive turbo equalizer with the temporal distortion compensator iteratively decoded the received data sequence. A prior work demonstrated that LMS direct- adaptive turbo equalization can dramatically outperform the conventional decision-feedback equalizer. Figures 8 and 9 summarize a Bit Error Rate (BER) performance of the receiver on the MACE data set. For al l transmissions, the receiver converged to the right code word in two or less iterations. Figure 10 shows a relative speed between transmitter and receiver as estimated by the proposed temporal distortion compensator during three example transmissions. The following relationship between relative speed v(t) and the derivative r ^~7 ( ) was used to generate the Figure 10 plot:

(r (nT)) ^A = T (r - ¹ (fiT)) = 1 - v(r (nT))/c Equation 21

[0042] After synchronization, v(r ^(nT ) is initialized with 0. According to the

Equation 21 formula this corresponds to an initialization of r ^{_ i} («7)) with 1 . It should be noted that an insertion of two chirps at the beginning of the data transmission and the measurement of their time dilation can be used to find a much better initial value. Due to the shal low water at the experiment site, the channel exhibited severe multi-path as illustrated in Figure 5.

Simulation Results [0043] To simulate the effects of motion of transmitter 1 02 and receiver 1 04 in the underwater channel, one may suppose that, for any time t, the transmitter position x _tx (t) and the receiver position x _rx (t), and the signal r(t) are provided, in order to evaluate the received signal r _PB (t). One goal is then to determine the function τ(ή for each t corresponding with the ADC samples in receiver 1 04. To this end, one can note that Equation 4, discussed above, determines an amount of time the transmitted signal travels times the speed of sound equals the distance traveled. Alternatively, Equation 4 can be rewritten as fol lows:

I |jc,(r(f)) - x _r (t)\ \ ₂ + Cz(t) - C t = 0 Equation 22

In order to solve for z(t), the following function is establ ished:

F _t (z) = I + Cz - C t = 0 Equation 23

[0044] Hence, for each t, Z is determined such that F _t (z)= 0. To guarantee that a unique solution to Equation 23 exists, reasonable conditions are needed such that F _t (z) is a strictly increasing and continuous function of Z for any given t. There are a number of conditions that can be applied to accomplish this. For example, one may specify that χ ₍ (τ) is Lipsch itz continuous for Lipschitz constant K < c, or more simply that \ \ ₂ < K < C. It is then a simple matter to compute τ(ΐ) as the unique root of the continuous and strictly increasing function F _t (z).

[0045] Intuitively, one can expect the estimation error of our temporal distortion compensator to increase as the relative acceleration between transmitter 1 02 and receiver 1 04 increases. In order to evaluate the performance of an embodiment of the proposed temporal distortion compensator for different levels of acceleration, a line of sight channel is assumed and the simulator discussed above is used to compute r _PB (t) for a relative acceleration that l inearly increases from 0 m/s ² to 2 m/s ² . As shown, Figure 1 2 compares the estimates of temporal distortion compensator with actual motion values when white Gaussian noise is added at an SNR of 0 dB. Again, on ly 1 0% of the transmitted data was used for training.

[0046] Each of processing units 1 1 8 and 1 34 can be implemented on a single- chip, multiple chips or multiple electrical components. For example, various architectures can be used including dedicated or embedded processor or microprocessor (μΡ), single purpose processor, control ler or a microcontrol ler (μθ, digital signal processor (DSP), or any combination thereof. In most cases, each of processing units 1 1 8 and 1 34 together with an operating system operates to execute computer code and produce and use data. Each of memory units 1 20 and 1 36 may be of any type of memory now known or later developed including but not limited to volati le memory (such as RAM), non-volati le memory (such as ROM, flash memory, etc.) or any combination thereof, which may store software that can be accessed and executed by processing units 1 1 8 and 1 34, respectively, for example.

[0047] In some embodiments, the disclosed method may be implemented as computer program instructions encoded on a computer-readable storage media in a machine-readable format. Figure 1 3 is a schematic i l lustrating a conceptual partial view of an example computer program product 1 300 that includes a computer program for executing a computer process on a computing device, arranged according to at least some embodiments presented herein. In one embodiment, the example computer program product 1 300 is provided using a signal bearing medium 1 301 . The signal bearing medium 1 301 may include one or more programming instructions 1 302 that, when executed by a processing unit may provide functional ity or portions of the functionality described above with respect to Figures 2 and 6. Thus, for example, referring to the embodiment shown in Figures 2 and 6, one or more features of blocks 202-214, and blocks 602-624 may be undertaken by one or more instructions associated with the signal bearing medium 1 301 .

[0048] In some examples, signal bearing medi um 1 301 may encompass a non- transitory computer-readable medium 1 303, such as, but not limited to, a hard disk drive, memory, etc. In sonic implementations, the signal bearing medium 1 301 may encompass a computer recordable medium 1 304, such as, but not limited to, memory, read/write (R/W) CDs, R/W DVDs, etc. In some implementations, signal bearing medi um 1 301 may encompass a communications medium 1 305, such as, but not limited to, a digital and/or an analog communication medium (e.g., a fiber optic cable, a waveguide, a wired communications link, etc.).

[0049] Whi le various aspects and embodiments have been disclosed herein, other aspects and embodiments wi l l be apparent to those ski l led in the art. The various aspects and embodiments disclosed herein are for purposes of i l l ustration and are not intended to be limiting, with the true scope and spirit being indicated by the fol lowing claims, along with the fu l l scope of equivalents to wh ich such claims are entitled. It is also to be understood that the terminology used herein is for the purpose of describing particular embodiments on ly, and is not intended to be limiting.