Effects of acoustic nonlinearity on communication performance in soft tissues

The use of wireless implanted medical devices (IMDs) is growing because they reduce the discomfort of patients and risk of infection associated with trailing wires. Currently, radio-frequency (RF) electromagnetic waves are most frequently used for wireless communication applications for television, radio, or mobile devices. However, there are various drawbacks of using RF waves with wireless IMDs to transmit data through the body, such as limited penetration depth (Sayrafian-Pour et al., 2009) and strict regulations on signal power and bandwidth [Federal Communications Commission (FCC), 2018, 2019]. These limitations restrict achievable data rates and set a significant barrier against possible wireless IMDs to include advanced data-rich features such as video transmission.

Several studies proposed using acoustic waves to transmit information to and from IMDs (Demirors et al., 2016; Keramatzadeh and Sodagar, 2018; Kondapalli et al., 2018; Kou et al., 2021; Santagati and Melodia, 2014; Singer et al., 2016; Tabak et al., 2021b). In Bos et al. (2019), small, biocompatible transducers were used for ultrasonic communication through tissue-mimicking phantoms. Data rates of 200 kbit/s were reported without channel equalization. The channel equalizer is a critical component of the receiver architecture for an unknown, dispersive communication channel. The equalizer enables high data rates by learning to handle the effects of dispersion introduced by the unknown impulse response of the channel (Proakis, 2001). In Tabak et al. (2021b) and Tabak et al. (2021a), adaptive equalization enabled video-capable transmission rates (>3200 kbit/s) and video streaming through real biological tissues and live animals using millimeter-sized, unfocused transducers. One of the factors that prevented even higher data rates in Tabak et al. (2021b) was the low signal-to-noise ratio (SNR) as the equalizer used in that work typically operates poorly in the low SNR regime due to error propagation (Madhow, 2008).

The data rate in a digital communication system that employs linear modulation with an M-ary alphabet is $R=fbη$⁠, where f_b is the symbol rate and $η=log2 M$ is the number of bits in a transmitted symbol. For a band-limited additive white Gaussian noise (AWGN) channel, f_b and η are limited by the available bandwidth and SNR. To achieve high data rates, it is desirable to use the highest transmit signal amplitude and bandwidth possible given the available channel bandwidth, safety, and power constraints. However, these two factors, increased SNR and bandwidth, might not always yield better performances when the channel exhibits nonlinearity.

Acoustic propagation is a nonlinear phenomenon, often approximated with linear equations for better understanding and convenient design (Hamilton and Blackstock, 1998). The field of diagnostic ultrasound had initially been developed without considering nonlinear effects. Later, nonlinear acoustic effects in biological tissues were predicted and demonstrated at biomedical frequencies and intensities (Muir and Carstensen, 1980; Starritt et al., 1986). More recently, tissue harmonic imaging (THI) has been implemented as a biomedical imaging mode that exploits the nonlinear propagation of ultrasonic signals in tissues to construct an image with better definition and reduced clutter using the second harmonic (Averkiou et al., 1997). Since then, studies have demonstrated that nonlinear behaviour of the tissue could be observed at signal pressure levels as low as 372 kPa (Averkiou et al., 1997), whereas a typical diagnostic peak pressure could be as high as 5 MPa (Szabo, 2004a), and piezoelectric arrays in high-intensity focused ultrasound (HIFU) systems can produce acoustic pressure disturbances with peak pressures on the order of tens of MPas (Zhou, 2015).

Some of the factors affecting nonlinear distortion of a pure harmonic wave have been examined extensively in the nonlinear acoustics literature (Hamilton and Blackstock, 1998). It is well-known that the distortion of the signal in soft tissues is mostly determined by the intricate balance between frequency-dependent attenuation and amplitude-dependent harmonic formation (Szabo, 2004a). However, the effect of acoustic nonlinearity on wideband waveforms, such as communication signals, is not well-studied in the research literature. Moreover, the effects of acoustic nonlinearity on communication performance have not been investigated.

Consider the following to illustrate the possible effect of nonlinear distortion in a communication application: Small-signal (linear) acoustic communication through soft tissues can achieve data rates as high as $6fb$ bits/s (Tabak et al., 2021b). On the other hand, the second harmonic amplitude in soft tissues can be within 14 dB of the fundamental in finite-amplitude (nonlinear) ultrasound applications under similar conditions (Christopher, 1998). If the second harmonic level were to be treated as an AWGN floor when transmitting a communication signal with 200% bandwidth, Shannon's capacity theorem would dictate the upper bound of $2.6 fb$ bits/s on the achievable data rates, which is lower than the demonstrated data rates without nonlinear effects. These estimates illustrate that data rates might suffer significantly from nonlinear distortion when it is not addressed and treated as noise. Therefore, it is important to analyze the nonlinear distortion of an acoustic communication signal, examine its effects on communication performance, and explore the benefits and drawbacks of different equalization techniques when dealing with nonlinear acoustic wave propagation.

In this work, we first analyze the frequency spectrum of the received communication signal when it propagates in a lossless nonlinear medium. We show that when the signal bandwidth is wide enough (more than 2/3 of center frequency), eliminating the effects of nonlinear distortion becomes nontrivial. Then, we examine how communication variables (carrier frequency, signal bandwidth, and signal amplitude), tissue properties, and tissue thickness affect the distortion of a communication signal at biomedical frequencies and intensities. We, then, demonstrate that high SNR and high bandwidth, two contributors of high data rates in linear systems, may reduce communication performance when the distortion is not compensated. Finally, we compare the performance of three commonly used equalizers [linear, decision feedback, and maximum likelihood sequence estimation (MLSE) equalizers] in addressing nonlinear distortion.

The paper is organized as follows: In Sec. II, we introduce the analytical form of a communication signal propagating in a lossless, nonlinear media. In Sec. III, we present our experimental setup used to simulate the propagation of acoustic waves in lossy media such as soft tissues. In Sec. IV, we examine how different variables affect the severity of distortion. Then, in Sec. V, we examine the effect of acoustic nonlinearity on the error rates when decoding a communication signal. Finally, we conclude our findings in Sec. VI.

Propagation of small-signal acoustic waves is modeled with linear differential equations. However, at biomedical frequencies and intensities, signal amplitudes are known to be high enough to violate the linear propagation assumption. In that case, the (positive) peaks of the signal travel faster than the (negative) troughs, i.e., the peaks arrive earlier and the troughs are delayed [red vs black curves in Fig. 1(a)], which causes nonlinear distortion of the received signal.

Let the continuous signal, $p0(t)=p(x=0,t)$⁠, denote the initial perturbations generated by an acoustic source. For plane waves with small amplitudes, the acoustic propagation in a lossless medium can be represented with the linear wave equation (Hamilton and Blackstock, 1998),

where c₀ is the small-signal sound speed. Then, the received signal measured at x m from the source is

where $τ=t−x/c0$ is retarded time. Note that for the rest of this paper, $τ(x,t)$ is denoted by τ to simplify notation.

When the signal amplitude is not small, propagation dynamics can no longer be modeled by the linear wave equation in Eq. (1). Instead, the propagation can be modeled by the simple wave equation,

where $β=1+B/2A$ is the coefficient of nonlinearity, B / A is the acoustic nonlinearity parameter, ρ₀ is the medium density, and u denotes the particle velocity, which, for plane waves, is related to the acoustic pressure by $u=p/c0ρ0$⁠. For an initial arbitrary perturbation, $u0(t)=u(x=0,t)$⁠, the implicit solution to the simple wave equation is (Rudenko et al., 2010)

and an approximate solution that includes up to second degree (quadratic) nonlinearity is (Rudenko et al., 2010)

where $θ=βx/c03ρ0$ and $pd(t)=p(x=d,t)$ is the pressure variations measured at d m from the source. Note that Eq. (5) is valid in lossless media when $d<xsh$⁠, i.e., in the region before shock waves form. The shock distance, x_sh, for a sinusoidal waveform can be computed as

where u_i denotes the initial amplitude of the waveform and f is the frequency. Note that the model in Eq. (6) does not take attenuation, dispersion, or spreading into account. In the presence of such losses, shocks are less likely to form. Nevertheless, the effect of nonlinearity is still important to consider as it causes distortion of the signal.

The complex baseband representation of an uncoded, quadrature amplitude modulated communication signal is

where g(t) is the pulse shaping filter with roll-of factor γ, $T=1/fb$ is the symbol duration, $sI(t)$ and $sQ(t)$ are the real (in-phase) and imaginary (quadrature) parts of $sB(t)$⁠, respectively, and b_k is the kth transmitted complex symbol, representing M bits. Each symbol belongs to a predefined symbol set, $S$⁠, where $|S|=2M$⁠.

The energy of a complex baseband signal is concentrated in a band around zero frequency. However, physical communication channels can rarely accommodate baseband signals because channel responses are usually centered at higher frequencies determined by physical characteristics and limitations of the system. Therefore, passband waveforms, centered at higher frequencies, are used for communicating with passband channels. The passband waveform is obtained by upconverting the baseband signal by multiplying it with a complex carrier waveform. In this work, we denote the complex passband signal by

where f_c is the passband center frequency and G is a scaling factor accounting for a possible gain. The real-valued passband signal,

is then transmitted through the channel. In an acoustic communication system, the transmission signal in Eq. (9) is generated by a time-varying acoustic source located at x = 0, and it is measured at x = d by an acoustic sensor. Based on the model in Eq. (5), the transmitted signal, s(t), and the received signal, y(t), correspond to $s(t)=p0(t)$ and $y(t)=pd(t)$⁠, respectivey, and the received signal can be modeled by

The passband communication signal in Eq. (9) is band-limited to the frequency interval $[fc−(1+γ)fb/2,fc+(1+γ)fb/2]$⁠. The time-derivative of Eq. (9) is

which yields

where time indexing of functions is omitted for brevity. Then, the received signal in Eq. (10) becomes

In the frequency domain, the Fourier transform of y(t), denoted by $F{y(t)}$⁠, is

where $SA(f):=F{sA(τ)}, HAB(f):=F{sA(τ)sB(τ)}$ for $A,B∈{I,Q}$⁠, assuming that $lim|t|→∞s2(t)=0$⁠. Note that $∀A,B∈{I,Q}, HAB(f)=SA(f)∗SB(f)$ is band-limited to $[−(1+γ)fb,(1+γ)fb]$⁠.

One implication of Eq. (13) is that the second harmonic frequency band has double the bandwidth of the fundamental band. For sufficiently low signal bandwidth (⁠$BW<23fc$⁠), harmonic bands do not overlap. In that case, it is possible to filter out the harmonics and proceed with linear equalization without much damage to the signal, except for some energy lost in the filtered out harmonics [Figs. 2(a) and 2(b)]. However, as the signal bandwidth increases, the harmonic bands interfere with their respective upper and lower bands [Figs. 2(c) and 2(d)].

Equation (13) is a one-dimensional model (without spreading) that is valid for lossless media. In lossy media, such as soft tissues, nonlinearity, attenuation, and dispersion are coupled. The complex nature of wave equations representing these phenomena often prevents an analytical solution for an arbitrary disturbance such as a communication signal. Nevertheless, various methods have been developed to numerically obtain the pressure disturbances as measured at particular points in time and space (Alford et al., 1974; Jiménez, 2015; Saenger et al., 2000; Varslot and Taraldsen, 2005).

In this work, the k-space pseudospectral method is used to simulate the nonlinear propagation of an ultrasonic communication signal as plane wave pressure disturbances in soft tissues. We use the k-wave acoustic toolbox (Treeby and Cox, 2010) to implement the k-space pseudospectral method. The open-source toolbox is able to simulate nonlinear propagation of ultrasound signals in nonhomogeneous media with power-law absorption. It is faster than conventional numerical methods such as finite difference methods (Treeby et al., 2012). The accuracy of the k-wave toolbox when simulating nonlinear propagation in soft tissues has been verified with several experimental studies (Martin et al., 2020; Treeby et al., 2012; Wang et al., 2012).

The effects of five parameters on the propagation of a communication signal are examined: signal amplitude, center frequency, bandwidth, tissue type, and tissue thickness. Quadrature amplitude modulated (QAM) communication signals, centered at various frequencies, with different amplitudes and bandwidths are considered as the pressure disturbances generated by a time-varying acoustic source. Unless stated otherwise, the parameters in Table I are used for simulating communication signals for the analysis of distortion in liver and fat tissues. Note that the shock distances in liver and fat are 5.1 cm and 2.6 cm, respectively, for a sinusoidal tone with these parameters.

The k-wave toolbox function, kspaceFirstOrder1D, is used to simulate the received signal when pressure disturbances propagate through liver and fat tissues of various thicknesses. The function simulates nonlinear plane wave propagation in homogeneous axisymmetric medium with power-law absorption. Diagnostic ultrasound transducers are typically focused, which results in more nonlinear distortion at the focus. However, focused transmission might not be practical for small IMDs in motion such as video capsule endoscopy devices. Therefore, nonlinearity resulting from focal effects is not considered in this work.

The parameters used in simulations are listed in Table II. Acoustic properties of the tissues are obtained from Azhari (2010) and Hasgall et al. (2018). Parameters in Table II enable simulating frequencies up to $10fc$ Hz with the simulation sampling rate, $fs=80fc$ Hz, where f_c is the center frequency of the communication signal. Reflections and scattering are not considered in this study. The instruments are assumed to have flat response in the frequency band of interest.

To analyze the distortion caused by the acoustic nonlinearity, propagation of a communication signal is simulated with and without nonlinear effects included. The corresponding differential equations used for modeling acoustic propagation with and without nonlinearity are Eqs. (2.7) and (2.4) in Treeby et al. (2016). The received passband waveform, denoted by y(t), is simulated at discrete time points, $nTs=n/fs, n∈ℤ+$⁠. Then, it is downconverted to baseband by multiplying with the complex conjugate of the carrier waveform, resulting in

Next, it is filtered with the receiving low-pass filter,

where $g∗$ denotes complex conjugate operation, and $gMF[n]$⁠, which is also called the matched filter, is the time-reversed complex conjugate of the pulse shaping filter, $g[n]$⁠, in Eq. (7). The resulting baseband signal, denoted by $yB[n]$⁠, is then

and the received symbols are obtained by downsampling $yB[n]$ by L, where $L=T/Ts∈ℤ$⁠.

Figure 3(a) shows the spectrogram of the transmitted 256-QAM passband communication signal centered at 1 MHz with 1.3 MHz bandwidth and 5 MPa amplitude. Figure 3(b) shows the spectrogram of the received signal measured at 1 cm away from the source in fat tissue. Because of nonlinear propagation, the received passband signal contains out-of-band energy at high frequencies. The baseband signal [Fig. 3(c)] is obtained by downconverting and low-pass filtering the passband signal. Note that even though the out-of-band distortion caused by nonlinear propagation is suppressed by the low-pass filter, attenuation and in-band harmonic distortion still affect the baseband communication signal.

The destructive effects of acoustic nonlinearity on a communication signal are more evident in the constellation diagram in Fig. 1(b). The black scatterplot shows the complex symbols sent by the transmitter, and the red scatterplot shows the received symbols after the signal propagates 1 cm in liver tissue. Note that the received symbols are obtained by sampling the received baseband signal at the symbol rate. Attenuation, dispersion, and noise are not considered when generating this plot to isolate the effect of nonlinear distortion. The grid lines in the constellation diagram represent decision boundaries. To decode a transmitted symbol, a decision is made at the receiver such that a received symbol is assigned to the closest transmitted symbol out of 256 possible values. In general terms, nonlinear distortion causes received symbols to move away from the center of the decision regions. When nonlinear distortion is high enough, the received symbol moves to the neighboring region and this results in a decision error if the distortion is not compensated. Note that the nonlinear distortion is more pronounced in the outer symbols compared to the symbols closer to the center of the constellation because nonlinear distortion increases with signal amplitude.

In Secs. IV A–IV E, we will examine how various factors contribute to distortion. The main contributors to distortion in homogeneous soft tissues are acoustic nonlinearity and frequency-dependent attenuation. We measure the distortion as the normalized total difference between transmitted and received signals after the signals are synchronized to compensate for the time delay induced by retarded time. More formally, the passband distortion is defined as

and baseband distortion is defined as

Note that no noise is added to the signals when computing distortion in simulations.

Passband distortion quantifies the distortion of the acoustic signal propagating in the tissue, i.e., the passband signal. Baseband distortion is computed for the baseband signal obtained by downconverting and low-pass filtering the received passband signal. It quantifies the distortion of the communication signal used for symbol detection and recovery. In the case of nonlinear propagation, the baseband signal is expected to have lower distortion compared to the distortion of the passband signal because low-pass filtering removes some of the high frequency components resulting from nonlinear distortion. Increased distortion of the baseband signal yields lower SNR and higher probability of error when not compensated.

Nonlinear distortion is known to increase with signal amplitude (Muir and Carstensen, 1980). Figures 4(a) and 4(e) demonstrate passband and baseband signal distortion, respectively, with respect to source signal amplitude. Distortion without nonlinear effects (dashed lines) is caused mostly by frequency-dependent attenuation and it is amplitude independent. However, when nonlinear effects are considered (solid lines), passband and baseband distortions increase, highlighting the effect of nonlinearity. Nonlinear distortion is lower in the baseband signal compared to the passband signal because most of the harmonic bands are filtered out with the baseband low-pass filter. The overall distortion value and rate of increase in nonlinear distortion depend on tissue type as it affects the frequency-dependent attenuation and nonlinearity. The effect of tissue type on distortion will be discussed more in Sec. IV E.

The loss for monochromatic plane waves caused by acoustic attenuation in lossy media over a distance, d, is modeled by $l(d)=exp (−αd)$⁠. The attenuation coefficient, α, depends on the frequency and obeys the following power-law equation:

where $ω=2πf$ is the angular frequency and α₀ and y are tissue-dependent constants (Szabo, 2004b). For soft tissue, y is typically in the 1–1.7 range (Szabo, 1994).

The frequency-dependent attenuation causes a fundamental band centered at a higher frequency to have lower power over the same distance. As a result, distortion of passband and baseband signals in Figs. 4(b) and 4(f), respectively, increases with center frequency, whether nonlinear propagation is present or not. Moreover, the shock distance is shorter at a higher frequency. This results in increased nonlinear distortion of a signal with higher frequency when the signals are measured at 1 cm away from the source. In Figs. 4(b) and 4(f), the increasing differences between distortion curves with (solid) and without (dashed) nonlinear propagation in both tissues highlight this effect. On the other hand, attenuation causes faster decay of harmonic frequencies compared to the fundamental, which reduces nonlinear distortion at longer relative distances for high frequencies. Figure 5 shows the amplitude of the second harmonic with respect to the fundamental. Harmonic distortion is higher for high frequency tones near the source as nonlinear propagation is more pronounced and the shock formation distance is shorter. However, it is lower at further distances as a result of increased attenuation.

Frequency-dependent attenuation and phase dispersion cause non-flat channel frequency response and contribute to increased distortion. Moreover, as introduced in Sec. II B, second-order nonlinearity causes the fundamental and second harmonic bands to overlap if $BW=(1+γ)fb>23fc$ in the nonlinear scheme. The effect of overlapping bands in the passband signal is minimal with respect to overall distortion [Fig. 4(c)]. However, the baseband signal clearly demonstrates increased nonlinear distortion for $BW>23fc$⁠, especially in fat tissue [Fig. 4(g)].

Acoustic nonlinearity is a cumulative phenomenon. Therefore, nonlinear distortion can increase as the distance between the transmitter and receiver increases (Anvari et al., 2015). On the other hand, in lossy media, such as soft tissues, attenuation of the signal increases with distance. First, this causes the signal amplitude to decrease and, consequently, reduces the nonlinear effects that depend on the signal amplitude. Second, harmonics themselves are attenuated more because of frequency-dependent attenuation (Hedrick and Metzger, 2005). Cumulative nonlinearity and frequency-dependent attenuation causes nonlinear distortion to be low near the source, increase up to a distance, and decrease as the distance increases further.

Figures 4(d) and 4(h) demonstrate increasing distortion with distance. Note that for both media and with or without nonlinear propagation, the distortion increases because of increased attenuation. The difference between dashed and solid curves indicates the effect of nonlinear distortion. Note that the difference is low near the source and increases with distance as shown in the plots. The cumulative nonlinearity and attenuation trade-off is evident in Fig. 5, which shows the amplitude of harmonic tones with respect to the fundamental across distance for different frequencies. The harmonic distortion increases up to a point and decreases as the distance increases further due to attenuation. Peak harmonic distortion is observed at a shorter distance for a higher-frequency tone because the shock formation distance is shorter.

The coefficient of nonlinearity, $β=1+B/2A$⁠, and the attenuation coefficient, α, are tissue dependent. A high coefficient of nonlinearity yields more nonlinear distortion in tissues with similar attenuation profiles. On the other hand, high attenuation limits the presence of nonlinearity to a shorter distance, but it also reduces the received signal power.

Liver tissue has a larger attenuation coefficient, whereas fat tissue has a larger parameter of nonlinearity. In Figs. 4(a)–4(h), distortion without nonlinear propagation (represented with dashed lines) is higher in liver than fat due to the higher attenuation. On the other hand, Fig. 4(a) reveals that with nonlinear propagation, distortion in fat exceeds that of the liver as nonlinearity increases with signal amplitude. The difference between solid and dashed curves for a tissue in a given plot in Fig. 4 indicates the effect of nonlinearity. The difference is larger in all plots for fat compared to the difference in liver as the coefficient of nonlinearity, β, is higher for fat than liver.

Finally, for baseband signals (Fig. 4, bottom plots), overall distortion is higher for liver than fat, whereas for passband signals (Fig. 4, top plots), the opposite is true. This indicates that the passband signal is affected more by nonlinear distortion than attenuation in these cases. Furthermore, it suggests that attenuation contributes to distortion of the baseband signal more than nonlinearity as most of the nonlinear distortion is filtered out with baseband low-pass filtering. However, as it will be discussed in Sec. V, this does not necessarily yield better communication performance because attenuation can be addressed by filter-based equalizers developed for linear channels but nonlinear distortion cannot.

In Sec. IV, we demonstrated that acoustic nonlinearity distorts passband and baseband signals. Nonlinear distortion increases with signal amplitude and it increases with bandwidth if the bandwidth is greater than 67% of the center frequency. Nonlinear distortion also increases with distance up to a point and then starts to decrease because of the trade-off between cumulative nonlinearity and attenuation. The severity of distortion depends on tissue properties. In this section, we quantify the effects of nonlinear distortion by examining the bit error rates (BERs) with linear, decision feedback, and MLSE equalizers.

A linear, band-limited channel with non-flat response can be modeled as a linear filter, and the associated dispersion causes inter-symbol interference (ISI), which results in high error rates if not adequately compensated. The acoustic through-tissue channel can be represented with the linear filter model if nonlinear propagation is not prominent. To compensate for or reduce the ISI, an equalizer is used at the receiver. A linear equalizer (LE) consists of a linear filter applied to a sequence of received symbols to estimate and decide on the transmitted symbol at a given time (Proakis, 2001). A decision feedback equalizer (DFE) consists of two linear filters, one filters the received signal and the other exploits the previously decided symbols to reduce the ISI (Belfiore and Park, 1979). The MLSE equalizer computes the most likely symbol sequences by using a state-space model for the ISI channel (Benedetto and Biglieri, 1999; Forney, 1972). MLSE is the optimum equalizer in the sense that it minimizes the probability of error for a sequence of transmitted symbols over an ISI channel, but its computational complexity is higher than LE and DFEs.

In this work, communication performance is quantified by calculating the BERs at various SNRs. AWGN is added to the received pressure disturbances (in pascals) after it is converted to an electrical signal (in volts) to capture both unmodeled phenomena as well as receiver electronics noise. Material properties of the transducer determine the conversion ratio $V=glP$⁠, where g is the voltage sensitivity factor and l is the thickness of the crystal (Usher, 1985). In this work, $g=0.0248 Vm/N$ and $l=1 mm$ are chosen as representative values for PZT-5 piezoelectric transducers commonly used in biomedical ultrasound (Ito and Uchino, 2005).

Figure 6 shows BER vs SNR per bit (⁠$Eb/N0$⁠) curves when a LE is employed at the receiver. The communication signals consists of 10 000 256-QAM-modulated symbols, centered at 1 MHz with 50% and 130% bandwidths, propagated 1 cm in liver with different signal amplitudes. The equalizer has ten filter coefficients that are updated with the recursive least squares algorithm. Out of 10 000 transmitted symbols, 10% are used in training mode and the rest were used in the decision-directed mode. Figure 6(a) demonstrates BERs at different SNR values. Note that all of the BER curves for nonlinear propagation (blue curves) are superimposed with the curve for linear propagation (red curve). This illustrates that when the signal bandwidth is low, linear and nonlinear propagation yields the same BER even at signal amplitudes where nonlinear propagation is prominent. This is because the second harmonic band does not interfere with the fundamental band. Therefore, after downconversion, the low-pass filter is able to suppress the harmonic distortion, and the LE is able to compensate for the distortion caused by attenuation. The performance in this case is close to the ideal channel, indicating successful equalization and symbol recovery. Figure 6(b) demonstrates that when high signal amplitude is employed together with high signal bandwidth, nonlinear distortion causes increased error rates because this time, the fundamental and second harmonic bands interfere. In-band nonlinear distortion cannot be compensated by the LE. As the signal amplitude increases, the severity of the nonlinear distortion also increases, causing an increased difference between the error rates with and without nonlinear propagation. Moreover, even at high SNR, the resulting BERs are more than $1e−3$⁠, which is considered as a typical threshold for successful communication at the physical layer because error rates below $1e−3$ could be compensated with appropriate error correction coding with minimal overhead.

Figure 7 compares the effect of signal amplitude and bandwidth on the BERs in 1 cm liver and fat tissues. The upper right corner, where the amplitude and bandwidth are high, yields the highest BERs in both tissues. When the signal amplitude and bandwidth are high enough, the resulting BERs are $1e−3$ at 25 dB SNR. The degradation in BER with bandwidth for a given amplitude or with amplitude for a given bandwidth is larger in fat than liver because fat exhibits more nonlinearity than liver as it has a higher coefficient of nonlinearity.

Figure 8(a) shows the amplitude of the second harmonic with respect to the fundamental of a 1 MHz sinusoidal tone with 5 MPa amplitude propagating in liver. Harmonic distortion increases up to 7 cm and decreases after that because of attenuation. Figure 8(b) shows the BERs achieved with a LE when a 256-QAM communication signal, centered at 1 MHz with 100% bandwidth and 5 MPa amplitude, propagates various distances in liver. Although the harmonic distortion decreases after 7 cm, the BER still increases and communication performance does not improve. This, again, indicates that linear equalization is able to correct for attenuation and dispersion but is not able to correct for nonlinear distortion. As a result, even though the attenuation and nonlinearity yield a trade-off on the signal distortion, communication performance may degrade because of the cumulative behaviour of nonlinear distortion.

Finally, the effectiveness of linear, decision feedback and MLSE receivers against nonlinear distortion are compared. The DFE used in these simulations has a feedforward filter with ten $T/2$-spaced taps (i.e., two taps for each symbol duration) and a feedback filter with five symbol-spaced taps. Out of 10 000 transmitted symbols, 10% are used for adapting the equalizer filter coefficients in training mode using the recursive least squares (RLS) algorithm. The MLSE receiver consists of a bank of K^L matched filters, where K is the size of the symbol alphabet and L is the channel memory, followed by a maximum likelihood sequence estimator. The filters are matched to the simulated received waveforms generated for each symbol sequence. Note that generating these matched filters requires the knowledge of channel properties and propagation distance. A Viterbi decoder with K^L states is used to iteratively decode the received symbols. To keep the complexity tractable, a 64-QAM signal is used for modulation (i.e., K = 64). The pulse shaping filter span is reduced to one symbol, and L is set to two.

Figure 9 shows that the DFE achieves lower error rates compared to the LE, possibly by addressing the ISI caused by frequency-dependent attenuation and dispersion. However, it does not eliminate the error floor at high SNR, and there is a relatively large gap between the ideal AWGN curve (10 dB at BER = $1e−3$⁠). These indicate that the DFE may not be able to compensate for the acoustic nonlinearity either. The BER curve with the MLSE decoder, on the other hand, is within a few dB of the AWGN curve, indicating successful equalization despite nonlinearity. However, the computational cost of MLSE increases exponentially with channel memory and modulation order. Also, the simulated matched filters depend on the knowledge of channel properties as well as the distance between the transmitter and receiver. These factors may limit the use of MLSE in real-time or near real-time applications.

Acoustic communication has been gaining traction as an alternative to RF-based communication through soft tissues, where attenuation hinders RF communication performance significantly. Acoustic propagation is known to be a nonlinear phenomenon but it is often approximated with linear models for easier analysis. However, soft tissues have been demonstrated to introduce nonlinearity so much so that higher-order harmonics have been successfully used for imaging applications clinically for several decades. Despite being widely used in imaging, the effects of nonlinear acoustic phenomena on communication signals have not been investigated.

This study analyzes the propagation of a communication signal in soft tissues in the finite-amplitude (nonlinear) regime. The results demonstrate that two of the factors that yield high data rates in linear channels, high signal amplitude and bandwidth, cause increased nonlinear distortion of a communication signal. Equalization methods used for linear channels, e.g., LE and DFEs, can address linear acoustic phenomena such as attenuation and dispersion, but they fail to compensate for the effects of acoustic nonlinearity. Cumulative acoustic nonlinearity degrades communication performance as the signal amplitude, bandwidth, transmission distance, and tissue nonlinearity increase. It prevents successful communication in some cases even when the SNR is high and there is no scattering or reflection. A MLSE receiver with a bank of filters matched to the simulated received waveform for each transmitted symbol sequence within the channel memory performs the best. However, the computational cost increases when high-order modulation is employed with channels having long memory. A suboptimal receiver that can address acoustic nonlinearity with reduced computational cost would be useful in such cases.

This work was supported in part by the National Institutes of Health (NIH) under Grant Nos. R21EB025327 and R21EB030743 and Office of Naval Research Grant No. N0004-12-1-2662.