Efficient Recovery of Linear Predicted Coefficients Based on Adaptive Steepest Descent Algorithm in Signal Compression for End-to-End Communications

Abstract

The efficiency of recovery and signal decoding efficacy at the receiver in end-to-end communications using linearly predicted coefficients are susceptible to errors, especially for highly compressed signals. In this paper, we propose a method to efficiently recover linearly predicted coefficients for high signal compression for end-to-end communications. Herein, the steepest descent algorithm is applied at the receiver to decode the affected linear predicted coefficients. This algorithm is used to estimate the unknown frequency, time, and phase. Subsequently, the algorithm facilitates down-conversion, time and carrier recovery, equalization, and correlation processes. To evaluate the feasibility of the proposed method, parameters such as multipath interference, additive white Gaussian noise, timing, and phase noise are modeled as channel errors in signal compression using the software-defined receiver. Our results show substantial recovery efficiency with noise variance between 0 and $y$ × 10E − 3, where $y$ lies between 0 and 10 using the modeled performance metrics of bit error rate, symbol error rate, and mean square error. This is promising for modeling software-defined networks using highly compressed signals in end-to-end communications.

This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

1. Introduction

People speak various languages; interestingly, in a single country, it is not unusual to speak more than one language. But, no matter how languages differ, the way speech is generated is the same. In humans, air is expelled from the lungs, enters the vocal tract, and is pushed out of the mouth to produce speech [1, 2]. Therefore, sound or speech originates from the lungs. In the speech production model, a synthesis filter simulates the vocal tract, glottal flow, and vibration of lips in human speech production. White noise and pulse train are used as inputs to the synthesis filter, depending on the voiced or unvoiced nature of the speech signal. For example, for voiced speech signals, a pulse train is used as excitation, and white noise or random noise is used for excitation when it comes to the unvoiced speech signal. The type of excitation or input to the synthesis filter is selected by the switch and depends on the desired output. Finally, the energy level of the output is determined by the gain parameters.

Relatedly, speech coding algorithms remove unstructured redundancy from speech signals and compress them to fewer bits at an acceptable quality [1, 3]. Specifically, speech coding algorithms input signals at a relatively high bit rate and output them at a low bit rate. Such algorithms include linear predictive coding (LPC) algorithms, waveform coding, and code-excited LPC algorithms [4, 5]. These coding algorithms are lossless or lossy, depending on the quality of the output signal at the decoder. The quality of signal produced at the output of the decoder diminishes for the lossy coding algorithm and is almost the same for the lossless coding algorithm. In this article, LPC algorithm is considered.

1.1. Prior and Related Works

The identification and classification of genomic signals from Coronavirus based on LPC and machine learning have been studied [6]. In this study, the LPC algorithm was used as a data compression technique. This is based on a support vector machine with an accuracy of 98% using the LPC algorithm as a compression technique. However, a significant challenge in analyzing these data arises from the varying sizes of genomic sequences. Addressing this issue can be facilitated by using artificial intelligence and adaptive signal processing techniques to manage and rectify errors. Furthermore, it is important to note that linear predictive coefficients are particularly sensitive to channel noise and errors. In [7], the focus is on solving the problem of signal detection in massive multiple-input multiple-output (MIMO). Moreover, a low-complexity signal detector was based on the steepest descent (SD) algorithm and the nonstationary Richardson iteration method. Their numerical findings illustrate that the suggested method of joint detection outperforms current iterative methods and offers reduced computational complexity compared to the traditional linear minimum mean square error (MMSE) detector. In [8], the evaluation of different coding methods for human listening in low-rate cellular communication standards was studied. The results obtained show that the coders modeled as a voice-excited LPC algorithm are very clear and have little distortion. However, any coder modeled as a plain LPC algorithm only facilitates signals other than speech signals, or else their quality would be very low. In [9], the size of a room was determined by measuring the reverberation time. Here, a LPC algorithm was used to extract the characteristics of the training data in the form of an audio signal. It was concluded that the detection of a room based on reverberation time using a LPC algorithm results in 83.33% accuracy. During data processing and speech signal reconstruction from coefficients, inaccuracies were observed due to noise that affects the integrity of linear predicted coefficients.

This research [10], delved into node deployment strategies to mitigate the complexities of target coverage, area coverage, and connectivity across homogeneous and heterogeneous wireless sensor networks (WSNs) with randomly distributed nodes. This paper introduced a fresh analytical deployment technique utilizing the SD algorithm along with Armijo and Wolf rules, as opposed to the evolutionary methods traditionally used. Results indicated that this approach surpasses genetic algorithm (GA) in efficiently directing sensor movement toward targets, ensures complete target coverage, and significantly reduces complexity of the algorithm, particularly by over 40% in the most challenging scenarios. Moreover, numerical findings demonstrated that while the complexity of this two-step approach closely aligns with existing GA methods, it offers superior trajectory planning for sensors and an enhanced accuracy in network coverage and connectivity.

1.2. Motivation and Contributions

A lot of research has been done on LPC algorithms [2–16]. However, most research focuses on the implementation of LPC algorithms as a speech compression technique. In particular, it is assumed that the coefficients extracted from the speech signals are already received at the decoder. Thereafter, the original speech signal can be modeled effectively and efficiently at the decoder without any errors in the transmitted coefficients (compressed speech signal). However, in the real communication system, the pre-decoding blocks are very vital. In addition, as the signal propagates from the transmitter to the receiver, it is affected by channel errors such as multipath interference and additive noise [3, 9, 10, 15, 16]. In a communication system, linear predicted parameters extracted from the speech signal are sent to the decoder instead of transmitting the whole speech samples. However, if the received parameters are affected by channel errors, the decoder cannot model the original speech signal from the received parameters [1, 2, 4]. Therefore, linear predicted coefficients are highly susceptible to errors, which affect the recovery efficiency and signal decoding at the receiver.

In this study, the concentration is put on mitigating errors that affect linear predictive parameters by implementing the adaptive SD algorithm in the receiver’s block. More specifically, the adaptive SD algorithm is implemented in carrier recovery, timing recovery, and equalization blocks. Once the receiver and the transmitter are synchronized, phase, frequency, and timing offsets in the received erroneous signal could be estimated. This eases demodulation, sampling at correct instants, and also efficient decoding of linearly predicted parameters, which enabled efficient modeling of the original speech signal. The main contributions of this paper can be summarized as follows:

• The SD algorithm is implemented in the receiver’s algorithms to effectively recover the speech signal from its parameters.

• The effect of channel errors on linearly predicted coefficients for highly compressed speech signals is studied.

• A comprehensive study on how receivers can be affected by channel errors during reconstructing highly compressed speech signals has been carried out.

• This study expands and enriches the researcher’s knowledge as far as end-to-end communication is concerned, thereby promoting innovation and creativity in software-defined networks for end-to-end communications.

The results simulated in terms of measurement parameters such as bit error rate (BER), symbol error, and mean square error (MSE) are analyzed to investigate the impacts of implementing adaptive SD algorithms in a software-defined receiver’s blocks while reconstructing a highly compressed speech signal from its parameters.

2. The Principle

2.1. SD Algorithm

The SD algorithm starts with initializing (guessing) the location of a minimum [11, 12]. Thereafter, assess which direction from an initial guess is the steepest as going down the hill, and then make a new guess along the same direction. In the same sense, ascending a hill also starts with an initial estimate or guess of a maximum location, assessing which direction climbs very fast, and then a new guess is made along that direction [17, 18]. In most cases, a new guess is more accurate than the previous one. The process repeats, ideally getting closer to an optimal location at each step. Most importantly, the direction of the current location is assessed by the gradient. The upward direction is determined by a positive gradient and a negative gradient is for the downward direction. Consider a polynomial given in the following equation [19, 20]:$$\begin{array}{}\text{(1)}& Jx=x^2+bx+c,\end{array}$$and the SD algorithm can be implemented to minimize it [12]. Let $x$ be the initial estimate at time $k$, which can then be represented by $xk$. A new estimate of $x$ at time $k+1$ obtained by SD algorithm is shown in the following equation:$$\begin{array}{}\text{(2)}& xk+1=xk-\mu \frac{\mathrm{\partial }Jx}{\mathrm{\partial }x},\end{array}$$where $\mu$ is an adaptive step size and $Jx$ is the objective function which is to be optimized at a current estimate $xk$. This process is repeated as the as the value of $k$ increases. Figure 1 illustrates this procedure. When the current estimate $xk$ is to the right of the minimum, the negative gradient points to the left. Conversely, when $xk$ is to the left of the minimum, the negative gradient points to the right. In both cases, if the step size is sufficiently small, the updated estimate $xk+1$ will be closer to the minimum than the previous estimate $xk$; that is, $Jxk+1$ will be smaller than $Jxk$ [21].

[figure(s) omitted; refer to PDF]

However, even though the descent algorithm may converge to a minimum, it might not necessarily be the absolute lowest one. Once it settles in a minimum, it cannot ascend to a peak and then descend to the true lowest minimum. Hence, it is essential to initialize the SD algorithm near the vicinity of the true minimum to ensure accurate convergence.

The determination of the step size is an important factor while implementing the SD algorithm. For smooth convergence, a smaller step size is required. In addition, the step size can be obtained using the golden search, Fibonacci search, and bisection method. In this research, the step size is obtained by applying the golden search method as explained in [22]. For example, determining the minimum point for the objective function in the form shown in equation (2) using the SD algorithm, the step size value of 0.05 obtained from the golden search method provided the smooth convergence as shown in Figure 2. In our system, the decision to choose the golden section search method was primarily driven by the specific characteristics of the communication environment, particularly the presence of noise and interference, which are inherent in real-world communication systems as explained in [22, 23].

[figure(s) omitted; refer to PDF]

2.2. LPC Algorithm

One of the oldest coding techniques is LPC. It works at a relatively low bit rate and is frequently employed in audio signal processing. The United States Department of Defense first created it in federal standard 1015, which was issued in 1984, to enhance security in communication for military uses [1]. A source encoder uses LPC as a compression algorithm. LPC algorithm at the source encoder extracts linear predicted parameters from speech signals in terms of pitch period, filter coefficients, and voicing parameters as shown by a block diagram in Figure 3.

[figure(s) omitted; refer to PDF]

Figure 3 shows a linear predictive encoder where the human speech signal acts as the input to this block. The whole speech samples are not transmitted; however, the parameters extracted are rather transmitted up to the decoder. Speech signals frequently vary over time, and thus it is important to segment them into frames using a short time duration, such as 25 ms–30 ms [1, 2, 4, 5]. For speech analysis, speech signals frequently vary over time, and thus it is important to segment them into frames using a short time duration, such as 25 ms–30 ms. However, signal framing causes discontinuous changes which are smoothed by windowing [13, 14]. The pre-emphasis filter passes components with high frequencies that adjust the spectrum of the input signal.

The linear predicted coefficients from the speech signal are useful in modeling the prediction error filter, which is used in filtering the pre-emphasized speech frames with a predictive error signal. For a voiced speech frame, the error signal acts as input to an algorithm to estimate pitch periods, and this leads to more accurate values of estimated pitch periods [2]. The parameters obtained which include filter coefficients, pitch period parameters, power, and voicing parameters are combined to form one bit stream and transmitted to the decoder shown in Figure 4 to reconstruct the speech signal from its parameters [1].

[figure(s) omitted; refer to PDF]

2.3. End-to-End Communication

A communication channel is the path a signal takes as it travels from a transmitter to a receiver. To add structured redundancy bits, the channel encoder receives the linear predicted parameters from the source encoder. A pulse-shaping filter then converts the message bits or symbols into an analog signal. The modulated signal travels through a noisy channel up to the receiver. Before reaching the receiver, the signal experiences numerous impairments or interferences as it travels through the channel. The receiver is equipped with blocks or algorithms to fully decode the transmitted signal. The transmitter–receiver algorithms are explained as follows.

2.3.1. Quadrature Amplitude Modulation (QAM)

QAM is a quadrature modulation system that efficiently utilizes bandwidth since it carries more information than other nonquadrature modulation schemes like amplitude modulation (AM). In QAM, two signals can be transmitted simultaneously on the same bandwidth using orthogonal cosine and sine carriers [24–26]. By the help of a pulse shaping filter $pt$, signals to be transmitted can be obtained by pulse shaping the digital symbols. The message symbols are represented by a signal $m_{i}t$ as indicated in the following equation [21]:$$\begin{array}{}\text{(3)}& m_{i}t={\displaystyle \sum }_k{s}_{i}kpt-kT,\end{array}$$given that $s_{i}k$ is taken as $i\text{th}$ symbol at time $k$ and $T$ is interval between adjacent symbol. The corresponding modulated signal $rt$ is expressed as$$\begin{array}{}\text{(4)}& rt=\text{Re}{\displaystyle \sum }_{k}pt-kTsk{e}^{j2\pi f_{c}t+\varphi },\end{array}$$where$$\begin{array}{}\text{(5)}& sk=s_{1}k+j{s}_{2}k,\end{array}$$$sk$ is a complex-valued symbol obtained by combining the two real-valued messages.

The received signal r(t) in equation (4) is mixed with the complex sinusoid $e^{-j2\pi ft+\theta }$ as shown in equation (6) where θ is the phase and f is the frequency at the receiver.$$\begin{array}{}\text{(6)}& yt=e^{-j2\pi ft+\theta }rt.\end{array}$$

The demodulated signal $yt$ is low-pass filtered to remove components with high frequencies. We assume that $f_c\sim f$ and the cutoff frequency $f_l$ of the low pass filter is less than $2f_c$ but greater than $f_c,\text{i.e.,}{f}_c<f_l<2f_c$. This means that the low-pass filter will only pass frequency components between $f_c$ and $2f_c$ otherwise removed. The down converted signal $st$ is expressed as$$\begin{array}{}\text{(7)}& st=\frac{1}{2}mt{e}^{-j2\pi f-f_{c}t+j\theta -\varphi },\end{array}$$given that $mt=m_{1}t+j{m}_{2}t$. If $f=f_c$, and, $\varphi =\theta$, the down converted signal $st$ is the attenuated copy of the transmitted signal $mt$ by a factor of $1/2$ as indicated in equation (8).$$\begin{array}{}\text{(8)}& st=\frac{1}{2}mt.\end{array}$$

In practice, a signal experiences several impairments as it travels from the transmitter to the receiver, including multipath interference, broad and narrow-band noise, inter-symbol interferences (ISIs), and many others. The frequencies and phases used at the transmitter may change from those at the receiver as a result of these interferences. If the transmitter’s frequency and phase are unknown at the receiver, it is difficult to down-convert the received signal. But, there are several algorithms available that can be used to determine these unknown parameters. Costas loop, phase-locked loop, and decision-directed algorithms are some of these algorithms. The Costas loop algorithm is taken into account in this investigation.

2.3.2. Costas Loop Algorithm

The 4-QAM Costas loop algorithm is shown in Figure 5 [12, 27]. In Figure 5, the four out-of-phase cosines are joined with the received signal $rt$. A low-pass filter is applied on each path and multiplied to generate an update term. Finally, they are added for an update at each iteration with a step size $\mu$.

[figure(s) omitted; refer to PDF]

2.3.3. Timing Recovery

A transmitted signal goes through the channel, up to the receiver, where it is received as an analog signal. The received signal is down-converted and sampling must be done at exact times to extract the message sent. The specification of the correct times upon when to sample is completed by time recovery. From equation (4), two message symbols $s_1$ and $s_2$ are received; hence, two adaptive timing parameters ${\tau }_1$ and ${\tau }_2$ are used to obtain optimal instants to determine when should sampling take place. However, these two messages are similar and must be sampled at the same time. Otherwise, one adaptive parameter $\tau$ is enough to determine when to sample message symbols $s_1$ and $s_2$ at the same time. The core objective is to estimate an adaptive parameter $\tau$ that optimizes the objective function shown in equation (9) [18, 25]. An objective function $J_T\tau$ is determined by summing the fourth power of the received signal.$$\begin{array}{}\text{(9)}& J_T\tau =\text{avg}{s}_1^4+s_2^4,\end{array}$$where $\text{avg}$ is the averaging operation, $s_1=m_{1}kT+\tau$, $s_2=m_{2}kT+\tau$, and $T$ is the interval between adjacent symbols. Thus,$$\begin{array}{}\text{(10)}& J_T\tau =\text{avg}{m}_1^{4}kT+\tau +m_2^{4}kT+\tau .\end{array}$$

From the SD (equation (2)) algorithm [7, 10, 21],$$\begin{array}{}\text{(11)}& \tau k+1=\tau k-\mu \frac{\mathrm{\partial }{J}_T\tau }{\mathrm{\partial }\tau },\text{(12)}& \tau k+1=\tau k-\stackrel{̿}{\mu }{\frac{\mathrm{\partial }{m}_1^{4}kT+\tau +m_2^{4}kT+\tau }{\mathrm{\partial }\tau }}_{\tau =\tau k}.\end{array}$$

Equation (12) can be expressed as shown in equation (13).$$\begin{array}{}\text{(13)}& \tau k+1=\overline{\mu }{m}_1^{3}kT+\tau +m_2^{3}kT+\tau \times {\frac{\mathrm{\partial }{m}_{1}kT+\tau +m_{2}kT+\tau }{\mathrm{\partial }\tau }}_{\tau =\tau k}.\end{array}$$

Numerically, equation (13) can be expressed as equation (14) [21].$$\begin{array}{}\text{(14)}& \tau k+1=\tau k-\mu m_1^{3}kT+\tau k+m_2^{3}kT+\tau k\times m_{1}kT+\tau +ϵ-m_{1}kT+\tau -ϵ+m_{2}kT+\tau +ϵ-m_{2}kT+\tau -ϵ,\end{array}$$given that $\mu =\overline{\mu }/ϵ$, where $ϵ$ is a small positive integer. The values for $m_{i}t$ for the timing offsets $\tau k$ and $\tau k\pm ϵ$ can be interpolated from the over-sampled $m_i$ [21].

2.3.4. Equalization for QAM

A signal experiences many impairments as it travels across the transmission medium, which causes symbols to interfere with one another. ISI is a result of the interaction of symbols, which can also be brought on by multipath interferences, the overlapping or interaction of pulse forms at the receiver, and the nonlinearity of the channel. Specifically, the equalization undoes channel effects, or it is considered as a filter at the receiver which cancels out the negative effects of the channel. The core objective is to estimate the adaptive parameter of the equalizer coefficient vector $\mathrm{g}$. The main purpose of this adaptive parameter is to optimize the objective function. The equalizer output $yk$ can be obtained as the inner product of the filter coefficient gi with a vector of the received signals $rk-i$. The equalizer output can be obtained as shown in equation (15) [18].$$\begin{array}{}\text{(15)}& yk=F^{T}k\mathrm{g},\end{array}$$where$$\begin{array}{c}\text{(16)}& F^{T}k=rk,rk-1,\dots ,rk-m,\mathrm{g}={g_1,g_2,\dots ,g_m}^T.\end{array}$$

Equation (17) shows the difference between the equalizer output $yk$ and a known training sequence $pk$.$$\begin{array}{}\text{(17)}& dk=pk-yk.\end{array}$$

The least mean squares (LMS) algorithm is an adaptive filtering technique used to optimize the filter coefficients related to the difference between the desired signal and the actual signal. By minimizing the LMS algorithm $J_{\text{LMS}}\mathrm{g}$, the equalizer coefficient vector $\mathrm{g}$ can be optimized. The LMS objective function can be expressed as shown in equation (18) [18].$$\begin{array}{}\text{(18)}& J_{\text{LMS}}\mathrm{g}=\frac{1}{2}\text{avg}dk{d}^{\mathrm{\ast }}k,\end{array}$$given that $d^{\mathrm{\ast }}k$ is complex conjugate. However, these terms are complex-valued, having real and imaginary parts. This is because QAM has two signals, which include real $R$ and imaginary $I$ parts of a single complex-valued signal [18]. Therefore,$$\begin{array}{c}\text{(19)}& dk=d_{R}k+j{d}_{I}k,pk=p_{R}k+j{p}_{I}k,\\ Fk=F_{R}k+j{F}_{I}k,\\ \mathrm{g}={\mathrm{g}}_{R}k+j{\mathrm{g}}_{I}k.\end{array}$$

The adaptive element $\mathrm{g}$ taken as an equalizer coefficient is optimized by minimizing the objective function derived from the SD algorithm (equation (2)).$$\begin{array}{}\text{(20)}& \mathrm{g}k+1=\mathrm{g}k-\overline{\mu }{\frac{\mathrm{\partial }dk{d}^{\mathrm{\ast }}k}{{\mathrm{g}}_{\mathrm{R}}}}_{\mathrm{g}=\mathrm{g}k}+j{\frac{\mathrm{\partial }dk{d}^{\mathrm{\ast }}k}{\mathrm{\partial }{\mathrm{g}}_{\mathrm{I}}}}_{\mathrm{g}=\mathrm{g}k},\end{array}$$given that ${}^{\mathrm{\ast }}$ represents complex conjugation. The equalizer output $yk=F^{T}K\mathrm{g}$ can then be expressed as$$\begin{array}{c}\text{(21)}& F^{T}k\mathrm{g}=F_R^{T}k+j{F}_I^{T}k{\mathrm{g}}_R+j{\mathrm{g}}_I=F_R^{T}k{\mathrm{g}}_R-F_I^{T}k{\mathrm{g}}_I+j{F}_R^{T}k{\mathrm{g}}_I+F_I^{T}k{\mathrm{g}}_R,\end{array}$$and its conjugate is given as$$\begin{array}{}\text{(22)}& {F^{T}k\mathrm{g}}^{\mathrm{\ast }}=F_R^{T}k{\mathrm{g}}_R-F_I^{T}k{\mathrm{g}}_I-j{F}_I^{T}k{\mathrm{g}}_R+F_R^{T}k{\mathrm{g}}_I.\end{array}$$

Consider $z=x+jy$ and $z=x-jy$; then $z{z}^{\mathrm{\ast }}=x^2+y^2$. Therefore,$$\begin{array}{}\text{(23)}& dk{d}^{\mathrm{\ast }}k=d_R^{2}k+d_I^{2}k,\end{array}$$which can be augmented as$$\begin{array}{c}\text{(24)}& d_{R}k=\text{Re}dk=\text{Re}pk-\text{Re}{F}^{T}k\mathrm{g}\\ =d_{R}k-F_R^{T}k{\mathrm{g}}_R+F_I^T{\mathrm{g}}_I,\\ d_{I}k=\text{Im}dk\\ =\text{Im}pk-\text{Im}{F}^{T}k\mathrm{g}\\ =d_{I}k-F_I^{T}k{\mathrm{g}}_R-F_R^T{\mathrm{g}}_I.\end{array}$$

However, the training sequence $p$ does not depend on filter coefficients $\mathrm{g}$, and hence$$\begin{array}{}\text{(25)}& \frac{\mathrm{\partial }p}{\mathrm{\partial }\mathrm{g}}=0.\end{array}$$

Therefore,$$\begin{array}{c}\text{(26)}& \frac{\mathrm{\partial }dk{d}^{\mathrm{\ast }}k}{\mathrm{\partial }{\mathrm{g}}_R}=\frac{\mathrm{\partial }{d}_R^{2}k+d_I^{2}k}{\mathrm{\partial }{\mathrm{g}}_R}=\frac{\mathrm{\partial }{d}_R^{2}k}{\mathrm{\partial }{d}_{R}k}\frac{\mathrm{\partial }{d}_{R}k}{\mathrm{\partial }{\mathrm{g}}_R}+\frac{\mathrm{\partial }{d}_I^{2}k}{\mathrm{\partial }{d}_{I}k}\frac{\mathrm{\partial }{d}_{I}k}{\mathrm{\partial }{\mathrm{g}}_R}\\ =-2d_{R}k{F}_{R}k-2d_{I}k{F}_{I}k.\end{array}$$

Then the adaptive LMS equalization algorithm can be implemented by the following recursive formula shown in equation (27) as explained in [21].$$\begin{array}{}\text{(27)}& \mathrm{g}k+1=\mathrm{g}k+\overline{\mu }{d}_{R}k{F}_{R}K+d_{I}k{F}_{I}k+j-d_{R}k{F}_{I}k+d_{I}k{F}_{R}k.\end{array}$$

The above equation can be deduced as shown in equation (28) [21].$$\begin{array}{}\text{(28)}& \mathrm{g}k+1=\mathrm{g}k-\mu dk{F}^{\mathrm{\ast }}k.\end{array}$$

3. Methodology

The human voice is recorded in MATLAB for 10 s, and it is considered as the signal of interest for this research. The linear predicted parameters are extracted from the recorded voice signal (refer to Subsection 2.2), and subjected to channel errors. The software receiver’s algorithms is implemented using the SD algorithm (refer to Subsection 2.3) to overcome channel errors such as phase and frequency offset, ISI and multipath interference, and phase and timing noise which are implemented in MATLAB as explained below:

• The timing offset is assumed to be 0.25. Timing and phase noise are implemented as random walk in MATLAB.

• The phase offset is assumed to be a constant value of $\pi /7$.

• The received signal at the receiver is subjected to additive noise, which is modeled as a white Gaussian random process. The noise variance is determined by specifying the channel’s signal-to-noise ratio (SNR) during simulation.

• ISI is from the multipath channel, which is implemented using three different configurations, a unity channel [1], a two-tap channel [0, 1], and a three-tap channel [−0.4, 1, 0.8]. This is done to investigate how each channel affects a software receiver while reconstructing the affected linear predicted parameters. In this study, the block diagram shown in Figure 6 is considered for transmitter–receiver simulation.

[figure(s) omitted; refer to PDF]

Channel encoding is completed by (7, 4) linear block algorithm such that the bit stream can be protected before transmission to the communication channel. The square root raised to cosine pulse with a roll-off factor of 0.3 is used to convert information bits into an analog signal. The analog signal is modulated using a QAM scheme and then sent to the receiver. For perfect down conversion, the phase and the frequency at the transmitter must be the same at the receiver; hence, carrier synchronization is by the adaptive Costas loop algorithm. The received signal is down-converted and low-pass filtered to stay with the signal in the band of interest. The timing recovery synchronization block predicts the correct times upon when to sample the demodulated signal. To train the equalizer, a correlation is performed. The sampled signal is sent to the equalization block, which is completed by both the trained LMS and decision-directed equalizer. Recall that the details of the carrier and timing synchronization algorithms are introduced and elucidated in Section 2.3.

4. Results and Analysis

LPC algorithm completes source encoding and decoding blocks. The source encoding block shown in Figure 3 is implemented in MATLAB, and linearly predicted parameters were extracted from the recorded voice signal. Therefore, the decoder reconstructs the original recorded voice from the linearly predicted parameters that are transmitted. For a LPC algorithm, the quality of the reconstructed speech signal is reduced at the output of the decoder because of the lossy compression technique. However, due to the flat nature of white noise, the power-spectrum density of the recorded voice signal is close to that of the reconstructed signal at the output of the decoder as shown in Figure 7.

[figure(s) omitted; refer to PDF]

To implement an adaptive Costas loop in MATLAB, the phase offset is taken to be $\pi /7$. When the algorithm is subjected to a unit channel using a step size of 0.5 $\mu =0.5$, the adaptive Costas loop algorithm converges to the correct value as shown in Figure 8. The minimum step size value is obtained using a golden search method. The algorithm is further subjected to a unit delay channel with the same step size, and again the adaptive Costas loop converges to the correct assumed value. This is because the offset is constantly and adaptively estimated over time until it is obtained by the algorithm. However, divergence is noted when the algorithm is subjected to a three-tap channel even though the same step value of step size is used. Since the phase offset is not correctly estimated by the algorithm, the received analog signal is not properly down-converted. But estimating the new step size value $\mu =5\times {10}^{-2}$ using the same method for a three-tap channel, the value of phase offset is correctly estimated and the received analog signal is demodulated. Therefore, while implementing the algorithm from the SD algorithm, the choice of the value of the step size is very important [11, 12, 17]. The value of timing offset is assumed to be 0.25 and the estimated step size value is 0.5, and the fourth power maximization algorithm converges to the correct value (Figure 9) when subjected to a unit delay channel. Nevertheless, when the algorithm is subjected to a two and a three-tap channel, the algorithm deviates from the assumed timing offset value. The divergence is attributed to too much noise in the channel such as ISI, which constantly changes the convergence value of the algorithm. However, when the step size value is again estimated at $5\times {10}^{-2}$ by a golden search method for a three-tap channel $-0.4,1,0.8$, the fourth power maximization algorithm converges to the correct timing offset.

[figure(s) omitted; refer to PDF]

4.1. Simulation Results of Software Receiver

For analysis, the measurement parameters, including BER, MSE, and symbol error rate (SER), are used herein. In Figure 6, the output of the decision-directed equalizer is compared with the input to the channel encoding block to estimate the MSE of the equalizer. In addition, the output of the channel encoder is compared to the channel decoder’s output to estimate SER. The output of the source encoder is compared to the input of the source decoder to estimate the BER. The measurement parameters, in terms of SER, bit error, and MSE for different channels and noise variances, are used to evaluate how channel errors influence a software receiver in regenerating highly compressed speech signal. Setting the variance value for phase and timing noise to zero while maintaining that of ISI to be $2\times {10}^{-8}$, the results obtained for different channel taps are shown in Table 1. The speech signal is reconstructed from its parameters in all three cases. This is because the channel errors are not too much in addition to the receiver’s adaptive algorithms which adaptively estimated all the offsets and the equalization block is able to mitigate the channel effects, and finally, the transmitted symbols are perfectly decoded at the source decoder. Setting the variance value for phase and timing noise to $1\times {10}^{-3}$ while maintaining that of ISI to $2\times {10}^{-8}$, the results obtained for different channel taps are shown in Table 2. The speech signal is reconstructed from its parameters in all three cases as indicated by the small values of measurement metrics (BER, SER, and MSE). Setting the variance value for phase and timing noise to $1\times {10}^{-4}$ while maintaining that of ISI at $2\times {10}^{-8}$, the results obtained for different channel taps are shown in Table 3. The receiver can reconstruct a compressed speech signal when subjected to a unit channel, a unit delay, and a three-tap channel. This informed the range of noise variance to be chosen for a receiver to reconstruct a speech signal from its parameters using the adaptive SD algorithm. Therefore, for a software-defined receiver to efficiently recover a speech signal from its transmitted parameters, its algorithm must be derived from the adaptive steepest algorithm provided that the noise variance value is between 0 and $y\times {10}^{-3}$, where $y$ is an integer between 0 and 10. However, a noisy channel affects a software-defined receiver while reconstructing a transmitted signal. Additionally, it is investigated to determine how BER varies with SNR for different channels. The theoretical (approximated) BER is estimated by the probability of an incorrect decision for an M-QAM system, given by the following equation [21]:$$\begin{array}{}\text{(29)}& \text{BER}=1-{1-1-\frac{1}{\sqrt{M}}\text{erfc}\sqrt{\frac{3{\sigma }_2^2}{2M-1{\sigma }_1^2}}}^2,\end{array}$$where ${\sigma }_1^2$ is the Gaussian noise variance, ${\sigma }_2^2$ is the sequence variance, and $\text{erfc}$ is the complementary error function. Figures 10, 11, and 12 represent BER versus SNR for the one tap [1], two tap [0,1], and three tap channel [−0.4,1,0.8]. It can be noticed that there is no significant difference among the plots. This is attributed to the adaptive nature of the receiver’s algorithms that adaptively estimate the offset irrespective of differences in the channel errors. The theoretical curve indicates the expected BER based on theoretical calculations as shown in equation (29) for 16-QAM system under ideal conditions. The measured curves (red, blue, and magenta triangles) indicate the actual BER obtained from simulations under practical conditions. Generally, both theoretical and measured curves show a decrease in BER as SNR increases. This means that as the signal becomes stronger relative to the noise, fewer errors occur in the transmitted data.

Table 1

The noise variances for both phase and timing noise are set to 0 and those of ISI are set to $2\times {10}^{-8}$.

Channel	Description	Results
[1]	BER	$1.20\times {10}^{-9}$
	SER	$1.90\times {10}^{-5}$
	MSE	$1.90\times {10}^{-4}$
[0, 1]	BER	$1.50\times {10}^{-8}$
	SER	$1.90\times {10}^{-4}$
	MSE	$3.40\times {10}^{-3}$
$-0.4,1,0.8$	BER	$1.40\times {10}^{-6}$
	SER	$1.70\times {10}^{-3}$
	MSE	$2.60\times {10}^{-2}$

Table 2

The noise variances for both phase and timing noise are set to $1\times {10}^{-3}$ and those of ISI are set to $2\times {10}^{-8}$.

Channel	Description	Results
[1]	BER	$3.68\times {10}^{-4}$
	SER	$7.04\times {10}^{-4}$
	MSE	$1.60\times {10}^{-4}$
[0, 1]	BER	$3.88\times {10}^{-3}$
	SER	$6.44\times {10}^{-3}$
	MSE	$1.33\times {10}^{-3}$
$-0.4,1,0.8$	BER	$4.50\times {10}^{-3}$
	SER	$6.61\times {10}^{-3}$
	MSE	$0.54\times {10}^{-3}$

Table 3

The noise variances for both phase and timing noise are set to $1\times {10}^{-4}$ and those of ISI are set to $2\times {10}^{-8}$.

Channel	Description	Results
[1]	BER	$1.12\times {10}^{-8}$
	SER	$5.99\times {10}^{-4}$
	MSE	$4.83\times {10}^{-4}$
[0, 1]	BER	$1.2\times {10}^{-7}$
	SER	$3.4\times {10}^{-3}$
	MSE	$5.70\times {10}^{-4}$
$-0.4,1,0.8$	BER	$1.25\times {10}^{-3}$
	SER	$2.04\times {10}^{-3}$
	MSE	$4.51\times {10}^{-4}$

[figure(s) omitted; refer to PDF]

However, the measured BER is higher than the theoretical BER across the SNR range. This is typically because practical systems often experience additional imperfections and noise sources not accounted for in theoretical models. At lower SNR values (e.g., around 5 dB), the BER is relatively high (close to 1) as shown in Figures 10, 11, and 12, indicating poor performance with a high error rate. As SNR increases to around 15–20 dB, the BER drops significantly, indicating improved performance with fewer errors. At high SNR values (e.g., 20 dB), the BER approaches very low values (close to ${10}^{-4}$), showing that the system performs very well with minimal errors. In addition, the BER versus SNR graphs demonstrate that the SD algorithm is highly effective in recovering highly compressed speech signals derived from linear predictive coefficients in the following ways:

• Low BER at high SNRs: As the SNR increases (beyond 15 dB), the BER approaches zero, which indicates the algorithm’s ability to accurately recover the signal in less noisy environments. This aligns with the theoretical performance, confirming its reliability.

• Robustness at low SNRs: Even in challenging conditions (SNR between 5 and 10 dB), the algorithm maintains a relatively low BER compared to typical benchmarks. This highlights its robustness in handling high levels of noise, which is critical for recovering signals after compression and transmission.

• Close match to theoretical performance: The measured BER shown in Figures 10, 11, and 12 closely aligns with the theoretical BER, validating the optimality of the SD algorithm for correcting channel errors and recovering compressed LPC-based speech signals.

• Overall, the graphs (Figures 10, 11, and 12) demonstrate that the SD algorithm effectively minimizes reconstruction errors, even under noisy conditions, making it a reliable choice for recovering highly compressed speech signals.

A software-defined receiver is highly influenced by channel errors while generating a compressed speech signal. This is due to a very small range of noise variance which must be selected for a receiver to perfectly reconstruct a transmitted signal. However, employing the SD algorithm in the receiver’s system significantly diminishes channel errors.

5. Conclusion

An investigation is done to investigate how an adaptive SD algorithm can be implemented at the receiver to decode linear predicted coefficients. In this study, the speech signal is encoded at the linear predictive encoder by extracting linear predicted coefficients from the author’s voice and then subjected to a noisy channel. The fundamental algorithms of transmitter–receiver communication are also theoretically explained and simulated in MATLAB. Using measurement parameters including SER, BER, and MSE, investigations are conducted to confirm the impacts of the adaptive SD algorithm while reconstructing a speech signal from its coefficients. The findings led to the following conclusions:

• A compressed speech signal is highly affected by channel errors.

• To perfectly decode a signal by a receiver, the transmitter must be synchronized with the receiver. Therefore, if the channel errors are too immense, adaption can be implemented in the receiver’s algorithms to estimate phase, frequency, and timing offsets to achieve synchronization between a transmitter and a receiver, thus reducing the negative effects of the channel.

• When choosing the step size value while implementing an algorithm derived from the SD algorithm, care must be taken. In other words, the step size value should not be too huge or too small.

Finally, this study has the potential to contribute to the existing body of literature in today’s ongoing research efforts. It addresses the persistent demand for advanced technologies characterized by enhanced capacity, lowered costs and size, and increased reliability crucial for integration into communication systems and networks such as 5G, thus enabling more dependable communications.

Funding

This work was funded by UNESCO TWAS Seed Grant for New African Principal Investigators (SG-NAPI) no. 3240337117. Kyambogo University Competitive Research Grants Scheme No. 8 of 2023.

Acknowledgments

The authors would like to acknowledge contribution from the thesis by Abbas Kagudde titled “Effects of Channel Errors on Coded Speech Communication in Software Defined Radio” published at İzmir Institute of Technology, 2022.

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