Linear Block Coding and Discrete Wavelet Transform Based Audio Signal Transmission over AWGN and Rician Fading Channels

Mobile communication has become an important part of our daily lives for voice communication, data sharing and access over the Internet. Mobile communication is an open network, so maintaining the privacy and reliability of data has always been anxiety. The reliability of the data against channel noise can be achieved by various error correction codes. The purpose of the channel coding process is to reduce the effect of some disruptive elements that the data is influenced in the transmission phase as much as possible. This procedure ensures that the data is delivered to the receiver with minimum error. In this study, a communication system was established in MATLAB environment for the transmission of five second audio signal. In this communication system, BER and MSE performances of four phase shift keying methods are obtained first.  In the second phase of the study, the AWGN and Rician fading channels were individually coded with four different LBC types. Performance evaluation of these coding types was done with BER and MSE criteria. In the last part of the study, it was investigated which wavelet family is suitable for which level, in order to ensure audio transmission over AWGN and Rician fading channels with the least possible error. Four wavelet transform families at different levels were applied to the audio signal for LBC (7, 4), LBC (15, 8), LBC (17, 8) and LBC (23, 12) encoded channels, and the Mean Squared Error (MSE) performances were compared.


I. INTRODUCTION
RADITIONAL ANALOG communication techniques do not perform well in mobile environments because appropriate techniques for reducing the effects of multi-path propagation fading have not been developed [1]. Here, extra bits (parity bits) are added to the raw data before modulation. On the receiver side, the channel coding is called the decoder. Channel coding allows the receiver to detect and correct errors caused by effects such as noise, parasite and fading that occur during transmission [2]. The key benefits of channel coding to communication systems are as follows: maximizing data transmission efficiency to provide high bit rate, low encoder / decoder complexity by limiting the size and cost of transceivers, minimizing the amount of energy required for transmission at the desired signal to noise ratio and thus achieving high Bit Error Rate (BER) performance. Lower-rate channel codes can usually correct more errors [3]. In recent years, many studies have been carried out using different channel coding methods in order to transmit data with the least possible loss. In [3], Dhaliwal et al. analyzed convolutional code's BER performance over Additive White Gaussian Noise (AWGN) channel using fixed Viterbi decoder, different code rates, different restriction lengths and Quadrature Phase Shift Keying (QPSK) modulation. In [4], Baviskar et al. proposed an adaptive Low Density Parity Control Code (LDPC) supported audio processing algorithm to transmit the speech signal over the wireless AWGN channel. Binary Phase Shift Keying (BPSK) technique have helped to achieve better BER performance. In [5], the robustness of the turbo decoding algorithm in a generalized Gaussian noise has been researched. In [6], Haque et al. studied the effect of various combined FEC codes on the performance of a wireless Orthogonal Frequency Division Multiplexing (OFDM) system. In this study, the authors used Reed-Solomon (RS) (255, 239, 8) encoder, Cyclic (15,11)  In the literature, Wavelet Transform (WT) is also used in some applications related to audio signal processing and channel coding. In [9], Romano et al. proposed a technique for analyzing and compressing speech signals using biorthogonal wavelet filters. This technique compared human voice with a typical Voice Over Internet Protocol (VoIP) encoding. This study has been emphasized how the wavelet filters can be suitable, especially in terms of the compression ratio, without causing significant degradation of the signal quality for the audiences. In [10], Kudumakis et al. investigated the performance of some different wavelet families for low bit rate coding of audio signals. For evaluating the coding gains of these wavelets, both octave and uniform sub-band coding schemes, fixed and dynamic bit-shares with entropy noiseless Huffman coding are used. In [11], Luo presented a new ultra-low latency wavelet voice encoder for real-time wireless transmission using non-overlapping short block processing and embedded coding. To investigate the correlation of the audio signal, a limited and minimized 2D fast lift WT is developed. In [12], Sinha et al. used a wavelet-based method that uses the median function threshold. By selecting a suitable main wavelet, a nearly unchanged signal quality and a flexible compression ratio are obtained and the performance of the proposed method is evaluated using various analysis methods. In [13], Sarnin et al. presented a performance evaluation of BPSK and QPSK techniques using error correcting code. BCH, Cyclic and Hamming codings are used as a coding method. Random data is passed through the AWGN channel and the BER is calculated. In [14], Bhatti recorded a speech signal of .wav format, represented by 8-kHz sampling frequency, 5-sec and 8 bits per sample in MATLAB environment. The audio signal is encoded in the AWGN channel using the Cyclic (17,8) Coding Method. BPSK was used for the modulation of the audio signal. In [15], the authors used RS coding and Code Division Multiple Access (CDMA) methods, unlike the study [14].
In this study, a communication system was established in MATLAB environment for the transmission of 5 second audio signal. In this communication system, BPSK modulation is used in the remainder of the study, because the modulation analysis with good performance is performed first and the performance of BPSK modulation is obtained better than the other modulations. In the second phase of the study, the AWGN and Rician fading channels were individually coded with LBC (7, 4), LBC (15,7), LBC (17,8) and LBC (23,12) respectively. Performance evaluation of these coding types was done with BER and MSE criteria. In the last part of the study, it was investigated which wavelet family is suitable for which level, in order to ensure audio transmission over AWGN and Rician fading channels with the least possible error. Four wavelet transform families at different levels were applied to the audio signal for four different types of LBC encoded channels, and the Mean Squared Error (MSE) performances were compared.

II. AUDIO TRANSMISSION STEPS
In this study, the five-second length audio signal obtained from the external environment is passed through the steps of sampling, quantization and A/D conversion and then given to AWGN noisy (or AWGN noisy with Rician fading) LBC coded channel. In this section, the basic concepts used in the system for the transmission of the audio signal will be mentioned.

A. Channel Coding
In the transmitter part of the digital communication system, the data like sound, image etc. produced by the source is converted into binary sequences with the shortest possible length, with no loss of information (or acceptable loss of information) with the aid of the source encoder. The binary kbit sequences at the output of the source encoder are converted to n-bit sequences by adding n-k control bits with the aid of the channel encoder (n>k). The intent of channel coding is to protect the data sequences from the disturbing effect of the channel. The coded arrays are then divided into blocks of a certain length. These sequences are then mapped to one of the elements of the signal set of the modulation to be used and modulated to give the channel. The type of modulation used varies depending on the channel's feature. On the receiver side, the demodulator allows distorted waves to be processed in the decoder due to disturbing effects in the channel. The channel and source decoders operate inversely to the channel and source encoders, respectively, to find the transmitted data and transfer it to the user. Because of the redundancy control bit insertion in the channel coding, the increase in sequence length reduces the data rate. One way to avoid this fall is to increase the modulation speed. The increase in modulation speed means to shorten the duration of the channel signal, which means that the bandwidth of the channel signal increases. Therefore, this can be done if the bandwidth of the channel is appropriate. The expansion of the signal set (for example, using 8-PSK modulation instead of 4-PSK) is another way of avoiding the drop in the transmission rate in band-limited channels. However, in this method, since the signs are closer to each other, the error performance of the system is reduced. Therefore, coding and modulation together in the design of efficient systems in terms of both error performance and bandwidth are now considered as a whole [16].
In this study, LBC is used for channel coding process and BPSK is used as modulation method.

B. Linear Block Coding
LBC is an encoding method characterized by an (n, k) notation, where k is the length of the message vector and n is the length of the codeword, and LBC is a subclass of similarity control codes. Here, the encoder converts the message block of length k into a longer code word block. The code is a set of vectors created by vectors called code words. k-bit messages constitute 2k different message arrays, referred to as k-arrays. Blocks of n bits form 2n different arrays, expressed as n-arrays As a result of the encoding process, each of the 2k k-arrays is assigned to each of 2n n-arrays. With a block code, each of the 2k message arrays is mapped to 2k different code words in the n-array. This mapping is an one to one correspondence. V i and V j are two separate code words from a set of (n, k) binary block codes. In order for any of these two words to be linear, the necessary and sufficient condition is that V i ⨁V j is also a code vector at the same time. In general, a G generator matrix is defined as a k×n matrix: m message vector is a 1 × k row vector consisting of k message bits: The U code vector is expressed as the product of the m and G sequences as follows: Hence, it can be said that the code vector of a message is a linear combination of the components of the G generator matrix. With a G generator matrix, a vector of length k is translated into a vector of length n. Here, the (n-k) bits are the parity bits. A systematic linear block code has a generator matrix in the form: Here, since the unit matrix is not required to be stored, the systematic generator matrix will reduce the coding complexity. Eq. (5) could be obtained by combining Eq. (3) and (4).
The systematic code vector can be expressed as follows.
The parity check matrix (H) provides us with a simple way to know if an error has occurred. It is necessary for decoding in receiver. GH T has to be equal to zero. Since U = mG, UH T = 0 condition will be provided for all valid codes. U can be said to be a code word generated by the G generator matrix if UH T = 0 is satisfied. The vector r= [r 1 r 2 …r n ] is the signal at the receiver and this vector is the consequent of the transmission of the U= [u 1 u 2 …u n ] vector over the channel. Eq. (7) gives the vector r.

= + (7)
Here the vector e = [e 1 e 2 ... e n ] is the error vector. The syndrome of vector r is S and it is given with Eq. (8) [17].

C. Types of Channels
It is much more convenient to construct mathematical models that reflect the characteristics of transmission media during the design of communication systems for the transmission of information over physical channels. Thus, this mathematical model of the channel can be used to design channel coders and modulators in transmitters and channel decoders and demodulators in receivers.

1) AWGN Channel
One of the simplest mathematical models of a communication channel is the additive noise channel shown in Fig. 1. In this model, the transmitted signal s(t) is distorted by the additive noise process n(t). Physically, additive noise process is produced by electronic components and amplifiers in the receiver end of the communication systems as well as in the case of transmission of radio signals. If the noise is mainly produced by the electronic components and the amplifier on the receiver side of the communication system, this noise can be characterized as thermal noise. This type of noise is statistically characterized as a Gaussian noise process. The mathematical model to be obtained for this channel is therefore called Additive White Gaussian Noise (AWGN) Channel. Since this channel model is a valid model for most of the physical communication channels and can be mathematically examined, it is a channel model that is often used in the analysis and design of communication systems. Channel attenuation components can easily be integrated into this model.
If a signal is attenuated during transmission along the channel, the received signal will be in the form of r(t) = αs(t) + n(t) (9) Here, α represents the attenuation factor [18]. It can be said that a random X variable has a normal distribution if it has a probability density function (pdf) with  average and  2 variance values as in the following Eq. (10) [19].

2) Rician Channel
The amplitude envelope of the small-scale faded channel is Rician distributed if there is a predominantly non-attenuated signal component (such as LOS) [20]. At the output of the envelope detector, the dominant signal causes the formation of a direct current component. The Rayleigh distribution is obtained when the dominant signal component in the Rician distribution is damped [21]. This channel can be seen in Fig. 2 [22] The Rician distribution is as follows.
Here Ais the maximum value (envelope) of a dominant signal I 0 (.)is the degree 0, first modified Bessel function  2is the power of the perpendicular components of the channel's complex damping variable.
In Fig. 3, the Rician distribution is given for different A values [22]. Fig. 3. Probability density function of Rician distribution [22] The Rician distribution is usually expressed using the K parameter. The K parameter is calculated in dB: The average power of the Rician distribution is as follows: Using these parameters, the probability density function can be rewritten as in Eq. (14).
If K = 0 is written in Eq. (14), the Rician distribution becomes the Rayleigh distribution. The increase in the value of the K parameter means that the effect of the fading in the channel is reduced. In the case of K  ∞, there is no fading in the channel and in this case, this channel is converged to the AWGN channel [22].

D. Wavelet Transform
Wavelets are small waves with an average value of 0, with a limited time interval. Wavelets are mathematical functions that localize a set of data or a function in both the frequency domain and the time domain. It can be said that the beginning of wavelet functions is Haar's thesis in 1909. In the 1930's, scale-varying functions were developed in physics, mathematics, seismology and electronics engineering. The broad concept of wavelet was introduced by Grossman and Morlet (1984) in the mid 80s [23]. WT gives successful results in data analysis where the Fourier transform is used ahead of time and some aspects are inadequate. For example, signals with alternating small or irregular details are generally better analyzed using wavelets compared to the conventional Fourier transform. WT is used extensively in onedimensional and two-dimensional signal processing applications such as voice and audio signal processing, communication, geophysics, economics and medicine [24].
There are two types of wavelet analysis: Continuous Wavelet Transform (CWT) and Discrete Wavelet Transform (DWT). In CWT, the signal being processedi is matched and transformed with the wavelet fundamental function in continuous frequency and time increases. The original signal is expressed as a weighted integral of the continuous basis wavelet function. If f(t) is a square integrable function of time, t, then the CWT of f(t ) is defined as where , , ≠ 0 , -is the set of real numbers; '*'denotes the complex conjugation; and the wavelet function is defined as Here: ais called the scaling parameter which captures the local frequency content; bis called the translation parameter which localizes the wavelet basis function at time t = b and its neighborhood.
If the scale and translation parameters a and b are taken at discrete values, DWT is obtained. In this case, the parameters a and b are often based on powers of two and called dyadic scales and translations: In this situation, Eq. (15) becomes The DWT discards all redundant information in CWT by employing a set of orthogonal basis functions. The multiresolution decomposition formula for DWT of the original function as follows: ISSN where Φ(. )is the scaling function; ,are the wavelet coefficients are the scaling coefficients.
In the right hand side of Eq. (18), the first term represents an approximation to the general trend of the original function and the second term represents the local details in the original function. The wavelet coefficients , times the dilated and translated wavelet function can be interpreted as the local residual error between successive signal approximations at scales j-1 and j and Eq. (20) is the detail signal at scale j [24].
There are many wavelets in the literature proposed by different researchers. Daubechies (db), Symlet (sym), Coiflet (coif) and Biorthogonal (bior) wavelets are used in this study. Some types of these wavelets are shown in Fig. 4.

III. SIMULATION RESULTS
In this section, the performances of different coding types on different channels and in different conditions are obtained using MATLAB simulations. LBC (15,8) is used first to determine the performance of phase shift keying modulations. The created communication system is also explained for LBC (15,8). Similar logic applies for other types of LBC encoding. The speech signal (original audio signal) used in simulation is sampled at f s = 8 kHz frequency and recorded in MATLAB environment for 5 seconds with 8 bits resolution. The original audio signal is shown in Fig. 5.
In the second step, the amplitude values of -1 to 1 are shifted to the range of 0 to 255 and these values are expressed by the 8 bit binary number system. In the third step, the H parity check matrix and the G generator matrix are formed appropriately. The audio signal is encoded as a 40,000 x 8 matrix using these matrices. AWGN noise was first added in the fourth stage (then Rician fading). The S (7x15 matrix) syndrome test matrix, which will be used in the recovery phase of the audio signal, has been produced. Here, since the AWGN is added to the original audio signal at 0.8, the largest amplitude value increases from about 0.25 to about 1 and the smallest amplitude value decreases from about -0.25 to about -1. The difference between the signal obtained in the coding end resultant signal and the first original audio signal, namely error, is found for different modulation methods. In the course of determining the modulation with the best performance, AWGN channel and Rician fading channel are used for both coded and non-coded scenarios. BER-EbNo graphs for different EbNo values were traced and the performances of channel-coded and non-channel-coded modulation techniques were compared. For channel-coded scenario, LBC (15,8) coding technique is used. Fig. 6 shows the performance without channel coding of the modulations and Fig. 7 shows the channel-coded performances of the modulations in AWGN channel.  The BER varies inversely with the signal to noise ratio. As can be seen from the figures, when the BER performances of the modulations are compared, the performance of the BPSK modulation is better than that of the other modulations for both channel coded and non-channel coded situations and for both of the channel types. The MSE performances of the modulations for AWGN channel and for LBC (15,8) coded version are given in Table I. For simulations done on the rest of the study, BPSK is used as modulation only because BPSK has good performance as the above figures and table show.
In the second phase of the study, the performance of different types of LBC codes on AWGN and Rician fading channels was investigated. For this purpose, firstly, MSE performance evaluation of LBC coding types in AWGN channel was performed. Table II shows the MSE performances of the LBC coding types in the AWGN channel. As can be seen in the table, the best MSE performance is achieved with the LBC (23,12) coding type, which has a higher code length. The BER performance of the LBC (23,12) coding type is shown in Fig. 10. As can be seen from the figure, high BER performance is achieved even with very low EbNo values using LBC (23,12). Fig. 10. BER performance of LBC (23,12) in AWGN channel Secondly, MSE performance evaluation of LBC coding types in Rician fading channel was performed. Table III shows the MSE performances of the LBC coding types in the Rician fading channel. As can be seen in the table, the best MSE performance is achieved with the LBC (23, 12) coding type same as AWGN channel. As seen in the table, the coding types in the Rician fading channel converge on the AWGN channel with different K-factor values.
In the Rician fading channel, a not-channel coded BER curve is also drawn in order to see the difference between the channel coded version and the not-channel coded version for the best MSE performance (LBC (23,12)). Fig. 11 shows the BER performance for the LBC (23, 12), K-factor = 14 case in Rician fading channel. Both not-coded and coded situations can be seen from the figure. At the end of the study, four WT families at different levels were applied to the audio signal, and the performance of these families was first compared for the AWGN channel and then for the Rician fading channel. Using DWT, data compressed at different levels instead of the original data are put into the designed communication system and the performance obtained at the output is observed. For example, while the number of samples in the original dataset is 40,000, the number of samples in the dataset obtained from the first-level decompression is 20,000. In Table IV, the MSE performances of different wavelet transform families for LBC (15,8) obtained by compression in different levels are given in the AWGN channel.
In these conditions, MSE performance was best obtained at db1 part of the Daubechies wavelet family at level 5, sym1 part of the Symlet wavelet family at the 5th level and bior1.1 part of the biorthogonal wavelet family at the 5th level. As can be seen, the MSE performance improves as the compression level increases.
The performances of different LBC species in Rician fading channels with different K-factor values was then obtained. These performances were obtained using DWT at the first, second, and third level compression levels, respectively. Table  V, Table VI and Table VII show the performances of the wavelet families at these levels, respectively. As can be seen, as the code size increases, the K-factor value that converge to the AWGN channel of the Rician fading channel also increases.
And generally, as the DWT compression level increases, the Kfactor value converging the AWGN value also shrinks. As the compression level increases, the MSE performance improves, and as a result no other compression levels need to be provided. The relationship between compression level and MSE for LBC (7,4) is shown in Fig. 12.

IV. CONCLUSION
According to the obtained results, the LBC method is a suitable method which can be used for channel coding for both AWGN and Rician fading channels. For the channels and coding method considered, BER performance of 4 types of modulation is obtained for the best performance modulation detection. The BER performance of BPSK modulation is the best obtained from these modulations. The BER performance curve is obtained inversely proportional to the signal-to-noise ratio. The coding of the channel by the LBC method reduces the error effect of the channels in the transmitted signal. LBC (7,4), LBC (15,7), LBC (17,8) and LBC (23,12) used in this study and LBC (23,12) was obtained as the best transmission performance from the LBC types studied for both of the channel types. The coding types in the Rician fading channel converge on the AWGN channel with different K-factor values.
At the end of the study, four WT families at different levels were applied to the audio signal, and the performance of these families was first compared for the different length LBC coded AWGN channels and then for the different length of LBC coded Rician fading channel. And generally, as the DWT compression level increases, the K-factor value converging the AWGN value also shrinks. As the compression level increases, the MSE performance improves. If both LBC (15,8) and LBC (17,8) performances are compared for both with wavelet and nonwavelet conditions, it can be said that the increase in the number of parity bits does not have much effect on the transmission performance. As a general evaluation, it can be said that Biorthogonal Wavelet family is the most suitable family among the families handled for audio transmission.
The difference between the original audio signal transmitted from the channels and the audio signal received at the receiver is not observed except for some acoustic differences. In order to obtain better performance as a result of voice transmission from the used channels by LBC method, sampling frequency can be increased and each sample can be represented by more bits. However, these increases will lead to a longer transmission of the audio signal from the channel. ISSN: 2147-284X http://dergipark.gov.tr/bajece     (7,4) in Rician fading channel