Digital Signal Processing

The concept of digital signal processing has been widely used these days, as it has provided many benefits to the users with its capability of converting analog signals into digital signals to facilitate the process of transmission. The encoding techniques are available which were developed after a comprehensive research to support and enable the transmission of digital signals to meet the user requirements.
Apart from that, the process of ramification of every signal into many frequency bands where ever band has been digitally encoded by itself is termed as sub-band coding. Furthermore, it is better to encode lower frequency band with more bits than the higher frequency bands because the lower frequencies hold most of the speech energy. Sub-band coding method is mainly used to solve this particular problem. This paper depicts the benefits and the importance of sub-band coding, as well as it explains the steps involved in performing sub-band coding.
The method of sub-band coding has been widely practiced for the purpose of transmitting digital signals. For efficient signal encoding this particular method has provided many benefits. Moreover, sub-band coding has also been used for many years in audio industry for high quality digital audio transmission. At first, there is only one signal available which is then subdivided in many numbers of smaller sub-bands where every single number composed of a fractioned spectral of the actual spectrum from that actual signal. This process of dividing signal into sub-band will further assist each band to be transformed through distinct number of bits for every sample, and further every single band is classified according to its step size. By performing the above process resulted in a finer signal quality (Proakis and Manolakis, 2007).

After completing the above process, it will now be possible to encode every single band separately according to the following set of steps. The starting step of the digital signal processing is to apply the filtering required for the signal, which might be a high pass or a low pass signal. The purpose of filtering is to avoid the noise linked with the signal. The frequency of noise associated with the signal can be high or low, depends upon the actual signal requirement. Apart from that, to reduce the sub-band sampling rates, filters are used to minimize the bit rate in the signal encoding process. This method helps to reduce the signal on every band from a factor of two in sampling rate, which suggests that every second sample must be taken from the signal in the process of digital signal processing (Crochiere, 1981).
Furthermore, the above step can be elaborated as if the signal is x[0-6], the samples taken from this would be x[0], x[2], x[4], and x[6]. The major reason of sampling through this method is to make sure that the reduced number of samples which would be quantized based on the following phase, which makes the quantization step to be as efficient and as quick as possible.
The next step includes the quantization of signal on each band. In this process of quantization involves quantization noise to all the bits that are going to be sampled. However, at the  receiving end, all those signals which are acquired from the process of quantization  are to be  sampled from the  factor of two. By doing this method if the input signal is x[0-6], the output signal would now results in x[0], x[2], x[4], x[6]. Through performing this step, the identical number of samples before down sampling would be obtained where every substitute sample was missing (Veldhuis, Breeuwer, and Van Der Wall, 1989).
The following step in the process of sub-band coding is to apply filters on all signals located at every single band where every filter should be of similar type which are used in the previous steps. Moreover, all filters are now used to lessen the number of sub-band sampling rates. These signals already moved through the quantization and the up and down sampling stages which results in proper mode of decoding (Schaffer and Rabiner, 1973).
The final step requires amalgamating the signals from many sub-bands to achieve the output signal and to produce an altered version of the input signal. The following equation shows that there is only one band available from the two bands that will move in the equivalent process.
X1 (z) is the signal on the transmitting end, which was acquired after moving from the H1(z) also known as the high-pass filter which is:
X1 (z)= x1 (0)+ x1 (1) z-1+ x1 (2) z-2+ x1 (3) z-3  …_____ “1”
X1 (-z)= x1 (0)- x1 (1) z-1+ x1 (2) z-2- x1 (3) z-3  …_____ “1A”
It is therefore proved that Z-transformed is actually a result of passing a high pass filter to the signal which is the actual input x[n].
X1 (z)= X (z) H1(z) _____ “2”
The down sampling has been performed by the factor of two on the signal which is originally X1 (z) will now be presented by Y1(z) signal as shown in the following equation:
Y1(z) = y1 (0)+ y1 (1) z-1+ y1 (2) z-2+ y1 (3) z-3 …_____ “3”
Y1(z)  = x1 (0)+ x1 (2) z-1+ x1 (4) z-2+ …_____ “4”
The equation “4” mentioned above, explains that the down sampling effect has removed every single alternate sample available.
However, at the other end signals that were previously up sampled, will now be considered as U1(z) at which every single alternate sample is equal to zero value.
U1(z) = u1 (0)+ u1 (1) z-1+ u1 (2) z-2+ u1 (3) z-3  …
U1(z) = y1 (0)+0+ y1 (1) z-2+ 0+y1 (2) z-4+ …
U1(z) = y1 (0)+ y1 (1) z-2+ y1 (2) z-4+ …
from “4”  U1(z)= Y1(z2)
U1(z) = x1 (0)+ x1 (2) z-2+ x1 (4) z-4+ x1 (6) z-6 …
U1(z) = (X1(z)+X1(-z))/2 = X(z) H1(z)+X(-z) H1(-z)/2__”5”
Besides, the U1(z) signal now move towards the next high pass filter which is K1(z) which is positioned at the receiving point and specified as:
V1(z)= U1(z) K1(z) = ½ K1(z) [X(z) H1(z) )+X(-z) H1(-z)] “6”
Furthermore, the final output signal will now be created after each sub-band output is added which results in the subsequent equation:
X_out(z)=V1(z)+ V2(z)
X_out(z)= ½ K1(z) [X(z) H1(z) )+X(-z) H1(-z)] + ½ K2(z) [X(z) H2(z) )+X(-z) H2(-z)] .
X_out(z)=  ½ [H2 (z) –H2 (-z)] X(z).
When the final output is obtained, to make equation more expressive, it will now be altered into the frequency domain of w from the resulted Z domain, which will now be expressed in the following manner:
X out(w)= ½ [H2 (w)- H2 (w-p)] X(w).
X out(w)= [e-jw(m-1) Hr2 (w) – e-j(m-1)(w-pi) Hr2 (w-p)] X(w).
In the above equation, m represents that even number which measures the length of the filter.
After completing the above procedure, the next phase requires experimentation which includes the implementation of sub-band coding that can be accomplished through two methods. The first method of experimentation is MATLAB, which requires the theory section to be followed from the same phases outlined. There is a file named as ‘subband.dat’ is provided from the input signal in this particular method. This file consists of many values which expresses the file regarding the capacity of the signal in a given time. Moreover, H2(z) was also given as the coefficients of the low pass filters (Croisier, 1974).
It is also suggested that the high pass filters H1(z) are used with the low pass filters K2(z) which creates relationship among the filters explained below:
H1(z) = H(z),  H2(z)= H(-z), K1(z)= 2H(z) and K2(z)= -2H(-z).
Apart from that, there is one more value known as the SNR value that is required for the process of quantization where every single value of Q1 will be computed through the following equation:
SNRdB=[ å xi(n)2 /  å (xi(n)- xo(n))2 ]. Where the limits of the summation is from n=0 to N-1.
The next method for the implementation of the sub-band coding used is called C6711. It is a device that works as a converter and facilitate users in converting software implementations into the physical results. On the other hand, CRO is used on which the output will be connected for the verification of results. Moreover, the sine wave is also generated through connecting the frequency generator to the C6711 device (Rabiner and Gold, 1975).
Finally, the results generated through MATLAB for the sub-ban coding reveals that before performing any find of modifications on the signal, it highly requires the plotting of input signal. Apart from that, result has also shown that the low pass and high pass filters of sub-bands were moved towards an intersection point which exactly equals to 0.5 rad/sample.
The SNR values used in the process of quantization of distinct number of bits reached at a highest level of 16.5dB at the 5th bit. On the other hand, the SNR value has been calculated for 4 bit PCM system was almost 13.2dB. The value suggests that there is a 0.5dB variation from the value computed at the forth bit which is 12.7 dB and is acceptable after the comparison. However, the resulted output signal appeared on the CRO is quite similar to the input signal which explains that as the frequency increases the output signal will move towards zero (Kuester and Mize, 1973).
After reviewing the whole process, it is concluded that the sub-band coding is a method to encode the input signal successfully with maximum efficiency. The two methods used in the process known as MATLAB and C6711 endorse the theory presented in the preceding sections which are considered as valid and reliable.