To Study the Effects of Basis & Sensing Matrix on Compressive Sensing

Compressive sensing (CS) is the appropriate way to recover the compressible signal with very few observations or precisely very little number of measurements rather than the conventional methods. As per the theory given by Shannon for proper recovery of a signal the sampling frequency must be greater than or equal to the largest frequency component in that signal. So the storage requirement to store the data according to the Nyquist theorem is too high. So compressive sensing is used to reduce this storage requirement. There are two important parameters one is sensing matrix and another is measurement matrix by changing these two parameters we can change the quality of the recovered signal. There are various reconstruction algorithms which are used for proper reconstruction of signal. The work which is done in this paper comprises of various music signals on which the compressive sensing applied. As per the result the single tone music signal have less value of MSE than the multi tone and vocal song signal. The SNR value is quiet good for single tone than the multi tone & vocal song. This is due to the single frequency component in the single tone music signal.


Introduction
In 1949, a sampling theorem given by Claude Shannon in which proved a sampling a periodic band limited signal should be uniformly regular sampled at a rate.According to Shannon/ Nyquist theorem to avoid losing the information the sampling rate should be at least double to the signal bandwidth.This sampling rate is known as the Nyquist rate, named after Harry Nyquist searching the same experiences from his work in telegraph transmissions at Bell Labs in 1928.In the modern signal processing the Shannon sampling theorem has been realized.This phenomena is used in basically communication region and data acquisition from audio to video to even medical x-ray imaging.As technology has developed, computers have found their place in communications and many more regions, along with them, digital signal processing (DSP) has opened a new door of possibilities of progress.Analog to digital converter (DSP) is a heart of the DSP, as a bridging the gap between the past and the present.The ADC too searches its functional restrictions and tried back to Shannon's sampling theorem with newer technologies, such as radar detection and wideband communications and telecommunication area.Analog to digital converter is very expensive for Increasing the data rate in the medical scanners, radars and high speed analog to digital converter architecture can no longer achieve the Nyquist rate needed to meet these high demands.As we know that there are some limitations with ADC some new applications also included to overcome this problem.In case of adequate ADC, the typical computer trying to precede the huge amount of data information.Suppose if we want to process the data of 1GHz at 16 bits/sec then it generates 4 GB/sec, and this data can fill the hard disk of modern era in 1 or 2 minutes.The problem which observed is that many times the information which is required to transmit is very less but the sampling process requires too much data.But today we understand that the data which we generated or acquired from that bunch of data, the data which we have to keep is very small in size and large amount of data will be discarded because it is of no use.

Literature survey
As per the paper wrote by Mads Graesbøll Christensen, [1] they applied compressive sensing approach to the audio & speech signals.They used the compressive sensing for the sparse decomposition of speech signal by the help of window based complex sinusoidal functions.As the results discussed by the author the compressive sensing approach is valid for the sparse decomposition for audio & speech signals.But the problem they found is that the speech and audio signal have non stationary components and complex too when the variation in sparsity taken in to the picture.Another experiment conducted by Anthony Griffin and Panagiotis Tsakalides [2] on real, non sparse audio signals.They examine the behavior of compressive sensing by using different basis matrices and recovery methods.Then they provide some novel approaches based on multi sensor compressive sensing by which the performance of standard recovery algorithms can be improved.They did another experiment in which they used the compressive sensing technology to reduce the number of transmission in a sensor network.A tracking device is mounted on a person and then the real time tracking of that person is taken in to consideration; the tracker transmits a very sparse audio signal on a particular interval of time.R.G. Moreno-Alvarado [3] did an analysis on the audio signals.As we know that we can treat an audio signal as a sparse signal in the frequency domain, they used this approach for further study.They used Discrete cosine transform and spectral analysis for compressive sensing.They used the compressive sensing for the sparse recovery of an audio signal and they got that the recovery of audio signal after compression is satisfactory without noise.As per the paper written by Trifun Savic, [4] he used two domains for sparse recovery, first one is the Fourier transform and another one is the discrete cosine transform.He conducted the experiments for the http://www.ispacs.com/journals/ojids/2018/ojids-00022/International Scientific Publications and Consulting Services two type of signals, first one is the audio signal and second one is the music signal.After that he did the comparative analysis of the recovery based on the different signals and different basis functions.He got the result that the signal which is recovered by Discrete cosine transform required less number of observation but when the same analysis is done by the Fourier transform then the number of iterations are much than in case of Discrete Cosine transform.He also found that the recovery using DCT have much clarity then the Fourier transform.Azad M. Madni [5] discussed that how the artificial thinking of the system can improve the compressive sensing process and can also reduce the problems arises in the sparse reconstruction of a signal.He also discussed about the sparse reconstruction algorithm.He found that the sparse recovery problem is the problem based on reverse engineering and also distributed in various fields of information technology, like remote sensing & social networking.As per the paper written by Ping-Keng Jao, Chin-Chia Michael Yeh, [6] they proposed two modifications in the LASSO technique which is used for the assignment of different code words used in the compressive sensing technique.The two modifications which are proposed by them, the first one exploits the repetitive nature of music signal and the second one is to optimize the screening constraint for signal recovery.The experiments done by them shows clearly that the runtime which is required for the 10,000 codeword reduces up to the level of runtime required for 1000 code words.The conclusion which was proposed by them is that the larger dictionary size can improve the value of mean average precision.Urvashi P Shukla, Niteen B Patel and Mr. Amit M Joshi [7] tried to address the problem in compressive sensing.They summarized the various approaches which are used for the speech signal sparse representation.The Methods which were shown by the authors have their own merits and demerits. of speech signal sparse representation.According to the requirement one can opt any of the procedure from three.First one is simple and less complex form implementation point of view while there are too many ambiguities left untouched.While in the second approach those limitations have been overcome but complexity has increased to a greater extent.In third approach the success for obtaining the representation to the right degree is not achieved.It requires more improvement in the unvoiced representation part of the speech which can enhance the performance of CS in this domain.Vivek Upadhyaya and Amit M Joshi [8] proposed some work on the compressive sensing.They put emphasis on the compression ratio and the value of RIP constant.They considered three types of music signal and then applied compressive sensing algorithm on it.The author taken one combination of basis and measurement matrix and then applied this combination on to the three music signals which are considered by them.As per the literature survey we found that compressive sensing is the way which can easily sort out the problem associated with Nyquist theorem.In next section of the paper performance evaluation parameters of the compressive sensing discussed.

Performance evaluation parameters
Audio compression technique introduces some amount of distortion in the reconstructed audio signal; therefore, evaluation of the audio, music signal quality is an important issue.The quality of reconstructed signal can be evaluated in terms of objective measure and subjective measure.In objective evaluation, statistical properties are considered whereas, in subjective evaluation, listeners can listen and observe the music directly to determine the music quality [42].In this paper, both objective and subjective performance evaluation parameters for audio compression algorithm are presented.The objective evaluation measure includes Signal to noise ratio (SNR), Compression ratio (CR).Music having same SNR value may have different perceptual quality [43].By adjusting the parameters, tradeoffs can be achieved for compressed audio against reconstructed audio quality over wide a range.
Where Q = original input signal, S = recovered signal, n = total number of observation.

iii) Compression ratio per frame (CR):
Compression Ratio is the ratio of total number of measurements to the total number of observation per frame.
Where, K= Number of measurements per frame N= Total Number of observations per frame These all three parameters are very much crucial to calculate the effectiveness of compressive sensing.According to these three parameters the experiment is done using the l 1 reconstruction algorithm with different Basis and Sensing matrices.Different types of music signal taken into consideration.The result and analysis for these signal effected with compressive sensing shown in the next section.

Result and analysis
Here we consider different types of music signals with same sampling frequency and same time frame.Then we applied DCT and DST two types of sensing matrices with Gaussian as a measurement matrix and then find out the effects of this combination of compressive sensing to the music signal.Different plots for SNR and MSE values in variation with compression ratio shown below.http://www.ispacs.com/journals/ojids/2018/ojids-00022/International Scientific Publications and Consulting Services    As per shown in the figure above, two types of comparison done on the basis of Compression Ratio, MSE and SNR.The signal is taken is in the form of .wavformat and the analysis for the same is carried out.MATLAB is used as a simulation tool for comparing the results.The time frame which is used for each signal is 10 sec.As we know that compression ratio based on the total number of samples and the number of measurements which are taken for the compression, so the analysis is based on this concept.Fig ( 1), (3), http://www.ispacs.com/journals/ojids/2018/ojids-00022/International Scientific Publications and Consulting Services (5) shows the SNR value variation with respect to compression ratio for DCT as a basis matrix and fig ( 7), (9), (11) also shows the same variation with DST as a basis matrix.Comparison which is carried out is based on the numerical value of SNR for same compression ratio in different types of basis matrix with different types of signals like single tone, multi tone and vocal song.Similar analysis is done for the MSE too in fig ( 2), ( 4), (6) and fig ( 8), (10), (12).The conclusion which we get from this analysis is given in the next section with proper reason for all three types of signals.

Conclusion
Compressive sensing is one of the finest answers for the problem which is associated with data rate of Shannon's-Nyquist theorem.As per the analysis done by us in this paper we have taken three types of music signal.First one is single tone, second one is multi tone and third one is the vocal song.Five samples of each type of music signal taken then we get some results after the compression and reconstruction of the music signal.The conclusion which we get is when the compression ratio for the compression increases then the value of SNR also increases.The value of MSE decreases which increase in compression ratio as we know the MSE roughly shows the difference in original and reconstructed signal.So when the value of compression ratio increases then the number of samples which are taken from a particular frame also increases due to that there is very less difference observed in actual signal and reconstructed signal.The value of SNR for Single tone music signal is greater than multi tone and vocal song.The reason behind this is single tone signal having only one frequency component so it is easy to reconstruct it rather than the multi tone or vocal song because both have large amount of frequency components in it so it is difficult to proper reconstruct the signal.The future work which we want to do is to change the measurement matrix too to find out the result of this variation.

Figure 10 .
Figure 10.Compression Ratio v/s MSE Curve for Multi tone (DST)

Signal to Noise Ratio (SNR)
http://www.ispacs.com/journals/ojids/2018/ojids-00022/Signal to Noise Ratio (SNR) is well known parameter to measure the quality of various types of signal as audio and video.Evaluation of SNR is too easy.To calculate the SNR in DB following expression is used.Mean Square Error is second parameter for quality evaluation of music, audio signals.It is majorly used with signal to noise ratio analysis.The use of MSE is to analyze the difference between the actual signal and its recovered version after compression.Main objective of MSE is to find out the distortion level between actual and reconstructed signal.The relation for MSE between actual & reconstructed signal is given below: International Scientific Publications and Consulting Services i)