Open Access

Application of 1-D discrete wavelet transform based compressed sensing matrices for speech compression

SpringerPlus20165:2048

https://doi.org/10.1186/s40064-016-3740-x

Received: 25 June 2016

Accepted: 25 November 2016

Published: 30 November 2016

Abstract

Background

Compressed sensing is a novel signal compression technique in which signal is compressed while sensing. The compressed signal is recovered with the only few numbers of observations compared to conventional Shannon–Nyquist sampling, and thus reduces the storage requirements. In this study, we have proposed the 1-D discrete wavelet transform (DWT) based sensing matrices for speech signal compression. The present study investigates the performance analysis of the different DWT based sensing matrices such as: Daubechies, Coiflets, Symlets, Battle, Beylkin and Vaidyanathan wavelet families.

Results

First, we have proposed the Daubechies wavelet family based sensing matrices. The experimental result indicates that the db10 wavelet based sensing matrix exhibits the better performance compared to other Daubechies wavelet based sensing matrices. Second, we have proposed the Coiflets wavelet family based sensing matrices. The result shows that the coif5 wavelet based sensing matrix exhibits the best performance. Third, we have proposed the sensing matrices based on Symlets wavelet family. The result indicates that the sym9 wavelet based sensing matrix demonstrates the less reconstruction time and the less relative error, and thus exhibits the good performance compared to other Symlets wavelet based sensing matrices. Next, we have proposed the DWT based sensing matrices using the Battle, Beylkin and the Vaidyanathan wavelet families. The Beylkin wavelet based sensing matrix demonstrates the less reconstruction time and relative error, and thus exhibits the good performance compared to the Battle and the Vaidyanathan wavelet based sensing matrices. Further, an attempt was made to find out the best-proposed DWT based sensing matrix, and the result reveals that sym9 wavelet based sensing matrix shows the better performance among all other proposed matrices. Subsequently, the study demonstrates the performance analysis of the sym9 wavelet based sensing matrix and state-of-the-art random and deterministic sensing matrices.

Conclusions

The result reveals that the proposed sym9 wavelet matrix exhibits the better performance compared to state-of-the-art sensing matrices. Finally, speech quality is evaluated using the MOS, PESQ and the information based measures. The test result confirms that the proposed sym9 wavelet based sensing matrix shows the better MOS and PESQ score indicating the good quality of speech.

Keywords

Speech compressionCompressed sensing (CS)Discrete wavelet transform (DWT)Mean opinion score (MOS)Perceptual evaluation of speech quality (PESQ)

Introduction

Conventional signal processing methods such as Fourier transform and a short time Fourier transform (STFT) are inadequate for the analysis of non-stationary signals which have abrupt transitions superimposed on the lower frequency backgrounds such as the speech, music and bio-electric signals. The wavelet transform (WT) (Daubechie Ingrid 1992) overcomes these drawbacks and provides both the time resolution and frequency resolution of a signal. The basic idea of the wavelet transform is to represent the signal to be analyzed as a superposition of wavelets. The wavelet transform is the most popular signal analysis tool, and it is successfully used in different application areas such as speech or audio and image compression.

Given an input signal x of length N, the wavelet transform consists of log2 N decomposition levels. The input signal decomposition is accomplished through a series filtering and downsampling processes. The reconstruction of the original signal is accomplished through an upsampling, series filtering and adding all the sub-bands. Figure 1 shows the block diagram of 1-D forward wavelet transform with 2-level decomposition (Mallat 2009; Meyer 1993). The input signal is filtered using the low-pass filter (u) and the high-pass filter (v). A filtering is achieved by computing a linear convolution between the input signal and the filter coefficients. The two filters are chosen such that, they are orthogonal to each other and provides a perfect reconstruction of the original signal x. Therefore, the quadrature mirror filter (QMF) is commonly used for the perfect reconstruction of a two-channel filter bank.
Fig. 1

Block diagram of 1-D fast forward wavelet transform with 2-level decomposition

Wavelet analysis provides approximation coefficients and detail coefficients. The low frequency information about the signal is given by the approximation, while the high frequency information is given by the detail coefficients. Since the low frequency signal is of more importance than the high frequency signal, the output of the low-pass filter is used as an input for the next decomposition stages; whereas the output of high-pass filter is used at the time of signal reconstruction. The wavelet coefficients are computed by using a series filtering and downsampling processes. The wavelet coefficients (f) are given by:
$$f = {\mathbf{W}}x$$
(1)
where W is the N × N wavelet matrix and defined as: W = WI, where I is N × N identity matrix.

Thus, the classical approach of data compression is to employ the discrete wavelet transform (DWT) based methods (Skodras and Ebrahimi 2001) prior to the transmission. However, these methods includes the complicated multiplications, exhaustive coefficient search and sorting procedure along with the arithmetic encoding of the significant coefficients with their locations, which consequently results in a huge storage requirement and power consumption. Furthermore, the smooth oscillatory signals such as the speech or music signals will be compressed more efficiently in the wavelet packet basis compared to the wavelet representation. Coifman and Wickerhauser (1992) proposed the algorithm for an efficient data compression based on the Shannon entropy for the best basis selection. The orthogonal wavelet packets and localized trigonometric functions are exploited as a basis. This allows an efficient compression of a voice and image signals; however, at the cost of an additional computation in searching the best wavelet packet basis.

The research work presented on CS by Donoho (2006), Baraniuk (2007), Candes and Wakin (2008), and Donoho and Tsaig (2006) have energized the research in many application areas like medical image processing (Lustig et al. 2008), wireless sensor networks (Guan et al. 2011), analog-to-information converters (AIC) (Laska et al. 2007), communications and networks (Berger et al. 2010), radar (Qu and Yang 2012), etc.

In the paper Liu et al. (2014) successfully implemented the CS based compression and the wavelet based compression procedure on the field programmable gate array (FPGA). The result shows that the CS based procedure achieves the better performance compared to the wavelet compression in terms of power consumption and the number of computing resources required. Furthermore, the sparse binary sensing matrix achieves the desired signal compression, but at the price of the higher signal reconstruction time and the higher sensing matrix construction time.

Candes et al. (2006a, b) proposed an i.i.d. (independent identical distribution) Gaussian or Bernoulli random sensing matrices for the compressed sensing. However, the practical implementation of these sensing matrices requires the huge computational cost and memory storage requirements, and therefore considered as inappropriate for large scale applications.

Rauhut (2009), Haupt et al. (2010), Xu et al. (2014), Yin et al. (2010), and Sebert et al. (2008) exploited the Toeplitz and Circulant sensing matrices which effectively recover the original signal with the reduction in the computational cost and the memory requirement.

As an alternative to the random sensing matrices, the authors in Arash and Farokh (2011) proposed the deterministic construction of sensing matrices such as binary, bipolar and the ternary matrices. Several authors have proposed the deterministic construction of sensing matrices using the codes such as the sparse binary matrices based on the low density parity check (LDPC) code (Lu and Kpalma 2012), chirp sensing codes (Applebauma et al. 2009), scrambled block Hadamard matrices (Gan et al. 2008), Reed–Muller sensing codes (Howard et al. 2008) and the Vandermond matrices (DeVore 2007).

The restricted isometry property (RIP) is just a sufficient condition for an exact signal recovery. Even though, the deterministic sensing matrices are an incapable to satisfy RIP condition, they are very useful in practice because of the deterministic nature of the sampler and might be able to advance some features like compression ratio and computational complexity.

The successful implementation of the CS technique is depends on the efficient design of the sensing matrices which are used to compress the given signal. Since, the DWT shows a very good energy compaction property, it can be used for designing the sensing matrices. In this study, we have proposed the 1-D discrete wavelet transform (DWT) based sensing matrices for speech signal compression. The major contributions of the research paper are the proposed 1-D DWT sensing matrices based on different wavelet families such as the Daubechies, Coiflets, Symlets, Battle, Beylkin and the Vaidyanathan wavelet families. Furthermore, the proposed DWT based sensing matrices are compared with state-of-the-art random and the deterministic sensing matrices. Besides, the speech quality is evaluated using mean opinion sore (MOS) and the perceptual evaluation of speech quality (PESQ) measures.

The paper is organized as follows. Section two briefly introduces the compressed sensing (CS) theory with signal acquisition and reconstruction model. Section three describes the proposed methodology for the discrete wavelet transform (DWT) matrix. Experimental results and discussion are presented in section four. Finally, section five presents the conclusions.

Compressed sensing (CS) framework

Background

Compressed sensing is a novel signal compression technique in which signal is acquired and compressed simultaneously. The signal is recovered with the only few number of observations compared to the conventional Shannon–Nyquist sampling which requires observations that are twice the signal bandwidth. Compressed sensing is performed with two basic steps: signal acquisition and signal reconstruction.

CS signal acquisition model

Compressed sensing technique is illustrated as follows:
$$y = {\varvec{\Phi}}f$$
(2)
where f is the input signal of length N × 1, y is the compressed output signal of length M × 1, and Φ is M × N sensing matrix.
The input signal f is sparse in some sparsifying domain (Ψ) and given as:
$$f = {\varvec{\Psi}}x$$
(3)
where x is the non-sparse input signal. Combined form of Eqs. (2) and (3) is given as:
$$y = {\varvec{\Theta}}f = {\mathbf{\varPhi \varPsi }}x$$
(4)
The two basic conditions should be satisfied for the successful implementation of the CS.
  1. 1.

    Sensing matrix (Φ) and sparsity transform (Ψ) should be incoherent to each other.

     
  2. 2.
    The Φ should satisfy the restricted isometric property (RIP) (Candes and Tao 2006) and defined as follow:
    $$(1 - \mathop \delta \nolimits_{k} )\mathop {\left\| x \right\|}\nolimits_{2}^{2} \le \mathop {\left\| {{\varvec{\Phi}}x} \right\|}\nolimits_{2}^{2} \le (1 + \mathop \delta \nolimits_{k} )\mathop {\left\| x \right\|}\nolimits_{2}^{2}$$
    (5)
    where δ k   (0, 1) is called as restricted isometric constant of the matrix and k is the number of non-zero coefficients.
     

CS signal reconstruction model

Since, the compressed sensing technique use only a few number of observations, there are large number of solutions. Therefore, the different optimization based algorithms are used to find the exact sparse solution. The basic algorithms are based on the norm minimization such as L0-norm, L1-norm and L2-norm. Out of these three, L1-norm is widely used, because of its ability to recover the exact sparse solution along with the efficient reconstruction speed. Presently, there are different recovery algorithm available such as the basis pursuit (BP) (Chen et al. 2001), orthogonal matching pursuit (OMP) (Tropp and Gilbert 2007), etc.

The proposed 1-D discrete wavelet transform (DWT) matrix

1-D DWT matrix

For a signal x of length N = 2 n and a low-pass filter (u), the ith level wavelet decomposition (Vidakovic 1999; Wang and Vieira 2010) is given by an Eqs. (6) and (7). Where, v is the high-pass filter.
$$\mathop f\nolimits_{u}^{(i)} (j) = \sum\limits_{k = 1}^{{\mathop 2\nolimits^{n - i + 1} }} {u(k - 2j)\mathop f\nolimits_{u}^{(i - 1)} } (k)\quad {\text{where,}}\quad j = 1,2, \ldots ,\mathop 2\nolimits^{n - i}$$
(6)
And
$$\mathop f\nolimits_{v}^{(i)} (j) = \sum\limits_{k = 1}^{{\mathop 2\nolimits^{n - i + 1} }} {v(k - 2j)\mathop f\nolimits_{u}^{(i - 1)} } (k)\quad {\text{where}},\quad j = 1,2, \ldots ,\mathop 2\nolimits^{n - i}$$
(7)
The reconstruction of \(f_{u}^{i - 1}\) from f u i and f v i can be obtained by
$$\mathop f\nolimits_{u}^{(i - 1)} (j) = \sum\limits_{k = 1}^{{\mathop 2\nolimits^{n - i} }} {u(j - 2k)} \mathop f\nolimits_{u}^{(i)} (k) + \sum\limits_{k = 1}^{{\mathop 2\nolimits^{n - i} }} {v(j - 2k)} \mathop f\nolimits_{v}^{(i)} (k)$$
(8)
The 1-D DWT matrix forms are given as below:
$$\mathop f\nolimits_{u}^{(i)} = \mathop U\nolimits^{(i)} \mathop f\nolimits_{u}^{(i - 1)}$$
(9)
and
$$\mathop f\nolimits_{v}^{(i)} = \mathop V\nolimits^{(i)} \mathop f\nolimits_{v}^{(i - 1)}$$
(10)
where, \(f_{u}^{(i)}\) is the 2 ni dimensional low pass vector in the ith level and \(f_{v}^{(i)}\) the high-pass, while \(f_{u}^{(i - 1)}\) is the 2 ni+1 dimensional low-pass vector in the (i − 1)th level. The two 2 ni by 2 ni+1 wavelet filter matrices are given below.
$$\mathop U\nolimits^{(i)} = \left[ {\begin{array}{*{20}c} {u( - 1)} & 0 & 0 & {\begin{array}{*{20}c} 0 & \cdots & {u( - 3)} & {u( - 2)} \\ \end{array} } \\ {u( - 3)} & {u( - 2)} & {u( - 1)} & {\begin{array}{*{20}c} 0 & \cdots & {u( - 5)} & {u( - 4)} \\ \end{array} } \\ \vdots & \vdots & \vdots & {\begin{array}{*{20}c} \vdots & \ddots & \vdots & \vdots \\ \end{array} } \\ 0 & 0 & 0 & {\begin{array}{*{20}c} 0 & \cdots & {u( - 1)} & 0 \\ \end{array} } \\ \end{array} } \right]$$
(11)
And
$$\mathop V\nolimits^{(i)} = \left[ {\begin{array}{*{20}c} {v( - 1)} & 0 & 0 & {\begin{array}{*{20}c} 0 & \cdots & {v( - 3)} & {v( - 2)} \\ \end{array} } \\ {v( - 3)} & {v( - 2)} & {v( - 1)} & {\begin{array}{*{20}c} 0 & \cdots & {v( - 5)} & {v( - 4)} \\ \end{array} } \\ \vdots & \vdots & \vdots & {\begin{array}{*{20}c} \vdots & \ddots & \vdots & \vdots \\ \end{array} } \\ 0 & 0 & 0 & {\begin{array}{*{20}c} 0 & \cdots & {v( - 1)} & 0 \\ \end{array} } \\ \end{array} } \right]$$
(12)
Thus, the ith scale wavelet transform can be represented as:
$$\left[ {\begin{array}{*{20}c} {\mathop f\nolimits_{u}^{(i)} } \\ {\mathop f\nolimits_{v}^{(i)} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\mathop U\nolimits^{(i)} } \\ {\mathop V\nolimits^{(i)} } \\ \end{array} } \right]\mathop f\nolimits_{u}^{(i - 1)}$$
(13)
This gives the wavelet matrix of 1-level decomposition. The wavelet matrix for different levels of decomposition is given as below.
$$\mathop f\nolimits_{u}^{(i - 1)} = \mathop U\nolimits^{(i - 1)} \mathop f\nolimits_{u}^{(i - 2)}$$
(14)
Above equation can be represented as,
$$\left[ {\begin{array}{*{20}c} {\mathop f\nolimits_{u}^{(i)} } \\ {\mathop f\nolimits_{v}^{(i)} } \\ {\mathop f\nolimits_{v}^{(i - 1)} } \\ {\begin{array}{*{20}c} \vdots \\ {\mathop f\nolimits_{v}^{(2)} } \\ {\mathop f\nolimits_{v}^{(1)} } \\ \end{array} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\mathop U\nolimits^{(i)} \mathop U\nolimits^{(i - 1)} \cdots \mathop U\nolimits^{(1)} } \\ {\mathop V\nolimits^{(i)} \mathop U\nolimits^{(i - 1)} \cdots \mathop U\nolimits^{(1)} } \\ {\mathop V\nolimits^{(i)} \mathop U\nolimits^{(i - 2)} \cdots \mathop U\nolimits^{(1)} } \\ {\begin{array}{*{20}c} \vdots \\ {\mathop V\nolimits^{(2)} \mathop U\nolimits^{(1)} } \\ {\mathop V\nolimits^{(1)} } \\ \end{array} } \\ \end{array} } \right]x$$
(15)
Here, the numbers of signal decomposition levels are restricted to 2 ni+1 ≥ L. Where, L is the length of the filter.
Thus, the final wavelet transform matrix is given by an Eq. (16).
$${\mathbf{W}} = \left[ {\begin{array}{*{20}c} {\mathop U\nolimits^{(i)} \mathop U\nolimits^{(i - 1)} \cdots \mathop U\nolimits^{(1)} } \\ {\mathop V\nolimits^{(i)} \mathop U\nolimits^{(i - 1)} \cdots \mathop U\nolimits^{(1)} } \\ {\mathop V\nolimits^{(i)} \mathop U\nolimits^{(i - 2)} \cdots \mathop U\nolimits^{(1)} } \\ {\begin{array}{*{20}c} \vdots \\ {\mathop V\nolimits^{(2)} \mathop U\nolimits^{(1)} } \\ {\mathop V\nolimits^{(1)} } \\ \end{array} } \\ \end{array} } \right]$$
(16)

Design procedure for the proposed 1-D DWT based sensing matrices

Following are the procedural steps to construct 1-D DWT based sensing matrices.
  1. 1.
    Create a desired quadrature mirror filters (QMF) such as Daubechies, Coiflets, Symlets, Beylkin, Vaidyanathan and Battle filters. For example db1 (Haar) filter is given as f = [1 1] and the db2 filter is formed as follows:
    $$f = \left[ {\begin{array}{*{20}c} {0.482962913145} & {0.836516303738} \\ {0.224143868042} & { - 0.129409522551} \\ \end{array} } \right]$$
    (17)
     
  2. 2.

    Create the N × N Identity matrix.

     
  3. 3.

    Perform 1-D forward wavelet transform on the N × N Identity matrix. Thus, the N × N wavelet transform matrix is generated.

     
  4. 4.

    Select the first m number of rows to form the m × N DWT sensing matrix. Where, m is the minimum number of measurements.

     

Experimental results and discussion

Methodology

The proposed work is evaluated on the CMU/CSTR KDT US English TIMIT database for speech synthesis by Carnegie Mellon University and Edinburgh University (Edinburgh 2002). The details of the database used are as follows: File name: Kdt_001.wav, channel: 1(Mono), bit rate: 256 kbps, audio sample rate: 16 kHz, total duration: 3 s. The number of samples (N) selected are 2048 and the total duration of analyzed speech signal is 0.128 s for simulation. The experimental work is performed using MATLAB 7.8.0 (R2009a) software with Intel (R) CORE 2 Duo CPU, 3 GB RAM system specifications. The discrete cosine transform (DCT) is used as the sparsifying basis for speech signal because of its high sparsity. The speech compression is performed using the sensing matrices based on the different DWT families (Donoho et al. 2007). The basis pursuit (BP) (Chen et al. 2001) is used as signal recovery algorithm for speech signal.

The performance of the reconstructed speech signal is evaluated using the metrics like compression ratio (CR), root mean square error (RMSE), relative error, signal to noise ratio (SNR), signal reconstruction time and sensing matrix construction time.

CR is obtained using relation,
$$CR = \frac{M}{N}$$
(18)
where N is the length of speech signal and M is the number of measurements taken from sensing matrix.
RMSE is given as below:
$${\text{RMSE}} = \sqrt {\frac{{\sum\nolimits_{n = 1}^{N} {\mathop {(x(n) - \tilde{x}(n))}\nolimits^{2} } }}{N}}$$
(19)
where x(n) is the original signal and \(\tilde{x}(n)\) is the reconstructed signal.
Relative error is defined as:
$$Rel.Error = \frac{{\left\| {\tilde{x}(n) - x(n)} \right\|_{2} }}{{\left\| {x(n)} \right\|_{2} }}$$
(20)
where x(n) is the original signal and \(\tilde{x}(n)\) is the reconstructed signal.
SNR is obtained as,
$$SNR(db) = 20\log \left( {\frac{{\left\| {x(n)} \right\|_{2} }}{{\left\| {x(n) - \tilde{x}(n)} \right\|_{2} }}} \right)$$
(21)
where x(n) is the original signal and \(\tilde{x}(n)\) is the reconstructed signal.

Besides, signal reconstruction time is computed to provide the amount of time required to recover the original signal using reconstruction algorithm. The amount of time required to construct the sensing matrix is also an important parameter and should be minimum.

Performance analysis of the Daubechies wavelet family based sensing matrices

This section demonstrates the performance analysis of the different DWT sensing matrices based on Daubechies wavelet family such as db1, db2, db3, db4, db5, db6, db7, db8, db9, db10. The speech signal of length 2048 is taken with 50% sparsity level, preserving the only 1024 number of non-zeros. For a different number of measurements (m), corresponding compression ratios (CR), signal reconstruction time (s), relative error, root mean square error (RMSE) and signal-to-noise ratio (SNR) are calculated (Tables 1, 2, 3, 4, 5, 6, 7, 8, 9, 10).
Table 1

Performance analysis of the proposed db1 (Haar) wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

16

1.281382

0.0585

0.9774

0.1985

2.584058

2048

410

0.2

50

1024

16

2.331942

0.0521

0.8711

1.1984

2.523152

2048

512

0.25

50

1024

16

3.752369

0.0521

0.8707

1.2024

2.514315

2048

614

0.3

50

1024

16

4.242975

0.0488

0.8153

1.7740

2.570737

2048

849

0.4

50

1024

14

5.853003

0.0479

0.7997

1.9418

2.581421

2048

1024

0.5

50

1024

14

9.486588

0.0478

0.7993

1.9454

2.581856

2048

1229

0.6

50

1024

14

11.080416

0.0471

0.7877

2.0723

2.593704

2048

1434

0.7

50

1024

13

14.440796

0.0468

0.7824

2.1310

2.546346

2048

1536

0.75

50

1024

12

17.799127

0.0468

0.7816

2.1406

2.523620

2048

1638

0.8

50

1024

11

16.568279

0.0468

0.7816

2.1406

2.484885

2048

1843

0.9

50

1024

11

20.629636

0.0468

0.7816

2.1407

2.518902

2048

2048

1.0

50

1024

9

22.525032

0.0468

0.7815

2.1409

2.612855

Table 2

Performance analysis of the proposed db2 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

15

1.336137

0.0596

0.9955

0.0391

2.159816

2048

410

0.2

50

1024

15

2.444694

0.0552

0.9224

0.7021

2.187808

2048

512

0.25

50

1024

15

3.622582

0.0552

0.9222

0.7034

2.103215

2048

614

0.3

50

1024

14

3.919803

0.0517

0.8635

1.2751

2.337719

2048

849

0.4

50

1024

13

5.740394

0.0502

0.8387

1.5278

2.105989

2048

1024

0.5

50

1024

12

8.508652

0.0501

0.8376

1.5391

2.319550

2048

1229

0.6

50

1024

12

10.622007

0.0475

0.7931

2.0137

2.055366

2048

1434

0.7

50

1024

12

13.716035

0.0469

0.7838

2.1160

2.378571

2048

1536

0.75

50

1024

11

15.991534

0.0468

0.7817

2.1395

2.135815

2048

1638

0.8

50

1024

11

17.085937

0.0468

0.7816

2.1404

2.078314

2048

1843

0.9

50

1024

11

27.217349

0.0468

0.7816

2.1405

2.423662

2048

2048

1.0

50

1024

9

29.832311

0.0468

0.7815

2.1409

2.365919

Table 3

Performance analysis of the proposed db3 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

18

1.626908

0.0567

0.9483

0.4615

2.253346

2048

410

0.2

50

1024

15

2.434635

0.0545

0.9106

0.8137

2.160987

2048

512

0.25

50

1024

15

3.636753

0.0545

0.9099

0.8199

2.252568

2048

614

0.3

50

1024

15

4.033450

0.0502

0.8383

1.5323

2.201634

2048

849

0.4

50

1024

13

5.929769

0.0496

0.8282

1.6373

2.285777

2048

1024

0.5

50

1024

13

9.180948

0.0496

0.8282

1.6372

2.309093

2048

1229

0.6

50

1024

13

12.480212

0.0477

0.7963

1.9788

2.427516

2048

1434

0.7

50

1024

13

16.554802

0.0469

0.7829

2.1258

2.509123

2048

1536

0.75

50

1024

12

27.233861

0.0468

0.7817

2.1396

2.175306

2048

1638

0.8

50

1024

11

16.821100

0.0468

0.7816

2.1406

2.283585

2048

1843

0.9

50

1024

11

28.673976

0.0468

0.7816

2.1407

2.192527

2048

2048

1.0

50

1024

9

30.905552

0.0468

0.7815

2.1409

2.262969

Table 4

Performance analysis of the proposed db4 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

15

2.301267

0.0545

0.9099

0.8198

2.260742

2048

410

0.2

50

1024

15

2.332355

0.0524

0.8749

1.1607

2.507263

2048

512

0.25

50

1024

15

3.696636

0.0523

0.8744

1.1657

2.370597

2048

614

0.3

50

1024

15

4.308034

0.0496

0.8283

1.6363

2.161184

2048

849

0.4

50

1024

13

5.856042

0.0484

0.8092

1.8387

2.199477

2048

1024

0.5

50

1024

13

19.062773

0.0484

0.8093

1.8376

2.219947

2048

1229

0.6

50

1024

13

11.001844

0.0472

0.7887

2.0622

2.080746

2048

1434

0.7

50

1024

12

13.359806

0.0468

0.7823

2.1325

2.134800

2048

1536

0.75

50

1024

12

16.748983

0.0468

0.7816

2.1405

2.337097

2048

1638

0.8

50

1024

11

16.197748

0.0468

0.7816

2.1407

2.025083

2048

1843

0.9

50

1024

11

20.308890

0.0468

0.7816

2.1407

2.074552

2048

2048

1.0

50

1024

9

21.916145

0.0468

0.7815

2.1409

2.149653

Table 5

Performance analysis of the proposed db5 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

15

1.308421

0.0550

0.9192

0.7316

2.240430

2048

410

0.2

50

1024

15

2.164346

0.0508

0.8496

1.4162

2.211380

2048

512

0.25

50

1024

15

3.399650

0.0509

0.8498

1.4138

2.250745

2048

614

0.3

50

1024

15

3.924939

0.0491

0.8201

1.7221

2.412286

2048

849

0.4

50

1024

14

5.935914

0.0481

0.8030

1.9057

2.270019

2048

1024

0.5

50

1024

14

9.353480

0.0481

0.8031

1.9048

2.337383

2048

1229

0.6

50

1024

13

11.118477

0.0470

0.7848

2.1053

2.216672

2048

1434

0.7

50

1024

13

17.975014

0.0468

0.7822

2.1335

2.245930

2048

1536

0.75

50

1024

11

15.538426

0.0468

0.7816

2.1405

2.440602

2048

1638

0.8

50

1024

11

16.539879

0.0468

0.7816

2.1407

2.182187

2048

1843

0.9

50

1024

11

20.442859

0.0468

0.7816

2.1407

2.235392

2048

2048

1.0

50

1024

9

22.641664

0.0468

0.7815

2.1409

2.210355

Table 6

Performance analysis of the proposed db6 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

15

1.257052

0.0554

0.9260

0.6677

2.291140

2048

410

0.2

50

1024

15

2.232353

0.0506

0.8458

1.4544

2.356109

2048

512

0.25

50

1024

16

3.710359

0.0506

0.8454

1.4584

2.357269

2048

614

0.3

50

1024

15

4.001119

0.0494

0.8259

1.6617

2.306622

2048

849

0.4

50

1024

15

6.425953

0.0487

0.8142

1.7855

2.367848

2048

1024

0.5

50

1024

14

9.85939

0.0487

0.8138

1.7893

2.389422

2048

1229

0.6

50

1024

13

15.540625

0.0469

0.7845

2.1080

2.698509

2048

1434

0.7

50

1024

13

18.172868

0.0468

0.7827

2.1286

2.627686

2048

1536

0.75

50

1024

12

21.322045

0.0468

0.7816

2.1404

2.699761

2048

1638

0.8

50

1024

12

30.749301

0.0468

0.7816

2.1406

2.608611

2048

1843

0.9

50

1024

12

39.662306

0.0468

0.7816

2.1406

2.706883

2048

2048

1.0

50

1024

9

37.150618

0.0468

0.7815

2.1409

2.581647

Table 7

Performance analysis of the proposed db7 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

16

1.544091

0.0564

0.9417

0.5216

2.557882

2048

410

0.2

50

1024

16

2.310240

0.0514

0.8584

1.3260

2.503921

2048

512

0.25

50

1024

16

3.655857

0.0514

0.8585

1.3247

2.371664

2048

614

0.3

50

1024

15

4.050385

0.0511

0.8537

1.3739

2.391605

2048

849

0.4

50

1024

14

5.772598

0.0501

0.8379

1.5362

2.502802

2048

1024

0.5

50

1024

14

9.536010

0.0501

0.8375

1.5399

2.450485

2048

1229

0.6

50

1024

12

10.281689

0.0470

0.7848

2.1049

2.458048

2048

1434

0.7

50

1024

12

13.554986

0.0468

0.7827

2.1278

2.505254

2048

1536

0.75

50

1024

12

17.109362

0.0468

0.7816

2.1404

2.616728

2048

1638

0.8

50

1024

12

18.134841

0.0468

0.7816

2.1406

2.543920

2048

1843

0.9

50

1024

11

21.115266

0.0468

0.7816

2.1406

2.512582

2048

2048

1.0

50

1024

9

22.753802

0.0468

0.7815

2.1409

2.492282

Table 8

Performance analysis of the proposed db8 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

16

1.379793

0.0562

0.9397

0.5406

2.445416

2048

410

0.2

50

1024

16

2.319429

0.0514

0.8597

1.3132

2.595614

2048

512

0.25

50

1024

16

3.672272

0.0514

0.8594

1.3161

2.396296

2048

614

0.3

50

1024

16

4.352767

0.0504

0.8417

1.4973

2.473640

2048

849

0.4

50

1024

14

5.844768

0.0496

0.8283

1.6365

2.437463

2048

1024

0.5

50

1024

13

8.951393

0.0495

0.8274

1.6458

2.359380

2048

1229

0.6

50

1024

13

11.074836

0.0469

0.7845

2.1084

2.573343

2048

1434

0.7

50

1024

13

14.574213

0.0468

0.7824

2.1318

2.609980

2048

1536

0.75

50

1024

11

15.649662

0.0468

0.7816

2.1405

2.559536

2048

1638

0.8

50

1024

11

16.879158

0.0468

0.7816

2.1406

2.556170

2048

1843

0.9

50

1024

11

20.535287

0.0468

0.7816

2.1406

2.569219

2048

2048

1.0

50

1024

9

22.892901

0.0468

0.7815

2.1409

2.513244

Table 9

Performance analysis of the proposed db9 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

15

1.252129

0.0583

0.9750

0.2203

2.649478

2048

410

0.2

50

1024

17

2.527130

0.0518

0.8660

1.2492

2.557930

2048

512

0.25

50

1024

17

3.972292

0.0518

0.8652

1.2582

2.531926

2048

614

0.3

50

1024

16

4.221050

0.0490

0.8182

1.7425

2.572460

2048

849

0.4

50

1024

15

6.270259

0.0478

0.7990

1.9495

2.683571

2048

1024

0.5

50

1024

14

9.593764

0.0478

0.7983

1.9571

2.533085

2048

1229

0.6

50

1024

13

11.075619

0.0470

0.7853

2.0992

2.499444

2048

1434

0.7

50

1024

13

14.518974

0.0468

0.7824

2.1309

2.539711

2048

1536

0.75

50

1024

12

17.178638

0.0468

0.7816

2.1405

2.489187

2048

1638

0.8

50

1024

12

18.043134

0.0468

0.7816

2.1406

2.506067

2048

1843

0.9

50

1024

12

22.370874

0.0468

0.7816

2.1407

2.569387

2048

2048

1.0

50

1024

9

22.751732

0.0468

0.7815

2.1409

2.497201

Table 10

Performance analysis of the proposed db10 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

16

1.281382

0.0585

0.9774

0.1985

2.584058

2048

410

0.2

50

1024

16

2.331942

0.0521

0.8711

1.1984

2.523152

2048

512

0.25

50

1024

16

3.752369

0.0521

0.8707

1.2024

2.514315

2048

614

0.3

50

1024

16

4.242975

0.0488

0.8153

1.7740

2.570737

2048

849

0.4

50

1024

14

5.853003

0.0479

0.7997

1.9418

2.581421

2048

1024

0.5

50

1024

14

9.486588

0.0478

0.7993

1.9454

2.581856

2048

1229

0.6

50

1024

14

11.080416

0.0471

0.7877

2.0723

2.593704

2048

1434

0.7

50

1024

13

14.440796

0.0468

0.7824

2.1310

2.546346

2048

1536

0.75

50

1024

12

17.799127

0.0468

0.7816

2.1406

2.523620

2048

1638

0.8

50

1024

11

16.568279

0.0468

0.7816

2.1406

2.484885

2048

1843

0.9

50

1024

11

20.629636

0.0468

0.7816

2.1407

2.518902

2048

2048

1.0

50

1024

9

22.525032

0.0468

0.7815

2.1409

2.612855

It is noted from Fig. 2 that the db1 (Haar) wavelet based sensing matrix requires less reconstruction time compared to all other Daubechies wavelet based sensing matrices. The second best choice will be db2 or db10, closely followed by the db9 wavelet based sensing matrix. From Fig. 3, it can be observed that the db10 wavelet based sensing matrix shows the minimum relative error compared to all other matrices. From Fig. 4, it can be observed that the db10 wavelet sensing matrix exhibits the high SNR (particularly from CR = 0.3 to CR = 1) compared to other sensing matrices.
Fig. 2

Effect of compression ratio on signal reconstruction time for different Daubechies wavelet sensing matrices

Fig. 3

Effect of compression ratio on relative error for different Daubechies wavelet sensing matrices

Fig. 4

Effect of compression ratio on signal-to-noise ratio for different Daubechies wavelet sensing matrices

Thus, it is evident from Figs. 2, 3 and 4 that overall the db10 wavelet based sensing matrix shows the good balance between signal reconstruction error and signal reconstruction time. Moreover, the db9 also shows a close performance to the db10 and may be the second best choice.

Performance analysis of the Coiflets wavelet family based sensing matrices

This section demonstrates the performance analysis of the different DWT sensing matrices based on Coiflets wavelet family such as coif1, coif2, coif3, coif4 and coif5 (Tables 11, 12, 13, 14, 15).
Table 11

Performance analysis of the proposed coif1 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

17

1.354238

0.0579

0.9668

0.2930

2.098881

2048

410

0.2

50

1024

15

2.874402

0.0550

0.9197

0.7271

2.041493

2048

512

0.25

50

1024

15

10.468670

0.0550

0.9196

0.7279

2.375475

2048

614

0.3

50

1024

14

5.955084

0.0519

0.8675

1.2346

2.308189

2048

849

0.4

50

1024

14

8.202050

0.0513

0.8579

1.3315

2.200196

2048

1024

0.5

50

1024

13

11.575710

0.0513

0.8568

1.3425

2.111921

2048

1229

0.6

50

1024

13

18.579869

0.0476

0.7952

1.9901

2.127905

2048

1434

0.7

50

1024

12

15.692999

0.0469

0.7830

2.1245

2.487333

2048

1536

0.75

50

1024

12

17.816849

0.0468

0.7817

2.1396

2.396088

2048

1638

0.8

50

1024

12

18.998337

0.0468

0.7816

2.1406

2.226200

2048

1843

0.9

50

1024

12

23.612591

0.0468

0.7816

2.1406

2.379104

2048

2048

1.0

50

1024

9

23.507435

0.0468

0.7815

2.1409

2.343294

Table 12

Performance analysis of the proposed coif2 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

16

1.344408

0.0589

0.9847

0.1338

2.656092

2048

410

0.2

50

1024

15

2.311920

0.0542

0.9050

0.8674

2.634235

2048

512

0.25

50

1024

15

3.535317

0.0542

0.9052

0.8649

2.354271

2048

614

0.3

50

1024

15

4.017180

0.0531

0.8869

1.0428

2.595762

2048

849

0.4

50

1024

14

6.107924

0.0527

0.8812

1.0990

2.515127

2048

1024

0.5

50

1024

13

8.927963

0.0525

0.8773

1.1375

2.531853

2048

1229

0.6

50

1024

13

11.323331

0.0481

0.8035

1.9008

2.503729

2048

1434

0.7

50

1024

13

21.738856

0.0469

0.7832

2.1229

2.657620

2048

1536

0.75

50

1024

12

27.116146

0.0468

0.7817

2.1392

2.598041

2048

1638

0.8

50

1024

12

26.312156

0.0468

0.7816

2.1405

2.358481

2048

1843

0.9

50

1024

11

32.646767

0.0468

0.7816

2.1406

2.564254

2048

2048

1.0

50

1024

9

34.162602

0.0468

0.7815

2.1409

2.617750

Table 13

Performance analysis of the proposed coif3 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

16

1.313659

0.0561

0.9375

0.5604

3.076415

2048

410

0.2

50

1024

16

2.300134

0.0529

0.8835

1.0761

2.883624

2048

512

0.25

50

1024

16

3.836235

0.0529

0.8834

1.0770

2.977264

2048

614

0.3

50

1024

15

4.095096

0.0492

0.8228

1.6944

2.991658

2048

849

0.4

50

1024

14

6.307728

0.0485

0.8105

1.8244

2.734366

2048

1024

0.5

50

1024

14

12.427927

0.0485

0.8105

1.8247

3.000826

2048

1229

0.6

50

1024

13

14.180720

0.0475

0.7931

2.0136

2.732994

2048

1434

0.7

50

1024

13

26.695054

0.0468

0.7824

2.1319

2.674142

2048

1536

0.75

50

1024

12

23.379583

0.0468

0.7816

2.1403

2.967887

2048

1638

0.8

50

1024

11

19.402939

0.0468

0.7816

2.1406

2.882567

2048

1843

0.9

50

1024

11

25.077965

0.0468

0.7816

2.1406

2.741203

2048

2048

1.0

50

1024

9

30.924540

0.0468

0.7815

2.1409

2.949021

Table 14

Performance analysis of the proposed coif4 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

16

1.262131

0.0567

0.9480

0.4643

2.843727

2048

410

0.2

50

1024

17

2.439706

0.0524

0.8756

1.1538

2.830243

2048

512

0.25

50

1024

17

3.704519

0.0524

0.8757

1.1529

2.791644

2048

614

0.3

50

1024

16

4.097077

0.0491

0.8211

1.7123

2.743363

2048

849

0.4

50

1024

15

6.153716

0.0485

0.8105

1.8248

2.783660

2048

1024

0.5

50

1024

14

9.041550

0.0485

0.8103

1.8267

2.812775

2048

1229

0.6

50

1024

13

10.712781

0.0473

0.7911

2.0354

2.633953

2048

1434

0.7

50

1024

13

14.213775

0.0468

0.7824

2.1318

2.631650

2048

1536

0.75

50

1024

12

16.368278

0.0468

0.7816

2.1404

2.785540

2048

1638

0.8

50

1024

12

17.142833

0.0468

0.7816

2.1406

2.690370

2048

1843

0.9

50

1024

11

19.494505

0.0468

0.7816

2.1406

2.606268

2048

2048

1.0

50

1024

9

21.432247

0.0468

0.7815

2.1409

2.659983

Table 15

Performance analysis of the proposed coif5 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

18

1.503436

0.0552

0.9229

0.6966

2.866005

2048

410

0.2

50

1024

17

2.491811

0.0509

0.8498

1.4132

2.895638

2048

512

0.25

50

1024

17

3.852817

0.0509

0.8500

1.4114

3.007931

2048

614

0.3

50

1024

16

4.188601

0.0489

0.8164

1.7616

3.009424

2048

849

0.4

50

1024

14

5.971268

0.0482

0.8060

1.8737

2.845210

2048

1024

0.5

50

1024

14

9.498384

0.0482

0.8058

1.8753

3.033461

2048

1229

0.6

50

1024

13

11.012061

0.0472

0.7889

2.0598

3.033341

2048

1434

0.7

50

1024

13

14.348884

0.0468

0.7824

2.1313

3.105129

2048

1536

0.75

50

1024

12

17.088544

0.0468

0.7816

2.1404

2.821626

2048

1638

0.8

50

1024

12

18.211057

0.0468

0.7816

2.1406

3.005033

2048

1843

0.9

50

1024

11

20.533120

0.0468

0.7816

2.1406

3.095259

2048

2048

1.0

50

1024

9

23.047740

0.0468

0.7815

2.1409

2.936779

It is noted from Fig. 5 that the coif5 and coif4 wavelet based sensing matrix shows a close performance and requires the less reconstruction time compared to all other Coiflets wavelet based sensing matrices. From Fig. 6, it can be observed that coif5 wavelet based sensing matrix shows the minimum relative error compared to all other matrices. Also, from Fig. 7, it is seen that coif5 wavelet based sensing matrix exhibits the high SNR compared to other sensing matrices.
Fig. 5

Effect of compression ratio on signal reconstruction time for different Coiflets wavelet sensing matrices

Fig. 6

Effect of compression ratio on relative error for different Coiflets wavelet sensing matrices

Fig. 7

Effect of compression ratio on signal-to-noise ratio for different Coiflets wavelet sensing matrices

Thus, overall the coif5 wavelet based sensing matrix shows the good performance, since it requires the less reconstruction time, minimum relative error and the high SNR compared to other Coiflets wavelet based sensing matrices. In addition, the coif4 may be the second choice of sensing matrix.

Performance analysis of the Symlets wavelet family based sensing matrices

This section demonstrates the performance analysis of the different DWT sensing matrices based on Symlets wavelet family such as sym4, sym5, sym6, sym7, sym8, sym9 and sym10 (Tables 16, 17, 18, 19, 20, 21, 22).
Table 16

Performance analysis of the proposed sym4 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

15

1.426156

0.0606

1.0128

0.1101

2.219932

2048

410

0.2

50

1024

15

2.466473

0.0550

0.9186

0.7376

2.287469

2048

512

0.25

50

1024

15

4.034720

0.0550

0.9188

0.7360

2.408524

2048

614

0.3

50

1024

15

4.520797

0.0510

0.8516

1.3956

2.262042

2048

849

0.4

50

1024

14

6.883726

0.0503

0.8412

1.5016

2.461359

2048

1024

0.5

50

1024

14

9.275989

0.0503

0.8412

1.5015

2.600820

2048

1229

0.6

50

1024

13

10.699366

0.0479

0.8001

1.9374

2.083195

2048

1434

0.7

50

1024

13

13.702683

0.0468

0.7828

2.1269

2.117664

2048

1536

0.75

50

1024

12

16.142659

0.0468

0.7817

2.1393

2.303114

2048

1638

0.8

50

1024

12

17.024746

0.0468

0.7816

2.1405

2.259328

2048

1843

0.9

50

1024

12

21.118064

0.0468

0.7816

2.1406

2.369993

2048

2048

1.0

50

1024

9

21.227179

0.0468

0.7815

2.1409

2.248962

Table 17

Performance analysis of the proposed sym5 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

18

1.562108

0.0587

0.9813

0.1644

2.474725

2048

410

0.2

50

1024

16

2.410647

0.0525

0.8767

1.1428

2.422995

2048

512

0.25

50

1024

15

3.363769

0.0524

0.8764

1.1460

2.414940

2048

614

0.3

50

1024

14

3.633501

0.0487

0.8139

1.7885

2.401602

2048

849

0.4

50

1024

14

5.901087

0.0485

0.8100

1.8307

2.285970

2048

1024

0.5

50

1024

13

8.629663

0.0484

0.8094

1.8370

2.270979

2048

1229

0.6

50

1024

12

10.436658

0.0476

0.7951

1.9910

2.295386

2048

1434

0.7

50

1024

12

12.997505

0.0468

0.7825

2.1299

2.416593

2048

1536

0.75

50

1024

12

16.543217

0.0468

0.7816

2.1400

2.226996

2048

1638

0.8

50

1024

11

15.839222

0.0468

0.7816

2.1407

2.255090

2048

1843

0.9

50

1024

11

19.887183

0.0468

0.7816

2.1407

2.431380

2048

2048

1.0

50

1024

9

21.319450

0.0468

0.7815

2.1409

2.266367

Table 18

Performance analysis of the proposed sym6 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

16

1.337284

0.0568

0.9486

0.4587

2.278876

2048

410

0.2

50

1024

16

2.354315

0.0519

0.8665

1.2450

2.291421

2048

512

0.25

50

1024

15

3.403486

0.0520

0.8696

1.2132

2.495518

2048

614

0.3

50

1024

15

3.900882

0.0506

0.8450

1.4632

2.513799

2048

849

0.4

50

1024

14

5.894927

0.0495

0.8274

1.6455

2.444608

2048

1024

0.5

50

1024

14

9.191334

0.0495

0.8274

1.6452

2.282627

2048

1229

0.6

50

1024

13

10.874663

0.0470

0.7858

2.0939

2.354739

2048

1434

0.7

50

1024

13

13.986553

0.0468

0.7825

2.1300

2.269408

2048

1536

0.75

50

1024

12

16.289155

0.0468

0.7816

2.1404

2.324362

2048

1638

0.8

50

1024

12

17.002074

0.0468

0.7816

2.1406

2.325569

2048

1843

0.9

50

1024

12

21.603990

0.0468

0.7816

2.1407

2.409376

2048

2048

1.0

50

1024

9

21.488590

0.0468

0.7815

2.1409

2.344651

Table 19

Performance analysis of the proposed sym7 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

19

1.604935

0.0549

0.9178

0.7450

2.385164

2048

410

0.2

50

1024

16

2.270355

0.0530

0.8860

1.0514

2.368900

2048

512

0.25

50

1024

16

3.611425

0.0530

0.8857

1.0539

2.390055

2048

614

0.3

50

1024

14

3.719657

0.0494

0.8259

1.6620

2.400246

2048

849

0.4

50

1024

14

5.943942

0.0486

0.8126

1.8025

2.379669

2048

1024

0.5

50

1024

14

9.231755

0.0486

0.8125

1.8037

2.626292

2048

1229

0.6

50

1024

14

11.697056

0.0477

0.7968

1.9732

2.395632

2048

1434

0.7

50

1024

13

14.038048

0.0468

0.7824

2.1319

2.547235

2048

1536

0.75

50

1024

12

16.379478

0.0468

0.7816

2.1398

2.541034

2048

1638

0.8

50

1024

12

17.139381

0.0468

0.7816

2.1406

2.299743

2048

1843

0.9

50

1024

11

19.813613

0.0468

0.7816

2.1406

2.628218

2048

2048

1.0

50

1024

9

21.393741

0.0468

0.7815

2.1409

2.435649

Table 20

Performance analysis of the proposed sym8 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

16

1.337804

0.0591

0.9881

0.1037

2.398959

2048

410

0.2

50

1024

16

2.289342

0.0525

0.8775

1.1354

2.449329

2048

512

0.25

50

1024

16

3.605457

0.0525

0.8774

1.1361

2.413914

2048

614

0.3

50

1024

15

3.899311

0.0490

0.8194

1.7296

2.367131

2048

849

0.4

50

1024

14

5.839426

0.0489

0.8165

1.7613

2.414094

2048

1024

0.5

50

1024

14

9.219034

0.0489

0.8167

1.7590

2.477653

2048

1229

0.6

50

1024

13

10.916775

0.0476

0.7953

1.9899

2.412281

2048

1434

0.7

50

1024

13

14.105790

0.0468

0.7823

2.1324

2.411482

2048

1536

0.75

50

1024

12

16.337553

0.0468

0.7816

2.1403

2.425477

2048

1638

0.8

50

1024

11

15.876267

0.0468

0.7816

2.1406

2.452373

2048

1843

0.9

50

1024

11

19.838644

0.0468

0.7816

2.1406

2.439456

2048

2048

1.0

50

1024

9

21.412296

0.0468

0.7815

2.1409

2.481033

Table 21

Performance analysis of the proposed sym9 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

17

1.433657

0.0582

0.9731

0.2364

2.566095

2048

410

0.2

50

1024

16

2.362390

0.0520

0.8695

1.2149

2.566405

2048

512

0.25

50

1024

16

3.703235

0.0520

0.8690

1.2191

2.529453

2048

614

0.3

50

1024

15

3.947898

0.0493

0.8243

1.6788

2.580208

2048

849

0.4

50

1024

14

5.824850

0.0481

0.8038

1.8975

2.566482

2048

1024

0.5

50

1024

13

8.527638

0.0481

0.8038

1.8973

2.627600

2048

1229

0.6

50

1024

12

9.961474

0.0470

0.7850

2.1023

2.621438

2048

1434

0.7

50

1024

12

12.963035

0.0468

0.7822

2.1333

2.582570

2048

1536

0.75

50

1024

12

16.468013

0.0468

0.7816

2.1406

2.628998

2048

1638

0.8

50

1024

11

15.820224

0.0468

0.7816

2.1406

2.705263

2048

1843

0.9

50

1024

11

19.915216

0.0468

0.7816

2.1407

2.695629

2048

2048

1.0

50

1024

9

21.477969

0.0468

0.7815

2.1409

2.618640

Table 22

Performance analysis of the proposed sym10 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

17

1.479121

0.0560

0.9351

0.5831

2.708358

2048

410

0.2

50

1024

16

2.348326

0.0513

0.8578

1.3323

2.607475

2048

512

0.25

50

1024

16

3.762705

0.0513

0.8580

1.3298

2.729382

2048

614

0.3

50

1024

16

4.328904

0.0490

0.8192

1.7321

2.819432

2048

849

0.4

50

1024

14

6.057941

0.0479

0.8003

1.9353

2.719236

2048

1024

0.5

50

1024

14

9.516700

0.0479

0.8010

1.9273

2.715742

2048

1229

0.6

50

1024

13

11.268634

0.0470

0.7853

2.0992

2.484238

2048

1434

0.7

50

1024

12

13.607226

0.0468

0.7822

2.1333

2.680826

2048

1536

0.75

50

1024

12

17.225086

0.0468

0.7816

2.1405

2.634090

2048

1638

0.8

50

1024

12

18.192019

0.0468

0.7816

2.1406

2.628795

2048

1843

0.9

50

1024

12

22.716723

0.0468

0.7816

2.1406

2.709115

2048

2048

1.0

50

1024

9

22.909145

0.0468

0.7815

2.1409

2.612016

It is noted from Fig. 8 that the sym9 wavelet based sensing matrix requires the less reconstruction time compared to all other Symlets wavelet based sensing matrices. Furthermore, the sym5 also shows a very close performance to that of the sym9 wavelet based sensing matrix. From Fig. 9, it can be observed that the sym9 and the sym10 wavelet based sensing matrices almost demonstrate similar performance with minimum relative error compared to all other matrices. Also, from Fig. 10, it is observed that the sym9 and the sym10 wavelet based sensing matrices nearly shows similar performance and exhibits the high SNR compared to other sensing matrices.
Fig. 8

Effect of compression ratio on signal reconstruction time for different Symlets wavelet sensing matrices

Fig. 9

Effect of compression ratio on relative error for different Symlets wavelet sensing matrices

Fig. 10

Effect of compression ratio on signal-to-noise ratio for different Symlets wavelet sensing matrices

Thus, it is evident from Figs. 9 and 10 that overall the sym9 wavelet sensing matrix demonstrates the less reconstruction time and the less relative error, and thus exhibits the good performance compared to other Symlets wavelet based sensing matrices. Moreover, the sym10 may be the second choice of sensing matrix followed by the sym5.

Performance analysis of the Beylkin, Vaidyanathan and Battle wavelet family based sensing matrices

This section shows the performance analysis of the different DWT sensing matrices based on Beylkin, Vaidyanathan, and Battle1, Battle3 and Battle5 wavelet families (Tables 23, 24, 25, 26, 27).
Table 23

Performance analysis of the proposed Beylkin wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

17

1.351917

0.0551

0.9205

0.7196

2.635292

2048

410

0.2

50

1024

16

2.313493

0.0522

0.8726

1.1837

2.500319

2048

512

0.25

50

1024

16

3.655763

0.0522

0.8722

1.1878

2.582726

2048

614

0.3

50

1024

16

4.267978

0.0490

0.8193

1.7316

2.668384

2048

849

0.4

50

1024

14

5.865898

0.0483

0.8064

1.8686

2.501577

2048

1024

0.5

50

1024

14

9.541513

0.0482

0.8056

1.8774

2.626348

2048

1229

0.6

50

1024

13

11.113448

0.0469

0.7838

2.1163

2.464044

2048

1434

0.7

50

1024

13

14.567693

0.0468

0.7822

2.1341

2.504371

2048

1536

0.75

50

1024

12

17.19308

0.0468

0.7816

2.1405

2.527696

2048

1638

0.8

50

1024

12

18.081755

0.0468

0.7816

2.1406

2.516479

2048

1843

0.9

50

1024

12

22.526654

0.0468

0.7816

2.1407

2.523531

2048

2048

1.0

50

1024

9

22.950703

0.0468

0.7815

2.1409

2.466272

Table 24

Performance analysis of the proposed Vaidyanathan wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

19

1.707644

0.0702

1.1724

1.3812

2.912996

2048

410

0.2

50

1024

19

3.459463

0.0580

0.9684

0.2785

3.332869

2048

512

0.25

50

1024

19

5.666164

0.0580

0.9685

0.2782

3.036002

2048

614

0.3

50

1024

18

5.760958

0.0513

0.8578

1.3320

3.034769

2048

849

0.4

50

1024

15

7.316646

0.0485

0.8112

1.8177

2.969049

2048

1024

0.5

50

1024

14

10.855239

0.0486

0.8123

1.8053

3.075962

2048

1229

0.6

50

1024

13

13.417422

0.0474

0.7924

2.0214

3.441779

2048

1434

0.7

50

1024

13

19.093624

0.0468

0.7822

2.1339

3.483836

2048

1536

0.75

50

1024

12

22.608453

0.0468

0.7816

2.1405

3.022221

2048

1638

0.8

50

1024

12

34.970415

0.0468

0.7816

2.1406

3.586396

2048

1843

0.9

50

1024

12

49.314450

0.0468

0.7816

2.1407

3.515476

2048

2048

1.0

50

1024

9

34.943702

0.0468

0.7815

2.1409

3.060519

Table 25

Performance analysis of the proposed Battle1 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

15

1.453478

0.0633

1.0577

0.4873

2.794529

2048

410

0.2

50

1024

14

2.113667

0.0556

0.9283

0.6459

2.953615

2048

512

0.25

50

1024

15

3.530676

0.0560

0.9363

0.5713

2.769843

2048

614

0.3

50

1024

14

3.957949

0.0514

0.8598

1.3125

2.621663

2048

849

0.4

50

1024

14

7.280928

0.0513

0.8571

1.3399

2.846945

2048

1024

0.5

50

1024

14

9.777801

0.0514

0.8587

1.3230

2.858507

2048

1229

0.6

50

1024

13

11.850497

0.0480

0.8013

1.9241

2.643100

2048

1434

0.7

50

1024

13

15.195628

0.0468

0.7827

2.1278

2.621185

2048

1536

0.75

50

1024

12

17.503087

0.0468

0.7817

2.1387

2.804361

2048

1638

0.8

50

1024

11

17.421545

0.0468

0.7816

2.1406

2.835200

2048

1843

0.9

50

1024

11

28.179929

0.0468

0.7816

2.1406

2.869269

2048

2048

1.0

50

1024

9

32.640965

0.0468

0.7815

2.1409

3.447333

Table 26

Performance analysis of the proposed Battle3 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

21

2.229166

0.0776

1.2960

2.2520

4.184386

2048

410

0.2

50

1024

17

3.000747

0.0619

1.0350

0.2991

4.037021

2048

512

0.25

50

1024

17

6.762783

0.0619

1.0350

0.2984

4.214151

2048

614

0.3

50

1024

17

5.495381

0.0576

0.9626

0.3313

3.779477

2048

849

0.4

50

1024

15

7.920600

0.0538

0.8989

0.9261

3.706719

2048

1024

0.5

50

1024

14

11.331212

0.0538

0.8995

0.9199

3.945231

2048

1229

0.6

50

1024

13

13.909190

0.0485

0.8106

1.8235

3.764331

2048

1434

0.7

50

1024

12

19.831917

0.0468

0.7823

2.1321

3.777555

2048

1536

0.75

50

1024

12

34.010388

0.0468

0.7816

2.1404

4.464107

2048

1638

0.8

50

1024

12

30.235744

0.0468

0.7816

2.1406

3.855705

2048

1843

0.9

50

1024

11

40.190816

0.0468

0.7816

2.1406

4.003610

2048

2048

1.0

50

1024

9

45.789195

0.0468

0.7815

2.1409

3.726738

Table 27

Performance analysis of the proposed Battle5 wavelet based sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

16

1.547147

0.0587

0.9809

0.1672

4.355811

2048

410

0.2

50

1024

16

2.776326

0.0517

0.8633

1.2764

4.523638

2048

512

0.25

50

1024

17

4.876061

0.0518

0.8648

1.2619

4.258686

2048

614

0.3

50

1024

16

5.068130

0.0492

0.8226

1.6961

4.216043

2048

849

0.4

50

1024

15

8.173479

0.0479

0.7999

1.9398

4.653944

2048

1024

0.5

50

1024

13

22.056223

0.0480

0.8013

1.9240

4.196677

2048

1229

0.6

50

1024

13

13.576984

0.0470

0.7858

2.0936

4.174972

2048

1434

0.7

50

1024

13

18.405100

0.0468

0.7821

2.1350

4.355072

2048

1536

0.75

50

1024

12

22.430252

0.0468

0.7816

2.1404

4.268081

2048

1638

0.8

50

1024

11

28.325501

0.0468

0.7816

2.1406

4.269212

2048

1843

0.9

50

1024

11

38.805249

0.0468

0.7816

2.1407

4.623429

2048

2048

1.0

50

1024

9

41.186350

0.0468

0.7815

2.1409

4.283169

Figure 11 shows that the Beylkin wavelet based sensing matrix requires the less reconstruction time compared to all other Symlets wavelet based sensing matrices. From Fig. 12, it can be observed that the Beylkin and the Battle5 wavelet based sensing matrices shows a very close performance with minimum relative error compared to all other matrices. Also, from Fig. 13, it can be seen that the Beylkin and the Battle5 wavelet based sensing matrices shows a very comparable performance and exhibits the high SNR compared to other sensing matrices.
Fig. 11

Effect of compression ratio on signal reconstruction time for Beylkin, Vaidyanathan and Battle wavelet sensing matrices

Fig. 12

Effect of compression ratio on relative error for Beylkin, Vaidyanathan and Battle wavelet sensing matrices

Fig. 13

Effect of compression ratio on signal-to-noise ratio for Beylkin, Vaidyanathan and Battle wavelet sensing matrices

Thus, it can be noted from Figs. 11, 12 and 13 that overall the Beylkin wavelet sensing matrix demonstrates the less reconstruction time and relative error, and thus exhibits the good performance compared to other wavelet based sensing matrices. However, the Battle5 shows a close performance and may be the second best choice of sensing matrix.

Performance analysis of the best-proposed DWT based sensing matrices namely: Beylkin, db10, coif5 and sym9 wavelet family

This section illustrates the performance analysis of the best-proposed DWT sensing matrices namely: Beylkin, db10, coif5 and sym9 wavelet families.

Figure 14 shows that the sym9 wavelet based sensing matrix clearly outperforms the Beylkin, db10, and the coif5 wavelet based sensing matrices in terms of signal reconstruction time. From Fig. 15, it can be observed that the db10 shows the good performance over CR = 0.3–0.5; however overall the sym9 wavelet based sensing matrices shows the good (from CR = 0.5–1.0) and comparable performance with db10. Also, from Fig. 16, it can be observed that the db10 and sym9 wavelet based sensing matrices shows a comparable performance and exhibits the high SNR compared to other sensing matrices. In addition, the sym9 wavelet based sensing matrix shows an edge over db10 from the CR = 0.5–1.0.
Fig. 14

Effect of compression ratio on signal reconstruction time for the best DWT based sensing matrices

Fig. 15

Effect of compression ratio on relative error for the best DWT based sensing matrices

Fig. 16

Effect of compression ratio on signal-to-noise ratio for the best DWT based sensing matrices

Thus, it can be evident from Figs. 14, 15 and 16 that overall the sym9 wavelet based sensing matrix shows the superior performance compared to the Beylkin, db10 and the coif5 wavelet based sensing matrices in views of signal reconstruction time and relative error. Furthermore, the db10 may be the second best choice of sensing matrix.

Performance analysis of the best-proposed sym9 wavelet based sensing matrix with state-of-the-art random and deterministic sensing matrices

This section illustrates the comparative analysis of the proposed sym9 wavelet based sensing matrix and state-of-the-art random sensing matrices such as Gaussian, Uniform, Toeplitz, Circulant and Hadamard matrix along with deterministic sensing matrices such as the DCT and the sparse binary sensing matrices for speech signal compression (Tables 28, 29, 30, 31, 32, 3334).
Table 28

Performance analysis of the random Gaussian sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

21

1.614613

0.0529

0.8846

0.6667

1.096733

2048

410

0.2

50

1024

22

3.101767

0.0506

0.8459

1.4526

0.748022

2048

512

0.25

50

1024

21

4.639830

0.0488

0.8163

1.6802

1.942504

2048

614

0.3

50

1024

20

5.067599

0.0488

0.8149

1.8149

2.142537

2048

849

0.4

50

1024

21

8.240858

0.0476

0.7949

2.0516

4.909329

2048

1024

0.5

50

1024

20

12.792057

0.0470

0.7847

2.1202

11.635789

2048

1229

0.6

50

1024

21

17.375626

0.0468

0.7817

2.1431

14.081669

2048

1434

0.7

50

1024

22

23.472323

0.0468

0.7816

2.1388

34.631693

2048

1536

0.75

50

1024

24

33.471279

0.0468

0.7816

2.1416

39.194676

2048

1638

0.8

50

1024

27

38.907176

0.0468

0.7816

2.1408

43.476185

2048

1843

0.9

50

1024

23

41.156261

0.0468

0.7816

2.1409

51.096753

2048

2048

1.0

50

1024

9

22.129988

0.0468

0.7815

2.1409

57.755398

Table 29

Performance analysis of the random Uniform sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

28

2.235539

0.0634

1.0600

1.0296

0.223625

2048

410

0.2

50

1024

25

3.600845

0.0537

0.8966

0.8152

0.738591

2048

512

0.25

50

1024

25

5.745004

0.0516

0.8616

0.8698

1.917257

2048

614

0.3

50

1024

25

6.434319

0.0496

0.8281

1.6951

2.113897

2048

849

0.4

50

1024

23

9.443467

0.0474

0.7924

1.9810

5.332054

2048

1024

0.5

50

1024

23

15.986578

0.0470

0.7857

2.0911

11.975395

2048

1229

0.6

50

1024

24

19.727464

0.0468

0.7823

2.1297

14.165440

2048

1434

0.7

50

1024

24

25.920229

0.0468

0.7816

2.1401

34.968949

2048

1536

0.75

50

1024

23

72.481383

0.0468

0.7816

2.1407

48.408631

2048

1638

0.8

50

1024

20

29.193737

0.0468

0.7816

2.1407

93.407060

2048

1843

0.9

50

1024

17

32.821700

0.0468

0.7816

2.1408

51.437677

2048

2048

1.0

50

1024

9

22.395440

0.0468

0.7815

2.1409

57.803101

Table 30

Performance analysis of the random Hadamard sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

24

2.160564

0.0543

0.9068

0.6816

0.175290

2048

410

0.2

50

1024

21

3.352213

0.0508

0.8486

1.5002

0.195649

2048

512

0.25

50

1024

21

5.245114

0.0488

0.8162

1.7910

0.227224

2048

614

0.3

50

1024

14

4.059570

0.0476

0.7959

1.7840

0.203726

2048

849

0.4

50

1024

14

7.103978

0.0483

0.8075

2.1509

0.238574

2048

1024

0.5

50

1024

17

12.420312

0.0469

0.7838

2.1742

0.312728

2048

1229

0.6

50

1024

20

18.202941

0.0468

0.7814

2.1341

0.262874

2048

1434

0.7

50

1024

24

28.330063

0.0468

0.7816

2.1400

0.281225

2048

1536

0.75

50

1024

21

32.017893

0.0467

0.7805

2.1406

0.400085

2048

1638

0.8

50

1024

23

47.091449

0.0467

0.7803

2.1407

0.304155

2048

1843

0.9

50

1024

26

70.684798

0.0467

0.7805

2.1408

0.318110

2048

2048

1.0

50

1024

9

31.953103

0.0468

0.7815

2.1408

0.463961

Table 31

Performance analysis of the random Toeplitz sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

21

1.756908

0.0536

0.8957

0.9104

0.409620

2048

410

0.2

50

1024

22

3.233689

0.0498

0.8315

1.5469

0.429934

2048

512

0.25

50

1024

21

4.788428

0.0493

0.8232

1.7366

0.469536

2048

614

0.3

50

1024

20

5.355458

0.0483

0.8068

1.8057

0.450123

2048

849

0.4

50

1024

20

8.429285

0.0474

0.7918

2.0918

0.471607

2048

1024

0.5

50

1024

20

13.439433

0.0469

0.7841

2.1270

0.544915

2048

1229

0.6

50

1024

21

17.961354

0.0467

0.7797

2.1501

0.503870

2048

1434

0.7

50

1024

21

23.162537

0.0467

0.7806

2.1523

0.524939

2048

1536

0.75

50

1024

23

32.312776

0.0467

0.7806

2.1511

0.626275

2048

1638

0.8

50

1024

24

35.325303

0.0467

0.7807

2.1490

0.549642

2048

1843

0.9

50

1024

27

49.591100

0.0467

0.7812

2.1453

0.562993

2048

2048

1.0

50

1024

6

18.351954

0.0468

0.7815

2.1409

0.695494

Table 32

Performance analysis of the random Circulant sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

26

2.507042

0.0597

0.9968

0.4222

2.757110

2048

410

0.2

50

1024

24

3.729785

0.0510

0.8516

1.5673

1.755148

2048

512

0.25

50

1024

22

5.278771

0.0491

0.8206

1.7278

1.819234

2048

614

0.3

50

1024

19

5.237320

0.0480

0.8021

1.9582

1.813663

2048

849

0.4

50

1024

18

8.220705

0.0470

0.7846

2.1030

1.845629

2048

1024

0.5

50

1024

17

11.916562

0.0468

0.7825

2.1335

1.917508

2048

1229

0.6

50

1024

16

14.656704

0.0468

0.7816

2.1404

1.857957

2048

1434

0.7

50

1024

15

17.577009

0.0468

0.7816

2.1408

1.852721

2048

1536

0.75

50

1024

16

23.864195

0.0468

0.7816

2.1408

2.020327

2048

1638

0.8

50

1024

15

23.451971

0.0468

0.7816

2.1409

1.893177

2048

1843

0.9

50

1024

12

25.954229

0.0468

0.7815

2.1409

1.914215

2048

2048

1.0

50

1024

8

24.190480

0.0468

0.7815

2.1409

2.065995

Table 33

Performance analysis of the Deterministic DCT sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

15

1.336874

0.0570

0.9532

0.4165

0.032836

2048

410

0.2

50

1024

16

2.309603

0.0537

0.8974

0.9405

0.060471

2048

512

0.25

50

1024

17

3.773514

0.0511

0.8543

1.3682

0.110442

2048

614

0.3

50

1024

16

4.122633

0.0510

0.8525

1.3858

0.087157

2048

849

0.4

50

1024

14

5.822766

0.0494

0.8259

1.6617

0.117717

2048

1024

0.5

50

1024

14

9.058112

0.0478

0.7981

1.9592

0.213083

2048

1229

0.6

50

1024

15

12.369809

0.0472

0.7889

2.0593

0.170241

2048

1434

0.7

50

1024

13

14.324836

0.0469

0.7832

2.1224

0.198500

2048

1536

0.75

50

1024

13

18.236878

0.0468

0.7825

2.1303

0.323367

2048

1638

0.8

50

1024

13

18.600745

0.0468

0.7820

2.1363

0.225549

2048

1843

0.9

50

1024

12

21.461995

0.0468

0.7816

2.1409

0.253978

2048

2048

1.00

50

1024

9

21.893476

0.0468

0.7815

2.1409

0.438310

Table 34

Performance analysis of the Deterministic Sparse Binary sensing matrix

Length of signal (N)

Number of measurements (m)

Compression ratio (CR = m/N)

Sparsity level = (k/N) × 100 (%)

No. of non-zeros (k)

No. of iterations required

Signal reconstruction time (s)

RMSE

Relative error

SNR (db)

Construction time for sensing matrix (s)

2048

205

0.1

50

1024

12

0.9278

0.0548

0.9154

0.7678

66.9190

2048

410

0.2

50

1024

13

1.8871

0.0503

0.8405

1.5090

207.1541

2048

512

0.25

50

1024

14

3.1431

0.0497

0.8308

1.6102

380.2697

2048

614

0.3

50

1024

15

4.8355

0.0487

0.8146

1.7809

502.1809

2048

849

0.4

50

1024

19

8.3895

0.0473

0.7906

2.0411

878.8051

2048

1024

0.5

50

1024

20

18.3753

0.0469

0.7843

2.1103

1271.5155

2048

1229

0.6

50

1024

23

25.4419

0.0468

0.7824

2.1313

1869.4049

2048

1434

0.7

50

1024

30

42.2472

0.0468

0.7816

2.1407

2913.4315

2048

1536

0.75

50

1024

28

57.3386

0.0468

0.7816

2.1408

2984.3214

2048

1638

0.8

50

1024

24

52.8342

0.0468

0.7816

2.1409

3575.2037

2048

1843

0.9

50

1024

17

50.7545

0.0468

0.7816

2.1409

4030.6333

2048

2048

1.0

50

1024

6

26.8845

0.0468

0.7815

2.1409

5856.4331

It is noted from Fig. 17 that the proposed sym9 wavelet based sensing matrix clearly outperforms the state-of-the-art random sensing matrices such as Gaussian, Uniform, Toeplitz, Circulant and Hadamard sensing matrices as well as the deterministic DCT and sparse binary sensing matrices in terms of signal reconstruction time. It can be observed from Figs. 18 and 19 that the proposed sym9 wavelet based sensing matrix demonstrates a close comparable performance compared to the state-of-the-art random and deterministic sensing matrices.
Fig. 17

Effect of compression ratio on signal reconstruction time for the proposed sym9 matrix and state-of-the-art random and deterministic sensing matrices

Fig. 18

Effect of compression ratio on relative error for the proposed sym9 matrix and state-of-the-art random and deterministic sensing matrices

Fig. 19

Effect of compression ratio on signal-to-noise ratio for the proposed sym9 matrix and state-of-the-art random and deterministic sensing matrices

The overall remark

Thus, it is evident from Figs. 17, 18 and 19 (Tables 28, 29, 30, 31, 32, 33, 34) that the proposed sym9 wavelet based sensing matrix exhibits the better performance compared to the state-of-the-art random and deterministic sensing matrices.

Subjective quality evaluation

Simple quality measures like SNR do not provide an accurate measure of the speech quality. Hence, speech quality assessment is performed by highly robust and accurate measures such as the mean opinion score (MOS) and perceptual evaluation of speech quality (PESQ) recommended by International Telecommunication Union Telephony (ITU-T) standards.

In this section, the performance of the proposed sensing matrices is evaluated using mean opinion score (MOS). The MOS is a subjective listening test to perceive the speech quality and one of the widely recommended method by ITU standard (ITU-T P.800) (ITU-T 1996).

Table 35 presents subjective evaluation of the reconstructed speech quality using the mean opinion score (MOS) test. The MOS test is performed on a group of seven male listeners and three female listeners. The listeners are required to train and evaluate the quality of the reconstructed speech signal with respect to the original signal. The speech quality is evaluated by rating to a signal within the range of 1–5. The MOS is computed by taking the average score of all the individual listeners and it ranges between 1 (bad speech quality) and 5 (excellent speech quality).
Table 35

Subjective evaluation of speech quality using Mean Opinion Score (MOS) test

Sr.

no.

Sensing

matrix

Listeners

MOS

Score

Male listener

Female listener

1

2

3

4

5

6

7

8

9

10

1.

db1

4

3

3

3

3

3

3

3

4

3

3.2

2.

db2

5

5

4

4

4

4

4

4

3

3

4.0

3.

db3

4

4

4

3

4

4

3

4

4

3

3.7

4.

db4

4

3

3

5

4

4

5

3

3

4

3.8

5.

db5

3

3

3

4

4

4

4

4

3

3

3.5

6.

db6

4

4

4

4

4

4

3

3

4

4

3.8

7.

db7

4

5

3

3

4

4

4

3

4

4

3.8

8.

db8

4

4

4

4

4

4

3

4

4

4

3.9

9.

db9

3

5

3

3

5

4

4

4

3

4

3.8

10.

db10

3

3

4

4

4

4

4

3

4

3

3.6

11.

coif1

4

5

4

5

3

3

4

3

4

4

3.9

12.

coif2

4

3

5

4

3

3

4

3

4

4

3.7

13.

coif3

3

3

3

3

3

3

4

4

3

4

3.3

14.

coif4

4

5

4

4

3

3

4

4

3

4

3.8

15.

coif5

4

5

3

3

3

4

5

4

5

4

4.0

16.

sym4

5

4

4

4

3

3

5

3

3

4

3.8

17.

sym5

5

4

4

4

4

3

4

3

3

4

3.8

18.

sym6

5

5

3

5

4

4

4

3

4

4

4.1

19.

sym7

4

3

4

3

4

4

5

4

3

4

3.8

20.

sym8

4

4

4

4

4

4

5

4

4

4

4.1

21.

sym9

5

5

5

5

5

4

5

4

3

3

4.4

22.

sym10

4

4

4

4

5

4

4

4

4

4

4.1

23.

Battle1

5

5

5

3

5

3

4

3

4

4

4.1

24.

Battle3

4

4

3

3

5

4

3

4

4

4

4.1

25.

Battle5

3

4

4

4

4

4

3

4

4

4

3.8

26.

Beylkin

3

3

4

3

3

3

3

3

4

4

3.3

27.

Vaidynathan

4

3

4

3

4

3

4

3

3

4

3.5

28.

Sparse Binary

4

3

3

3

4

4

5

3

3

3

3.5

29.

DCT matrix

4

5

4

4

5

4

4

4

4

4

4.2

30.

Random Gaussian

4

4

3

4

4

4

3

4

4

4

3.8

31.

Random

uniform

3

4

4

3

4

3

4

3

4

4

3.6

32.

Random

Toeplitz

4

4

3

3

4

3

4

3

4

4

3.6

33.

Random

Circulant

4

4

4

3

4

4

4

4

4

4

3.9

34.

Random

Hadamard

4

5

4

4

4

3

4

4

4

4

4.0

35.

Wavelet compression

3

4

3

3

5

5

5

4

3

4

3.9

The following conclusions can be drawn from Table 35.
  1. 1.

    Overall, the Symlets wavelet family achieves the good MOS scores compared to other proposed as well as state-of-the-art sensing matrices.

     
  2. 2.

    The highest MOS score of 4.4 is achieved by the sym9 wavelet family followed by the sym6, sym8, sym10, Battle1, Battle3 (MOS = 4.1) and followed by the db2, coif5 (MOS = 4.0) respectively. Thus, these MOS scores can be considered as an acceptable score for speech quality.

     
  3. 3.

    Moreover, the state-of-the-art DCT sensing matrix (MOS = 4.2) and the random Hadamard sensing matrix (MOS = 4.0) shows the good MOS score compared to other state-of-the-art sensing matrices.

     

However, MOS test frequently requires a sizeable number of listeners to accomplish stable results, and is also the time-consuming and expensive. Nevertheless, subjective quality measures are still one of the most decisive ways to estimate speech quality.

Objective quality evaluation

The PESQ is a most modern international ITU-T standard (P.862) (ITU-T 2005) for an automated prediction of speech quality by estimating quality scores ranging from −1 to 4.5. In other way, it estimates the MOS (Mean Opinion Score) from both the clean signal and its distorted signal. A higher quality score signifies the better speech quality. Moreover, since human listeners are not required; PESQ is less expensive, accurate and less time-consuming;

Table 36 presents the different objective speech quality metrics such as the PESQ, log-likelihood ratio (LLR) and weighted spectral slope (WSS) along with the three subjective rating scales namely: signal distortion, noise distortion, and overall quality. The ratings are based on the five-point (1–5) MOS scale (Hu and Loizou 2008).
Table 36

Objective evaluation of speech quality using measures such as Perceptual Evaluation of Speech Quality (PESQ), Log-Likelihood Ratio (LLR) and Weighted Spectral Slope (WSS)

Sr.

no.

Different sensing

matrices

Speech distortion

Background distortion

Overall quality

LLR

WSS

PESQ

MOS

Score

1.

db1

3.4959

2.3958

2.8940

0.618194

45.751366

2.4060

2.

db2

3.3878

2.3902

2.8228

0.735509

40.165614

2.3436

3.

db3

3.1887

2.2231

2.6572

0.791790

50.476578

2.2632

4.

db4

3.5157

2.4790

2.9531

0.701403

37.954092

2.4644

5.

db5

3.7339

2.5471

3.0838

0.535443

34.465134

2.4909

6.

db6

3.4648

2.3444

2.8207

0.612405

40.395098

2.2646

7.

db7

3.4839

2.3110

2.8473

0.619242

39.438455

2.2937

8.

db8

3.5478

2.3337

2.8925

0.560800

41.502149

2.3307

9.

db9

3.7200

2.5313

3.0680

0.521539

37.389013

2.4879

10.

db10

3.7502

2.5574

3.0842

0.508764

34.772191

2.4771

11.

coif1

3.4799

2.2788

2.8831

0.646498

42.964582

2.3862

12.

coif2

3.5241

2.3374

2.9602

0.705475

36.450763

2.4628

13.

coif3

3.8500

2.5787

3.2063

0.527448

29.371499

2.5938

14.

coif4

3.7495

2.5243

3.1167

0.551742

34.069359

2.5387

15.

coif5

3.7443

2.5377

3.1099

0.551411

34.169205

2.5310

16.

sym4

3.2751

2.1626

2.7240

0.658010

63.799345

2.3770

17.

sym5

3.7144

2.4995

3.0894

0.550669

38.139942

2.5395

18.

sym6

3.6323

2.4225

3.0189

0.603048

36.964962

2.4751

19.

sym7

3.8098

2.5829

3.1993

0.569036

31.505778

2.6300

20.

sym8

3.8135

2.5483

3.1875

0.548968

31.641938

2.6039

21.

sym9

3.8163

2.5878

3.1843

0.534342

32.757831

2.6003

22.

sym10

3.8335

2.6147

3.1985

0.536466

30.625240

2.6006

23.

Battle1

3.4479

2.3080

2.8621

0.664658

44.171851

2.3821

24.

Battle3

3.7297

2.4548

3.0899

0.532391

37.307250

2.5212

25.

Battle5

3.6017

2.4841

2.9674

0.578535

39.038937

2.4135

26.

Beylkin

3.6053

2.4451

2.9289

0.546456

35.907963

2.3180

27.

Vaidynathan

3.8044

2.5202

3.1186

0.461501

35.343833

2.4948

28.

Sparse Binary

3.6393

2.7576

3.0467

0.666517

29.706413

2.4868

29.

DCT matrix

3.7628

2.5854

3.1092

0.518874

34.672901

2.5138

30.

Random Gaussian

2.9737

2.6990

2.6855

1.276865

30.019462

2.4291

31.

Random

uniform

3.3255

2.7452

2.8794

0.966803

28.298916

2.4577

32.

Random

Toeplitz

2.6847

2.6565

2.5249

1.520660

33.026663

2.4108

33.

Random

Circulant

3.8147

2.7854

3.1549

0.529008

28.229330

2.5210

34.

Random

Hadamard

3.8147

2.7854

3.1549

0.529008

28.229330

2.4934

35.

Wavelet compression

3.6017

2.4841

2.9674

0.578535

39.038937

2.4135

The following conclusions can be drawn from Table 36.
  1. 1.

    The Symlets wavelet family shows the higher signal distortion rating (rating between: 3–4) indicating the fairly natural speech signal quality compared to other proposed and state-of-the art sensing matrices

     
  2. 2.

    The db5, db9, db10, coif3, coif4, coif5 and Symlets wavelet families shows the good background distortion rating (between rating: 2–3) indicating noticeable noise, but not intrusive and are close comparable to state-of-the art sensing matrices.

     
  3. 3.

    The db5, db9, db10, coif3, coif4, coif5 and Symlets wavelet families shows the higher signal quality rating (between rating: 3–4) indicating the good/fair speech quality compared to state-of-the art sensing matrices.

     
  4. 4.

    Overall, the sym9 and the sym10 wavelet family based sensing matrices exhibits good/fair overall quality (For db9 and db10 ratings are 3.1843 and 3.1985 respectively) compared to other proposed and state-of-the art sensing matrices.

     
  5. 5.

    In terms of objective measures, the sym9 and the sym10 wavelet family based sensing matrices exhibits the lower values of log-likelihood ratio (LLR) and weighted spectral slope (WSS) metrics, indicating the good speech quality and are close comparable with state-of-the art sensing matrices.

     
  6. 6.

    Finally, in views of PESQ measure, the sym9 and the sym10 wavelet family based sensing matrices exhibits the higher PESQ scores; PESQ = 2.6003 (sym9) and PESQ = 2.6006 (sym10) respectively, signifying the good/fair speech quality compared to other proposed and state-of-the art sensing matrices.

     

Information based evaluation

Entropy (H) is a measure of an average information content of a signal (x) and widely used in signal processing applications. It is defined as:
$$H(X) = - \sum\limits_{i = 1}^{N} {P(x_{i} )\log } P(x_{i} )$$
(22)
where X = {x 1, x 2,…,x N } is a set of random variable, P(x i ) is a probability of random variable x i and N is the length of a signal or possible outcomes. It is obvious that the higher signal entropy reflects more information content or more unpredictability of information content.
Table 37 presents the information based evaluation of speech quality. Furthermore, it also provides insights on the selection of the best basis sensing matrix.
Table 37

Information based evaluation of speech quality and selection of the best basis sensing matrices

Sr. no.

Different sensing matrices

Entropy of original speech signal

Entropy of reconstructed speech signal

Entropy of sensing matrix

1.

Daubechies

wavelet family

db1

10.2888

11.0000

0.1191

2.

db2

10.2888

11.0000

0.4663

3.

db3

10.2888

11.0000

0.6966

4.

db4

10.2888

11.0000

0.9066

5.

db5

10.2888

11.0000

1.0980

6.

db6

10.2888

11.0000

1.2699

7.

db7

10.2888

11.0000

1.4416

8.

db8

10.2888

11.0000

1.6132

9.

db9

10.2888

11.0000

1.7689

10.

db10

10.2888

11.0000

1.9047

11.

Coiflet wavelet family

coif1

10.2888

11.0000

0.6966

12.

coif2

10.2888

11.0000

1.2699

13.

coif3

10.2888

11.0000

1.7689

14.

coif4

10.2888

11.0000

2.1759

15.

coif5

10.2888

11.0000

2.5818

16.

Symmlet wavelet family

sym4

10.2888

11.0000

0.9066

17.

sym5

10.2888

11.0000

1.0980

18.

sym6

10.2888

11.0000

1.2699

19.

sym7

10.2888

11.0000

1.4416

20.

sym8

10.2888

11.0000

1.6132

21.

sym9

10.2888

11.0000

1.7689

22.

sym10

10.2888

11.0000

1.9047

23.

Battle wavelet family

Battle1

10.2888

11.0000

2.0789

24.

Battle3

10.2888

11.0000

3.1632

25.

Battle5

10.2888

11.0000

4.0745

26.

Other wavelet families

Beylkin

10.2888

11.0000

1.7689

27.

Vaidynathan

10.2888

11.0000

2.1759

28.

Random sensing matrices

Random Gaussian

10.2888

11.0000

21.0000

29.

Random uniform

10.2888

11.0000

21.0000

30.

Random Toeplitz

10.2888

11.0000

20.7505

31.

Random Circulant

10.2888

11.0000

11

32.

Random Hadamard

10.2888

11.0000

1

33.

Deterministic sensing matrices

DCT matrix

10.2888

11.0000

19.1415

34.

Sparse Binary

10.2888

11.0000

0.0659

35.

Classical approach

Wavelet compression

10.2888

9.7573

The following observations are evident from Table 37.
  1. 1.

    CS based sensing matrices, including proposed as well as state-of-the-art sensing matrices has the higher entropy (H = 11.0) compared to classical wavelet compression technique (H = 9.7573).

     
  2. 2.

    It is also evident that for the proposed sensing matrices the entropy of the reconstructed speech signal (H = 11.0) is very close to the original signal entropy (H = 10.2888).

     
  3. 3.

    Furthermore, we have computed the entropy of sensing matrices which shows that state-of-the-art random matrices like Gaussian, Uniform, Toeplitz, Circulant attains higher entropy due to its randomness, followed by deterministic DCT matrix.

     
  4. 4.

    The proposed sensing matrices such as the Battle (for Battle5, H = 4.0745) and the Symlets wavelet families (for sym9 and sym10, H = 1.7689 and H = 1.9047, respectively) shows the higher entropy compared to the sparse binary (H = 0.0659) and the random Hadamard sensing matrices (H = 1).

     

Spectrographic analysis

The spectrograms are used to visually investigate the joint time–frequency properties of speech signals with intensity or color representing the relative energy of contributing frequencies and it plays an important role in decoding the underlying linguistic massage. Figure 20 shows the spectrographic analysis of the original and the reconstructed speech signal for the proposed sym9 wavelet based sensing matrix (for CR = 0.5). Figure 20a shows the spectrogram of the original input speech signal and Fig. 20b shows the spectrogram of the reconstructed speech signal.
Fig. 20

Spectrographic analysis of original and reconstructed speech signal for the proposed sym9 wavelet based sensing matrix (For CR = 0.5). a Spectrogram of original speech signal and b spectrogram of reconstructed speech signal

Thus, the spectrographic analysis from Fig. 20 shows that the time–frequency characteristic of the reconstructed spectrogram is a very close to the original speech spectrogram, preserving most of the signal energy. Moreover, the red color shows energy at the highest frequency followed by the yellow, blue respectively, and the white area shows the absence of frequency components.

Furthermore, Fig. 21 shows the original and the reconstructed speech signal with the DCT basis for CR = 0.5 (N = 2048 and m = 1024). It can be observed that the original speech signal is successfully reconstructed using the proposed sym9 wavelet based sensing matrix.
Fig. 21

Original and reconstructed speech signal for proposed sym9 wavelet based sensing matrix (for CR = 0.5)

Conclusions

In this study, an attempt was made to investigate the DWT based sensing matrices for the speech signal compression. This study presents the performance comparison of the different DWT based sensing matrices such as the: Daubechies, Coiflets, Symlets, Battle, Beylkin and Vaidyanathan wavelet families. Further study presents the performance analysis of the proposed DWT based sensing matrices with state-of-the-art random and deterministic sensing matrices. The speech quality is evaluated using subjective and objective measures. The subjective evaluation of speech quality is performed by mean opinion sore (MOS). Moreover, the objective speech quality is evaluated using the PESQ and other measures such as the log-likelihood ratio (LLR) and weighted spectral slope (WSS). Besides, an attempt was made to evaluate the speech quality using the information based measure such as Shannon entropy. In addition, efforts are made to present an insight on the selection of the best basis sensing matrix using the information based measure.

The following major conclusions are drawn based on the investigation:
  • Overall, the db10 wavelet based sensing matrix shows the good balance between signal reconstruction error and signal reconstruction time compared to other Daubechies wavelet based sensing matrices. Moreover, the db9 also shows close performance to the db10 and may be the second best choice.

  • The coif5 wavelet based sensing matrix shows the good performance, since it requires less reconstruction time, minimum relative error and the high SNR compared to other Coiflets wavelet based sensing matrices. In addition, the coif4 may be the second choice of sensing matrix.

  • Overall, the sym9 wavelet sensing matrix demonstrates the less reconstruction time and the less relative error, and thus exhibits the good performance compared to other Symlets wavelet based sensing matrices. Moreover, the sym10 may be the second choice of sensing matrix followed by the sym9.

  • The Beylkin wavelet sensing matrix demonstrates the less reconstruction time and relative error, and thus exhibits the good performance compared to the Battle and the Vaidyanathan wavelet based sensing matrices. However, the Battle5 shows a close performance and may be the second best choice of sensing matrix.

  • When compared for the best of the DWT sensing matrix, the sym9 wavelet based sensing matrix shows the superior performance compared to the db10, coif5 and Beylkin wavelet based sensing matrices, in the views of signal reconstruction time and relative error. Furthermore, the db10 may be the second best choice of sensing matrix.

  • Finally, it is revealed that the proposed sym9 wavelet based sensing matrix exhibits the better performance compared to state-of-the-art random and deterministic sensing matrices in terms of signal reconstruction time and reconstruction error.

  • Overall, the Symlets wavelet family achieves good MOS scores compared to other proposed as well as state-of-the-art sensing matrices.

  • The highest MOS score of 4.4 is achieved by the sym9 wavelet family followed by the sym6, sym8, sym10, Battle1, Battle3 (MOS = 4.1) and followed by the db2, coif5 (MOS = 4.0) respectively. Thus, these MOS scores can be considered as an acceptable score for speech quality.

  • In terms of the PESQ measure, the sym9 and the sym10 wavelet family based sensing matrices exhibits the higher PESQ scores i.e. PESQ = 2.6003 (sym9) and PESQ = 2.6006 (sym10) respectively; signifying the good/fair speech quality compared to other proposed and state-of-the art sensing matrices.

  • The sym9 and the sym10 wavelet family based sensing matrices exhibits the lower values of Log-Likelihood Ratio (LLR) and Weighted Spectral Slope (WSS) metrics indicating the good speech quality, and are the close comparable with state-of-the art sensing matrices.

  • In views of information based evaluation, CS based sensing matrices, including the proposed DWT based as well as state-of-the-art sensing matrices, has the higher entropy (H = 11.0) compared to the classical wavelet compression technique (H = 9.7573).

  • The proposed sensing matrices such as the Battle (For the Battle5, H = 4.0745) and the Symlets wavelet families (For the sym9 and the sym10, H = 1.7689 and H = 1.9047 respectively) shows the higher entropy compared to the sparse binary (H = 0.0659) and the random Hadamard sensing matrices (H = 1).

  • Finally, the DWT based sensing matrices exhibits the good promise for speech signal compression.

Thus, this study shows the effectiveness of the DWT based sensing matrices for speech signal processing applications. The scope of this study can be further expanded by investigating the use of the DWT based sensing matrices in other application areas such as music signal processing, under water acoustics and the biomedical signal processing such as the ECG and EEG analysis.

Abbreviations

DWT: 

discrete wavelet transform

CS: 

compressed sensing

CR: 

compression ratio

RMSE: 

root mean square error

SNR: 

signal to noise ratio

MOS: 

mean opinion score

PESQ: 

perceptual evaluation of speech quality

Declarations

Authors’ contributions

YVP have made substantial contributions to design and development of DWT based sensing matrices and their application to speech signal processing. YVP formulated the problem with objective, performed the experimentation and wrote the paper. SLN has been involved in the critical testing and analysis of proposed DWT based sensing matrices, manuscript preparation and proof reading. Both authors read and approved the final manuscript.

Acknowledgements

The authors wish to acknowledge the Dr. Babasaheb Ambedkar Technological University, Lonere, Maharashtra, India for providing infrastructure for this research work. The authors would like to thank the anonymous reviewers for their constructive comments and questions which greatly improved the quality of article.

Competing interests

The authors declare that they have no competing interests.

Availability of data and materials

All datasets on which the conclusions of the manuscript are rely and the data supporting their findings are presented in the main paper.

Funding

The authors declare that they have no funding provided for the research reported in this paper.

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Electronics and Telecommunication Engineering, Dr. Babasaheb Ambedkar Technological University

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© The Author(s) 2016