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Application of 1‑D discrete wavelet transform based compressed sensing matrices for speech compressionYuvraj V. Parkale* and Sanjay L. Nalbalwar
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 compres-sion. The present study investigates the performance analysis of the different DWT based sensing matrices such as: Daubechies, Coiflets, Symlets, Battle, Beylkin and Vaidy-anathan 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 matri-ces. 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 perfor-mance 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 compression, Compressed sensing (CS), Discrete wavelet transform (DWT), Mean opinion score (MOS), Perceptual evaluation of speech quality (PESQ)
Open Access
© The Author(s) 2016. This 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.
RESEARCH
Parkale and Nalbalwar SpringerPlus (2016) 5:2048 DOI 10.1186/s40064‑016‑3740‑x
*Correspondence: yuvrajparkale@gmail.com Department of Electronics and Telecommunication Engineering, Dr. Babasaheb Ambedkar Technological University, Lonere, Maharashtra, India
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IntroductionConventional 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 popu-lar 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 decom-position 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 (Mal-lat 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 recon-struction of a two-channel filter bank.
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:
where W is the N × N wavelet matrix and defined as: W = WI, where I is N × N identity matrix.
(1)f = Wx
(i 1)vf+
(i 2)vf+
(i)uf
1-level Detail Coefficient (CD1)
Detail Coefficient (CD2)
Approximation Coefficient (CA2)
u
v
2
2
u
v 2
2
Approx. Coefficient (CA1)
(i 1)uf+
(i 2)uf+
Fig. 1 Block diagram of 1-D fast forward wavelet transform with 2-level decomposition
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Thus, the classical approach of data compression is to employ the discrete wavelet transform (DWT) based methods (Skodras and Ebrahimi 2001) prior to the transmis-sion. However, these methods includes the complicated multiplications, exhaustive coefficient search and sorting procedure along with the arithmetic encoding of the sig-nificant 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) pro-posed 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 appli-cation 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), commu-nications 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 com-pared 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) Gauss-ian 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, bipo-lar and the ternary matrices. Several authors have proposed the deterministic construc-tion 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 sig-nal 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
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sampler and might be able to advance some features like compression ratio and compu-tational 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) frameworkBackground
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:
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:
where x is the non-sparse input signal. Combined form of Eqs. (2) and (3) is given as:
The two basic conditions should be satisfied for the successful implementation of the CS.
1. Sensing matrix (Φ) and sparsity transform (Ψ) should be incoherent to each other.2. The Φ should satisfy the restricted isometric property (RIP) (Candes and Tao 2006)
and defined as follow:
(2)y = �f
(3)f = �x
(4)y = �f = ΦΨ x
(5)(1− δk) �x�22 ≤ ��x�22 ≤ (1+ δk) �x�
22
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where δk ∊ (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) matrix1‑D DWT matrix
For a signal x of length N = 2n and a low-pass filter (u), the ith level wavelet decomposi-tion (Vidakovic 1999; Wang and Vieira 2010) is given by an Eqs. (6) and (7). Where, v is the high-pass filter.
And
The reconstruction of f i−1u from fu
i and fvi can be obtained by
The 1-D DWT matrix forms are given as below:
and
where, f (i)u is the 2n−i dimensional low pass vector in the ith level and f (i)v the high-pass, while f (i−1)
u is the 2n−i+1 dimensional low-pass vector in the (i − 1)th level. The two 2n−i by 2n−i+1 wavelet filter matrices are given below.
(6)f (i)u (j) =2n−i+1∑
k=1
u(k − 2j) f (i−1)u (k) where, j = 1, 2, . . . , 2
n−i
(7)f (i)v (j) =2n−i+1∑
k=1
v(k − 2j) f (i−1)u (k) where, j = 1, 2, . . . , 2
n−i
(8)f (i−1)u (j) =
2n−i∑
k=1
u(j − 2k) f (i)u (k)+2n−i∑
k=1
v(j − 2k) f (i)v (k)
(9)f (i)u = U(i) f (i−1)
u
(10)f (i)v = V(i) f (i−1)
v
(11)U(i) =
u(−1) 0 0 0 · · · u(−3) u(−2)u(−3) u(−2) u(−1) 0 · · · u(−5) u(−4)
......
... .... . .
......
0 0 0 0 · · · u(−1) 0
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And
Thus, the ith scale wavelet transform can be represented as:
This gives the wavelet matrix of 1-level decomposition. The wavelet matrix for different levels of decomposition is given as below.
Above equation can be represented as,
Here, the numbers of signal decomposition levels are restricted to 2n−i+1 ≥ L. Where, L is the length of the filter.
Thus, the final wavelet transform matrix is given by an Eq. (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. Create a desired quadrature mirror filters (QMF) such as Daubechies, Coiflets, Sym-lets, Beylkin, Vaidyanathan and Battle filters. For example db1 (Haar) filter is given as f = [1 1] and the db2 filter is formed as follows:
2. Create the N × N Identity matrix.
(12)V(i) =
v(−1) 0 0 0 · · · v(−3) v(−2)v(−3) v(−2) v(−1) 0 · · · v(−5) v(−4)
......
... .... . .
......
0 0 0 0 · · · v(−1) 0
(13)
[
f (i)u
f (i)v
]
=
[
U (i)
V (i)
]
f (i−1)u
(14)f (i−1)u = U
(i−1) f (i−2)u
(15)
f (i)u
f (i)v
f (i−1)v
.
.
.
f (2)v
f (1)v
=
U (i)U (i−1) · · ·U (1)
V (i)U (i−1) · · ·U (1)
V (i)U (i−2) · · ·U (1)
.
.
.
V (2)U (1)
V (1)
x
(16)W =
U (i)U (i−1) · · ·U (1)
V (i)U (i−1) · · ·U (1)
V (i)U (i−2) · · ·U (1)
...
V (2)U (1)
V (1)
(17)f =
[
0.482962913145 0.8365163037380.224143868042 −0.129409522551
]
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3. Perform 1-D forward wavelet transform on the N × N Identity matrix. Thus, the N × N wavelet transform matrix is generated.
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 discussionMethodology
The proposed work is evaluated on the CMU/CSTR KDT US English TIMIT database for speech synthesis by Carnegie Mellon University and Edinburgh University (Edin-burgh 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 sig-nal 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,
where N is the length of speech signal and M is the number of measurements taken from sensing matrix.
RMSE is given as below:
where x(n) is the original signal and x̃(n) is the reconstructed signal.Relative error is defined as:
where x(n) is the original signal and x̃(n) is the reconstructed signal.SNR is obtained as,
where x(n) is the original signal and x̃(n) is the reconstructed signal.
(18)CR =M
N
(19)RMSE =
√
∑Nn=1 (x(n)− x̃(n))2
N
(20)Rel.Error =
∥
∥x̃(n)− x(n)∥
∥
2
�x(n)�2
(21)SNR(db) = 20 log
(
�x(n)�2∥
∥x(n)− x̃(n)∥
∥
2
)
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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 matri-ces 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), cor-responding 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).
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.
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 recon-struction 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 matri-ces based on Coiflets wavelet family such as coif1, coif2, coif3, coif4 and coif5 (Tables 11, 12, 13, 14, 15).
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 Coi-flets 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.
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 matri-ces based on Symlets wavelet family such as sym4, sym5, sym6, sym7, sym8, sym9 and sym10 (Tables 16, 17, 18, 19, 20, 21, 22).
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Tabl
e 1
Perf
orm
ance
ana
lysi
s of
the
prop
osed
db1
(Haa
r) w
avel
et b
ased
sen
sing
mat
rix
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n
time
(s)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2416
1.28
1382
0.05
850.
9774
0.19
852.
5840
58
2048
410
0.2
5010
2416
2.33
1942
0.05
210.
8711
1.19
842.
5231
52
2048
512
0.25
5010
2416
3.75
2369
0.05
210.
8707
1.20
242.
5143
15
2048
614
0.3
5010
2416
4.24
2975
0.04
880.
8153
1.77
402.
5707
37
2048
849
0.4
5010
2414
5.85
3003
0.04
790.
7997
1.94
182.
5814
21
2048
1024
0.5
5010
2414
9.48
6588
0.04
780.
7993
1.94
542.
5818
56
2048
1229
0.6
5010
2414
11.0
8041
60.
0471
0.78
772.
0723
2.59
3704
2048
1434
0.7
5010
2413
14.4
4079
60.
0468
0.78
242.
1310
2.54
6346
2048
1536
0.75
5010
2412
17.7
9912
70.
0468
0.78
162.
1406
2.52
3620
2048
1638
0.8
5010
2411
16.5
6827
90.
0468
0.78
162.
1406
2.48
4885
2048
1843
0.9
5010
2411
20.6
2963
60.
0468
0.78
162.
1407
2.51
8902
2048
2048
1.0
5010
249
22.5
2503
20.
0468
0.78
152.
1409
2.61
2855
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Tabl
e 2
Perf
orm
ance
ana
lysi
s of
the
prop
osed
db2
wav
elet
bas
ed s
ensi
ng m
atri
x
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m)
Com
pres
sion
ra
tio
(CR =
m/N
)
Spar
sity
le
vel =
(k/N
) ×
100
(%)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n
time
(s)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2415
1.33
6137
0.05
960.
9955
0.03
912.
1598
16
2048
410
0.2
5010
2415
2.44
4694
0.05
520.
9224
0.70
212.
1878
08
2048
512
0.25
5010
2415
3.62
2582
0.05
520.
9222
0.70
342.
1032
15
2048
614
0.3
5010
2414
3.91
9803
0.05
170.
8635
1.27
512.
3377
19
2048
849
0.4
5010
2413
5.74
0394
0.05
020.
8387
1.52
782.
1059
89
2048
1024
0.5
5010
2412
8.50
8652
0.05
010.
8376
1.53
912.
3195
50
2048
1229
0.6
5010
2412
10.6
2200
70.
0475
0.79
312.
0137
2.05
5366
2048
1434
0.7
5010
2412
13.7
1603
50.
0469
0.78
382.
1160
2.37
8571
2048
1536
0.75
5010
2411
15.9
9153
40.
0468
0.78
172.
1395
2.13
5815
2048
1638
0.8
5010
2411
17.0
8593
70.
0468
0.78
162.
1404
2.07
8314
2048
1843
0.9
5010
2411
27.2
1734
90.
0468
0.78
162.
1405
2.42
3662
2048
2048
1.0
5010
249
29.8
3231
10.
0468
0.78
152.
1409
2.36
5919
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Tabl
e 3
Perf
orm
ance
ana
lysi
s of
the
prop
osed
db3
wav
elet
bas
ed s
ensi
ng m
atri
x
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2418
1.62
6908
0.05
670.
9483
0.46
152.
2533
46
2048
410
0.2
5010
2415
2.43
4635
0.05
450.
9106
0.81
372.
1609
87
2048
512
0.25
5010
2415
3.63
6753
0.05
450.
9099
0.81
992.
2525
68
2048
614
0.3
5010
2415
4.03
3450
0.05
020.
8383
1.53
232.
2016
34
2048
849
0.4
5010
2413
5.92
9769
0.04
960.
8282
1.63
732.
2857
77
2048
1024
0.5
5010
2413
9.18
0948
0.04
960.
8282
1.63
722.
3090
93
2048
1229
0.6
5010
2413
12.4
8021
20.
0477
0.79
631.
9788
2.42
7516
2048
1434
0.7
5010
2413
16.5
5480
20.
0469
0.78
292.
1258
2.50
9123
2048
1536
0.75
5010
2412
27.2
3386
10.
0468
0.78
172.
1396
2.17
5306
2048
1638
0.8
5010
2411
16.8
2110
00.
0468
0.78
162.
1406
2.28
3585
2048
1843
0.9
5010
2411
28.6
7397
60.
0468
0.78
162.
1407
2.19
2527
2048
2048
1.0
5010
249
30.9
0555
20.
0468
0.78
152.
1409
2.26
2969
Page 12 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 4
Perf
orm
ance
ana
lysi
s of
the
prop
osed
db4
wav
elet
bas
ed s
ensi
ng m
atri
x
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ra
tio (C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n
time
(s)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2415
2.30
1267
0.05
450.
9099
0.81
982.
2607
42
2048
410
0.2
5010
2415
2.33
2355
0.05
240.
8749
1.16
072.
5072
63
2048
512
0.25
5010
2415
3.69
6636
0.05
230.
8744
1.16
572.
3705
97
2048
614
0.3
5010
2415
4.30
8034
0.04
960.
8283
1.63
632.
1611
84
2048
849
0.4
5010
2413
5.85
6042
0.04
840.
8092
1.83
872.
1994
77
2048
1024
0.5
5010
2413
19.0
6277
30.
0484
0.80
931.
8376
2.21
9947
2048
1229
0.6
5010
2413
11.0
0184
40.
0472
0.78
872.
0622
2.08
0746
2048
1434
0.7
5010
2412
13.3
5980
60.
0468
0.78
232.
1325
2.13
4800
2048
1536
0.75
5010
2412
16.7
4898
30.
0468
0.78
162.
1405
2.33
7097
2048
1638
0.8
5010
2411
16.1
9774
80.
0468
0.78
162.
1407
2.02
5083
2048
1843
0.9
5010
2411
20.3
0889
00.
0468
0.78
162.
1407
2.07
4552
2048
2048
1.0
5010
249
21.9
1614
50.
0468
0.78
152.
1409
2.14
9653
Page 13 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 5
Perf
orm
ance
ana
lysi
s of
the
prop
osed
db5
wav
elet
bas
ed s
ensi
ng m
atri
x
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2415
1.30
8421
0.05
500.
9192
0.73
162.
2404
30
2048
410
0.2
5010
2415
2.16
4346
0.05
080.
8496
1.41
622.
2113
80
2048
512
0.25
5010
2415
3.39
9650
0.05
090.
8498
1.41
382.
2507
45
2048
614
0.3
5010
2415
3.92
4939
0.04
910.
8201
1.72
212.
4122
86
2048
849
0.4
5010
2414
5.93
5914
0.04
810.
8030
1.90
572.
2700
19
2048
1024
0.5
5010
2414
9.35
3480
0.04
810.
8031
1.90
482.
3373
83
2048
1229
0.6
5010
2413
11.1
1847
70.
0470
0.78
482.
1053
2.21
6672
2048
1434
0.7
5010
2413
17.9
7501
40.
0468
0.78
222.
1335
2.24
5930
2048
1536
0.75
5010
2411
15.5
3842
60.
0468
0.78
162.
1405
2.44
0602
2048
1638
0.8
5010
2411
16.5
3987
90.
0468
0.78
162.
1407
2.18
2187
2048
1843
0.9
5010
2411
20.4
4285
90.
0468
0.78
162.
1407
2.23
5392
2048
2048
1.0
5010
249
22.6
4166
40.
0468
0.78
152.
1409
2.21
0355
Page 14 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 6
Perf
orm
ance
ana
lysi
s of
the
prop
osed
db6
wav
elet
bas
ed s
ensi
ng m
atri
x
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n
time
(s)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2415
1.25
7052
0.05
540.
9260
0.66
772.
2911
40
2048
410
0.2
5010
2415
2.23
2353
0.05
060.
8458
1.45
442.
3561
09
2048
512
0.25
5010
2416
3.71
0359
0.05
060.
8454
1.45
842.
3572
69
2048
614
0.3
5010
2415
4.00
1119
0.04
940.
8259
1.66
172.
3066
22
2048
849
0.4
5010
2415
6.42
5953
0.04
870.
8142
1.78
552.
3678
48
2048
1024
0.5
5010
2414
9.85
939
0.04
870.
8138
1.78
932.
3894
22
2048
1229
0.6
5010
2413
15.5
4062
50.
0469
0.78
452.
1080
2.69
8509
2048
1434
0.7
5010
2413
18.1
7286
80.
0468
0.78
272.
1286
2.62
7686
2048
1536
0.75
5010
2412
21.3
2204
50.
0468
0.78
162.
1404
2.69
9761
2048
1638
0.8
5010
2412
30.7
4930
10.
0468
0.78
162.
1406
2.60
8611
2048
1843
0.9
5010
2412
39.6
6230
60.
0468
0.78
162.
1406
2.70
6883
2048
2048
1.0
5010
249
37.1
5061
80.
0468
0.78
152.
1409
2.58
1647
Page 15 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 7
Perf
orm
ance
ana
lysi
s of
the
prop
osed
db7
wav
elet
bas
ed s
ensi
ng m
atri
x
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2416
1.54
4091
0.05
640.
9417
0.52
162.
5578
82
2048
410
0.2
5010
2416
2.31
0240
0.05
140.
8584
1.32
602.
5039
21
2048
512
0.25
5010
2416
3.65
5857
0.05
140.
8585
1.32
472.
3716
64
2048
614
0.3
5010
2415
4.05
0385
0.05
110.
8537
1.37
392.
3916
05
2048
849
0.4
5010
2414
5.77
2598
0.05
010.
8379
1.53
622.
5028
02
2048
1024
0.5
5010
2414
9.53
6010
0.05
010.
8375
1.53
992.
4504
85
2048
1229
0.6
5010
2412
10.2
8168
90.
0470
0.78
482.
1049
2.45
8048
2048
1434
0.7
5010
2412
13.5
5498
60.
0468
0.78
272.
1278
2.50
5254
2048
1536
0.75
5010
2412
17.1
0936
20.
0468
0.78
162.
1404
2.61
6728
2048
1638
0.8
5010
2412
18.1
3484
10.
0468
0.78
162.
1406
2.54
3920
2048
1843
0.9
5010
2411
21.1
1526
60.
0468
0.78
162.
1406
2.51
2582
2048
2048
1.0
5010
249
22.7
5380
20.
0468
0.78
152.
1409
2.49
2282
Page 16 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 8
Perf
orm
ance
ana
lysi
s of
the
prop
osed
db8
wav
elet
bas
ed s
ensi
ng m
atri
x
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m)
Com
pres
sion
ra
tio (C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al re
cons
truc
tion
time
(s)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2416
1.37
9793
0.05
620.
9397
0.54
062.
4454
16
2048
410
0.2
5010
2416
2.31
9429
0.05
140.
8597
1.31
322.
5956
14
2048
512
0.25
5010
2416
3.67
2272
0.05
140.
8594
1.31
612.
3962
96
2048
614
0.3
5010
2416
4.35
2767
0.05
040.
8417
1.49
732.
4736
40
2048
849
0.4
5010
2414
5.84
4768
0.04
960.
8283
1.63
652.
4374
63
2048
1024
0.5
5010
2413
8.95
1393
0.04
950.
8274
1.64
582.
3593
80
2048
1229
0.6
5010
2413
11.0
7483
60.
0469
0.78
452.
1084
2.57
3343
2048
1434
0.7
5010
2413
14.5
7421
30.
0468
0.78
242.
1318
2.60
9980
2048
1536
0.75
5010
2411
15.6
4966
20.
0468
0.78
162.
1405
2.55
9536
2048
1638
0.8
5010
2411
16.8
7915
80.
0468
0.78
162.
1406
2.55
6170
2048
1843
0.9
5010
2411
20.5
3528
70.
0468
0.78
162.
1406
2.56
9219
2048
2048
1.0
5010
249
22.8
9290
10.
0468
0.78
152.
1409
2.51
3244
Page 17 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 9
Perf
orm
ance
ana
lysi
s of
the
prop
osed
db9
wav
elet
bas
ed s
ensi
ng m
atri
x
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n
time
(s)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2415
1.25
2129
0.05
830.
9750
0.22
032.
6494
78
2048
410
0.2
5010
2417
2.52
7130
0.05
180.
8660
1.24
922.
5579
30
2048
512
0.25
5010
2417
3.97
2292
0.05
180.
8652
1.25
822.
5319
26
2048
614
0.3
5010
2416
4.22
1050
0.04
900.
8182
1.74
252.
5724
60
2048
849
0.4
5010
2415
6.27
0259
0.04
780.
7990
1.94
952.
6835
71
2048
1024
0.5
5010
2414
9.59
3764
0.04
780.
7983
1.95
712.
5330
85
2048
1229
0.6
5010
2413
11.0
7561
90.
0470
0.78
532.
0992
2.49
9444
2048
1434
0.7
5010
2413
14.5
1897
40.
0468
0.78
242.
1309
2.53
9711
2048
1536
0.75
5010
2412
17.1
7863
80.
0468
0.78
162.
1405
2.48
9187
2048
1638
0.8
5010
2412
18.0
4313
40.
0468
0.78
162.
1406
2.50
6067
2048
1843
0.9
5010
2412
22.3
7087
40.
0468
0.78
162.
1407
2.56
9387
2048
2048
1.0
5010
249
22.7
5173
20.
0468
0.78
152.
1409
2.49
7201
Page 18 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 10
Per
form
ance
ana
lysi
s of
the
prop
osed
db1
0 w
avel
et b
ased
sen
sing
mat
rix
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N)
× 1
00 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2416
1.28
1382
0.05
850.
9774
0.19
852.
5840
58
2048
410
0.2
5010
2416
2.33
1942
0.05
210.
8711
1.19
842.
5231
52
2048
512
0.25
5010
2416
3.75
2369
0.05
210.
8707
1.20
242.
5143
15
2048
614
0.3
5010
2416
4.24
2975
0.04
880.
8153
1.77
402.
5707
37
2048
849
0.4
5010
2414
5.85
3003
0.04
790.
7997
1.94
182.
5814
21
2048
1024
0.5
5010
2414
9.48
6588
0.04
780.
7993
1.94
542.
5818
56
2048
1229
0.6
5010
2414
11.0
8041
60.
0471
0.78
772.
0723
2.59
3704
2048
1434
0.7
5010
2413
14.4
4079
60.
0468
0.78
242.
1310
2.54
6346
2048
1536
0.75
5010
2412
17.7
9912
70.
0468
0.78
162.
1406
2.52
3620
2048
1638
0.8
5010
2411
16.5
6827
90.
0468
0.78
162.
1406
2.48
4885
2048
1843
0.9
5010
2411
20.6
2963
60.
0468
0.78
162.
1407
2.51
8902
2048
2048
1.0
5010
249
22.5
2503
20.
0468
0.78
152.
1409
2.61
2855
Page 19 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
20
25
30
35
40
Compression Ratio (CR)
Sign
al R
econ
stru
ctio
n Ti
me
(Sec
onds
) db1(Haar) Matrixdb2 Matrixdb3 Matrixdb4 Matrixdb5 Matrixdb6 Matrixdb7 Matrixdb8 Matrixdb9 Matrixdb10 Matrix
Fig. 2 Effect of compression ratio on signal reconstruction time for different Daubechies wavelet sensing matrices
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.75
0.8
0.85
0.9
0.95
1
Compression Ratio (CR)
Sign
al R
econ
stru
ctio
n Er
ror (
Rel
ativ
e er
ror) db1(Haar) Matrix
db2 Matrixdb3 Matrixdb4 Matrixdb5 Matrixdb6 Matrixdb7 Matrixdb8 Matrixdb9 Matrixdb10 Matrix
Fig. 3 Effect of compression ratio on relative error for different Daubechies wavelet sensing matrices
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
Compression Ratio (CR)
Sign
al-to
-Noi
se R
atio
(SN
R) (
db)
db1(Haar) Matrixdb2 Matrixdb3 Matrixdb4 Matrixdb5 Matrixdb6 Matrixdb7 Matrixdb8 Matrixdb9 Matrixdb10 Matrix
Fig. 4 Effect of compression ratio on signal-to-noise ratio for different Daubechies wavelet sensing matrices
Page 20 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 11
Per
form
ance
ana
lysi
s of
the
prop
osed
coi
f1 w
avel
et b
ased
sen
sing
mat
rix
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ra
tio (C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2417
1.35
4238
0.05
790.
9668
0.29
302.
0988
81
2048
410
0.2
5010
2415
2.87
4402
0.05
500.
9197
0.72
712.
0414
93
2048
512
0.25
5010
2415
10.4
6867
00.
0550
0.91
960.
7279
2.37
5475
2048
614
0.3
5010
2414
5.95
5084
0.05
190.
8675
1.23
462.
3081
89
2048
849
0.4
5010
2414
8.20
2050
0.05
130.
8579
1.33
152.
2001
96
2048
1024
0.5
5010
2413
11.5
7571
00.
0513
0.85
681.
3425
2.11
1921
2048
1229
0.6
5010
2413
18.5
7986
90.
0476
0.79
521.
9901
2.12
7905
2048
1434
0.7
5010
2412
15.6
9299
90.
0469
0.78
302.
1245
2.48
7333
2048
1536
0.75
5010
2412
17.8
1684
90.
0468
0.78
172.
1396
2.39
6088
2048
1638
0.8
5010
2412
18.9
9833
70.
0468
0.78
162.
1406
2.22
6200
2048
1843
0.9
5010
2412
23.6
1259
10.
0468
0.78
162.
1406
2.37
9104
2048
2048
1.0
5010
249
23.5
0743
50.
0468
0.78
152.
1409
2.34
3294
Page 21 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 12
Per
form
ance
ana
lysi
s of
the
prop
osed
coi
f2 w
avel
et b
ased
sen
sing
mat
rix
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2416
1.34
4408
0.05
890.
9847
0.13
382.
6560
92
2048
410
0.2
5010
2415
2.31
1920
0.05
420.
9050
0.86
742.
6342
35
2048
512
0.25
5010
2415
3.53
5317
0.05
420.
9052
0.86
492.
3542
71
2048
614
0.3
5010
2415
4.01
7180
0.05
310.
8869
1.04
282.
5957
62
2048
849
0.4
5010
2414
6.10
7924
0.05
270.
8812
1.09
902.
5151
27
2048
1024
0.5
5010
2413
8.92
7963
0.05
250.
8773
1.13
752.
5318
53
2048
1229
0.6
5010
2413
11.3
2333
10.
0481
0.80
351.
9008
2.50
3729
2048
1434
0.7
5010
2413
21.7
3885
60.
0469
0.78
322.
1229
2.65
7620
2048
1536
0.75
5010
2412
27.1
1614
60.
0468
0.78
172.
1392
2.59
8041
2048
1638
0.8
5010
2412
26.3
1215
60.
0468
0.78
162.
1405
2.35
8481
2048
1843
0.9
5010
2411
32.6
4676
70.
0468
0.78
162.
1406
2.56
4254
2048
2048
1.0
5010
249
34.1
6260
20.
0468
0.78
152.
1409
2.61
7750
Page 22 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 13
Per
form
ance
ana
lysi
s of
the
prop
osed
coi
f3 w
avel
et b
ased
sen
sing
mat
rix
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2416
1.31
3659
0.05
610.
9375
0.56
043.
0764
15
2048
410
0.2
5010
2416
2.30
0134
0.05
290.
8835
1.07
612.
8836
24
2048
512
0.25
5010
2416
3.83
6235
0.05
290.
8834
1.07
702.
9772
64
2048
614
0.3
5010
2415
4.09
5096
0.04
920.
8228
1.69
442.
9916
58
2048
849
0.4
5010
2414
6.30
7728
0.04
850.
8105
1.82
442.
7343
66
2048
1024
0.5
5010
2414
12.4
2792
70.
0485
0.81
051.
8247
3.00
0826
2048
1229
0.6
5010
2413
14.1
8072
00.
0475
0.79
312.
0136
2.73
2994
2048
1434
0.7
5010
2413
26.6
9505
40.
0468
0.78
242.
1319
2.67
4142
2048
1536
0.75
5010
2412
23.3
7958
30.
0468
0.78
162.
1403
2.96
7887
2048
1638
0.8
5010
2411
19.4
0293
90.
0468
0.78
162.
1406
2.88
2567
2048
1843
0.9
5010
2411
25.0
7796
50.
0468
0.78
162.
1406
2.74
1203
2048
2048
1.0
5010
249
30.9
2454
00.
0468
0.78
152.
1409
2.94
9021
Page 23 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 14
Per
form
ance
ana
lysi
s of
the
prop
osed
coi
f4 w
avel
et b
ased
sen
sing
mat
rix
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ra
tio (C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2416
1.26
2131
0.05
670.
9480
0.46
432.
8437
27
2048
410
0.2
5010
2417
2.43
9706
0.05
240.
8756
1.15
382.
8302
43
2048
512
0.25
5010
2417
3.70
4519
0.05
240.
8757
1.15
292.
7916
44
2048
614
0.3
5010
2416
4.09
7077
0.04
910.
8211
1.71
232.
7433
63
2048
849
0.4
5010
2415
6.15
3716
0.04
850.
8105
1.82
482.
7836
60
2048
1024
0.5
5010
2414
9.04
1550
0.04
850.
8103
1.82
672.
8127
75
2048
1229
0.6
5010
2413
10.7
1278
10.
0473
0.79
112.
0354
2.63
3953
2048
1434
0.7
5010
2413
14.2
1377
50.
0468
0.78
242.
1318
2.63
1650
2048
1536
0.75
5010
2412
16.3
6827
80.
0468
0.78
162.
1404
2.78
5540
2048
1638
0.8
5010
2412
17.1
4283
30.
0468
0.78
162.
1406
2.69
0370
2048
1843
0.9
5010
2411
19.4
9450
50.
0468
0.78
162.
1406
2.60
6268
2048
2048
1.0
5010
249
21.4
3224
70.
0468
0.78
152.
1409
2.65
9983
Page 24 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 15
Per
form
ance
ana
lysi
s of
the
prop
osed
coi
f5 w
avel
et b
ased
sen
sing
mat
rix
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2418
1.50
3436
0.05
520.
9229
0.69
662.
8660
05
2048
410
0.2
5010
2417
2.49
1811
0.05
090.
8498
1.41
322.
8956
38
2048
512
0.25
5010
2417
3.85
2817
0.05
090.
8500
1.41
143.
0079
31
2048
614
0.3
5010
2416
4.18
8601
0.04
890.
8164
1.76
163.
0094
24
2048
849
0.4
5010
2414
5.97
1268
0.04
820.
8060
1.87
372.
8452
10
2048
1024
0.5
5010
2414
9.49
8384
0.04
820.
8058
1.87
533.
0334
61
2048
1229
0.6
5010
2413
11.0
1206
10.
0472
0.78
892.
0598
3.03
3341
2048
1434
0.7
5010
2413
14.3
4888
40.
0468
0.78
242.
1313
3.10
5129
2048
1536
0.75
5010
2412
17.0
8854
40.
0468
0.78
162.
1404
2.82
1626
2048
1638
0.8
5010
2412
18.2
1105
70.
0468
0.78
162.
1406
3.00
5033
2048
1843
0.9
5010
2411
20.5
3312
00.
0468
0.78
162.
1406
3.09
5259
2048
2048
1.0
5010
249
23.0
4774
00.
0468
0.78
152.
1409
2.93
6779
Page 25 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
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. Fur-thermore, 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 wave-let based sensing matrices almost demonstrate similar performance with minimum rela-tive 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.
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 exhib-its 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.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
20
25
30
35
Compression Ratio (CR)
Sign
al R
econ
stru
ctio
n Ti
me
(Sec
onds
) coif1 Matrixcoif2 Matrixcoif3 Matrixcoif4 Matrixcoif5 Matrix
Fig. 5 Effect of compression ratio on signal reconstruction time for different Coiflets wavelet sensing matri-ces
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.75
0.8
0.85
0.9
0.95
1
Compression Ratio (CR)
Sign
al R
econ
stru
ctio
n Er
ror (
Rel
ativ
e er
ror)
coif1 Matrixcoif2 Matrixcoif3 Matrixcoif4 Matrixcoif5 Matrix
Fig. 6 Effect of compression ratio on relative error for different Coiflets wavelet sensing matrices
Page 26 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Performance analysis of the Beylkin, Vaidyanathan and Battle wavelet family based
sensing matrices
This section shows the performance analysis of the different DWT sensing matri-ces based on Beylkin, Vaidyanathan, and Battle1, Battle3 and Battle5 wavelet families (Tables 23, 24, 25, 26, 27).
Figure 11 shows that the Beylkin wavelet based sensing matrix requires the less recon-struction 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 matri-ces 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 com-pared to other 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 recon-struction time. From Fig. 15, it can be observed that the db10 shows the good perfor-mance 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
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
Compression Ratio (CR)
Sign
al-to
-Noi
se R
atio
(SN
R) (
db)
coif1 Matrixcoif2 Matrixcoif3 Matrixcoif4 Matrixcoif5 Matrix
Fig. 7 Effect of compression ratio on signal-to-noise ratio for different Coiflets wavelet sensing matrices
Page 27 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 16
Per
form
ance
ana
lysi
s of
the
prop
osed
sym
4 w
avel
et b
ased
sen
sing
mat
rix
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N)
× 1
00 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2415
1.42
6156
0.06
061.
0128
0.11
012.
2199
32
2048
410
0.2
5010
2415
2.46
6473
0.05
500.
9186
0.73
762.
2874
69
2048
512
0.25
5010
2415
4.03
4720
0.05
500.
9188
0.73
602.
4085
24
2048
614
0.3
5010
2415
4.52
0797
0.05
100.
8516
1.39
562.
2620
42
2048
849
0.4
5010
2414
6.88
3726
0.05
030.
8412
1.50
162.
4613
59
2048
1024
0.5
5010
2414
9.27
5989
0.05
030.
8412
1.50
152.
6008
20
2048
1229
0.6
5010
2413
10.6
9936
60.
0479
0.80
011.
9374
2.08
3195
2048
1434
0.7
5010
2413
13.7
0268
30.
0468
0.78
282.
1269
2.11
7664
2048
1536
0.75
5010
2412
16.1
4265
90.
0468
0.78
172.
1393
2.30
3114
2048
1638
0.8
5010
2412
17.0
2474
60.
0468
0.78
162.
1405
2.25
9328
2048
1843
0.9
5010
2412
21.1
1806
40.
0468
0.78
162.
1406
2.36
9993
2048
2048
1.0
5010
249
21.2
2717
90.
0468
0.78
152.
1409
2.24
8962
Page 28 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 17
Per
form
ance
ana
lysi
s of
the
prop
osed
sym
5 w
avel
et b
ased
sen
sing
mat
rix
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n
time
(s)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2418
1.56
2108
0.05
870.
9813
0.16
442.
4747
25
2048
410
0.2
5010
2416
2.41
0647
0.05
250.
8767
1.14
282.
4229
95
2048
512
0.25
5010
2415
3.36
3769
0.05
240.
8764
1.14
602.
4149
40
2048
614
0.3
5010
2414
3.63
3501
0.04
870.
8139
1.78
852.
4016
02
2048
849
0.4
5010
2414
5.90
1087
0.04
850.
8100
1.83
072.
2859
70
2048
1024
0.5
5010
2413
8.62
9663
0.04
840.
8094
1.83
702.
2709
79
2048
1229
0.6
5010
2412
10.4
3665
80.
0476
0.79
511.
9910
2.29
5386
2048
1434
0.7
5010
2412
12.9
9750
50.
0468
0.78
252.
1299
2.41
6593
2048
1536
0.75
5010
2412
16.5
4321
70.
0468
0.78
162.
1400
2.22
6996
2048
1638
0.8
5010
2411
15.8
3922
20.
0468
0.78
162.
1407
2.25
5090
2048
1843
0.9
5010
2411
19.8
8718
30.
0468
0.78
162.
1407
2.43
1380
2048
2048
1.0
5010
249
21.3
1945
00.
0468
0.78
152.
1409
2.26
6367
Page 29 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 18
Per
form
ance
ana
lysi
s of
the
prop
osed
sym
6 w
avel
et b
ased
sen
sing
mat
rix
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2416
1.33
7284
0.05
680.
9486
0.45
872.
2788
76
2048
410
0.2
5010
2416
2.35
4315
0.05
190.
8665
1.24
502.
2914
21
2048
512
0.25
5010
2415
3.40
3486
0.05
200.
8696
1.21
322.
4955
18
2048
614
0.3
5010
2415
3.90
0882
0.05
060.
8450
1.46
322.
5137
99
2048
849
0.4
5010
2414
5.89
4927
0.04
950.
8274
1.64
552.
4446
08
2048
1024
0.5
5010
2414
9.19
1334
0.04
950.
8274
1.64
522.
2826
27
2048
1229
0.6
5010
2413
10.8
7466
30.
0470
0.78
582.
0939
2.35
4739
2048
1434
0.7
5010
2413
13.9
8655
30.
0468
0.78
252.
1300
2.26
9408
2048
1536
0.75
5010
2412
16.2
8915
50.
0468
0.78
162.
1404
2.32
4362
2048
1638
0.8
5010
2412
17.0
0207
40.
0468
0.78
162.
1406
2.32
5569
2048
1843
0.9
5010
2412
21.6
0399
00.
0468
0.78
162.
1407
2.40
9376
2048
2048
1.0
5010
249
21.4
8859
00.
0468
0.78
152.
1409
2.34
4651
Page 30 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 19
Per
form
ance
ana
lysi
s of
the
prop
osed
sym
7 w
avel
et b
ased
sen
sing
mat
rix
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2419
1.60
4935
0.05
490.
9178
0.74
502.
3851
64
2048
410
0.2
5010
2416
2.27
0355
0.05
300.
8860
1.05
142.
3689
00
2048
512
0.25
5010
2416
3.61
1425
0.05
300.
8857
1.05
392.
3900
55
2048
614
0.3
5010
2414
3.71
9657
0.04
940.
8259
1.66
202.
4002
46
2048
849
0.4
5010
2414
5.94
3942
0.04
860.
8126
1.80
252.
3796
69
2048
1024
0.5
5010
2414
9.23
1755
0.04
860.
8125
1.80
372.
6262
92
2048
1229
0.6
5010
2414
11.6
9705
60.
0477
0.79
681.
9732
2.39
5632
2048
1434
0.7
5010
2413
14.0
3804
80.
0468
0.78
242.
1319
2.54
7235
2048
1536
0.75
5010
2412
16.3
7947
80.
0468
0.78
162.
1398
2.54
1034
2048
1638
0.8
5010
2412
17.1
3938
10.
0468
0.78
162.
1406
2.29
9743
2048
1843
0.9
5010
2411
19.8
1361
30.
0468
0.78
162.
1406
2.62
8218
2048
2048
1.0
5010
249
21.3
9374
10.
0468
0.78
152.
1409
2.43
5649
Page 31 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 20
Per
form
ance
ana
lysi
s of
the
prop
osed
sym
8 w
avel
et b
ased
sen
sing
mat
rix
Leng
th o
f sig
nal
(N)
Num
ber
of m
easu
re‑
men
ts (m
)
Com
pres
sion
ra
tio (C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
100
(%)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al re
con‑
stru
ctio
n tim
e (s
)
RMSE
Rela
tive
erro
rSN
R (d
b)Co
nstr
uctio
n tim
e fo
r sen
sing
m
atri
x (s
)
2048
205
0.1
5010
2416
1.33
7804
0.05
910.
9881
0.10
372.
3989
59
2048
410
0.2
5010
2416
2.28
9342
0.05
250.
8775
1.13
542.
4493
29
2048
512
0.25
5010
2416
3.60
5457
0.05
250.
8774
1.13
612.
4139
14
2048
614
0.3
5010
2415
3.89
9311
0.04
900.
8194
1.72
962.
3671
31
2048
849
0.4
5010
2414
5.83
9426
0.04
890.
8165
1.76
132.
4140
94
2048
1024
0.5
5010
2414
9.21
9034
0.04
890.
8167
1.75
902.
4776
53
2048
1229
0.6
5010
2413
10.9
1677
50.
0476
0.79
531.
9899
2.41
2281
2048
1434
0.7
5010
2413
14.1
0579
00.
0468
0.78
232.
1324
2.41
1482
2048
1536
0.75
5010
2412
16.3
3755
30.
0468
0.78
162.
1403
2.42
5477
2048
1638
0.8
5010
2411
15.8
7626
70.
0468
0.78
162.
1406
2.45
2373
2048
1843
0.9
5010
2411
19.8
3864
40.
0468
0.78
162.
1406
2.43
9456
2048
2048
1.0
5010
249
21.4
1229
60.
0468
0.78
152.
1409
2.48
1033
Page 32 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 21
Per
form
ance
ana
lysi
s of
the
prop
osed
sym
9 w
avel
et b
ased
sen
sing
mat
rix
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2417
1.43
3657
0.05
820.
9731
0.23
642.
5660
95
2048
410
0.2
5010
2416
2.36
2390
0.05
200.
8695
1.21
492.
5664
05
2048
512
0.25
5010
2416
3.70
3235
0.05
200.
8690
1.21
912.
5294
53
2048
614
0.3
5010
2415
3.94
7898
0.04
930.
8243
1.67
882.
5802
08
2048
849
0.4
5010
2414
5.82
4850
0.04
810.
8038
1.89
752.
5664
82
2048
1024
0.5
5010
2413
8.52
7638
0.04
810.
8038
1.89
732.
6276
00
2048
1229
0.6
5010
2412
9.96
1474
0.04
700.
7850
2.10
232.
6214
38
2048
1434
0.7
5010
2412
12.9
6303
50.
0468
0.78
222.
1333
2.58
2570
2048
1536
0.75
5010
2412
16.4
6801
30.
0468
0.78
162.
1406
2.62
8998
2048
1638
0.8
5010
2411
15.8
2022
40.
0468
0.78
162.
1406
2.70
5263
2048
1843
0.9
5010
2411
19.9
1521
60.
0468
0.78
162.
1407
2.69
5629
2048
2048
1.0
5010
249
21.4
7796
90.
0468
0.78
152.
1409
2.61
8640
Page 33 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 22
Per
form
ance
ana
lysi
s of
the
prop
osed
sym
10 w
avel
et b
ased
sen
sing
mat
rix
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n
time
(s)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2417
1.47
9121
0.05
600.
9351
0.58
312.
7083
58
2048
410
0.2
5010
2416
2.34
8326
0.05
130.
8578
1.33
232.
6074
75
2048
512
0.25
5010
2416
3.76
2705
0.05
130.
8580
1.32
982.
7293
82
2048
614
0.3
5010
2416
4.32
8904
0.04
900.
8192
1.73
212.
8194
32
2048
849
0.4
5010
2414
6.05
7941
0.04
790.
8003
1.93
532.
7192
36
2048
1024
0.5
5010
2414
9.51
6700
0.04
790.
8010
1.92
732.
7157
42
2048
1229
0.6
5010
2413
11.2
6863
40.
0470
0.78
532.
0992
2.48
4238
2048
1434
0.7
5010
2412
13.6
0722
60.
0468
0.78
222.
1333
2.68
0826
2048
1536
0.75
5010
2412
17.2
2508
60.
0468
0.78
162.
1405
2.63
4090
2048
1638
0.8
5010
2412
18.1
9201
90.
0468
0.78
162.
1406
2.62
8795
2048
1843
0.9
5010
2412
22.7
1672
30.
0468
0.78
162.
1406
2.70
9115
2048
2048
1.0
5010
249
22.9
0914
50.
0468
0.78
152.
1409
2.61
2016
Page 34 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
matrices. In addition, the sym9 wavelet based sensing matrix shows an edge over db10 from the CR = 0.5–1.0.
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
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
20
25
Compression Ratio (CR)
Sign
al R
econ
stru
ctio
n Ti
me
(Sec
onds
) sym4 Matrixsym5 Matrixsym6 Matrixsym7 Matrixsym8 Matrixsym9 Matrixsym10 Matrix
Fig. 8 Effect of compression ratio on signal reconstruction time for different Symlets wavelet sensing matri-ces
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.75
0.8
0.85
0.9
0.95
1
1.05
1.1
Compression Ratio (CR)
Sign
al R
econ
stru
ctio
n Er
ror (
Rel
ativ
e er
ror)
sym4 Matrixsym5 Matrixsym6 Matrixsym7 Matrixsym8 Matrixsym9 Matrixsym10 Matrix
Fig. 9 Effect of compression ratio on relative error for different Symlets wavelet sensing matrices
Page 35 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
such as the DCT and the sparse binary sensing matrices for speech signal compression (Tables 28, 29, 30, 31, 32, 33, 34).
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.
The overall remark
Thus, it is evident from Figs. 17, 18 and 19 (Tables 28, 29, 30, 31, 32, 33, 34) that the pro-posed 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
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
Compression Ratio (CR)
Sign
al-to
-Noi
se R
atio
(SN
R) (
db)
sym4 Matrixsym5 Matrixsym6 Matrixsym7 Matrixsym8 Matrixsym9 Matrixsym10 Matrix
Fig. 10 Effect of compression ratio on signal-to-noise ratio for different Symlets wavelet sensing matrices
Page 36 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 23
Per
form
ance
ana
lysi
s of
the
prop
osed
Bey
lkin
wav
elet
bas
ed s
ensi
ng m
atri
x
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2417
1.35
1917
0.05
510.
9205
0.71
962.
6352
92
2048
410
0.2
5010
2416
2.31
3493
0.05
220.
8726
1.18
372.
5003
19
2048
512
0.25
5010
2416
3.65
5763
0.05
220.
8722
1.18
782.
5827
26
2048
614
0.3
5010
2416
4.26
7978
0.04
900.
8193
1.73
162.
6683
84
2048
849
0.4
5010
2414
5.86
5898
0.04
830.
8064
1.86
862.
5015
77
2048
1024
0.5
5010
2414
9.54
1513
0.04
820.
8056
1.87
742.
6263
48
2048
1229
0.6
5010
2413
11.1
1344
80.
0469
0.78
382.
1163
2.46
4044
2048
1434
0.7
5010
2413
14.5
6769
30.
0468
0.78
222.
1341
2.50
4371
2048
1536
0.75
5010
2412
17.1
9308
0.04
680.
7816
2.14
052.
5276
96
2048
1638
0.8
5010
2412
18.0
8175
50.
0468
0.78
162.
1406
2.51
6479
2048
1843
0.9
5010
2412
22.5
2665
40.
0468
0.78
162.
1407
2.52
3531
2048
2048
1.0
5010
249
22.9
5070
30.
0468
0.78
152.
1409
2.46
6272
Page 37 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 24
Per
form
ance
ana
lysi
s of
the
prop
osed
Vai
dyan
atha
n w
avel
et b
ased
sen
sing
mat
rix
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2419
1.70
7644
0.07
021.
1724
1.38
122.
9129
96
2048
410
0.2
5010
2419
3.45
9463
0.05
800.
9684
0.27
853.
3328
69
2048
512
0.25
5010
2419
5.66
6164
0.05
800.
9685
0.27
823.
0360
02
2048
614
0.3
5010
2418
5.76
0958
0.05
130.
8578
1.33
203.
0347
69
2048
849
0.4
5010
2415
7.31
6646
0.04
850.
8112
1.81
772.
9690
49
2048
1024
0.5
5010
2414
10.8
5523
90.
0486
0.81
231.
8053
3.07
5962
2048
1229
0.6
5010
2413
13.4
1742
20.
0474
0.79
242.
0214
3.44
1779
2048
1434
0.7
5010
2413
19.0
9362
40.
0468
0.78
222.
1339
3.48
3836
2048
1536
0.75
5010
2412
22.6
0845
30.
0468
0.78
162.
1405
3.02
2221
2048
1638
0.8
5010
2412
34.9
7041
50.
0468
0.78
162.
1406
3.58
6396
2048
1843
0.9
5010
2412
49.3
1445
00.
0468
0.78
162.
1407
3.51
5476
2048
2048
1.0
5010
249
34.9
4370
20.
0468
0.78
152.
1409
3.06
0519
Page 38 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 25
Per
form
ance
ana
lysi
s of
the
prop
osed
Bat
tle1
wav
elet
bas
ed s
ensi
ng m
atri
x
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ra
tio (C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n
time
(s)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2415
1.45
3478
0.06
331.
0577
0.48
732.
7945
29
2048
410
0.2
5010
2414
2.11
3667
0.05
560.
9283
0.64
592.
9536
15
2048
512
0.25
5010
2415
3.53
0676
0.05
600.
9363
0.57
132.
7698
43
2048
614
0.3
5010
2414
3.95
7949
0.05
140.
8598
1.31
252.
6216
63
2048
849
0.4
5010
2414
7.28
0928
0.05
130.
8571
1.33
992.
8469
45
2048
1024
0.5
5010
2414
9.77
7801
0.05
140.
8587
1.32
302.
8585
07
2048
1229
0.6
5010
2413
11.8
5049
70.
0480
0.80
131.
9241
2.64
3100
2048
1434
0.7
5010
2413
15.1
9562
80.
0468
0.78
272.
1278
2.62
1185
2048
1536
0.75
5010
2412
17.5
0308
70.
0468
0.78
172.
1387
2.80
4361
2048
1638
0.8
5010
2411
17.4
2154
50.
0468
0.78
162.
1406
2.83
5200
2048
1843
0.9
5010
2411
28.1
7992
90.
0468
0.78
162.
1406
2.86
9269
2048
2048
1.0
5010
249
32.6
4096
50.
0468
0.78
152.
1409
3.44
7333
Page 39 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 26
Per
form
ance
ana
lysi
s of
the
prop
osed
Bat
tle3
wav
elet
bas
ed s
ensi
ng m
atri
x
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2421
2.22
9166
0.07
761.
2960
2.25
204.
1843
86
2048
410
0.2
5010
2417
3.00
0747
0.06
191.
0350
0.29
914.
0370
21
2048
512
0.25
5010
2417
6.76
2783
0.06
191.
0350
0.29
844.
2141
51
2048
614
0.3
5010
2417
5.49
5381
0.05
760.
9626
0.33
133.
7794
77
2048
849
0.4
5010
2415
7.92
0600
0.05
380.
8989
0.92
613.
7067
19
2048
1024
0.5
5010
2414
11.3
3121
20.
0538
0.89
950.
9199
3.94
5231
2048
1229
0.6
5010
2413
13.9
0919
00.
0485
0.81
061.
8235
3.76
4331
2048
1434
0.7
5010
2412
19.8
3191
70.
0468
0.78
232.
1321
3.77
7555
2048
1536
0.75
5010
2412
34.0
1038
80.
0468
0.78
162.
1404
4.46
4107
2048
1638
0.8
5010
2412
30.2
3574
40.
0468
0.78
162.
1406
3.85
5705
2048
1843
0.9
5010
2411
40.1
9081
60.
0468
0.78
162.
1406
4.00
3610
2048
2048
1.0
5010
249
45.7
8919
50.
0468
0.78
152.
1409
3.72
6738
Page 40 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 27
Per
form
ance
ana
lysi
s of
the
prop
osed
Bat
tle5
wav
elet
bas
ed s
ensi
ng m
atri
x
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ra
tio (C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2416
1.54
7147
0.05
870.
9809
0.16
724.
3558
11
2048
410
0.2
5010
2416
2.77
6326
0.05
170.
8633
1.27
644.
5236
38
2048
512
0.25
5010
2417
4.87
6061
0.05
180.
8648
1.26
194.
2586
86
2048
614
0.3
5010
2416
5.06
8130
0.04
920.
8226
1.69
614.
2160
43
2048
849
0.4
5010
2415
8.17
3479
0.04
790.
7999
1.93
984.
6539
44
2048
1024
0.5
5010
2413
22.0
5622
30.
0480
0.80
131.
9240
4.19
6677
2048
1229
0.6
5010
2413
13.5
7698
40.
0470
0.78
582.
0936
4.17
4972
2048
1434
0.7
5010
2413
18.4
0510
00.
0468
0.78
212.
1350
4.35
5072
2048
1536
0.75
5010
2412
22.4
3025
20.
0468
0.78
162.
1404
4.26
8081
2048
1638
0.8
5010
2411
28.3
2550
10.
0468
0.78
162.
1406
4.26
9212
2048
1843
0.9
5010
2411
38.8
0524
90.
0468
0.78
162.
1407
4.62
3429
2048
2048
1.0
5010
249
41.1
8635
00.
0468
0.78
152.
1409
4.28
3169
Page 41 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
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).
The following conclusions can be drawn from Table 35.
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. 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 accept-able score for speech quality.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
20
25
30
35
40
45
50
Compression Ratio (CR)
Sign
al R
econ
stru
ctio
n Ti
me
(Sec
onds
) Beylkin MatrixVaidyanathan MatrixBattle1 MatrixBattle3 MatrixBattle5 Matrix
Fig. 11 Effect of compression ratio on signal reconstruction time for Beylkin, Vaidyanathan and Battle wave-let sensing matrices
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.7
0.8
0.9
1
1.1
1.2
1.3
1.4
Compression Ratio (CR)
Sign
al R
econ
stru
ctio
n Er
ror (
Rel
ativ
e er
ror)
Beylkin MatrixVaidyanathan MatrixBattle1 MatrixBattle3 MatrixBattle5 Matrix
Fig. 12 Effect of compression ratio on relative error for Beylkin, Vaidyanathan and Battle wavelet sensing matrices
Page 42 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
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
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
Compression Ratio (CR)
Sign
al-to
-Noi
se R
atio
(SN
R) (
db)
Beylkin MatrixVaidyanathan MatrixBattle1 MatrixBattle3 MatrixBattle5 Matrix
Fig. 13 Effect of compression ratio on signal-to-noise ratio for Beylkin, Vaidyanathan and Battle wavelet sens-ing matrices
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
5
10
15
20
25
Compression Ratio (CR)
Sign
al R
econ
stru
ctio
n Ti
me
(Sec
onds
) Beylkin Matrixdb10 Matrixcoif5 Matrixsym9 Matrix
Fig. 14 Effect of compression ratio on signal reconstruction time for the best DWT based sensing matrices
Page 43 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
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).
The following conclusions can be drawn from Table 36.
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
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.78
0.8
0.82
0.84
0.86
0.88
0.9
0.92
0.94
0.96
0.98
Compression Ratio (CR)
Sign
al R
econ
stru
ctio
n Er
ror (
Rel
ativ
e er
ror)
Beylkin Matrixdb10 Matrixcoif5 Matrixsym9 Matrix
Fig. 15 Effect of compression ratio on relative error for the best DWT based sensing matrices
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
Compression Ratio (CR)
Sign
al-to
-Noi
se R
atio
(SN
R) (
db)
Beylkin Matrixdb10 Matrixcoif5 Matrixsym9 Matrix
Fig. 16 Effect of compression ratio on signal-to-noise ratio for the best DWT based sensing matrices
Page 44 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 28
Per
form
ance
ana
lysi
s of
the
rand
om G
auss
ian
sens
ing
mat
rix
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ra
tio (C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2421
1.61
4613
0.05
290.
8846
0.66
671.
0967
33
2048
410
0.2
5010
2422
3.10
1767
0.05
060.
8459
1.45
260.
7480
22
2048
512
0.25
5010
2421
4.63
9830
0.04
880.
8163
1.68
021.
9425
04
2048
614
0.3
5010
2420
5.06
7599
0.04
880.
8149
1.81
492.
1425
37
2048
849
0.4
5010
2421
8.24
0858
0.04
760.
7949
2.05
164.
9093
29
2048
1024
0.5
5010
2420
12.7
9205
70.
0470
0.78
472.
1202
11.6
3578
9
2048
1229
0.6
5010
2421
17.3
7562
60.
0468
0.78
172.
1431
14.0
8166
9
2048
1434
0.7
5010
2422
23.4
7232
30.
0468
0.78
162.
1388
34.6
3169
3
2048
1536
0.75
5010
2424
33.4
7127
90.
0468
0.78
162.
1416
39.1
9467
6
2048
1638
0.8
5010
2427
38.9
0717
60.
0468
0.78
162.
1408
43.4
7618
5
2048
1843
0.9
5010
2423
41.1
5626
10.
0468
0.78
162.
1409
51.0
9675
3
2048
2048
1.0
5010
249
22.1
2998
80.
0468
0.78
152.
1409
57.7
5539
8
Page 45 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 29
Per
form
ance
ana
lysi
s of
the
rand
om U
nifo
rm s
ensi
ng m
atri
x
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n
time
(s)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2428
2.23
5539
0.06
341.
0600
1.02
960.
2236
25
2048
410
0.2
5010
2425
3.60
0845
0.05
370.
8966
0.81
520.
7385
91
2048
512
0.25
5010
2425
5.74
5004
0.05
160.
8616
0.86
981.
9172
57
2048
614
0.3
5010
2425
6.43
4319
0.04
960.
8281
1.69
512.
1138
97
2048
849
0.4
5010
2423
9.44
3467
0.04
740.
7924
1.98
105.
3320
54
2048
1024
0.5
5010
2423
15.9
8657
80.
0470
0.78
572.
0911
11.9
7539
5
2048
1229
0.6
5010
2424
19.7
2746
40.
0468
0.78
232.
1297
14.1
6544
0
2048
1434
0.7
5010
2424
25.9
2022
90.
0468
0.78
162.
1401
34.9
6894
9
2048
1536
0.75
5010
2423
72.4
8138
30.
0468
0.78
162.
1407
48.4
0863
1
2048
1638
0.8
5010
2420
29.1
9373
70.
0468
0.78
162.
1407
93.4
0706
0
2048
1843
0.9
5010
2417
32.8
2170
00.
0468
0.78
162.
1408
51.4
3767
7
2048
2048
1.0
5010
249
22.3
9544
00.
0468
0.78
152.
1409
57.8
0310
1
Page 46 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 30
Per
form
ance
ana
lysi
s of
the
rand
om H
adam
ard
sens
ing
mat
rix
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2424
2.16
0564
0.05
430.
9068
0.68
160.
1752
90
2048
410
0.2
5010
2421
3.35
2213
0.05
080.
8486
1.50
020.
1956
49
2048
512
0.25
5010
2421
5.24
5114
0.04
880.
8162
1.79
100.
2272
24
2048
614
0.3
5010
2414
4.05
9570
0.04
760.
7959
1.78
400.
2037
26
2048
849
0.4
5010
2414
7.10
3978
0.04
830.
8075
2.15
090.
2385
74
2048
1024
0.5
5010
2417
12.4
2031
20.
0469
0.78
382.
1742
0.31
2728
2048
1229
0.6
5010
2420
18.2
0294
10.
0468
0.78
142.
1341
0.26
2874
2048
1434
0.7
5010
2424
28.3
3006
30.
0468
0.78
162.
1400
0.28
1225
2048
1536
0.75
5010
2421
32.0
1789
30.
0467
0.78
052.
1406
0.40
0085
2048
1638
0.8
5010
2423
47.0
9144
90.
0467
0.78
032.
1407
0.30
4155
2048
1843
0.9
5010
2426
70.6
8479
80.
0467
0.78
052.
1408
0.31
8110
2048
2048
1.0
5010
249
31.9
5310
30.
0468
0.78
152.
1408
0.46
3961
Page 47 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 31
Per
form
ance
ana
lysi
s of
the
rand
om T
oepl
itz
sens
ing
mat
rix
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2421
1.75
6908
0.05
360.
8957
0.91
040.
4096
20
2048
410
0.2
5010
2422
3.23
3689
0.04
980.
8315
1.54
690.
4299
34
2048
512
0.25
5010
2421
4.78
8428
0.04
930.
8232
1.73
660.
4695
36
2048
614
0.3
5010
2420
5.35
5458
0.04
830.
8068
1.80
570.
4501
23
2048
849
0.4
5010
2420
8.42
9285
0.04
740.
7918
2.09
180.
4716
07
2048
1024
0.5
5010
2420
13.4
3943
30.
0469
0.78
412.
1270
0.54
4915
2048
1229
0.6
5010
2421
17.9
6135
40.
0467
0.77
972.
1501
0.50
3870
2048
1434
0.7
5010
2421
23.1
6253
70.
0467
0.78
062.
1523
0.52
4939
2048
1536
0.75
5010
2423
32.3
1277
60.
0467
0.78
062.
1511
0.62
6275
2048
1638
0.8
5010
2424
35.3
2530
30.
0467
0.78
072.
1490
0.54
9642
2048
1843
0.9
5010
2427
49.5
9110
00.
0467
0.78
122.
1453
0.56
2993
2048
2048
1.0
5010
246
18.3
5195
40.
0468
0.78
152.
1409
0.69
5494
Page 48 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 32
Per
form
ance
ana
lysi
s of
the
rand
om C
ircu
lant
sen
sing
mat
rix
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2426
2.50
7042
0.05
970.
9968
0.42
222.
7571
10
2048
410
0.2
5010
2424
3.72
9785
0.05
100.
8516
1.56
731.
7551
48
2048
512
0.25
5010
2422
5.27
8771
0.04
910.
8206
1.72
781.
8192
34
2048
614
0.3
5010
2419
5.23
7320
0.04
800.
8021
1.95
821.
8136
63
2048
849
0.4
5010
2418
8.22
0705
0.04
700.
7846
2.10
301.
8456
29
2048
1024
0.5
5010
2417
11.9
1656
20.
0468
0.78
252.
1335
1.91
7508
2048
1229
0.6
5010
2416
14.6
5670
40.
0468
0.78
162.
1404
1.85
7957
2048
1434
0.7
5010
2415
17.5
7700
90.
0468
0.78
162.
1408
1.85
2721
2048
1536
0.75
5010
2416
23.8
6419
50.
0468
0.78
162.
1408
2.02
0327
2048
1638
0.8
5010
2415
23.4
5197
10.
0468
0.78
162.
1409
1.89
3177
2048
1843
0.9
5010
2412
25.9
5422
90.
0468
0.78
152.
1409
1.91
4215
2048
2048
1.0
5010
248
24.1
9048
00.
0468
0.78
152.
1409
2.06
5995
Page 49 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 33
Per
form
ance
ana
lysi
s of
the
Det
erm
inis
tic
DCT
sen
sing
mat
rix
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ratio
(C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2415
1.33
6874
0.05
700.
9532
0.41
650.
0328
36
2048
410
0.2
5010
2416
2.30
9603
0.05
370.
8974
0.94
050.
0604
71
2048
512
0.25
5010
2417
3.77
3514
0.05
110.
8543
1.36
820.
1104
42
2048
614
0.3
5010
2416
4.12
2633
0.05
100.
8525
1.38
580.
0871
57
2048
849
0.4
5010
2414
5.82
2766
0.04
940.
8259
1.66
170.
1177
17
2048
1024
0.5
5010
2414
9.05
8112
0.04
780.
7981
1.95
920.
2130
83
2048
1229
0.6
5010
2415
12.3
6980
90.
0472
0.78
892.
0593
0.17
0241
2048
1434
0.7
5010
2413
14.3
2483
60.
0469
0.78
322.
1224
0.19
8500
2048
1536
0.75
5010
2413
18.2
3687
80.
0468
0.78
252.
1303
0.32
3367
2048
1638
0.8
5010
2413
18.6
0074
50.
0468
0.78
202.
1363
0.22
5549
2048
1843
0.9
5010
2412
21.4
6199
50.
0468
0.78
162.
1409
0.25
3978
2048
2048
1.00
5010
249
21.8
9347
60.
0468
0.78
152.
1409
0.43
8310
Page 50 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
Tabl
e 34
Per
form
ance
ana
lysi
s of
the
Det
erm
inis
tic
Spar
se B
inar
y se
nsin
g m
atri
x
Leng
th o
f si
gnal
(N)
Num
ber o
f m
easu
rem
ents
(m
)
Com
pres
sion
ra
tio (C
R =
m/N
)Sp
arsi
ty
leve
l = (k
/N) ×
10
0 (%
)
No.
of n
on‑
zero
s (k
)N
o. o
f ite
ratio
ns
requ
ired
Sign
al
reco
nstr
uctio
n tim
e (s
)
RMSE
Rela
tive
er
ror
SNR
(db)
Cons
truc
tion
time
for s
ensi
ng
mat
rix
(s)
2048
205
0.1
5010
2412
0.92
780.
0548
0.91
540.
7678
66.9
190
2048
410
0.2
5010
2413
1.88
710.
0503
0.84
051.
5090
207.
1541
2048
512
0.25
5010
2414
3.14
310.
0497
0.83
081.
6102
380.
2697
2048
614
0.3
5010
2415
4.83
550.
0487
0.81
461.
7809
502.
1809
2048
849
0.4
5010
2419
8.38
950.
0473
0.79
062.
0411
878.
8051
2048
1024
0.5
5010
2420
18.3
753
0.04
690.
7843
2.11
0312
71.5
155
2048
1229
0.6
5010
2423
25.4
419
0.04
680.
7824
2.13
1318
69.4
049
2048
1434
0.7
5010
2430
42.2
472
0.04
680.
7816
2.14
0729
13.4
315
2048
1536
0.75
5010
2428
57.3
386
0.04
680.
7816
2.14
0829
84.3
214
2048
1638
0.8
5010
2424
52.8
342
0.04
680.
7816
2.14
0935
75.2
037
2048
1843
0.9
5010
2417
50.7
545
0.04
680.
7816
2.14
0940
30.6
333
2048
2048
1.0
5010
246
26.8
845
0.04
680.
7815
2.14
0958
56.4
331
Page 51 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
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. 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. 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 respec-tively) compared to other proposed and state-of-the art sensing matrices.
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.
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
10
20
30
40
50
60
70
80
Compression Ratio (CR)
Sign
al R
econ
stru
ctio
n Ti
me
(Sec
onds
) Random Gaussian MatrixRandom Uniform MatrixRandom Toeplitz MatrixRandom Circulant MatrixRandom Hadamard MatrixDeterministic DCT MatrixSparse Binary MatrixProposed sym9 Matrix
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
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10.75
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
Compression Ratio (CR)
Sign
al R
econ
stru
ctio
n Er
ror (
Rel
ativ
e er
ror)
Random Gaussian MatrixRandom Uniform MatrixRandom Toeplitz MatrixRandom Circulant MatrixRandom Hadamard MatrixDeterministic DCT MatrixSparse Binary MatrixProposed sym9 Matrix
Fig. 18 Effect of compression ratio on relative error for the proposed sym9 matrix and state-of-the-art ran-dom and deterministic sensing matrices
Page 52 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
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 com-pared 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:
where X = {x1, x2,…,xN} is a set of random variable, P(xi) is a probability of random vari-able xi 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 informa-tion 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.
The following observations are evident from Table 37.
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 compres-sion technique (H = 9.7573).
2. It is also evident that for the proposed sensing matrices the entropy of the recon-structed speech signal (H = 11.0) is very close to the original signal entropy (H = 10.2888).
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.
(22)H(X) = −
N∑
i=1
P(xi) logP(xi)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0.5
1
1.5
2
2.5
3
Compression Ratio (CR)
Sign
al-to
-Noi
se R
atio
(SN
R) (
db)
Random Gaussian MatrixRandom Uniform MatrixRandom Toeplitz MatrixRandom Circulant MatrixRandom Hadamard MatrixDeterministic DCT MatrixSparse Binary MatrixProposed sym9 Matrix
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
Page 53 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
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, respec-tively) shows the higher entropy compared to the sparse binary (H = 0.0659) and the random Hadamard sensing matrices (H = 1).
Table 35 Subjective evaluation of speech quality using Mean Opinion Score (MOS) test
Sr.no.
Sensingmatrix
Listeners MOSScore
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. Randomuniform
3 4 4 3 4 3 4 3 4 4 3.6
32. RandomToeplitz
4 4 3 3 4 3 4 3 4 4 3.6
33. RandomCirculant
4 4 4 3 4 4 4 4 4 4 3.9
34. RandomHadamard
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
Page 54 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
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.
Table 36 Objective evaluation of speech quality using measures such as Perceptual Evalu‑ation of Speech Quality (PESQ), Log‑Likelihood Ratio (LLR) and Weighted Spectral Slope (WSS)
Sr.no.
Different sensingmatrices
Speech distortion
Background distortion
Overall quality
LLR WSS PESQMOSScore
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. Randomuniform
3.3255 2.7452 2.8794 0.966803 28.298916 2.4577
32. RandomToeplitz
2.6847 2.6565 2.5249 1.520660 33.026663 2.4108
33. RandomCirculant
3.8147 2.7854 3.1549 0.529008 28.229330 2.5210
34. RandomHadamard
3.8147 2.7854 3.1549 0.529008 28.229330 2.4934
35. Wavelet compres-sion
3.6017 2.4841 2.9674 0.578535 39.038937 2.4135
Page 55 of 60Parkale and Nalbalwar SpringerPlus (2016) 5:2048
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 spec-trogram of the reconstructed speech signal.
Thus, the spectrographic analysis from Fig. 20 shows that the time–frequency charac-teristic of the reconstructed spectrogram is a very close to the original speech spectro-gram, preserving most of the signal energy. Moreover, the red color shows energy at the
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. Daubechieswavelet 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 –
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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 sens-ing matrix.
ConclusionsIn 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, Bat-tle, 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
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 140
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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 recon-structed speech signal
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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 informa-tion 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 sig-nal reconstruction error and signal reconstruction time compared to other Daube-chies wavelet based sensing matrices. Moreover, the db9 also shows close perfor-mance 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 com-pared 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 sec-ond 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.
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Original Speech signalReconstructed Speech signal
Fig. 21 Original and reconstructed speech signal for proposed sym9 wavelet based sensing matrix (for CR = 0.5)
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• 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 sens-ing 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 accept-able 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 com-pared 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, includ-ing 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 inves-tigating 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.
AbbreviationsDWT: 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.
Authors’ contributionsYVP have made substantial contributions to design and development of DWT based sensing matrices and their applica-tion 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.
AcknowledgementsThe 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 interestsThe authors declare that they have no competing interests.
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Availability of data and materialsAll datasets on which the conclusions of the manuscript are rely and the data supporting their findings are presented in the main paper.
FundingThe authors declare that they have no funding provided for the research reported in this paper.
Received: 25 June 2016 Accepted: 25 November 2016
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