International Journal of Engineering Trends and Technology (IJETT) – Volume 14 Number 3 – Aug 2014
Sub Band Coding of ECG Signal via Quantized
Coefficient QMF Bank
Sachin Bhaiji1, Jyotsna ogale2
1
2
PG Student, Department of Electronics and Communication Engineering, S.A.T.I.(Degree), Vidisha,(M.P.)
Associate Professor, Department of Electronics and Communication Engineering, S.A.T.I. (Degree), Vidisha (M.P.)
Abstract— This paper presents a comparative study of Blackman
window family for the design of near perfect reconstruction
(NPR) quadrature mirror filter bank (QMF) bank for subband
coding of ECG signal. The design method employs Blackman
window family and rounding operation to generate quantized
coefficients of the prototype low pass finite impulse response
filter (FIR). To get optimum result in term of distortion
parameter, linear algorithm is used . The designed filter bank is
computationally more efficient and has good frequency
selectivity. Future its application is extended to subband coding
of ECG signal.
Keywords— Filter bank,
optimization, rounding.
near
perfect
reconstruction,
I. INTRODUCTION
Processes in the design of filter banks and their applications
have been made since last two decades. Among different
family of filter banks quadrature mirror filter bank (QMF) a
two-channel filter bank as shown in Fig. 1 was the first type of
filter bank used in signal processing for splitting the speech
and image signal into subband signal [1-4] with uniform
frequency bands so that each subband can be independently
carried out and processed .
The researchers now a days give a lot of attention to design
the QMF bank because of there wide application in many
signal processing fields such as filter banks with adjustable
stop band attenuation and efficient resolution are useful for
analysing ECG signal of different patients[5], .in trans
multiplexers [6-8],equalization of wireless communication
channels[9],perceptual audio coding and filter bank with high
stopband attenuation ,small channel overlap and efficient
resolution can improve sound quality.
( )
2
2
( )
in this paper, we consider the design of symmetric QMF bank
with low arithmetic complexity at both analysis and synthesis
section. This filter bank comprised of a pair of lowpass and
high pass filters, which satisfy some complementary
properties. By choosing proper combinations of the filters at
both analysis and synthesis section aliasing can be eliminated
completely. Only distortion at the output is in amplitude. The
early phase of research and design methods were based on
direct minimization of error function either in frequency
domain [10] or in time domain [11].In this paper we use a
simple iterative linear algorithm to design parent filter of the
filter bank. Future the application of design filter bank has
been extended to subband coding of ECG signal.
In the analysis section, filters are used to split the signal into
two equal-width frequency segments, the resulting signal are
decimated by factor of two which reduces the total rate by 2.
At the receiver, the decimated signal are interpolated and
recombined such that, theoretically there is no aliasing.
Hence, the channel signals can be processed at a half of the
original signal sampling rate.
Section II describes different performance parameters of the
QMF bank. Section III describes design method of prototype
quantized coefficients filter. Section IV describes optimization
algorithm .Section V describes quality assessment parameters
of the ECG signal. Section VI carries design examples with
discussion. Finally conclusions have been made in section VII.
The rounding technique is applied on window based FIR filter
to satisfy the given specification. The technique is described
in the equation
II. PERFORMANCE PARAMETER OF QMF BANK
Consider two-band QMF bank with system architecture as
shown in Fig (1). The Reconstructed output signal is given as
y[n] ( ) = [ ( ) ( ) + ( ) (− )]
x[n]
( )
2
2
( )
Fig.1 Quadrature mirror filter bank
Consequently, fast flexible and efficient filter bank design
method that yield low complexity high stop band attenuation
and channel overlap are highly desirable.
ISSN: 2231-5381
( ) = {
( ) = {
( )
(− )
( )+
( )+
(1)
( )
( )}(2)
(− )
( )} (3)
Where Y(z) is the reconstructed signal and X(z) is the
original signal in QMF bank , T(z) is called the distortion
http://www.ijettjournal.org
Page 135
International Journal of Engineering Trends and Technology (IJETT) – Volume 14 Number 3 – Aug 2014
transfer function and A(z) is called the aliasing error function.
Which is completely eliminated by setting as Eq.(4).
( ) = −2
(− ),
( )=2
( )(4)
For aliasing free function ( ) = (− ) only one proto
( ) exists in the system response which state
type filter
that the overall w=
design problem of filter banks reduces
to determine of the filter type coefficient of prototype filter.
The overall transfer function of QMF bank is reducing to
equation.
(
) = (
) − (−1)
(
)
(5)
The Blackman window family is used in the work with its
variant. Window with 3 and 4 non zero term achieve a
minimum side lobe are called as Blackman Harris Window
Family These window w[n] are define for the DFT [15]-[17]
3
( )= 0− 1
(6 / ).
| ( )| = |
( )| + |
( − )| (6)
The amplitude distortion ( ) and peak reconstruction error
(PRE )are computed by the respective Eq.10 and Eq. 11.
= max | ( )| − min | ( )|(7)
PRE = max {20log (
(
)
+
(
) } (8)
The performance of proposed method is evaluated in term of
amplitude distortion (
) and fidelity assessment parameter
of ECG signal.
III. DESIGN METHOD OF PROTOTYPE FILTER
The filter design technique employs window technique, a
finite duration weighting function that called a window
function w[n], which satisfying w(N-n)=w(n)
for n =
0,1 … … … N and exactly zero outside the interval − ≤ ≤
. A prototype filter h[n] of length N, and cut off frequency
is generated by convolving window function w(n) with
ideal impulse response of the filter given as
h(n) = h (n)w(n)(9)
where
h (n) =
sin wc n−0.5N
π n−0.5N
(10)
are the impulse response value of ideal filter with cut off
frequency located at = .
(11)
TABLE I
Window
Coefficients
a2
a0
a1
7938/1
8608
0.42
9240/1860
8
0.50
7938/186
08
0.08
0.00010
Modified
Blackman 1
0.5600
0
0.44000
0.56000
0.01000
modified
Blackman2
0.5000
0.46000
0.03000
0.01000
BlackmanNuttell
0.3635
8
0.4495
9
0.4232
3
0.4021
7
0.3587
5
0.48918
0.36358
0.01064
0.49364
0.44959
0.00000
0.49755
0.42323
0.00000
0.49703
0.09392
0.01168
0.48829
0.35875
0.01168
Exact
Blackman
BlackmanHarris(-61)
BlackmanHarris(-67)
BlackmanHarris(-74)
BlackmanHarris(-92)
a3
0.00
Then coefficients h(n) are quantized using rounding
operation as described below –
h (n) =
×
( )=
×
(ℎ( )/ )
(12)
Where h(n) is an impulse response of the FIR filter which
satisfies the given specification ( ) is the new impulse
response derived by rounding all the coefficients of h(n) to
the nearest integer .The rounding coefficient is chosen in the
form of = 2
where N is the integer,determine the
precision of the approximation. This process introduce some
null coefficients in the rounding impulse response the number
of non zero integer coefficients corresponds to the number of
sums and the number of multiplier corresponds to number of a
different positive integer coefficients . Computational
complexity is expressed in terms of number of integer
multiplication, which itself depends on rounding constant.
Now the final filter coefficients are h (n)
OPTIMIZATIONOF ERROR FUNCTION
In QMF Filter bank, perfect reconstruction (PR) is possible if
H (e )
ISSN: 2231-5381
/ ) + 2 (4 / ) −
n = 0,1,……… N-1
COEFFICIENTS OF BLACKMAN WINDOW FAMILY
Black man
It is mandatory N is taken as even and transfer function must
be a delay function (
),If the order of the filter is odd at
w=0.5π ( ) is reduced to zero which is not adaptive for
the perfect reconstruction of the signal. If the perfect
reconstruction condition are satisfied, the reconstructed output
signal is an exact replica of the original input is signal with
some delay, that is ( ) = ( − ) output. The overall
amplitude response of filter bank is
(2
http://www.ijettjournal.org
+ H (e (
/
) =1
(13)
Page 136
International Journal of Engineering Trends and Technology (IJETT) – Volume 14 Number 3 – Aug 2014
If it is evaluated at w=pi/4, then it’s reduced to
H (e
/
Step9: Step size = Step size/2 till the tolerance is not satisfied .
) = 0.707
(14)
IV. QUALITY ASSESSMENT PARAMETER OF ECG SIGNAL
Hence design problem of QMF filter bank is condensed to
design a prototype filter whose magnitude response at
frequency w=pi/4 is 0.707 .in the proposed methodology , cutoff frequency is optimized to minimized the error
function .this can be accomplished by solving the optimization
problem
= [ H
e
− 1/√2]
(15)
The main step discusses the computational process of the
proposed algorithm:
step1: initialize the parameter ‘ρ’ is called the roll of factor
(RF) .
step2: calculate the stop band frequency and pass band edge
frequency is set slightly smaller than
.
= (1+ρ)F
8
(16)
step3: initialize the counter ,tolerance ,step size and stop band
attenuation .
step4: Estimate the order of the filter and cut-off frequency by
using given specification.
=
−7.95
14.95⧍
=
=
(fs +fp)
2
(18)
− 0.707
(19)
step5: if tolerance (tol.) is not satisfied, then cut-off frequency
is varied in two ways:
a.
b.
If error is positive , increase
Otherwise = −
=
+
Step6: If tolerance is satisfied then, design filters using
equation.
Step7: Redesign the prototype filter using new
filter order. Calculate F and also Error.
Step8: Increment the counter by 1.
ISSN: 2231-5381
=
∑ =1( [ ]− [ ])2
2
∑ =1( [ ])
× 100
(20)
Mean square value Error (MSE) is also one of the important
parameter to evaluate the quality of reconstructed signal . it is
expressed as :
= ∑
( [ ] − [ ])
(21)
The signal to noise ratio
= −20 log(0.01
1)(22)
Where
1 =
∑
∑
( [ ]
( [ ]
[ ])
[ ])
× 100(23)
(17)
step5: evaluate the prototype filter coefficients using
Blackman window family with
intial iteration algorithm
start with finding the magnitude response of design filter at
w=pi/4 . Also compute error or deviation of magnitude
response of designed ( / ) filter from the ideal magnitude
response (MR) given by equ (12).
1. Performance parameters
The quality of retrieved signal is measured using the
Percentage Root Mean-square Difference (PRD), which is
define as :
and same
V. DESIGN EXAMPLE
In this section design of the two-bands linear phase QMF
bank with Blackman window family and with and without
coefficients quantization have been considered for the given
value, roll of factor = 0.20, the pass band and stop band
frequency are chosen to be 0.60π and 0.25π respectively. The
filter with 32 filter order cut off frequencies 0.5333π, 0.5328π,
0.5247π,
0.5380π
,0.5380π
0.53450π,
0.5385π,0.5311π,0.5330π have been designed for Exact
Blackman, Blackman, Modified blackman1, modified
blackman2,
Blackman-Nuttell,
Blackman-Harris(-1),
Blackman-Harris(-67), Blackman-Harris(-74), BlackmanHarris(-92). The simulation result obtained in each case is
tabulated in table II table III. As it can be seen from the
simulation result the proposed method yields better
performance in term of
and PRE at redused arithmetic
complexity then the previously reported works. Numbers of
multipliers
and
adders
in
algorithms
[26],[27],[28],[29],[30],[31] are 16 and 32 as compared to 14
and 30 with two zero coefficients in the proposed work this
clearly shows the seniority of the proposed work over the
existing work . When different Blackman window function
are compared, the proposed method with Blackman window
gives betters performance in terms of and PRE with and
without coefficients quantization. The Blackman harries 4-
http://www.ijettjournal.org
Page 137
termalso competed value .In subband coding of ECG signal
several records have been taken from MIT BIH data base.
The quality of reconstructed ECG signal are evaluated by
considering fidelity assessment parameters discuss in section
().it is observed that in both the designs Blackman window
function offered superior reconstructed signal quality , due to
their better performance measures .
Magnitude
International Journal of Engineering Trends and Technology (IJETT) – Volume 14 Number 3 – Aug 2014
VI. RESULT AND CONCLUSION
An improved method is presented for the design of twoband QMF bank with Blackman window family, coefficients
quantization and linear iterative optimization algorithmSimulation result have shown
.
N
PRE
Algorithm
in[26]
32
35.29
Algorithm
in [27]
32
36.87
Algorithm
in [28]
32
38.67
Algorithm
in[29]
32
35.67
Algorithm
in [30]
32
36.59
Algorithm
in [31]
32
34.38
Proposed
32
78.32
9.60×
10
7.72
× 10
6.60
× 10
4.60
× 10
4.50×
10
4.10×
10
1.10×
10
0..270
0.0223
Normalized Frequency
0.0196
(c)
0.0114
0.0102
0.0089
0.0095
Magnitude
Reported
result
Magnitude
TABLEII
RELATIVE PERFORMANCE OF PREPOSED WITH RESPECT TO OTHER ALGORITHM.
Normalized Frequency
(b)
That Blackman window function yields smallest amplitude
distortion, peak reconstruction error , MSE, PRD and good
SNR at lowest arithmetic complexity .Therefore the proposed
method is highly suitable for subband coding of ECG signal
with quantized coefficient .
Rounding factor
Magnitude
Magnitude
(d)
Rounding factor
(e)
Normalized Frequency
(a)
Fig.1 Two-channel QMF bank by proposed method with N=32 and = 0.2.
(a) Amplitude distortion functions of Blackman window. (b) Magnitude
response of prototype filter (c) magnitude response of QMF filters (d)
variation in PRD with respect to Rounding Factor.(e)variation in MSE with
respect to Rounding Factor
1. Subband coding of ECG signal
Record have been taken from MIT BIH data base. The quality
of reconstructed ECG signal is evaluated by considering
several fidelity assessment parameters. Blackman window
ISSN: 2231-5381
http://www.ijettjournal.org
Page 138
International Journal of Engineering Trends and Technology (IJETT) – Volume 14 Number 3 – Aug 2014
give the improved performance as compared to the other
variants. Hence it offers superior reconstruction quality.
Amplitude
TABLE.IV
.PERFORMANCE MEASURES OF VARIOUS WINDOWS WITH ROUNDING
CONSTANT.
Sample index
Fig:-2 superimposed original and reconstructed signal
TABLE. III
PERFORMANCE MEASURES OF VARIOUS WINDOWS
window
Exact
Blackman
Blackman
Modified
Blackman-1
Blackman Nuttall
Blackman –
Harries (-74)
Blackman –
Harries (-92)
Blackman –
Harries (-67)
Blackman –
Harries (-61)
Modified
Blackman2
Fig:- Computational complexity of design 32 order filter bank
in term of no of multiplication and no adders for different
value of rounding constant.
MSE
7.377e09
1.701e09
34.26e09
5.904e08
4.267e09
7.597e08
5.001e09
4.0078
e-09
4.8525
e-008
Maxim
um
Error
PRD
SNR
Amplitude
Distortion
3.2658e004
1.4642e004
1.9834e004
7.0724e004
1.9834e004
8.0124e004
2.0889e004
2.2997e004
7.5432e004
0.0550
65.166
0.0032
0.0264
71.536
0.0011
0.0418
67.544
0.0057
0.1556
56.133
0.0062
0.0418
67.544
0.0015
0.1765
55.038
0.0071
0.0453
66.854
0.0024
0.0405
67.816
0.0023
0.1410
56.985
0.0067
VII CONCLUSION
A improved method is presented for design of coefficient
quantized quadrature mirror filter bank by using Blackman
window families foe subband of ECG signal .The result
better in term of fidelity assessment parameters.
Reconstructed signal show no significant loss in
diagnostically important features and morphologies. Hence
the proposed design of QMF banks gives simple and
straightforward design with low computational burden.
window
Roun
dingconst
ant
MSE
Maxi
mum
Error
PRD
Exact
Blackman
18
5.2570
e-009
00.0464
0.0028
Blackman
18
1.6512
e-009
0.0260
0.0011
Modified
Blackman-1
17
1.9371
e-007
2.573
8e004
1.472
9e004
0.002
3
0.2818
0.0057
Blackman Nuttall
18
5.8892
e-008
0.1554
0.0062
Blackman –
Harries (-74)
19
3.7583
e-009
0.0393
0.0015
Blackman –
Harries (-92)
18
7.6783
e-008
0.1774
0.0071
Blackman –
Harries (-67)
18
3.9975
e-009
0.0405
0.0024
Blackman –
Harries (-61)
18
3.5211
e-009
0.0380
0.0019
Modified
Blackman2
19
4.8509
e-008
7.074
9e 004
1.758
7e004
7.829
4e004
2.112
8e004
2.267
7e004
7.541
e-004
0.1410
0.0067
REFERENCES
[1]
[2]
[3]
[4]
Number
[5]
[6]
[7]
[8]
Rounding constant
ISSN: 2231-5381
Amplitu
de
Distortio
n
[9]
D.Esteban, C. Galand, Application of quadrature mirror filter to split
band voice coding scheme, in : Proc. IEEE International Conference on
Acoustics, Speech, and Signal Processing (ASSP), 1977,pp. 191-195.
R.E.Crochiere, Sub-band coding, Bell Syst.Tech.J.9 (1981) 1633-1654.
M.J.T. Smith .S.L.Eddins ,Analysis/synthesis techniques for sub-band
image coding , IEEE Trans. Acoust.Speech Signal Process. ASSP-38
(8) (1990) 1446-1456.Signal process.
F. Delgosa, F.Fekri, Results on the factorization of multidimensional
matrices for paraunitary filter banks over the complex field ,IEEE
Trans. Signal Process. 52 52 (5) (2004) 1289-1303.
M. Vetterli, Multidimensional sub-band coding: Some theory and
algorithms, Signal Process. 6 (1984) 97–112.
J.W. Woods, S.D. O’Neil, Sub-band coding of images, IEEE Trans.
Acoust. Speech Signal Process. ASSP-34 (10) (1986) 1278–1288.
Valtino X. Afonso, Member, IEEE, Willis J. Tompkins,* Fellow, IEEE,
“ ECG Beat Detection Using Filter Banks’’1999.
M.G.Bellanger , J.L. Daguet. TMD-FDM transmuitiplexer : Digital
Polyphase and FFT , IEEE Trans. Commun. 22 (9) (1974) 1199-1204.
H. Scheurmann, H. Gockler, A comprerative survey on digital transmultiplexers method ,Proc. IEEE 69(11) (1981) 1419-1450
http://www.ijettjournal.org
Page 139
International Journal of Engineering Trends and Technology (IJETT) – Volume 14 Number 3 – Aug 2014
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
M.Vetterly, perfect transmultiplexer, in : Proc IEEE International
Conference on Acoustics ,Speech and Signal Processing, , vol4, 1986,
pp. 2567-2570.
G. Gu., E.F. Barden , Optional design fo channel equalization visa the
filter bank approach ,IEEE Trans. Signal Process.52 (6) (2004) 536544;
Johnston ,J.D:A filter family designed for use in quadrature mirror
filter banks , In: Proceedings IEEE international conference on
acoustics ,speech ,and signal processing ,April 1980,pp.291-294(1980).
Jain V.K., Crochiere , R.E.: “Quaderature mirror filter design in time
domain” . IEEE Trans .acoust Speech Signal Processing .ASSP 32(2),353-361 (1984). 1170–1191.
Datar ,A,. Jain, A,. Sharma, P.,C.: Performance of Blackman window
family in M-channel cosine modulated filter banks for ECG signal ,
international Conf. on Multimedia , Signal Modulated , Signal
Processing and Communication technologies (IMPACT09), pp . 98101(2009) .
F.j.Harries “On the use of window for harmonics analysis with the
Discrete Fourier Transform, “Proc. IEEE , vol 66,no.1 pp 51-83
January .
Digital Signal Process. 45 (7) (1998) 922–927.Ashraf M. Aziz
“ Subband Coding of Speech Signals Using Decimation and
Interpolation’’,2009.
A.H. Nuttel : “some window with very good sidelobe behavior IEEE
Trans. Acoustics , Speech ,and Signal Processing , vol ASSI 29 ,
no1 ,pp, 84-91, February 1981.
P. Martin Martin, F. Cruz-Rodman, and T. Sarawak, “A new window
for the design of cosine modulated multirate systems,” Proc.Int. Symp.
Circuits and Systems (ISCAS’04) 2004, vol. 3,pp. 529-532, Vancouver,
Canada, May 2004.
Vaidyanathan.P,,Multiand and filter banks , Englewood Cliffs , NJ :
Prentice-Hall .1993.
C.K. Chen and J.H. Lee , “Design of quaderature mirror filter with
linear phase in the frequency domain” IEEE Transsaction on Circuit
and System II , VOL 39, no 9 ,September 1992,pp.593-605.
ISSN: 2231-5381
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
M. Bhattacharya and T, Saramaki, “Some observations leading to
multiplier less implementation of linear phase filters,” Proc. ICASSP,
pp.517-520, 2003.
A. Bartolo, B. D. Clymer, R. C. Burges, and J.P. Turnbull,“An efficient
method of FIR filtering based on Impulse responserounding,” IEEE
Trans. on Signal Processing, vol. 46, No. 8, pp.2243-2248, August
1998.
Tomasz Rusc ,Ulrich Heute , ”Nearly–Perfect Reconstruction filter
banks with critical sampling achievable quality with quantized
coefficients .
John G. Proakis and Dimitris G. Manolakis. Digital Signal Processing,
Principles, Algorithms, and Applications. Prentice Hall. New Jersey,
2008.
A. Croisier ,E Esteban ,and c. galand “perfect channel splitting by use
of interpolation /decimation /tree decomposition technique’’ int. symp
info circuit and sustem ,patras grecce,1976.
O.P. Sahu M.K. Soni,I.M. Talwar,Marquardt optimization method to
design of multidimensional quardrature mirror filter banks ,Digital
Signal Process. 16(6) (2006)
J.Upendra ,C>P. Gupta ,G.K. Singh , Design of two –channel
quadrature mirror filter bank using partial swarm optimization ,Digital
Signal Proceds.. .20 (2010 304-313) .
A.Ghose ,R Giri A Chowdhury ,S, Das< a. Abraham ,Two –channel
quadrature mirror filter banks ,design using a fitness-adaptive
differencual evoiution algorithum in: 2010 Second World Congress on
nature and Biologically Inspired computing , Fukuoka , Japan , 2010
pp. 634-641.
A. Kumrar , G.K. singh, RS Anand , An improved method for
designing
quadrature mirror filter banks
via unconstructed
optimization , J. Math Model Algorithm 9 (1) (2010) 99-111.
A. Kumar G.K. Singh R.S. Anand , An improved method for the
design of V using Talwar,Marquardt optimization technique Signal
image video Process , doi: 10.100760-110-9.
A. Kumrar , S.M. Rafi ,G.K. singh , A hybride method for design
linear –phase quadrature mirror filter bank 22 (2012) 453-462.
http://www.ijettjournal.org
Page 140