Journal of Signal and Information Processing, 20 10 , 1, 63 -76
doi:10.4236/jsip.2010.11007 Published Online November 2010 (
Copyright © 2010 SciRes. JSIP
A Discrete Cosine Adaptive Harmonic Wavelet
Packet and Its Application to Signal
Nandini Basumallick, S. V. Narasimhan
Aerospace Electronics and Systems Division, National Aerospace Laboratories, Council of Scientific and Industrial Research (CSIR),
Bangalore, India.
Received October 29th, 2010; revised November 13th, 2010; accepted November 15th, 2010.
A new adaptive Packet algorithm based on Discrete Cosine harmonic wavelet transform (DCHWT), (DCAHWP) has
been proposed. This is realized by the Discrete Cosine Harmonic Wavelet transform (DCHTWT) which exploits the
good properties of DCT viz., energy compaction (low leakag e), frequency resolution and computational simplicity due
its real nature, compared to those of DFT and its harmonic wavelet version. Hence the proposed wavelet packet is ad-
vantageous both in terms of performance and computational efficiency compared to those of existing DFT harmonic
wavelet packet. Further, the new DCAHWP also enjoys the desirable properties of a Harmonic wavelet transform over
the time domain WT, viz., built in decimation without any explicit antialiasing filtering and easy interpolation by mere
concatenation of differen t scales in frequency (DCT) domain with out any image rejection filter and with out laborious
delay compensa tion required. Further, the compression by the proposed DCAHWP is much better compared to that by
adaptive WP based on Daubechies-2 wavelet (DBAWP). For a compression factor (CF) of 1/8, the ratio of the per-
centage error energy by proposed DCAHWP to that by DBAWP is about 1/8 and 1/5 for considered 1-D signal and
speech signal, respectively. Its compression performance is better than that of DCHWT, both for 1-D and 2-D signals.
The improvement is more significant for signals with abrupt changes or images with rapid variations (textures). For
compression factor of 1/8, the ratio of the percentage error energy by DCAHWP to that by DCHWT, is about 1/3 and
1/2, for the considered 1-D signal and speech signal, respectively. This factor for an image considered is 2/3 and in
particular for a textural image it is 1/5.
Keywords: Adaptive Harmonic Wavelet Packets, Discrete Cosine Transform, Signal Compression
1. Introduction
The wavelet transform (WT) provides a frequency de-
pendent resolution so that the high and low frequencies
have a coarse and fine frequency resolution. This is
based on the assumption that high and low frequencies
require, fine time resolution and coarse frequency resolu -
tion; respectively. But to achieve for high frequencies, a
finer frequency resolution and for low frequencies, a
finer time resolution; the wavelet packet (WP) system is
used [1]. It also allows flexibility of selection of wavelet
tree structure that enables the WP to select an optimum
time-frequency tiling for a given data. This is achieved
by adaptive WP and compared to the normal WT, it is
more attractive, for applications like signal compression
and transient detection [2-5]. This is because the adaptive
nature of WP facilitates better energy compaction.
Basically, the WP is a generalization of the WT. In
WT, only the coarse/approximation is split at each stage
and the detail is carried down as it is. Unlike this, in WP
at each stage, the detail/highpass filter branch is also fur-
ther split and decimated similar to the approximation/
lowpass branch. Thus the complexity of the WP algo-
rithm is more as compared to that of the conventional
WT. Hence filterbank realization of WP is cumbersome
as it involves additional filtering and sampling rate con-
To simplify the adaptive WP, a Harmonic Wavelet
Packet based on Discrete Four ier Transform (DFAHW P)
A Discrete Cosine Adaptive Harmonic Wavelet Packet and Its Application to Signal Compression
Copyright © 2010 SciRes. JSIP
has been used [4]. This is possible as the Harmonic WT
(HWT) due to its built in decimation and interpolation
and absence of any explicit antialiasing and image rejec-
tion filtering. However for such a HWT, its coefficients
not only suffer from leakage effect but also are of com-
plex nature. Further for an image (2D signal), the HWT
complex nature involves approximation which results in
neglection of the imaginary part of a coefficient once
along the rows and columns during each LL stage split-
ting [9]. These drawbacks are carried over to the
DFAHWP also. To overcome these limitations, a HWT
based on Discrete Cosine Transform (DCHWT) has been
used [8,9]. This is based on the fact that DCT provides
real coefficients, which are also less affected by leakage
[8,12]. The DCHWT has been used for computationally
efficient and better quality signal compression and sub-
band spectral estimation [9]. This has also been applied
for reducing the cross-term effect, which occurs in
Wigner-Ville distribution [10]. Further, a shift invariant
version of DCHWT has been considered for applications
like signal denoising which reduc es glitches in th e recon-
structed signals [11].
In this paper, an adaptive Harmonic Wavelet Packet
based on DCT (DCAHWP) is proposed. The method is
computationally simple, as it is based on grouping of real
DCT coefficients and the WP decomposition up to the
last level is readily available in DCT domain with out any
repeated filtering and decimation like in a time domain
filter bank. Further, the reconstruction also does not in-
volve any explicit interpolation and its associated filter-
ing. The new algorithm has a better performance than the
DFAHWP as its scales are less affected by leakage effect
and does not involve any approximation for 2D signals.
The DCAHWP has been applied both for 1-D and 2-D
signal compression and its performance is better com-
pared to that of DCHWT. Further its performance is su-
perior to that of DBAWP for 1D signals.
2. Adaptive Wavelet Packet
The WP decomposition for a signal, showing the split of
detail/ highpass branch in addition to coarse/lowpass
branch, is illustrated in Figure 1(a). For a signal of
Npoints, this generates an array of
N coefficients
is the total number of decomposition stages,
given by 2
. From this array, N coefficients
can be selected to represent the signal. These N coeffi-
cients can be that of normal WT decomposition or any
other combination.
Wavelet packet facilitates selection of an optimal basis
for a given signal. In an adaptive wavelet packet, a spe-
cific combination of all the branches is selected to repre-
sent a particular signal based on some predefined crite-
rion. The criterion generally aims at having maximum
infor mat ion wi th min imu m po ssib le nu mber of coef fic ien ts.
A Discrete Cosine Adaptive Harmonic Wavelet Packet and Its Application to Signal Compression
Copyright © 2010 SciRes. JSIP
Figure 1. (a) Wavelet packet showing the split of detail/high pass branch in addition to coarse/lowpass branch (b)
Optimum scale tree selection, (c) Optimum scale tree.
That is, it minimizes the number of nonzero coef ficients
of the resulting wavelet transform. The criterion or cost
function used in practice is the Shannon’s entropymeas-
ure, give n by
Epp p
, (1)
log 0
pp if0
Ep is the entropy for a particular scale, and
L is the number of wavelet packet coefficients in that
scale. i
p are the normalized energies of the wavelet
packet coefficients to be considered and are given by [4]
, 1
where, i
are the absolute values of wavelet packet
coefficients in that scale, 2
is the normalization fac-
tor and
n is the original signal of length N.
be the Hilbert space, vHand i
be an orthogonal decomposition of
. The entropy of
v relative to the decomposition i
is defined
[3,6] as a measure of the distance between v and the
orthogonal decomposi t i on.
vHv v
is characterized by the Shannon equation.
If ij
 
 , i
the orthogonal decompo sitions i
. Then the Shannon entropy is given by
; ,lnln
vHH vvv
  
This equation enables searching for the smallest en-
tropy expansion of a signal. The best basis algorithm
minimizes the cost function for the transform coefficients.
This involves complete decomposition according to the
wavelet packet transform.
For the wavelet packet decomposition of a signal
shown in Figure 1a, the entropies for different scales are
as given below:
The entropy for scale0
The entropy for scale1
LAj Aj
And the entropy for scale
AA k
Here 0
and AAk
are the wavelet packet
coefficients in the scales 0
A, having 0A
and AA
L samples; respectively. The entropies of
other scales can also be found in a similar manner.
The entropy will be low when energy is concentrated
only at few locations, furth er it will be zero only when all
the energy of the signal is at one coefficient. On the other
A Discrete Cosine Adaptive Harmonic Wavelet Packet and Its Application to Signal Compression
Copyright © 2010 SciRes. JSIP
hand, the entropy will be maximum when energy is
equally distributed over all coefficients. Hence the set of
N wavelet packet coefficients for which en trop y is least
provides the efficient representation of the signal. This
enables to select those coefficients set which has maxi-
mum concentration of the energy.
To choose such a set, at each scale, each pair of parti-
tioned coefficient sets is compared with those of their
parent from which the partitions are derived. If the com-
bined partition coefficients have entropy smaller than
that of their parent, the partitions are considered, other-
wise the parent.To make further comparisons while pro-
ceeding upwards (towards parent), if partitions are con-
sidered, then their entropy is assigned to their parent. In
this way the complete wave packet array is scanned and
this provides an optimal time-frequ ency tiling in terms of
location of coefficient energy. The Figure 1b shows the
method of optimum splitting for an arbitrary case. Here,
the combined entropy of S and
is less than the
signal spectrum entropy. Hence, this signal spectrum
splitting into S and
is valid. Further for S, its
splitting into SS and SA is not valid as the combined
entropy of SS and SA is greater than that of S. But
splitting of
S and
A is valid as their
combined entropy is less than that of
. Further, split of
S into 3
and 2
is not valid as their combined
entropy is larger than that of
S. But
A split into 1
and 0
is valid as their combined entropy is less than
that of
A. The optimum scale tree selected is shown in
the Figure 1c.
The decimation of the scales in the scale tree selected
depends up on the bandwidth of the individual scales
resulting in a time-frequency tiling and this may totally
differ from the normal dyadic tiling. Such a tiling results
in a maximum energy compaction for a given number of
coefficients. It is not only the time frequency tiling but
also the wavelet function used, also determines the per-
formance of compression. That is, a good wavelet for a
particular signal results in a good energy compaction
with least number of coefficients. Thus the wavelet
packet can result in an efficient lossless compression.
This type of scale tree selection is also known as best
basis selection.
3. Discrete Cosine Harmonic Wavelet
The filter bank realizatio n of WT, invo lves decimation of
the scale components. The restoration of the processed
overall spectrum corresponding to the original sampling
rate, involves interpolation and summation of the inter-
polated scale outputs in time. The harmonic wavelet
transform based on DFT (DFHWT) realizes the subband
decomposition in the frequency domain by grouping the
Fourier transform (FT) coefficients and the inverse of
these groups results in decimated signals [7]. Further
after processing, the FT of the scales can be repositioned
in their corresponding positions to recover the overall
spectrum, with the original sampling rate. Therefore, this
will not involve explicit decimation and interpolation
operations. As a consequence, no band limiting and im-
age rejection filters are necessary. Also, while recon-
struction, there are no delay compensations as the scales
are combined in the frequency domain by repositioning
them. In view of this, the harmonic subband decomposi-
tion is very attractive due to its simplicity. Further, the
decomposition being done in frequency domain, it is well
suited for those processing methods which are performed
in frequency domain, like group delay processing [8 ].
For a 1-D signal in the DFHWT, the grouping of the
DFT coefficients with possible conjugate symmetry
though makes the WT coefficients complex; this will not
pose any problem for reconstruction as after concatena-
tion of the groups, the conjugate symmetry is restored to
get the real signal.
The DFHWT is very attractive as long as no process-
ing of the components is involved prior to inverse trans-
formation. However, for a signal segment obtained
without using any window fun ctio n, ther e can be a seve re
leakage effect from one scale to another scale. If differ-
ent scales have to be processed differently, this is not
achieved as the signal energy from one to another has
already leaked. The DFHWT may be tolerable for a sig-
nal with well-separated frequency components of suffi-
ciently high magnitude. But for closely spaced compo-
nents of significantly different magnitudes, during the
computation of the FT itself, the energy will leak from
the higher amplitude component to the lower one (and
vice versa). This results in a large bias in the spectral
magnitude and may even totally eclipse smaller ampli-
tude spectral peaks. In such a case, decomposing the
signal based on DFHWT and processing the subbands
may not be very effective. Further leakage in DFHWT
will also limit its use in signal or image compression
application. The reason for this is that it is not p ossible to
get a good signal reconstruction by omitting the lower
scales (corresponding to high frequencies) in WT as the
leaked energy cannot be recovered unless all the scales
are considered
Therefore to utilize the attractive features of the har-
monic wavelet transform, DCT is used instead of DFT,
which has a comparatively reduced leakage effect. This
is due to symmetrical data extension which results in a
smooth transition from one DCT period to the other
without any discontinuity [8].
The wavelet transform
Wabcharacterizes the cor-
A Discrete Cosine Adaptive Harmonic Wavelet Packet and Its Application to Signal Compression
Copyright © 2010 SciRes. JSIP
relation or similarity between the signal
tto be ana-
lyzed and the wavelet function
. Such a
correlation is given by
 
Wab xtdt
is the prototype/mother wavelet. By shifting
and scaling
by the parameters b and a, respec-
tively; all the basis funct ions
,ab ta tba
are obtained. Eqn. (3) can be realized in the frequency
domain using Parseval’s theorem as [13]
  
1/2 *
WabXa ed
Therefore, the wavelet transform can be derived by
windowing the spectrum
inverse Fourier transforming the product.
 
1/2 1*
Wab aFXa
are the FT of the mother wave-
and the signal
t. That is,
for a
particular scale ''a can be computed by the Eqn. (4b)
by FFT algorithm.
For a real symmetric signal
t and a real sym-
metric wavelet
function, Eqn.(2a) becomes [9]
 
CabX a bd
are the Fourier transfo rm of
respectively. (Generally the wavelet function
is a symmetrical one but to have con sistency in the nota-
is used). In other words, they are the cosine
transforms of
t and the mother wavelet
Cab is the wavelet transform in cosine domain in-
stead of Fourier domain. Hence the corresponding equa-
tion for Eqn.(4b) is
 
1/2 1
CabaC Xa
Therefore the cosine wavelet transform coefficient
Cabfor a particular scale ''a can be computed by
the Eqn. (5b) using
by a fast
cosine transform algorithm which indirectly uses FFT
is very simple for the Harmonic
cosine wavelet transform (CHWT), and it is zero at all
frequencies except constant over a small frequency band.
1, ,
() ,
For this the wavelet)(t
sin cos
Representing sin c
sin c
cos sin
Hence the mother wavelet is a cosine modulated sinc
function. Here the decomposition of the signal in the
frequency domain is simple but suffers from the problem
of poor time localization due to slow decaying of the sinc
function. Though a spectral weighing other than rectan-
gular improves the localization in time it results in a
non-orthogonal wavelet set. The type of spectral weigh-
ing will determine the wavelet as it is the cosine trans-
form of the wavelet.
For the cosine harmonic wavelet transform, the spec-
tral weighing is a symmetrical rectangular function and
for a discrete signal it is zero except over symmetrical
finite bands
 where
,pq can be real numbers, not necessarily integers.
For an orthogon al CHWT, the wavelet func tion is fixed
and corresponds to a rectangular weighing in the fre-
quency domain which results in such a wavelet trans-
The Discrete cosine transform (DCT) enables the im-
plementation of the above cosine transform discussed as
it forms the symmetric signals
t and
itself (for the given non-symmetric
t and
For a sampled signal
0,1,2, 1nN, the
DCT of N points, is defined as the DFT of a 2N
point symmetrically extended signal
)( NnNnNx
nis even symmetric with respect to the point
. This leads to DCT and is given by
  
2cos,0 1
2, 21
xnk N
Ck N
 
Using the above
in the CHWT, the subband
decomposition is done in frequency domain unlike in
time domain by a filter bank. This is achieved by group-
ing the 2N coefficients of a discrete cosine transform
(DCT) of length 2Nand this is equivalent to applying a
window or weighing by a constant in the frequency do-
The DCT coefficients can be grouped in a way similar
to that of DFT coefficients and the DCT being r eal, there
A Discrete Cosine Adaptive Harmonic Wavelet Packet and Its Application to Signal Compression
Copyright © 2010 SciRes. JSIP
is no necessity to do the conjugate operation in placing
the coefficients symmetrically. The symmetrical place-
ment is also not necessary due to the very definition of
the DCT as it provides only half the number of coeffi-
cients and the inverse DCT definition takes care of the
symmetry. The grouped coefficients for each band have
to be treated as if they are the DCT coefficients of that
subband Figure 2a.
For the reconstruction, the DCTs of the subband sig-
nals are concatenated to get the DCT of the fullband sig-
nal. For the first stage of inverse DCHWT illustrated in
Figure 2a, the DCTs of the subband signals correspond-
ing to groups C3 and C4 are concatenated. The resulting
group of coefficients is concatenated with the DCT of
subband signal corresponding to group C2, in the next
stage. Again, the resulting group of coefficients is con-
catenated with the DCT of subband signal corresponding
to group C1, to form the DCT of the fullband signal.
The 1-D DFHWT or DCHWT can be extended to 2-D
signals (images) by applying it to rows and columns
separately. Fo r images, the DFHWT performan ce is poor,
even considering 100% coefficient (without omitting any
coefficient). This is due to the approximation in the algo
rithm which takes into account only the real part of a
complex coefficient, both during row and column wise
coefficient grouping which repeats for every scale.
The DCHWT does not pose such a problem for a 2D
signal. This is due to the fact that the DCT is a real
transform and the grouping does not involve conjugate
symmetry to get real signals. Here for the image, the
DCT coefficients of each column are grouped and their
inverse DCT results in DCHWT for that column. For
each scale (along columns), the DCT coefficients along
each row are taken and grouped. The inverse DCT of
these groups will result in 2D DCHWT. This procedure
is repeated for further scales considering the low-low
(LL) subband as input Figure 2b. Considering from one
LL to next LL as one stage, it is essential that the LL
Figure 2. (a) DCHWT for a 1-D signal, N=16, Subbands: C4, C3, C2, C1, (b) DCHWT for a 2-D signal.
A Discrete Cosine Adaptive Harmonic Wavelet Packet and Its Application to Signal Compression
Copyright © 2010 SciRes. JSIP
subband which goes as input should be in data domain
and not in frequency (DCT) domain. This is essential as
it enables to split DCT of columns in to H and L and
proceed further. Since no approximations are made, there
will not be any error when the image is constructed with
100% coefficients [9] wh ich is not so with DFHWT.
4. Discrete Cosine Adaptive Harmonic
Wavelet Packet (DCAHWP)
The DCAHWP for both 1-D and 2 -D sign als will be con-
sidered in this section.
4.1 For 1-D Signal
Adaptive wavelet packets can be realized in the fre-
quency domain using the approach of grouping of fre-
quency domain coefficients (harmonic wavelet packet)
[4]. The DCT based harmonic wavelet packet is simpler
because the DCT coefficients are nothing but d ecomposi-
tion up to the last possible level and they are directly
available withou t any filtering unlike th e discrete wa velet
transform realized by a filter bank. Further compared to
DFAHWP, their low leakage effect and real nature con-
tribute to the quality and simplicity of DCAHWP. The
DCAHWP decomposition is illustrated in Figure3.
Unlike in DCHWT Figure2a, the high subband is also
split into two subbands at each decomposition level,
apart from the low subband like in any WP. Ev en though
the decomposition up to the last level, to apply the treat-
ment of adaptive WP, it is indicated in Figure 3 as if all
the DCT coefficients are together and its splitting takes
place as in time domain filter bank approach. Here rather
levels 2 and 1 are formed by combining the coefficients
available at level-3.
Maintaining the total number of coefficients equal to
the signal length, different combinations of subbands
may be possible for reconstruction. For DCAHWP also,
the scales/subbands to be used for reconstruction are
selected such that their combined entropy is minimum, as
described in Section-2. Here a detailed algorithm for this
is given.
Algorith m for DAHWP implem entation:
1) Compute the N-point Discrete cosine transform
(DCT) of the N-point signal.
2) The DCT coefficients correspond to the level (last
level) 2
. Obtain the other
1L decompo-
sitions of WP by grouping the DCT coefficients. The
number of scales/subbands at level l is 2land each is
of length 2
points; 1ltoL
3) Starting from the last level, the sum of entropy lev-
els of two adjacent subbands/scales (children) in that
level, are comp ared to the entropy of the band in the pre-
vious level (parent) from which they were formed (actu-
ally the parent is formed from children here !). If the
combined entropy of th e children is found to be less than
that of the parent, the decomposition is valid. Each time
the children are selected, the combined entropy of the
children is assigned to their parent for subsequent com-
4) The selected scales/subbands at different levels are
used to reconstruct the signal, and the total number of
coefficients is N. For compression, the number of coef-
ficients to be considered is decided by the compression
factor (CF). The selection of these coefficients is based
on their magnitude values starting from the largest [2].
For 2-D signal:
The DCAHWP implementation for a 2-D signal (image)
is an extension of the above method. At each decomposi-
tion level, each of the subband images (LL, HL, LH and
HH achieved by the block D in Figure 4) is further di-
vided into four equal subbands. This is repeated till the last
possible decomposition level, that is when each subband
consists of only one pixel Fi gure 4. For reconstruction, t he
Figure 3. DCT harmonic wavelet packet.
A Discrete Cosine Adaptive Harmonic Wavelet Packet and Its Application to Signal Compression
Copyright © 2010 SciRes. JSIP
Figure 4. (a) Decomposition of an image into four subbands by DCHWT. (b) DCT Harmonic Wavelet Packet Decomposition
for an image.
combined entropy of four children is compared to that of
the parent at each level and a combination with lowest
entropy is selected. The detailed al gorit hm is given.
Algorithm for DCAHWP implementation:
1) First take the DCT along columns and then along
rows, i.e. compute the 2-D DCT of the NN pixel
2) Perform the WP decomposition up to the last possi-
ble level, L
logLN. The number of
scales/subbands at level l is 4l and each is of size
pixels; 1ltoL. The decomposition is car-
ried out by grouping the DCT coefficients.
3) Starting from the last level, the entropy comparisons
are made between the children and their parent and the
scales/subband decomposition tree with the lowest en-
tropy is selected.
4) For reconstructing the image, the selected scales /
subbands with a total of NN
coefficients, are used.
For compression a specified number of largest coeffi-
cients amongst the NN coefficients across the scales
are selected. This specific number is decided by the CF.
6. Simulation Results
To illustrate the performance of the proposed DCAHWP
adaptive wavelet packet, four examples: 1)1-D signal, 2)
a speech signal, 3) a 2-D image (512 x 512) and 4) a tex-
tural image (64 x 64); are considered. Its performance is
compared with that of DCHWT. For the examples 1 and
2, the performance of the proposed method is compared
with those of Daubechies-2 based WT (DBWT) and its
adaptive WP (DBAWP). For all examples, the perform-
ance of proposed WP method is compared with that of
DCHWT. Since the performance of DFHWT is inferior
to that of DCHWT [7], the proposed DCAHWP has not
been compared with those of DFAHWP, the WP based
on DFT. The compression performance is evaluated by
the index,
100% error energy error energys ignal energy
Example 1 [2]: A signal of 64 samples, having a sinu-
soid and a spike Figure 5, is decomposed into subbands
using both DCHWT and DCAHWP. The signal was re-
constructed with and without compression in both the
The decomposition tree structures for DCHWT and
DCAHWP are depicted in Figures 6(a), 6(b) respec-
Figure 5. 1-D signal having a sinusoid and a spike.
A Discrete Cosine Adaptive Harmonic Wavelet Packet and Its Application to Signal Compression
Copyright © 2010 SciRes. JSIP
Figure 6. Decomposition tree for the signal in Figure 5 by: (a) DCHWT (b) DCAHWP.
The frequency band splittin g in each decompositio n level
is indicated by vertical lines. The selected groups are
represented by shaded blocks. In DCHWT, the high sub-
band is selected at each level and in the last level the low
subband is also selected Figure 6a. But the tree structure
for DCAHWP Figure 6b differs from that of DCHWT
and depends on the signal characteristics
The DCHWT time-frequency tiling is such that lower
scales have larger frequency spacing and smaller time
spacing, whereas higher scales have smaller frequency
spacing and larger time spacing Figure 7a. But the
DCAHWP time-frequency tiling Figure 7b is according
to the best basis selected for the signal on hand. In the
Figures 7a, 7b, larger coefficients are represented by
darker shades.
The WP coefficient values at various decomposition
levels are shown in Figure 8a. At each decomposition
level, each subband is further split into two subbands.
For example for the level 2, Figure 8a shows the WP
coefficients for the four scales; the different scales are
put side by side. The same is valid for other levels also.
The selected coefficients for DCHW T and D CAHW P are
shown in Figures. 8b, 8c respectively. For DCHWT, a
fixed set of the coefficients is selected; whereas for
DCAHWP, the set of coefficients varies according to the
signal. For DCAHWP the frequency band splitting is
denser, providing higher frequency resolution in the low
frequency range (Figure 8b). This is also found in the
time-frequency plot (Figure 7b) and the tree structure
(Figure 6b) for DCAHWP. However, this is not so for
DCHWT (Figures. 6a, 7a, 8a). For compression, only 8
largest coefficients from the selected 64 Figure 8(b, c),
are used to reconstruct the signal.
Signal compression performance of DCAHWP was
compared to that of DCHWT and also with that of
DBWT and DBAWP. The reconstructed signal by
DCAHWP with CF=1/8, contains a well defined spike in
the same position as in the original (Figure 9d). But for
the same CF, the spike is not clear in the reconstructed
signal by DCHWT (Figure 9c). The performance of
DBAWP is poor and that of DBWT is even poorer, as
neither the spike nor the sinusoid are clearly brought out
(Figure. 9a, b). The error plots corresponding to Figure
9(a-d) are shown in Figure 9(e-h) and it is seen that the
magnitude of the error for compression by DCAHWP is
significantly less than that by other methods. Further, the
percentage error energy by DCAHWP is about 1/3rd of
that by DCHWT and about 1/8th that of DBAWP Table
1. Example 2:
For a speech signal (Figure 10) also, the results ob-
tained are similar to those for example-1. For the com-
pression factor 1/8, the performance of DBWT and of
DBAWP are inferior compared to that of DCHWT. Be-
tween DBAWP and DBWT, the former is better as is
optimal for the Daubechies-2 wavelet. But DCAHWP
resulted in best performance compression compared to
the other methods, which is clear from comparing the
reconstruction errors for the various methods (Figure
11e-h). The percentage error energy by DCAHWP is
about 1/5th of that by DBAWP and about 1/2 of that by
DCHWT Table 2.
Table 1. % error energy (% EE) for reconstruction of the
signal in Fig 5 with 1/8 coefficients (CF= 1/8).
Method % EE with CF=1/8
DBWT 13.4947
DBAWP 5.8149
DCHWT 1.9550
DCAHWP 0.7396
A Discrete Cosine Adaptive Harmonic Wavelet Packet and Its Application to Signal Compression
Copyright © 2010 SciRes. JSIP
igure 7. Time-freque nc y tiling for the signal in Figure 5 by: (a) DCHWT (b) DCAHWP.
Figure 8. WP decomposition coefficients for the signal in Figure 5.
Table 2. % error energy (% EE) for reconstruction of the
speech in Figure 10 with 1/8 coefficients (CF= 1/8).
Method % EE with CF= 1/8
DBWT 4.2695
DBAWP 2.7425
DCHWT 1.1542
DCAHWP 0.5886
Example – 3: The performances of DCAHWP and
DCHWT were compared for a 512 512 ‘Barbara’
image. The compression results for a CF of 1/32 clearly
Indicate the superior performance of DCAHWP over
DCHWT (Figure 12b, 12c). There is a significant blur-
ring in the reconstructed image by DCHWT and hence
the edges like eyes, nose and stripes ar e not distinct Fig-
ure 12c. In contrast, the edge information is preserved to
a large extent in reconstructed image by DCAHWP,
which is manifested as the sharpness of the nose, eyes
and stripes Figure 12c.
The above results are supported by the percentage er-
ror energy index as for DCAHWP, it is only about 2/3rd
that of DCHWT Table 3.
Example 4: To illustrate the performance superiority of
DCAHWP due to its best basis selection over DCHWT,
A Discrete Cosine Adaptive Harmonic Wavelet Packet and Its Application to Signal Compression
Copyright © 2010 SciRes. JSIP
Figure 9. Reconstruction of the signal in Figure 5 a. (a-d), Reconstructed signal for CF=1/8 by, (a) DBWT, (b) DBAWP, (c)
DCHWT, (d) DCAHWP (e-h), error corresponding to (a-d).
Figure 10. Speech signal.
Figure 11. (a-d): Reconstructed speech for CF=1/8 by (a) DBWT (b) DBAWP, (c) DCHWT, (d) DCAHWP , (e-h): error
corresponding to (a-d).
A Discrete Cosine Adaptive Harmonic Wavelet Packet and Its Application to Signal Compression
Copyright © 2010 SciRes. JSIP
(a) (b) (c)
Figure 12 Image reconstruction of the ‘Barbara’ image (512×512), (a) Original, (b) DCHWT (CF=1/32), (c) DCAHWP
Figure 12. Image reconstruction of the ‘Barbara’ image (512×512), a) Original Barbara image, (b) Part of original w ith in the
square(64×64), (c) DCHWT (CF=1/32), (d) DCAHWP (CF=1/32).
(a) (b) (c)
(d) (e) (f)
Figure 14 Adaptive wavelet packet decomposition tree for the image in Figure 13a. (a-f): Decomposition levels 1-6 respec-
tively. Shaded blocks indicate selected wavelet coefficients at each decomposition level.
A Discrete Cosine Adaptive Harmonic Wavelet Packet and Its Application to Signal Compression
Copyright © 2010 SciRes. JSIP
Table 3. % error energy (% EE) for reconstruction of the
image in Figure 12a with 1/32 coefficients (CF=1/32).
Method % EE with CF = 1/32
DCHWT 1.1310
DCAHWP 0.7517
Table 4. % error energy (% EE) for reconstruction of the
image in Figure 13(a) with 1/32 coefficients (CF= 1/32).
Method % EE with CF= 1/32
DCHWT 4.3492
DCAHWP 0.9691
the two methods are applied to Barbara’ image Figure 13
aselecting a portion of it of size
64 64, which is
marked by the square box in Figure 13 b. This small
portion corresponds to an image with repetitive varia-
tions in intensity (stripes).
The best basis selection by the adaptive wavelet packet
for this image is shown in the Figure 14. The subbands
/scales in each wavelet packet decomposition level are
depicted by square demarcations. The shaded squares are
the selected subbands/scales for reconstruction.
The percentage error energy by DCAHWP is about
1/5th of that by DCHWT in this case Table 4. The com-
pressed image by DCHWT is a very crude representation
of the image as the stripes are almost not visible (Figure
13c). But the stripes are clearly visible in the compressed
image by DCAHWP Figure 13(d).
6. Conclusions
An Adaptive Harmonic Wavelet Packet based on Dis-
crete Cosine Transform (DCAHWP) was proposed. The
implementation is simple as the DCT coefficients are real.
The DCAHWP was achieved by using the entropy crite-
rion realized by comparing the normalized energies of
parents and children. The proposed DCAHWP performed
better compared to Daubechies-2 based adaptive WP
(DBAWP). For a compression factor (CF) of 1/8, for a
1-D signal and speech signal considered, the ratio of the
percentage error energy (PEE) by DCAHWP to that by
DBAWP was about 1/ 8 a n d 1/ 5, res pect ivel y .
Compared to DCT based harmonic wavelet transform
(DCHWT), as expected the proposed DCAHWP per-
formed better as the ratio of the percentage error energies
by DCAHWP to that by DCHWT better for 1D and
speech, signals respectively were about 1/3 and 1/2, for a
compression factor 1/8.
The ratio of PEE for DCAHWP and that for DCHWT
for an image ((512 512) Barbara) was 2/3 for a CF of
1/32. The compression by DCAHWP is superior in
bringing out the edge information (high frequency con-
tent) such as eyes, nose and stripes (texture), whereas
that by DCHWT results in blurred images. Further for a
textural image ((64 64)
, a portion of the Barbara image)
this normalized energy ratio between DCAHWP and that
for DCHWT it was 1/5 for a CF of 1/32. But in all the
cases, the reconstructed signals/images without compres-
sion were almost identical to the original.
The results clearly indicate that for signals character-
ized by abrupt changes or images with rapid variations,
the proposed DCAHWP provides significantly better
compression performance compared to that of DCHWT.
It has been found that the DCAHWP has a significantly
better compression performance over the adaptive wave-
let packet based on Daubechies-2, DBAWPT.
[1] C. Sidney Burrus, R. A. Gopinath and H. T. Guo, “Intro-
duction to Wavelets and Wavelet Transforms: A Primer,”
Prentice-Hall, Inc., Upper Saddle River, 1998.
[2] P. S. Addison, “The Illustrated Wavelet Transform Hand-
Book,” Institute of Physics Publishing, London, 2002.
[3] R. R. Coifman and M. V. Wickerhauser, “Entropy-Based
Algorithms for Best Basis Selection,” IEEE Transactions
on Information Theory, Vol. 38, No. 2, March 1992, pp.
[4] B. Liu, “Adaptive Harmonic Wavelet Transform with
Applications in Vibration Analysis,” Journal of Sound
and Vibration, Vol. 262, No. 1, 2003, pp. 45-64.
[5] M. Kivanc Mihcak, K. Ramchandran and P. Moulin,
“Adaptive Wavelet Packet Image Coding Using an Esti-
mation-Quantization Framework,” Proceedings of Inter-
national Conference on Image Processing, Chicago, USA,
Vol. 1, 1998, pp. 92-96.
[6] M. Musaruddin and A. Z. Kouzani, “Embedding Data in
Images Using Wavelet Packets,” International Confer-
ence IEEE TENCO, Vol. 2, 21-24 November 2004, pp.
120- 123.
[7] D. E. Newland, “Harmonic Wavelet Analysis,” Proceed-
ings: Mathematical and Physical Sciences, Vol. 444, No.
1917, 1993, pp. 203-225.
[8] S. V. Narasimhan and M. Harish, “A New Spectral Esti-
mator Based on Discrete Cosine Transform and Modified
Group Delay,” Signal Processing, Vol. 86, No. 7, 2006,
pp. 1586-1596.
[9] S. V. Narasimhan, M. Harish, A. R. Haripriya and N.
Basumallick, “Discrete Cosine Harmonic Wavelet Trans-
form and Its Application to Signal Compression and
Spectral Estimation Using Modified Group Delay,” Sig-
nal, Image and Video Processing, Vol. 3, No. 1, February
2009, pp. 85-99.
[10] S. V. Narasimhan, A. R. Haripriya and B. K. Shreyamsha
Kumar, “Improved Wigner-Ville Distribution Perform-
ance Based on Dct/Dft Harmonic Wavelet Transform and
Modified Magnitude Group Delay,” Signal Processing,
A Discrete Cosine Adaptive Harmonic Wavelet Packet and Its Application to Signal Compression
Copyright © 2010 SciRes. JSIP
Vol. 88, No. 1, 2008, pp. 1-18.
[11] S. V. Narasimhan and A. Adiga, “Shift Invariant Discrete
Cosine Harmonic Wavelet Transform and Its Application
to Denoising,” Proceedings, IEEE International Confer-
ence INDICON, Bangalore, 2007.
[12] S. V. Narasimhan and S. Veena, “Signal Processing Prin-
ciples and Implementation,” Alpha Science, Harrow, UK
and Narosa publishers, New Delhi, India, 2008.
[13] T. Gulzow, T. Ludwig and U. Heute “Spectral-Subtrac-
tion Speech Enhancement in Multirate Systems with and
without Non-Uniform and Adaptive Bandwidths,” Signal
Processing, Vol. 83, No. 8, 2003, pp. 1613-1631.