Wavelets analyze data at different scales or resolutions by decomposing it into different frequency components, providing advantages over Fourier methods for non-stationary signals. This document provides an overview of the history and development of wavelets, starting with Fourier's work in the 1800s, then developments in the 1900s that led to modern wavelet theory and applications across many fields including image processing, data compression, and more. Key people who contributed include Haar, Littlewood, Paley, Stein, Grossman, Morlet, Mallat, Meyer, and Daubechies.

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Wavelets are mathematical functions that cut up data into different frequency compo-
nents, and then study each component with a resolution matched to its scale. They have
advantages over traditional Fourier methods in analyzing physical situations where the
signal contains discontinuities and sharp spikes. Wavelets were developed independently
in the fields of mathematics, quantum physics, electrical engineering, and seismic geology.
Interchanges between these fields during the last years have led to many new wavelet
applications such as image compression, turbulence, human vision, radar, and earthquake
prediction. This course introduces wavelets to the interested technical person outside of
the digital signal processing field. We describe the history of wavelets beginning with
Fourier, compare wavelet transforms with Fourier transforms, state properties and other
special aspects of wavelets, and provide with some interesting applications such as image
compression, interpolation, solution to differential equations, and de-noising noisy data.
The fundamental idea behind wavelets is to analyze according to scale. Indeed, some
researchers in the wavelet field feel that, by using wavelets, one is adopting a whole new
mindset or perspective in processing data.
Wavelets are functions that satisfy certain mathematical requirements and are used in
representing data or other functions. This idea is not new. Approximation using superpo-
sition of functions has existed since the early 1800’s, when Joseph Fourier discovered that
he could superpose sines and cosines to represent other functions. However, in wavelet
analysis, the scale that we use to look at data plays a special role. Wavelet algorithms pro-
cess data at different scales or resolutions. If we look at a signal with a large “window,”
we would notice gross features. Similarly, if we look at a signal with a small “window,”
we would notice small features.
This makes wavelets interesting and useful. For many decades, scientists have wanted
more appropriate functions than the sines and cosines which comprise the bases of Fourier
analysis, to approximate choppy signals. By their definition, these functions are non-local
(and stretch out to infinity). They therefore do a very poor job in approximating sharp
spikes. But with wavelet analysis, we can use approximating functions that are con-
tained neatly in finite domains. Wavelets are well-suited for approximating data with
sharp discontinuities. The wavelet analysis procedure is to adopt a wavelet prototype
function, called an analyzing wavelet or mother wavelet. Temporal analysis is per-
formed with a contracted, high-frequency version of the prototype wavelet,
while frequency analysis is performed with a dilated, low-frequency version of
the same wavelet. Because the original signal or function can be represented in terms of
a wavelet expansion (using coefficients in a linear combination of the wavelet functions),
data operations can be performed using just the corresponding wavelet coefficients. And
if you further choose the best wavelets adapted to your data, or truncate the coefficients
below a threshold, your data is sparsely represented. This sparse coding makes wavelets
an excellent tool in the field of data compression. Other applied fields that are making
use of wavelets include astronomy, acoustics, nuclear engineering, sub-band coding, signal
and image processing, neurophysiology, music, magnetic resonance imaging, speech dis-
crimination, optics, fractals, turbulence, earthquake-prediction, radar, human vision, and
pure mathematics applications such as solving partial differential equations.
HISTORICAL PERSPECTIVE

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a0 +
∞
X
k=1
(ak cos kx + bk sin kx)
where
a0 =
1
2π
Z 2π
0
f(x)dx, ak =
1
π
Z 2π
0
f(x) cos(kx)dx, ak =
1
2π
Z π
0
f(x) sin(kx)dx,
Fourier’s assertion played an essential role in the evolution of the ideas mathematicians
had about the functions. He opened up the door to a new functional universe.
After 1807, by exploring the meaning of functions, Fourier series convergence, and
orthogonal systems, mathematicians gradually were led from their previous notion of
frequency analysis to the notion of scale analysis. That is, analyzing f(x) by creating
mathematical structures that vary in scale. We construct a function, shift it by some
amount, and change its scale. Apply that structure in approximating a signal. Now
repeat the procedure. Take that basic structure, shift it, and scale it again. Apply it to
the same signal to get a new approximation. And so on. It turns out that this sort of
scale analysis is less sensitive to noise because it measures the average fluctuations of the
signal at different scales.
The first mention of wavelets appeared in an appendix to the thesis of A. Haar (1909).
One property of the Haar wavelet is that it has compact support, which means that it
vanishes outside of a finite interval. Unfortunately, Haar wavelets are not continuously
differentiable which somewhat limits their applications.
In the 1930s, several groups working independently researched the representation of
functions using scale-varying basis functions. Understanding the concepts of basis func-
tions and scale-varying basis functions is key to understanding wavelets; in below provides
a short detour lesson for those interested. By using a scale-varying basis function called
the Haar basis function (more on this later) Paul Levy, a 1930s physicist, investigated
Brownian motion, a type of random signal (2). He found the Haar basis function supe-
rior to the Fourier basis functions for studying small complicated details in the Brownian
motion.
Another 1930s research effort by Littlewood, Paley, and Stein involved computing the
energy of a function f(x):
energy =
1
2
Z 2π
0
|f(x)|2
dx
The computation produced different results if the energy was concentrated around a few
points or distributed over a larger interval. This result disturbed the scientists because it
indicated that energy might not be conserved. The researchers discovered a function that
can vary in scale and can conserve energy when computing the functional energy. Their
work provided David Marr with an effective algorithm for numerical image processing
using wavelets in the early 1980s.
What are Basis Functions?

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It is simpler to explain a basis function if we move out of the realm of analog (functions)
and into the realm of digital (vectors) (*).
Every two-dimensional vector (x; y) is a combination of the vector (1; 0) and (0; 1): These
two vectors are the basis vectors for (x, y): Why? Notice that x multiplied by (1; 0) is
the vector (x, 0); and y multiplied by (0; 1) is the vector (0; y) : The sum is (x, y) : The
best basis vectors have the valuable extra property that the vectors are perpendicular, or
orthogonal to each other. For the basis (1; 0) and (0; 1); this criteria is satisfied. Now
let’s go back to the analog world, and see how to relate these concepts to basis functions.
By choosing the appropriate combination of sine and cosine function terms whose inner
product add up to zero. The particular set of functions that are orthogonal and that
construct f(x) are our orthogonal basis functions for this problem.
What are Scale-varying Basis Functions?
A basis function varies in scale by chopping up the same function or data space using
different scale sizes. For example, imagine we have a signal over the domain from 0 to 1.
We can divide the signal with two step functions that range from 0 to 1/2 and 1/2 to 1.
Then we can divide the original signal again using four step functions from 0 to 1/4, 1/4
to 1/2, 1/2 to 3/4, and 3/4 to 1. And so on. Each set of representations code the original
signal with a particular resolution or scale.
Between 1960 and 1980, the mathematicians Guido Weiss and Ronald R. Coifman
studied the simplest elements of a function space, called atoms, with the goal of finding
the atoms for a common function and finding the “assembly rules” that allow the recon-
struction of all the elements of the function space using these atoms. In 1980, Grossman
and Morlet, a physicist and an engineer, broadly defined wavelets in the context of quan-
tum physics. These two researchers provided a way of thinking for wavelets based on
physical intuition.
In 1985, Stephane Mallat gave wavelets an additional jump-start through his work
in digital signal processing. He discovered some relationships between quadrature mirror
filters, pyramid algorithms, and orthonormal wavelet bases (more on these later). Inspired
in part by these results, Y. Meyer constructed the first non-trivial wavelets. Unlike the
Haar wavelets, the Meyer wavelets are continuously differentiable; however they do not
have compact support. A couple of years later, Ingrid Daubechies used Mallat’s work to
construct a set of wavelet orthonormal basis functions that are perhaps the most elegant,
and have become the cornerstone of wavelet applications today.
FOURIER ANALYSIS
Fourier’s representation of functions as a superposition of sines and cosines has become
ubiquitous for both the analytic and numerical solution of dierential equations and for
the analysis and treatment of communication signals. Fourier and wavelet analysis have
some very strong links.
FOURIER TRANSFORMS
The Fourier transform’s utility lies in its ability to analyze a signal in the time domain
for its frequency content. The transform works by first translating a function in the time
domain into a function in the frequency domain. The signal can then be analyzed for its
frequency content because the Fourier coefficients of the transformed function represent
the contribution of each sine and cosine function at each frequency. An inverse Fourier

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transform does just what you’d expect, transform data from the frequency domain into
the time domain.
DISCRETE FOURIER TRANSFORMS
The discrete Fourier transform (DFT) estimates the Fourier transform of a function
from a finite number of its sampled points. The sampled points are supposed to be typical
of what the signal looks like at all other times.
The DFT has symmetry properties almost exactly the same as the continuous Fourier
transform. In addition, the formula for the inverse discrete Fourier transform is easily
calculated using the one for the discrete Fourier transform because the two formulas are
almost identical.
WINDOWED FOURIER TRANSFORMS(WFT)
If f(t) is a nonperiodic signal, the summation of the periodic functions, sine and co-
sine, does not accurately represent the signal. You could artificially extend the signal to
make it periodic but it would require additional continuity at the endpoints. The win-
dowed Fourier transform (WFT) is one solution to the problem of better representing the
nonperiodic signal. The WFT can be used to give information about signals simultane-
ously in the time domain and in the frequency domain. With the WFT, the input signal
f(t) is chopped up into sections, and each section is analyzed for its frequency content
separately. If the signal has sharp transitions, we window the input data so that the
sections converge to zero at the endpoints. This windowing is accomplished via a weight
function that places less emphasis near the interval’s endpoints than in the middle. The
effect of the window is to localize the signal in time. The short-time Fourier transform
(STFT), or alternatively short-term Fourier transform, is a Fourier-related transform used
to determine the sinusoidal frequency and phase content of local sections of a signal as it
changes over time.
STFT {x(t)} ≡ X(τ, ω) =
Z ∞
−∞
x(t)w(t − τ)e−jωt
dt
where w(t) is the window function, commonly a Hann window or Gaussian “hill” centered
around zero, and x(t) is the signal to be transformed. X(τ, ω) is essentially the Fourier
Transform of x(t)w(t − τ), a complex function representing the phase and magnitude of
the signal over time and frequency. Often phase unwrapping is employed along either or
both the time axis, τ, and frequency axis, ω, to suppress any jump discontinuity of the
phase result of the STFT. The time index τ is normally considered to be ”slow” time and
usually not expressed in as high resolution as time t.
Discrete-time STFT
STFT {x[n]} ≡ X(m, ω) =
∞
X
n=−∞
x[n]w[n − m]e−jωn
In this case, m is discrete and ω is continuous, but in most typical applications the STFT
is performed on a computer using the Fast Fourier Transform(FFT), so both variables
are discrete and quantized.

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The magnitude squared of the STFT yields the spectrogram of the function:
spectrogram {x(t)} ≡ |X(τ, ω)|2
Inverse STFT
Given the width and definition of the window function w(t), we initially require the height
of the window function to be scaled so that
Z ∞
−∞
w(τ) dτ = 1 ⇒
Z ∞
−∞
w(t − τ) dτ = 1 ∀ t
x(t) = x(t)
Z ∞
−∞
w(t − τ) dτ =
Z ∞
−∞
x(t)w(t − τ) dτ.
X(ω) =
Z ∞
−∞
x(t)e−jωt
dt.
X(ω) =
Z ∞
−∞
Z ∞
−∞
x(t)w(t − τ) e−jωt
dt dτ (1)
=
Z ∞
−∞
Z ∞
−∞
x(t)w(t − τ) e−jωt
dt
dτ (2)
=
Z ∞
−∞
X(τ, ω) dτ. (3)
Since the inverse Fourier transform is
x(t) =
1
2π
Z ∞
−∞
X(ω)ejωt
dω =
1
2π
Z ∞
−∞
Z ∞
−∞
X(τ, ω)ejωt
dτ dω.
comparing to above that windowed x(t) or “wavelet” of x(t) is
x(t)w(t − τ) =
1
2π
Z ∞
−∞
X(τ, ω)ejωt
dω.
the inverse Fourier transform of X(τ, ω) for τ fixed.
To numerically evaluate the STFT, we sample the frequency ω in N equally spaced samples
from 0 ≤ ω ≤ 2π, ωk = 2πk/N, 0 ≤ k ≤ N − 1
X(k, m) = X(2πk/N, m) =
R−1
X
n=0
x(n − m)w(n)e−j2πk/Nn
, k = 0, 1, · · · , N − 1
= DFTN



x(n − m)w(n)|R−1
n=0 , 0, 0, · · · , 0
| {z }
N−R



FAST FOURIER TRANSFORMS
To approximate a function by samples, and to approximate the Fourier integral by
the discrete Fourier transform, requires applying a matrix whose order is the number

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sample points n : Since multiplying an n × n matrix by a vector costs on the order of
n2
arithmetic operations, the problem gets quickly worse as the number of sample points
increases. However, if the samples are uniformly spaced, then the Fourier matrix can be
factored into a product of just a few sparse matrices, and the resulting factors can be
applied to a vector in a total of order n log n arithmetic operations. This is the so-called
Fast Fourier transform or (FFT).
WAVELET TRANSFORMS VERSUS FOURIER TRANSFORMS
SIMILARITIES BETWEEN FOURIER AND WAVELET TRANSFORMS
The fast Fourier transform (FFT) and the discrete wavelet transform (DWT) are both
linear operations that generate a data structure that contains log2 n segments of various
lengths, usually filling and transforming it into a different data vector of length 2n
. The
mathematical properties of the matrices involved in the transforms are similar as well.
The inverse transform matrix for both the FFT and the DWT is the transpose of the
original. As a result, both transforms can be viewed as a rotation in function space to a
different domain. For the FFT, this new domain contains basis functions that are sines
and cosines. For the wavelet transform, this new domain contains more complicated basis
functions called wavelets, mother wavelets, or analyzing wavelets.
Both transforms have another similarity. The basis functions are localized in frequency,
making mathematical tools such as power spectra (how much power is contained in a
frequency interval) and scalegrams (to be defined later) useful at picking out frequencies
and calculating power distributions.
DISSIMILARITIES BETWEEN FOURIER AND WAVELET TRANSFORMS
The most interesting dissimilarity between these two kinds of transforms is that indi-
vidual wavelet functions are localized in space. Fourier sine and cosine functions are not.
This localization feature, along with wavelets’ localization of frequency, makes many func-
tions and operators using wavelets “sparse” when transformed into the wavelet domain.
This sparseness, in turn, results in a number of useful applications such as data compres-
sion, detecting features in images, and removing noise from time series. One way to see
the time-frequency resolution differences between the Fourier transform and the wavelet
transform is to look at the basis function coverage of the time-frequency plane. Figure
7 shows a windowed Fourier transform, where the window is simply a square wave. The
square wave window truncates the sine or cosine function to fit a window of a particular
width. Because a single window is used for all frequencies in the WFT(windowed Fourier
transform, the resolution of the analysis is the same at all locations in the time-frequency
plane.
One of the downfalls of the STFT is that it has a fixed resolution. The width of
the windowing function relates to how the signal is representedit determines
whether there is good frequency resolution (frequency components close to-
gether can be separated) or good time resolution (the time at which frequen-
cies change). A wide window gives better frequency resolution but poor time resolution.
A narrower window gives good time resolution but poor frequency resolution. These are
called narrowband and wideband transforms, respectively.

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, Fall-1390, WSP 7
Time
Frequency
Figure 1: Better time resolution
Example
Using the following sample signal x(t) that is composed of a set of four sinusoidal
waveforms joined together in sequence. Each waveform is only composed of one of four
frequencies (10, 25, 50, 100 Hz). The definition of x(t) is:
x(t) =







cos(2π10t),
cos(2π25t),
cos(2π50t),
cos(2π100t),
N=400; t1=linspace(0,1,N);
s1=cos(2*pi*10*t1);s2=cos(2*pi*20*t1);s3=cos(2*pi*40*t1);s4=cos(2*pi*80*t1);
plot(t1,s1,t1+1,s2,t1+2,s3,t1+3,s4)
plot(linspace(-200,200,4*400),abs(fftshift(fft([s1 s2 s3 s4])))); grid; figure(gcf)
Let’s determine the spectral behavior using FFT
plot(linspace(-200,200,4*400),abs(fftshift(fft([s1 s2 s3 s4])))); grid; figure(gcf)
From figure 3, as we observe, there is no information as to the spectral behavior of
sinusoidal signals(how many? what frequencies?). On the other hand, from figure 4, there
is no information as to the beginning and end of each sinusoids though the frequencies
are determined. In figure 6 we observe that each signal there is only one specific
frequency centered around (10,20,40,80) for all times.

8. AKUH· AMIN DAVAR
, Fall-1390, WSP 8
Time
Frequency
Figure 2: Better frequency resolution
subplot(221); spectrogram([s1],’’,0,256,400);title(’a’); figure(gcf)
subplot(222); spectrogram([s2],’’,0,256,400);title(’b’); figure(gcf)
subplot(223); spectrogram([s3],’’,0,256,400);title(’c’); figure(gcf)
subplot(224); spectrogram([s4],’’,0,256,400);title(’d’); figure(gcf)
Concatenation of the four sinusoids in time and frequency are distinguished using
spectrogram([s1 s2 s3 s4],’’,0,256,400); figure(gcf)
In mathematics, a wavelet series is a representation of a square-integrable (real- or
complex-valued) function by a certain orthonormal series generated by a wavelet.
A function ψ ∈ L2
(R) is called an orthonormal wavelet if it can be used to define
a Hilbert basis, that is a complete orthonormal system, for the Hilbert space L2
(R) of
square integrable functions. The Hilbert basis is constructed as the family of functions
{ψjk : j, k ∈ Z} by means of dyadic translations and dilations of ψ,
ψjk(x) = 2j/2
ψ(2j
x − k)
for integers j, k ∈ Z. This family is an orthonormal system if it is orthonormal under
the inner product
hψjk, ψlmi = δjlδkm
where δjl is the Kronecker delta and hf, gi is the standard inner product hf, gi =
R ∞
−∞
f(x)g(x)dx
on L2
(R). The requirement of completeness is that every function f ∈ L2
(R) may be ex-
panded in the basis as
f(x) =
∞
X
j,k=−∞
cjkψjk(x)

9. AKUH· AMIN DAVAR
, Fall-1390, WSP 9
0 0.5 1 1.5 2 2.5 3 3.5 4
−1
−0.8
−0.6
−0.4
−0.2
0
0.2
0.4
0.6
0.8
1
Time, sec.
Figure 3: Concatenation of some sinusoids in time domain.
−200 −150 −100 −50 0 50 100 150 200
0
50
100
150
200
250
freq., Hz
Figure 4: Spectral behavior of concatenation of the sinusoids in frequency domain.

10. AKUH· AMIN DAVAR
, Fall-1390, WSP 10
0 50 100 150 200
0.2
0.4
0.6
0.8
Frequency (Hz)
a
Time
0 50 100 150 200
0.2
0.4
0.6
0.8
Frequency (Hz)
b
Time
0 50 100 150 200
0.2
0.4
0.6
0.8
Frequency (Hz)
c
Time
0 50 100 150 200
0.2
0.4
0.6
0.8
Frequency (Hz)
d
Time
Figure 5: Time-frequency analysis of each individual sinusoids.
0 50 100 150 200
0.5
1
1.5
2
2.5
3
3.5
Frequency (Hz)
Time
Figure 6: Time-frequency analysis four concatenated sinusoids.

11. AKUH· AMIN DAVAR
, Fall-1390, WSP 11
Time
Frequency
Figure 7: Fourier basis functions, time-frequency tiles, and coverage of the time-frequency
plane
with convergence of the series understood to be convergence in norm. Such a repre-
sentation of a function f is known as a wavelet series. This implies that an orthonormal
wavelet is self-dual. The integral wavelet transform is the integral transform defined as
[Wψf] (a, b) =
1
p
|a|
Z ∞
−∞
ψ
x − b
a
f(x)dx
The wavelet coefficients cjk are then given by
cjk = [Wψf] (2j
, k2j
)
Here, a = 2j
is called the binary dilation or dyadic dilation, and b = k2j
is the binary
or dyadic position.
Introduction to the world of transform
A mathematical operation that takes a function or sequence and maps into another one
f(x) =
Z
F(w)K(w, x)dw

12. AKUH· AMIN DAVAR
, Fall-1390, WSP 12
fj =
X
j
KijFi
Examples:
Laplace, Fourier, DTFT, DFT, FFT, z-transform Fourier Transform
X(f) =
Z ∞
−∞
x(t)e−2jπft
dt
x(t) =
Z ∞
−∞
X(f)e2jπft
df
Fourier transform identifies all spectral components present in the signal, however it
does not provide any information regarding the temporal (time) localization of the
components.
Fourier Transform : Limitations
1. Signals are of two types
(a) Stationary
(b) Non-stationary
2. Non-stationary signals are those who have got time varying spectral components
. . . FT gives only provides the existence of the spectral components of the signal
. . . But does not provide any information on the time occurrence of spectral compo-
nents
3. The basis function e−jωt
stretches to infinity, Hence, only analyzes the signal globally.
4. In order to obtain time-localization of spectral components, the signal need to be
analyzed locally.
Time–Frequency Representation
1. Instantaneous frequency:
fx(t) =
1
2π
d
dt
∠x(t)
2. Group delay:
tx(f) = −
1
2π
d
df
∠X(f)
3. Disadvantages of above expressions: These equations though have a huge theoretical
significance but are not easy to implement.
4. All time-frequency-energy representations should be classified as time-frequency
analysis; thus, wavelet, Wigner-Ville Distribution and spectrogram should all be
included.

13. AKUH· AMIN DAVAR
, Fall-1390, WSP 13
5. Almost by default, the term, time-frequency analysis, was monopolized by the
Wigner-Ville distribution.
6. To have a valid time-frequency representation, we have to have frequency and energy
functions varying with time. Therefore, the frequency and energy functions should
have instantaneous values
Wigner-Ville Distribution
Wigner-Ville Distribution, W(ω, t), is defined as
W(ω, t) =
1
2π
Z ∞
−∞
x(t − τ/2)x(t + τ/2) e−jωτ
dτ
E{W(ω, t)} =
1
2π
Z ∞
−∞
E{x(t)x(t + τ)} e−jωτ
dτ
=
1
2π
Z ∞
−∞
Rxx(t, τ) e−jωτ
dτ
WV Instantaneous Frequency
ω(t) =
R ∞
−∞
ωW(ω, t) dω
R ∞
−∞
W(ω, t) dω
7. Therefore, at any given time, there is only one instantaneous frequency value.
8. What if there are two independent components? In this case, VW gives the weighted
mean.
If x(t) = x1(t) + x2 (t) then
S(ω) = S1(ω) + S2(ω) ,
where S(ω) = 1
√
2π
R
t
x(t) e−iωt
d t .
|S|2
= |S1 + S2|2
= |S1|2
+ |S2|2
+ 2ℜ [S∗
1S2]
6= |S1|2
+ |S2|2
.
Short-time Fourier Transform
STFTw
x (τ, w) =
Z
t
[x(t)w(t − τ)]e−jwt
dt

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, Fall-1390, WSP 14
• w(t) : windowing function generally a Gaussian pulse is used, other choices are
rectangular, elliptic etc . . .
• Maps 1D function to 2D time-frequency domain
• Advantages:
– Gives us time-frequency description of the signal
– Overcomes the difficulties of Fourier transform by use of windowing functions
STFT: Disadvantages
• Heisenberg Principle: One can not get infinite time and frequency resolution beyond
Heisenberg’s Limit
∆t ∆f ≥
1
4π
, ♣
where
∆t =
sR ∞
−∞
t2|g(t)|2dt
R ∞
−∞
|g(t)|2dt
, ∆f =
sR ∞
−∞
f2|G(f)|2df
R ∞
−∞
|G(f)|2df
Proof using Schwartz inequality:
Z ∞
−∞
g1(s)g2(s)ds
2
≤
Z ∞
−∞
g2
1(u)du
Z ∞
−∞
g2
2(u)du
Let’s g1(t) = tg(t), g2(t) = g′
(t) ⇒
Z ∞
−∞
tg(t)g′
(t)dt
2
≤
Z ∞
−∞
t2
g2
(t)dt
Z ∞
−∞
[g′
(t)]2
dt
[g′
(t)]2 F
←→[j2πfG(f)] ∗ [j2πfG(f)] = −4π2
Z ∞
−∞
νG(ν)(f − ν)G(f − ν)dν ⇒
Z ∞
−∞
[g′
(t)]2
dt = 4π2
Z ∞
−∞
ν2
|G(ν)|2
dν
g2
(t)
F
←→ G(f) ∗ G(f)
2g′
(t)g(t)
F
←→ j2πf[G(f) ∗ G(f)]
Z ∞
−∞
g(t)g′
(t)e−j2πft
dt = jπf
Z ∞
−∞
G(ν)G(f − ν)dν ⇒
Z ∞
−∞
(−j2πt)g(t)g′
(t) e−j2πft

16. f=0
dt = jπ
Z ∞
−∞
G(ν)G(f − ν)

20. f=0
dν+
jπf
Z ∞
−∞
G(ν)G′
(f − ν)dν

24. f=0
−2
Z ∞
−∞
tg(t)g′
(t)dt =
Z ∞
−∞
G(ν)G(−ν)dν =
Z ∞
−∞
|G(f)|2
df

25. AKUH· AMIN DAVAR
, Fall-1390, WSP 15
−2
Z ∞
−∞
tg(t)g′
(t)dt =
Z ∞
−∞
|G(f)|2
df
Z ∞
−∞
tg(t)g′
(t)dt
2
≤
Z ∞
−∞
t2
g2
(t)dt
Z ∞
−∞
4π2
f2
|G(f)|2
df
Divide both sides by E2
g =
Z ∞
−∞
g2
(t)dt
2
=
Z ∞
−∞
|G(f)|2
df
2
to have the desired
result ♣. The function that satisfies ♣ with equality is Gaussian:
g1(t) = K0g2(t) ⇒ tg(t) = K0g′
(t) ⇒ g(t) = K1et2/K0
• Trade offs:
– Wider window: Good frequency resolution, Poor time resolution
– Narrower window: Good time resolution, Poor frequency resolution
The definition of the class of bilinear (or quadratic) time-frequency distributions:
Cx(t, f) =
Z ∞
−∞
Z ∞
−∞
Ax(η, τ)Φ(η, τ) exp(−j2π(ηt + τf)) dη dτ,
where Ax (η, τ) is the ambiguity function .
Φ (η, τ) is the kernel function which is usually a low-pass function and is used to mask
out the interference.
The class of bilinear (or quadratic) time-frequency distributions can be most easily
understood in terms of the ambiguity function an explanation of which follows.
Consider the well known power spectral density Px (f) and the signal auto-correlation
function Rx (τ) in the case of a stationary process. The relationship between these
Px(f) =
Z ∞
−∞
Rx(τ)ej2πfτ
dτ, (4)
Rx(τ) = E{x(t + τ/2)x∗
(t − τ/2)} (5)
For a non-stationary signal x (t) , these relations can be generalized using a
time-dependent power spectral density or equivalently the famous Wigner
distribution function of x (t) as follows:
Wx(t, f) =
Z ∞
−∞
Rx(t, τ)e−j2πfτ
dτ, (6)
Rx (t, τ) = E{x(t + τ/2)x(t − τ/2)}. (7)
If the Fourier transform of the auto-correlation function is taken with respect to t
instead of τ, we get the ambiguity function as follows:
Ax(η, τ) =
Z ∞
−∞
x(t + τ/2)x∗
(t − τ/2)ej2πtη
dt.

26. AKUH· AMIN DAVAR
, Fall-1390, WSP 16
Ax(η, τ)
FtF−1
f
Fτ F−1
η
Wx(t, f)
Rx(t, τ)
Ft
F−1
η
F−1
f
Fτ
Figure 8: The relationship between the Wigner distribution function, the auto-
correlation function and the ambiguity function.
Linear vs. Quadratic Distributions
Linear: Quadratic
Pros: Pros:
Linear superposition Better time and frequency resolutions than linear
No interference terms for muti-component signals Shows the energy distribution
Cons: Cons:
Trade off between time and frequency resolutions Cross terms for multi-component signals
Heisenberg inequality
Wavelet Transform
• Overcomes the shortcoming of STFT by using variable length windows: i.e. Nar-
rower window for high frequency thereby giving better time resolution and Wider
window at low frequency thereby giving better frequency resolution
• Heisenberg’s Principle still holds
• Mathematical formula:
CWTΨ
x (τ, s) = Ψ(τ, s) =
1
p
|s|
Z
t
x(t)Ψ∗
(
t − τ
s
)dt
where

27. AKUH· AMIN DAVAR
, Fall-1390, WSP 17
Absolute Values of Ca,b Coefficients for a = 0.01 1.02 2.03 3.04 4.05 ...
time (or space) b
scales
a
100 200 300 400 500 600
0.01
5.06
10.11
15.16
20.21
25.26
30.31
35.36
40.41
45.46
50.51
55.56
60.61
65.66
70.71
75.76
80.81
85.86
90.91
95.96
50
100
150
200
Figure 9: Continuous wavelet of δ(t) for small scales(large frequencies) around 400 are
strong and as we move away from b = 400 there is conic relationship between scales and
delays.
x(t) = given signal
τ = translation parameter
s = scaling parameter = 1/f
Ψ(t) = Mother wavelet , All kernels are obtained by scaling
and/or translating mother wavelet
Example using Morlet wavelet Ψ(t) = cos(5t)e−t2/2
:
x=[zeros(1,400) 1 zeros(1,199)];
s=cwt(x,linspace(1e-2,100,100),’morl’,’plot’);
figure(gcf);pause;
colormap(’default’);
X=linspace(0,100,100);Y=0:599;
contour(Y,X,s);figure(gcf);pause;
mesh(Y,X,s);figure(gcf)
x=cos(2*pi*5*(0:199)/200);
s=cwt(x,linspace(1e-2,50,100),’morl’,’plot’);
figure(gcf);pause;colormap(’default’);
X=linspace(1e-2,50,100);Y=0:199;contour(Y,X,s);
figure(gcf);pause; mesh(Y,X,s);figure(gcf)

28. AKUH· AMIN DAVAR
, Fall-1390, WSP 18
0 100 200 300 400 500
0
10
20
30
40
50
60
70
80
90
100
−0.4
−0.3
−0.2
−0.1
0
0.1
0.2
0.3
0.4
Figure 10: Contour plot of wavelet of δ(t), for scales less than 30 wavelet vales of δ(t) are
large around b = 400.
Figure 11: Mesh plot of wavelet of δ(t).

29. AKUH· AMIN DAVAR
, Fall-1390, WSP 19
Absolute Values of Ca,b Coefficients for a = 0.01 0.51495 1.0199 1.5248 2.0298 ...
time (or space) b
scales
a
50 100 150 200
0.01
2.53475
5.05949
7.58424
10.109
12.6337
15.1585
17.6832
20.208
22.7327
25.2575
27.7822
30.307
32.8317
35.3565
37.8812
40.406
42.9307
45.4555
47.9802
50
100
150
200
Figure 12: Continuous wavelet of cos(2π5t) for small scales around 30 for almost all delays
is strong but we observe some repetition in x−axis.
0 50 100 150
5
10
15
20
25
30
35
40
45
50
−6
−4
−2
0
2
4
6
Figure 13: Contour plot of wavelet of cos(2π5t), for scales around 30 wavelet vales of δ(t)
are strong.

30. AKUH· AMIN DAVAR
, Fall-1390, WSP 20
0
50
100
150
200
0
20
40
60
−10
−5
0
5
10
−6
−4
−2
0
2
4
6
Figure 14: Mesh plot of wavelet of cos(2π5t).
close all
t=linspace(0,1,100); x=[sin(2*pi*5*t)];
figure(1)
cw1 = cwt(x,1:30,’morl’,’plot’);
title(’Continuous Transform, absolute coefficients.’)
ylabel(’Scale’)
figure(2)
%[cw1,sc] = cwt(x,1:40,’morl’,’scal’);
[cw1,sc] = cwt(x,1:30,’morl’,’scalCNT’);
title(’Scalogram’)
ylabel(’Scale’)
figure(gcf)
close all
t=linspace(0,1,100); x=[sin(2*pi*5*t) sin(2*pi*15*t) sin(2*pi*25*t)];
figure(1)
cw1 = cwt(x,1:30,’morl’,’plot’);
title(’Continuous Transform, absolute coefficients.’)
ylabel(’Scale’)
figure(2)
%[cw1,sc] = cwt(x,1:40,’morl’,’scal’);
[cw1,sc] = cwt(x,1:30,’morl’,’scalCNT’);
title(’Scalogram’)

31. AKUH· AMIN DAVAR
, Fall-1390, WSP 21
ylabel(’Scale’)
figure(gcf)
Scale values determine the degree to which the wavelet is compressed or stretched. Low
scale values compress the wavelet and correlate better with high frequencies. The low
scale CWT coefficients represent the fine-scale features in the input signal vector. High
scale values stretch the wavelet and correlate better with the low frequency content of
the signal. The high scale CWT coefficients represent the coarse-scale features in the
input signal.
Continuous Wavelet transform
• The kernel functions used in wavelet transform are all obtained from one prototype
function known as mother wavelet, by scaling and/or translating it.
Ψa,b(t) =
1
√
a
Ψ(
t − b
a
)
a = scale parameter
b = translation parameter
Ψ1,0(t) = Ψ(t)
• Continuous Wavelet transform
W(a, b) =
1
√
a
Z ∞
−∞
x(t)Ψa,b(t)dt
• In order to become a wavelet a function must satisfy the above two conditions
Z ∞
−∞
Ψ(t)dt = 0
Z ∞
−∞
|Ψ(t)|2
dt ∞
Inverse wavelet transform
x(t) =
1
C
Z ∞
−∞
Z ∞
−∞
1
a2
W(a, b)Ψa,b(t)dadb
where
C =
Z ∞
−∞
|Ψ(w)|
|w|
dw
provided
Z ∞
−∞
Ψ(t)dt = 0

32. AKUH· AMIN DAVAR
, Fall-1390, WSP 22
Proof:
f(ω) =
1
2π
Z ∞
−∞
f(t)e−jωt
dt, Fourier transform
F(ω, s) =
1
2π
Z ∞
−∞
f(t)g(t − s)e−jωt
dt, Windowed transform
For each s, we recover f(t)g(t − s) by
f(t)g(t − s) =
Z ∞
−∞
F(ω, s)ejωt
dω
Multiply both sides by g∗
(t − s) and integrate over s.
f(t)
Z ∞
−∞
g(t − s)g∗
(t − s)ds =
Z ∞
−∞
Z ∞
−∞
F(ω, s)g∗
(t − s)ejωt
dωds
f(t) =
1
||g(t)||2
Z ∞
−∞
Z ∞
−∞
F(ω, s)g∗
(t − s)ejωt
dωds
This is the reconstruction from time-frequency(space) transform.
So far we concentrated on windowed STFT.
Wavelets begin with one function Ψ(t). The position variable s still comes from
translation Ψ(t − s). Now comes the difference. Instead of the frequency variable we have
a scale variable.
Instead of modulating them we rescale them. The “time-scale atoms” are the translates
and dilates of Ψ(t) :
Ψa,b(t) =
1
√
a
Ψ((t − b)/a), Wavelet functions
The mother wavelet is Ψ(t) = Ψ1,0(t). The factor |a|−0.5
assures that rescaled wavelets
have equal energy ||Ψa,b(t)|| = ||Ψ(t)||. We normalize so that all these functions have
unit norm ||Ψa,b(t)|| = 1.
Scaling the time by aj
automatically scales the translation steps by a−j
Ψ(aj
t − k) = Ψ(aj
[t − a−j
k])
The mesh length at level j is scaled down by a−j
. The frequency is scaled up by aj
. This
hyperbolic scaling or dyadic scaling or octave scaling (a = 2) is a prime characteristic of
wavelet analysis.
Any smooth decaying function Ψ(t) is a mother wavelet provided that
Z
Ψ(t)dt = 0
F
n
Wf (a, b) = f(t) ∗
p
|a|−1Ψ(−t/a)
o
=
p
|a|F(ω)Ψ∗
(aω)

33. AKUH· AMIN DAVAR
, Fall-1390, WSP 23
F
nh
Wg(a, b) = g(t) ∗
p
|a|−1Ψ(−t/a)
i∗o
=
p
|a|G∗
(ω)Ψ(aω)
Wf (a, b) ←→
p
|a|F(ω)Ψ∗
(aω)
W∗
g (a, b) ←→
p
|a|G(ω)Ψ(aω)
Multiply Wf (a, b) and W∗
g (a, b) and integrate over b
Z ∞
−∞
Wf (a, b)W∗
g (a, b)db = 2π
Z ∞
−∞
|a|F(ω)G∗
(ω)|Ψ(aω)|2
dω
Integrate this equation with respect to da/|a|2
and reverse the order of integration:
Z ∞
−∞
Z ∞
−∞
Wf (a, b)W∗
g (a, b)
dadb
|a|2
= 2π
Z ∞
−∞
Z ∞
−∞
|a|F(ω)G∗
(ω)|Ψ(aω)|2 dωda
|a|2
, U
2π
Z ∞
−∞
Z ∞
−∞
F(ω)G∗
(ω)|Ψ(aω)|2 dωda
|a|
= 2π
Z ∞
−∞
F(ω)G∗
(ω)dω
Z ∞
−∞
|Ψ(aω)|2 da
|a|
2π
Z ∞
−∞
|Ψ(aω)|2 da
|a|
= 2π
Z ∞
−∞
|Ψ(ω)|2
|ω|
dω
Obtained by ν = aω, dν = ωda, da/|a| = dω/|ω|
Therefore,
Z ∞
−∞
Z ∞
−∞
Wf (a, b)W∗
g (a, b)
dadb
|a|2
= C
Z ∞
−∞
f(t)g∗
(t)dt, C = 2π
Z ∞
−∞
|Ψ(ω)|2
|ω|
dω
By assuming g(t − τ) = δ(t) ⇒ Wg(a, b) = Ψ((τ − b)/a)/
√
a, from equation U we have
Z ∞
−∞
Z ∞
−∞
Wf (a, b)Ψ((τ − b)/a)/
√
a
dadb
|a|2
= C
Z ∞
−∞
f(t)δ(t − τ)dt = Cf(τ) ⇒
f(t) =
1
C
Z ∞
−∞
Z ∞
−∞
Wf (a, b)Ψ((t − b)/a)/
√
a
dadb
|a|2
Example of Meyer wavelet and its Fourier transform
Meyer wavelet is defined in frequency domain, by refering to MATLAB for their
representation:
ψ(ω) =
(
ejω/2
√
2π
sin(0.5v(3|ω|/2π − 1)) 2π
3
≤ |ω| ≤ 4π
3
ejω/2
√
2π
sin(0.5v(3|ω|/4π − 1)) 4π
3
≤ |ω| ≤ 8π
3
ψ(ω) = 0, ω ∋
2π
3
,
8π
3
v(a) = a4
(35 − 84a + 70a2
− 20a3
), a ∈ [0, 1]
φ(x) =
1
√
2π
|ω| ≤
2π
3
φ(x) =
1
√
2π
cos(0.5v(3|ω|/2/π − 1))
2π
3
≤ |ω| ≤
4π
3
φ(x) = 0 |ω| ≥
4π
3

34. AKUH· AMIN DAVAR
, Fall-1390, WSP 24
−8 −6 −4 −2 0 2 4 6 8
−1
−0.5
0
0.5
1
1.5
Meyer wavelet: ψ(x)
x
ψ(x)
−600 −400 −200 0 200 400 600
−200
−150
−100
−50
0
50
f
F{ψ(x)}
Figure 15: Wavelet ψ and its Fourier transform, they are very much concentrated in time
and frequency domain.
[phi,psi,x] = wavefun(’meyr’,10);
figure(1)
subplot(211)
plot(x,psi);grid;figure(gcf);
title(’Meyer wavelet: $psi(x)$’)
xlabel(’$x$’);ylabel(’$psi(x)$’);
subplot(212)
f=linspace(-.5,.5,length(x))*length(x);
PSI=abs(fftshift(fft(psi)));
plot(f,10*log10(PSI));grid;figure(gcf);
xlabel(’$f$’);ylabel(’$mathcal{F}{psi(x)}$’);
figure(2)
subplot(211)
plot(x,phi);grid;figure(gcf);
title(’Meyer wavelet: $phi(x)$’)
xlabel(’$x$’);ylabel(’$phi(x)$’);
subplot(212)
f=linspace(-.5,.5,length(x))*length(x);
PHI=abs(fftshift(fft(phi)));
plot(f,10*log10(PHI));grid;figure(gcf);
xlabel(’$f$’);ylabel(’$mathcal{F}{phi(x)}$’);

35. AKUH· AMIN DAVAR
, Fall-1390, WSP 25
−8 −6 −4 −2 0 2 4 6 8
−0.5
0
0.5
1
1.5
Meyerwavelet: φ(x)
x
φ(x)
−600 −400 −200 0 200 400 600
−200
−100
0
100
f
F
{φ(x)}
Figure 16: Scaling function(father wavelet) φ and its Fourier transform, they are very
much concentrated in time and frequency domain.
Constant Q-filtering
• CWT can be rewritten as
W(a, b) = x(b) ∗ Ψ∗
a,0(−b)
Proof:
Ψa,0(t) =
1
√
a
Ψ(t/a) ⇒ Ψa,0(−b) =
1
√
a
Ψ(−b/a)
x(b) ∗ Ψa,0(−b) =
Z ∞
−∞
x(t)
1
√
a
Ψ(−(b − t)/a) = CWTΨ
x (a, b)
• A special property of the above filter defined by the mother wavelet is that they are
Constant-Q filters
• Q factor = Center frequency/Bandwidth
Q = 2π
Energy Stored
Energy dissipated per cycle
= 2πfr
Energy Stored
Power Loss
, fr = resonant frequency
• Hence the filter defined by wavelet increases their Bandwidth as scale increases (
i.e. center frequency increases )
• This leads to filter bank implementation of discrete wavelet transform

36. AKUH· AMIN DAVAR
, Fall-1390, WSP 26
I. Daubechies believes that the wavelet transform can be used to analyze time series
that contain nonstationary power at many different frequencies.
The term “wavelet function” is used generically to refer to either orthogonal or
nonorthogonal wavelets. The term “wavelet basis” refers only to an
orthogonal set of functions. The use of an orthogonal basis implies the use of the
discrete wavelet transform, while a nonorthogonal wavelet function can be used with
either the discrete or the continuous wavelet transform.
The continuous wavelet transform of a discrete sequence xn is defined as the convolution
of xn with a scaled and translated version of ψ0(t):
Wn(s) =
N−1
X
n′=0
xn′ ψ∗
((n′
− n)δt/s) (8)
in this formula ψ(·) is normalized:
Z ∞
−∞
|ψ̂(ω′
)|2
dω′
= 1
Use of DFT allows to do convolution (8) easily
x̂k =
1
N
N−1
X
n=0
xne−j2πkn/N
Using F{ψ(t/s)} = ψ̂(sω), By the convolution theorem, the wavelet transform is the
inverse Fourier transform of the product:
Wn(s) =
N−1
X
k=0
x̂kψ̂∗
(sωk)ejωknδt
(9)
angular frequency ωk :
ωk =







2πk
Nδt
, k ≤ N
−
2πk
Nδt
, k N
Using (9), the continuous wavelet transform (for a given s) at all n is calculated
simultaneously and efficiently.
To ensure that the wavelet transforms (9) at each scale s are directly comparable to
each other and to the transforms of other time series, the wavelet function at each scale
s is normalized to have unit energy:
ψ̂(sωk) =
r
2πs
δt
ψ̂0(sωk), ψ((n′
− n)δt/s) =
r
δt
s
ψ((n′
− n)δt/s) (10)

37. AKUH· AMIN DAVAR
, Fall-1390, WSP 27
Name ψ0(t) ψ̂0(sω)
Morlet
(ω0=frequency)
π−1/4
ejω0t
e−t2/2
π−1/4
H(ω)e−(sω−ω0)2/2
Paul
(m=order)
2mjmm!
√
π(2m)!
(1 − jt)−(m+1) 2m
√
m(2m−1)!
H(ω)(sω)m
e−sω
DOG
(m=derivative)
(−1)m+1
√
Γ(m+1/2)
dm
dtm e−t2/2 −jm
√
Γ(m+1/2)
(sω)m
e−(sω)2/2
Table 1: H(ω) = Heaviside step function, H(ω) = 1 if ω 0, H(ω) = 0 otherwise. DOG
= derivative of a Gaussian; m = 2 is the Marr or Mexican hat wavelet.
Some of the unscaled ψ̂0 are defined in Table 1 to have
Z ∞
−∞
|ψ̂(ω′
)|2
dω′
= 1
Using these normalizations, at each scale s :
N−1
X
k=0
|ψ̂(sωk)|2
= N
Discrete wavelet Transform
• Discrete domain counterpart of CWT
• Implemented using Filter banks satisfying PR(perfect reconstruction) condition
• Represents the given signal by discrete coefficients {dk,n}
• DWT is given by
Ψk,n(t) = 2−k/2
Ψ(2−k
t − n)
x(t) =
X
k
X
n
dk,nΨk,n(t), dk,n = hx, Ψk,ni =
Z
x(t)Ψ∗
k,n(t)dt
Scaling Function
• These are functions used to approximate the signal up to a particular level of detail
φk,n(t) = 2k/2
φ(2k
t − n)

38. AKUH· AMIN DAVAR
, Fall-1390, WSP 28
0 0.5 1
0
0.5
1
1.5
t
Φ(t)
0 0.5 1
−2
−1
0
1
2
Ψ(t)
t
Figure 17: Haar Scaling and Wavelet Function.
[PHI,PSI,XVAL] = wavefun(’db2’,9)
f=linspace(-.5,.5,length(XVAL))*length(XVAL)
G1=abs(fftshift(fft(PSI)));
G2=abs(fftshift(fft(PSI)));
Refinement equation and wavelet Equation
• Refinement equation is an equation relating to scaling function and filter coefficients
φ(t) =
√
2
N
X
k=0
h0(k)φ(2t − k), h0(k) is even length,
X
k
h0(k) =
√
2
• Wavelet equation is an equation relating to wavelet function and filter coefficients
ψ(t) = 2
N
X
k=0
h1(k)ψ(2t − k)
• By solving the above two we can obtain the scaling and wavelet function for a given
filter bank structure
Some Important properties of wavelets
• Compact Support:
– Finite duration wavelets are called compactly supported in time domain but
are not band-limited in frequency. They can be implemented using FIR filters
– Examples: Haar, Daubechies, Symlets , Coiflets
– Narrow band wavelets are called compactly supported in frequency domain.
Can be implemented using IIR filters

39. AKUH· AMIN DAVAR
, Fall-1390, WSP 29
0 1 2 3
−0.5
0
0.5
1
1.5
t
Φ(t)
0 1 2 3
−2
−1
0
1
2
t
Ψ(t)
−1000 0 1000
−60
−40
−20
0
f
10log
10
(F
{Φ(t)})
−1000 0 1000
−200
−150
−100
−50
0
f
10log
10
(F
{Ψ(t)})
Figure 18: Db2 Scaling and Wavelet Function.
– Examples: Meyer’s wavelet
• Symmetry
• Symmetric / Antisymmetric wavelets have linear-phase
• Orthogonal wavelets are asymmetric and have a non-linear phase
• Biorthogonal wavelets are asymmetric but have linear phase and can be implemented
using FIR filters
• Vanishing Moment:
R
tk
ψ(t)dt = 0, k = 0, 1, · · · , p − 1
• p-th vanishing moment is defined as
Mp =
Z
tp
Ψ(t)dt
• The more the number of moments of a wavelets are zero the more is its compressive
power
• Smoothness
• it is roughly the number of times a function can be differentiated at any given point
• This is closely related to vanishing Moments
• Smoothness provides better numerical stability

40. AKUH· AMIN DAVAR
, Fall-1390, WSP 30
• It also provides better reconstruction property
Orthogonal wavelet
An orthogonal wavelet is a wavelet where the associated wavelet transform is
orthogonal. That is the inverse wavelet transform is the adjoint of the wavelet
transform. If this condition is weakened you may end up with biorthogonal wavelets.
Orthogonal filters using MATLAB:
[H0,H1,F0,F1] = wfilters(’db2’)
sum(H0.*H1)=-6.9389e-018
sum(F0.*F1)= 6.9389e-018
Biorthogonal filters using MATLAB:
[Rf,Df] = biorwavf(’bior3.3’)
8 filters are obtained:
[H01,H11,F01,F11,H02,H12,F02,F12] = biorfilt(Df,Rf,’8’)
sum(H01.*H11)=1.0495e-016
sum(F01.*F11)=-1.0495e-016
sum(H02.*H12)=6.9389e-018
sum(F02.*F12)=-6.9389e-018
The scaling function is a refinable function. That is, it is the a fractal functional
equation, called refinement equation:
φ(x) =
N−1
X
k=0
akφ(2x − k),
where the sequence
(a0, . . . , aN−1)
of real numbers is called scaling sequence or scaling mask. The wavelet proper is
obtained by a similar linear combination,
ψ(x) =
M−1
X
k=0
bkφ(2x − k),
where the sequence
(b0, . . . , bM−1)
of real numbers is called wavelet sequence or wavelet mask.

41. AKUH· AMIN DAVAR
, Fall-1390, WSP 31
A necessary condition for the orthogonality of the wavelets is, that the scaling sequence
is orthogonal to any shifts of it by an even number of coefficients:
X
n∈Z
anan+2m = 2δm,0
X
(−1)k
km
h0(k) = 0, m = 0, 1, · · · , p − 1
In this case there is the same number M=N of coefficients in the scaling as in the
wavelet sequence, the wavelet sequence can be determined as
bn = (−1)n
aN−1−n.
In some cases the opposite sign is chosen.
Vanishing moments, polynomial approximation and smoothness
A necessary condition for the existence of a solution to the refinement equation is, that
some power
(1 + Z)A
, A 0,
divides the polynomial
a(Z) := a0 + a1Z + · · · + aN−1ZN−1
.
The maximally possible power A is called polynomial approximation order (or pol. app.
power) or number of vanishing moments. It describes the ability to represent
polynomials up to degree A − 1 with linear combinations of integer translates of the
scaling function.
In the biorthogonal case, an approximation order A of φ corresponds to A vanishing
moments of the dual wavelet ψ̃, that is, the scalar products of ψ̃ with any polynomial up
to degree A − 1 are zero. In the opposite direction, the approximation order à of φ̃ is
equivalent to à vanishing moments of ψ. In the orthogonal case, A and à coincide.
Biorthogonal filters using MATLAB can have linear phase
[PHI1,PSI1,PHI2,PSI2,XVAL] = wavefun(’bior3.3’,5);
plot(XVAL,abs(fftshift(fft(PSI1))),XVAL,abs(fftshift(fft(PSI2))))
plot(XVAL,angle(fft(PSI1)),XVAL,angle(fft(PSI2)))
A sufficient condition for the existence of a scaling function is the following: If one
decomposes a(Z) = 21−A
(1 + Z)A
p(Z), and the estimate holds
1 ≤ sup
t∈[0,2π]
|p(eit
)| 2A−1−n
for some n ∈ N,
holds, then the refinement equation has a n times continuously differentiable solution
with compact support.
Examples:

42. AKUH· AMIN DAVAR
, Fall-1390, WSP 32
• a(Z) = 21−A
(1+Z)A
, that is p(Z) = 1, has n = A−2. The solutions are Schoenbergs
B-splines of order A − 1, where the (A − 1)-th derivative is piecewise constant, thus
the (A − 2)-th derivative is Lipschitz-continuous. A = 1 corresponds to the index
function of the unit interval.
• A = 2 and p linear may be written as a(Z) = 1
4
(1 + Z)2
((1 + Z) + c(1 − Z)).
Expansion of this degree 3 polynomial and insertion of the 4 coefficients into the
orthogonality condition results in c2
= 3. The positive root gives the scaling sequence
of the Db4-wavelet.
Biorthogonal wavelet
A biorthogonal wavelet is a wavelet where the associated wavelet transform is invertible
but not necessarily orthogonal. Designing biorthogonal wavelets allows more degrees of
freedoms than orthogonal wavelets. One additional degree of freedom is the possibility
to construct symmetric wavelet functions.
In the biorthogonal case, there are two scaling functions φ, φ̃, which may generate
different multiresolution analyses, and accordingly two different wavelet functions ψ, ψ̃.
So the numbers M, N of coefficients in the scaling sequences a, ã may differ. The
scaling sequences must satisfy the following biorthogonality condition
X
n∈Z
anãn+2m = 2 δm,0.
Then the wavelet sequences can be determined as
bn = (−1)n
ãM−1−n, n = 0, . . . , M − 1
and
b̃n = (−1)n
aM−1−n, n = 0, . . . , N − 1.
Down-Sampling:
y[n] = x[2n]







.
.
.
y[−1]
y[0]
y[1]
.
.
.







=






· · · · · · · · · · · · · · · · · · · · ·
· · · 1 0 0 0 0 · · ·
· · · 0 0 1 0 0 · · ·
· · · 0 0 0 0 1 · · ·
· · · · · · · · · · · · · · · · · · · · ·















.
.
.
x[−1]
x[0]
x[1]
x[2]
.
.
.










43. AKUH· AMIN DAVAR
, Fall-1390, WSP 33
Y (ejω
) = 1
2
X ejω/2
+ X ej(ω−2π)/2
Y (z) = 1
2
X z1/2
+ X −z1/2
Up-Sampling:
y[n] =
x[n/2], n even
0, n odd









.
.
.
y[−1]
y[0]
y[1]
y[2]
.
.
.









=










· · · · · · · · · · · · · · ·
· · · 1 0 0 · · ·
· · · 0 0 0 · · ·
· · · 0 1 0 · · ·
· · · 0 0 0 · · ·
· · · 0 0 1 · · ·
· · · · · · · · · · · · · · ·



















.
.
.
x[−1]
x[0]
x[1]
x[2]
.
.
.









Y (ejω
) = X (ej2ω
)
Y (z) = X (z2
)
Filter Bank(FB):
First FB designed for speech coding, [Croisier-Esteban-Galand 1976]
Orthogonal FIR filter bank, [Smith-Barnwell 1984], [Mintzer 1985]
H0(z)
H1(z)
↓ 2
↓ 2
Q
Q is processing
↑ 2
↑ 2
F1(z)
F0(z)
+ X̂(z)
X(z)
Figure 19: FB analysis and synthesis.
Example using MATLAB
x=[zeros(1,100) ones(1,100) zeros(1,100)];
[xa,xd]=dwt(x,’db2’);
plot(xa);figure(gcf);pause
plot(xd);figure(gcf);pause
[xa,xd]=dwt(x,’bior3.3’);
plot(xa);figure(gcf);pause
plot(xd);figure(gcf);pause

44. AKUH· AMIN DAVAR
, Fall-1390, WSP 34
X̂(z) =
1
2
for Distortion Elimination, set to 2z−ℓ
z }| {
[F0(z)H0(z) + F1(z)H1(z)] X(z)
+
1
2
[F0(z)H0(−z) + F1(z)H1(−z)]
| {z }
for Aliasing Cancellation, set to 0
X(−z) = z−ℓ
X(z)
The filters H0(z) and H1(z) are still to be chosen, these choices are connected.
Historically, designers chose the lowpass filter coefficients {h0(0), · · · , h0(N)} and then
construct H0(z) and H1(z). There are three possibilites that produce equal length
filters:
Early choice: Alternating signs: H1(z) = H0(−z) to produce h1(k) = {h0(0), −h0(1), h0(2), −h0(3) · · ·}
Better choice: Alternating flip: H1(z) = −z−N
H0(−z−1
) to produce h1(k) = {h0(N), −h0(N − 1), · · ·}
General choice: F0(z)H0(z) is a halfband filter. This gives biorthogonality,
when aliasing is cancelled by relation of F0 to H0 and F1 to H1
Alternating-Sign Construction:
F0(z) = H1(−z)
F1(z) = −H0(−z)
Perfect Reconstruction:
With Aliasing Cancellation:
F0(z) = H1(−z)
F1(z) = −H0(−z)
Distortion Elimination becomes:
F0(z)H0(z) − F0(−z)H0(−z) = 2z−l
Half-band Filter:
⇒ P0(z) − P0(−z) = 2z−ℓ
where P0(z) ≡ F0(z)H0(z)
Half-band Filter:
P0(z) = a + bz−1
+ cz−2
+ dz−3
+ ez−4
P0(−z) = a − bz−1
+ cz−2
− dz−3
+ ez−4

45. AKUH· AMIN DAVAR
, Fall-1390, WSP 35
P0(z) − P0(−z) = 0 + 2bz−1
+ 0 + 2dz−3
+ 0 = 2z−ℓ
can only have one odd-power
Standard design procedure:
Design a good low-pass half-band filter P0(z) Factor P0(z) into H0(z) and F0(z)
Use the aliasing cancellation condition to obtain H1(z) and F1(z)
Examples:
[H0,H1,F0,F1] = wfilters(’db2’)
P0=conv(H0,F0)=[-0.0625,0.0000,0.5625,1.0000,0.5625,0.0000,-0.0625]
[H W]=freqz(P0);
HH=polyval(P0,exp(j*(W+pi)));
plot(W,abs(H)/16,W,abs(HH))
Spectral Factorization:
Orthogonality:
fi[n] = hi[−n], Fi(z) = Hi(z−1
)
Example:
[H0,H1,F0,F1] = wfilters(’db2’);
f_i [n] = h_i [ - n]:
H0 =[-0.1294,0.2241,0.8365,0.4830]
F0=[0.4830,0.8365,0.2241,-0.1294]
H1=[-0.4830,0.8365,-0.2241,-0.1294]
F1=[-0.1294,-0.2241,0.8365,-0.4830]
z and z−ℓ
must be separated Symmetry:
hi[n] = ±hi[L − 1 − n], Hi(z) = ±z−(L−1)
Hi(z−1
)
z and z−ℓ
must stay together
P0(z) = H0(z)F0(z) =
Y
n
(1 − znz−1
); {zn} = roots
H0(z) =
Y
k∈H
(1 − zkz−1
)
F0(z) =
Y
ℓ∈F
(1 − zℓz−1
)
Filter Banks(FB):
perfect reconstruction; halfband filters and possible factorizations.
Example: Product filter of degree 6

46. AKUH· AMIN DAVAR
, Fall-1390, WSP 36
multiple
zeros at
z = −1 ℜ(z)
ℑ(z)
Figure 20: Zeros of Half-band Filter.
[H0,H1,F0,F1] = wfilters(’db2’);
P0(z) = conv(H0, F0) =
1
16
(−1 + 9z−2
+ 16z−3
+ 9z−4
− z−6
)
P0(z) − P0(−z) = 2z−3
⇒
Expect perfect reconstruction with a 3 sample delay
Centered form:
P(z) = z3
P0(z) =
1
16
(−z3
+ 9z1
+ 16 + 9z−1
− z−3
)
P(z) + P(−z) = 2
In the frequency domain:
P(eiω
) + P(ei(ω+π)
) = 2, halfband condition

47. AKUH· AMIN DAVAR
, Fall-1390, WSP 37
ω
Halfband
π/2
1
Figure 21: Antisymmetry about ω = π/2.
How do we factor P0(z) into H0(z)F0(z)?
P0(z) =
1
16
(1 + z−1
)4
(−1 + 4z−1
− z−2
)
= −
1
16
(1 + z−1
)4
(2 +
√
3 − z−1
)(2 −
√
3 − z−1
))
4th order
zero at
z = −1 ℜ(z)
ℑ(z)
Some possible factorizations:
H0(z) or F0(z) F0(z) or H0(z)
(a) 1 − 1
16
(1 + z−1)4(2 +
√
3 − z−1)(2 −
√
3 − z−1)
(b) (1 + z−1
)/2 − 1
8
(1 + z−1
)3
(2 +
√
3 − z−1
)(2 −
√
3 − z−1
)
(c) (1 + z−1
)/4 − 1
4
(1 + z−1
)2
(2 +
√
3 − z−1
)(2 −
√
3 − z−1
)
(d) (1 + z−1
)(2 +
√
3 − z−1
)/2 − 1
8
(1 + z−1
)3
(2 −
√
3 − z−1
)
(e) (1 + z−1
)3
/8 − 1
2
(1 + z−1
)(2 +
√
3 − z−1
))(2 −
√
3 − z−1
)
(f)
√
3−1
4
√
2
(1 + z−1
)2
(2 +
√
3 − z−1
)) −
√
2
4(
√
3−1)
(1 + z−1
)2
(2 −
√
3 − z−1
)
(g) 1
16
(1 + z−1
)4
(2 +
√
3 − z−1
)(2 −
√
3 − z−1
)
case (b)– Symmetric filters (linear phase):
Case (c)–Symmetric filters (linear phase)
Case (f)–Orthogonal filters (minimum phase/maximum phase)
that, in this case, one filter is the flip (transpose) of the other:
f0(n) = h0(3 − n) ⇔ F0(z) = z−3
H0(z−1
)

48. AKUH· AMIN DAVAR
, Fall-1390, WSP 38
1st
order
2−
√
3 2+
√
3
3rd
order
filter length=2
1/2{1, 1}
filter length=6
1/4{−1, 1, 8, 8, 1, −1}
2nd
order
2−
√
3 2+
√
3
2nd
order
filter length=3
1/4{1, 2, 1}
filter length=5
1/4{−1, 2, 6, 2, −1}
General form of product filter (to be derived later):
P(z) = 2
1 + z−1
2
p
1 + z
2
p p−1
X
k=0
p + k − 1
k
1 − z
2
k
1 + z−1
2
k
P0(z) = z−(2p−1)
P(z) (11)
=
(1 + z−1
)2p
| {z }
Binomial
(spline filter)
1
22p−1
p−1
X
k=0
p + k − 1
k
(−1)k
z−(p−1)+k
1 − z−1
2
2k
| {z }
Q(z)
cancels all odd powers
except z−(2p−1)
(12)

49. AKUH· AMIN DAVAR
, Fall-1390, WSP 39
2−
√
3
2nd
order
2+
√
3
2nd
order
filter length=4
1
4
√
2
n
1 +
√
3, 3 +
√
3,
3 −
√
3, 1 −
√
3
o
filter length=4
1
4
√
2
n
1 −
√
3, 3 −
√
3,
3 +
√
3, 1 +
√
3
o
p = 1
P0(z) has degree 2 99K leads to Haar filter bank.
{1, 1, 1, 1}
H0(z)
H1(z)
↓2
↓2 {0, 0}
{1, 1}
H0(z)
H1(z)
↓2
↓2
1
0
F0(z) = 1 + z−1
, H0(z) = 1+z−1
2
, H1(z) = 1 − z−1
Synthesis lowpass filter has 1 zero at π Leads to cancellation of constant signals in
analysis highpass channel.
Additional zeros at π would lead to cancellation of higher order polynomials.

50. AKUH· AMIN DAVAR
, Fall-1390, WSP 40
p = 2
P0(z) has degree 4p − 2 = 6
From (11):
P0(z) =
1
16
(−1 + 9z−2
+ 16z−3
+ 9z−4
− z−6
)
Possible factorizations:
{1/8, 2/6, 3/5
| {z }
linear phase
, 4/4
|{z}
db4
}
2−
√
3 2+
√
3
4nd
order
% function [p0,b,q] = prodfilt(p)
%
% Generate the halfband product filter of degree 4p-2.
%
% p0 = coefficients of product filter of degree 4p-2.
% b = coefficients of binomial (spline) filter of degree 2p
% q = coefficients of filter of degree 2p-2 that produces the halfband
% filter p0 when convolved with b.
function [p0,b,q] = prodfilt(p)
% Binomial filter (1 + z^-1)^2p
tmp1 = [1 1];
b = 1;
for k = 0:2*p-1
b = conv(b, tmp1);
end
% Q(z)
tmp2 = [-1 2 -1] / 4;
q = zeros(1,2*p-1);

51. AKUH· AMIN DAVAR
, Fall-1390, WSP 41
vec = zeros(1,2*p-1);
vec(p) = 1;
for k=0:p-1
q = q + vec;
vec = conv(vec, tmp2) * (p + k) / (k + 1);
vec = wkeep(vec, 2*p-1);
end
q = q / 2^(2*p-1);
% Halfband filter, P0(z).
p0 = conv(b, q);
Spline Wavelet
Definition: Spline wavelet is a wavelet constructed using a spline function.
Definition: In mathematics, a spline function is a special function defined piecewise by
polynomials.
The simplest spline is a piecewise polynomial function, with each polynomial having a
single variable. The spline S takes values from an interval [a, b] and maps them to R, the
set of real numbers,
S : [a, b] → R. Since S is piecewise defined, choose k subintervals to partition [a, b] :
[ti, ti+1] , i = 0, . . . , k − 1 [a, b] = [t0, t1] ∪ [t1, t2] ∪ · · · ∪ [tk−2, tk−1] ∪ [tk−1, tk]
[a, b] = [t0, t1] ∪ [t1, t2] ∪ · · · ∪ [tk−2, tk−1] ∪ [tk−1, tk] a = t0 ≤ t1 ≤ · · · ≤ tk−1 ≤ tk = b
a = t0 ≤ t1 ≤ · · · ≤ tk−1 ≤ tk = b
Each of these subintervals is associated with a polynomial Pi,
Pi : [ti, ti+1] → R Pi : [ti, ti+1] → R.
On the ith subinterval of [a, b], S is defined by Pi,
S(t) = P0(t) , t0 ≤ t t1, S(t) = P0(t) , t0 ≤ t t1, S(t) = P1(t) , t1 ≤ t t2,
S(t) = P1(t) , t1 ≤ t t2, S(t) = Pk−1(t) , tk−1 ≤ t ≤ tk. The given k + 1 points
tj(0 ≤ j ≤ k), t = (t0, . . . , tk) is called a knot vector for the spline. If the knots are
equidistantly distributed in the interval [a, b] we say the spline is uniform, otherwise we
say it is non-uniform.
If the k polynomial pieces Pi each have degree at most n, then the spline is said to be of
degree ≤ n (or of order ≤ n + 1).
Example
func=@(x)sin(x.^2);
b=linspace(1,10,100);
for k=1:100;a=0;b(k);
int1(k) = integral(func,a,b(k),’RelTol’,0,’AbsTol’,1e-12);
x = a:.001:b(k); y = sin(x.^2);
pp = spline(x,y);
int2(k) = integral(@(x)ppval(pp,x),a,b(k),’RelTol’,0,’AbsTol’,1e-12);
end

52. AKUH· AMIN DAVAR
, Fall-1390, WSP 42
1 2 3 4 5 6 7 8 9 10
0.4
0.5
0.6
0.7
0.8
0.9
1
b
Z
b
0
sin(x
2
)
dx
Figure 22: Application of Spline interpolation.
Natural cubic splines
Cubic splines have polynomial pieces of the form
Pi(x) = ai + bi(x − xi) + ci(x − xi)2
+ di(x − xi)3
. Given k + 1 coordinates
(x0, y0), (x1, y1), . . . , (xk, yk), we find k polynomials Pi(x),Pi(x), which satisfy for
1 ≤ i ≤ k − 1 :
P0(x0) = y0 and Pi−1(xi) = yi = Pi(xi), P′
i−1(xi) = P′
i (xi), P′′
i−1(xi) = P′′
i (xi),
P′′
0 (x0) = P′′
k−1(xk) = 0.
One such polynomial Pi is given by a 5-tuple (a, b, c, d, x) where a, b, c and d correspond
to the coefficients as used above and x denotes the variable over the appropriate domain
[xi, xi+1]
Two dimensional interpolation
% original coordinates
% close all
% [x,y]=meshgrid(1:256,1:256);
% z=imread(’cameraman.tif’);
% figure(1); imshow(z1); figure(gcf)
% % new coordinates
% a=2; x1=x/a; y1=y/a;
% % Do the interpolation
% z1=interp2(x,y,double(z),x1,y1,’cubic’);
% figure(2);imshow(uint8(z1));figure(gcf)
%pause

53. AKUH· AMIN DAVAR
, Fall-1390, WSP 43
Figure 23: Original image.
%%%%%%%%%%%%%%%%
close all
z=double(imread(’cameraman.tif’));
imshow(uint8(z));figure(1); figure(gcf); pause(001)
% Do the interpolation
z1=uint8(interp2(z,3,’cubic’));
figure(2); imshow(z1); figure(gcf)
Spline Wavelets
Spline wavelet is a wavelet constructed using a spline function. There are different types
of spline wavelets. The interpolatory spline wavelets are based on a certain spline
interpolation formula. Though these wavelets are orthogonal,
they do not have compact supports. but, there is a certain class of wavelets, unique in
some sense, constructed using B-splines and having compact supports. Even though
these wavelets are not orthogonal they have some special properties that have made
them quite popular. The terminology spline wavelet is sometimes used to refer to the
wavelets in this class of spline wavelets. These special wavelets are also called B-spline
wavelets and cardinal B-spline wavelets. The Battle-Lemarie wavelets are also wavelets
constructed using spline functions.
A cardinal B-spline is a special type of cardinal spline. For any positive integer m the
cardinal B-spline of order m, denoted by Nm(x), is defined recursively as follows.

54. AKUH· AMIN DAVAR
, Fall-1390, WSP 44
Figure 24: Interpolated image three times in each dimension.
N1(x) =
(
1 0 ≤ x 1
0 otherwise
Nm(x) =
Z 1
0
Nm−1(x − t)dt, m 1.
fun = @(x,t) heaviside(x-t)-heaviside(x-t-1);
x=linspace(-10,10,501);
q=heaviside(x)-heaviside(x-1);
for k=1:501;
xx=x(k);
q(2,k)=integral(@(t)fun(xx,t),0,1);
end
for m=2:5;
pp=spline(x,q(m,:));
for k=1:501;
q(m+1,k)= - integral(@(x)ppval(pp,x),x(k),x(k)-1);
end
end

55. AKUH· AMIN DAVAR
, Fall-1390, WSP 45
Constant B-spline:
N1(x) =
(
1 0 ≤ x 1
0 otherwise
Linear B-spline:
N2(x) =





x 0 ≤ x 1
−x + 2 1 ≤ x 2
0 otherwise
Quadratic B-spline:
N3(x) =









1
2
x2
0 ≤ x 1
−x2
+ 3x − 3
2
1 ≤ x 2
1
2
x2
− 3x + 9
2
2 ≤ x 3
0 otherwise
Cubic B-spline:
N4(x) =















1
6
x3
0 ≤ x 1
−1
2
x3
+ 2x2
− 2x + 2
3
1 ≤ x 2
1
2
x3
− 4x2
+ 10x − 22
3
2 ≤ x 3
−1
6
x3
+ 2x2
− 8x + 32
3
3 ≤ x 4
0 otherwise
Bi-quadratic B-spline:
N5(x) =



















1
24
x4
0 ≤ x 1
−1
6
x4
+ 5
6
x3
− 5
4
x2
+ 5
6
x − 5
24
1 ≤ x 2
1
4
x4
− 5
2
x3
+ 35
4
x2
− 25
2
x + 155
24
2 ≤ x 3
−1
6
x4
+ 5
2
x3
− 55
4
x2
+ 65
2
x − 655
24
3 ≤ x 4
1
24
x4
− 5
6
x3
+ 25
4
x2
− 125
6
x + 625
24
4 ≤ x 5
0 otherwise
Quintic B-spline:
N6(x) =

























1
120
x5
0 ≤ x 1
− 1
24
x5
+ 1
4
x4
− 1
2
x3
+ 1
2
x2
− 1
4
x + 1
20
1 ≤ x 2
1
12
x5
− x4
+ 9
2
x3
− 19
2
x2
+ 39
4
x − 79
20
2 ≤ x 3
− 1
12
x5
+ 3
2
x4
− 21
2
x3
+ 71
2
x2
− 231
4
x + 731
20
3 ≤ x 4
1
24
x5
− x4
+ 19
2
x3
− 89
2
x2
+ 409
4
x − 1829
20
4 ≤ x 5
− 1
120
x5
+ 1
4
x4
− 3x3
+ 18x2
− 54x + 324
5
5 ≤ x 6
0 otherwise
properties

56. AKUH· AMIN DAVAR
, Fall-1390, WSP 46
−1 0 1 2 3 4 5 6
−0.2
0
0.2
0.4
0.6
0.8
1
1.2
x
N
i
(x)|
6
i=1
B−spline
Figure 25: The first six B-Splines.
1. Nm(x) is the closed interval [0, m].
2. The function Nm(x) is non-negative, Nm(x) 0 for 0 x m.
3.
∞
X
k=−∞
Nm(x − k) = 1, ∀x.
4. The cardinal B-splines of orders m and m − 1 are related by the identity:
Nm(x) =
x
m − 1
Nm−1(x) +
m − x
m − 1
Nm−1(x − 1)
5. Nm(x) is symmetrical about x =
m
2
, Nm
m
2
− x
= Nm
m
2
+ x
6. The derivative Nm(x) is given by N′
m(x) = Nm−1(x) − Nm−1(x − 1).
7.
Z ∞
−∞
Nm(x) dx = 1.
8. The cardinal B-spline of order m satisfies the following two-scale relation:
Nm(x) =
Pm
k=0 2−m+1 m
k
Nm(2x − k)
9. The cardinal B-spline of order m satisfies the following property, known as the Riesz
property:
A k{ck}k2
≤
P∞
k=−∞ ckNm(x − k)
2
≤ B k{ck}k2
, ∀
P∞
k=−∞ |ck|2
6 ∞, ∀x

57. AKUH· AMIN DAVAR
, Fall-1390, WSP 47
Orthogonal factorization
This leads to a minimum phase filter and a maximum phase filter, which may be a
better choice for applications such as audio.
biorwavf in MATLAB
The orthogonal factorization leads to the Daubechies family of wavelets example:
4/4 factorization:
H0(z) =
√
3 − 1
4
√
2
(1 + z−1
)2
(2 +
√
3 − z−1
)
F0(z) =
−
√
2
4(
√
3 − 1)
(1 + z−1
)2
(2 −
√
3 − z−1
)
F0(z) = z3
H0(z−1
)
X(z)
H0(z)
H1(z)
↓ 2
↓ 2
Low frequency component, X0(z)
High frequency component, X1(z)
Figure 26: Analysis bank.
X̄0(z) = X(z)H0(z)
X̄1(z) = X(z)H1(z)
X0(z) = 0.5X(z1/2
)H0(z1/2
) + 0.5X(−z1/2
)H0(−z1/2
)
X1(z) = 0.5X(z1/2
)H1(z1/2
) + 0.5X(−z1/2
)H1(−z1/2
)
In matrix form:
X0(z)
X1(z)
= 1
2
H0(z1/2
) H0(−z1/2
)
H1(z1/2
) H1(−z1/2
)
#
X(z1/2
)
X(−z1/2
)
X(z) = G0(z)X0(z2
) + G1(z)X1(z2
).

58. AKUH· AMIN DAVAR
, Fall-1390, WSP 48
X0(z)
X1(z) ↑ 2
↑ 2
G1(z)
G0(z)
+ X̂(z)
Figure 27: Synthesis filter bank.
In matrix form:
X(z) = [G0(z) G1(z)]
X0(z2
)
X1(z2
)
Quadrature Mirror Filter Banks
Quadrature mirror filter banks (QMF) are two-channel subband coding (SBC)
filter banks with power complementary frequency responses

61. H(ejΩ
)

64. 2
+

67. H(ej(Ω−π)
)

70. 2
= 1,∀Ω,
The frequency responses of this category are only approximately power complementary.
The standard QMF banks were the first published filter banks which were able to
reconstruct the original signal from the subband signals.
X(z)
H0(z)
H1(z)
↓ 2
↓ 2 ↑ 2
↑ 2
G1(z)
G0(z)
+ X̂(z)
Figure 28: Two-channel SBC filter bank.
X̂(z) = [G0(z) G1(z)]
1
2
H0(z) H0(−z)
H1(z) H1(−z)
X(z)
X(−z)

71. AKUH· AMIN DAVAR
, Fall-1390, WSP 49
X̂(z) =
1
2
[G0(z)H0(z) + G1(z)H1(z)] X(z)
+
1
2
[G0(z)H0(−z) + G1(z)H1(−z)] X(−z)
= F0(z)X(z) + F1(z)X(−z).
F1(z) denotes the alias components, which are produced by the overlapping of the
frequency responses If F1(z) equals zero, we would obtain an alias-free filter bank.
F1(z) =
1
2
G0(z)H0(−z) +
1
2
G1(z)H1(−z) = 0
F0(z) denotes the quality of the reconstruction. If this is merely a delay, i.e.
F0(z) = z−ℓ
, the filter bank performs perfect reconstruction:
F0(z) =
1
2
G0(z)H0(z) +
1
2
G1(z)H1(z) = z−ℓ
Two-channel SBC filter bank:
1
2
[G0(z) G1(z)]
H0(z) H0(−z)
H1(z) H1(−z)
=
z−ℓ
0
These filters were referred to as “Quadrature Mirror Filters (QMF)”. Starting with a
suitable low-pass prototype H(z), the following four filters are specified:
H0(z) = H(z), H1(z) = H(−z) (13)
G0(z) = 2H(z), G1(z) = −2H(−z) (14)
These selections leads to Aliasing Cancellation and Distortion Elimination
This result is remarkable, since the sampling theorem is violated in both of
individual filter bank channels, but is satisfied in the overall filter bank.
Using (13) and (14)
F0(z) =
1
2
G0(z)H0(z) +
1
2
G1(z)H1(z) = z−ℓ
Gives
H2
(z) − H2
(−z) = z−ℓ
For linear phase FIR filters with even number N of coefficients:
H(z) = A(z) z−(N−1)/2

72. AKUH· AMIN DAVAR
, Fall-1390, WSP 50
A2
(z)z−(N−1)
−A2
(−z)(−1)−(N−1)
z−(N−1)
= z−ℓ
The delay of the overall filter bank is N − 1 clock periods, so that the number ℓ can be
identified as N − 1.
We can then say
A2
(z) + A2
(−z) = 1.
In terms of frequency responses
A2
(ejΩ
) + A2
(ej(Ω−π)
) = 1, ∀Ω (15)

75. H(ejΩ
)

78. 2
+

81. H(ej(Ω−π)
)

84. 2
= 1, ∀Ω, (16)
A2
(ejΩ
) is a half-band low-pass filter
A2
(ej(Ω−π)
) is a half-band high-pass filter
The squared amplitude frequency responses are mirror images of each other about the
line Ω = π/2, which has led to the name quadrature mirror filter.
It can be shown that linear phase FIR filters cannot exactly satisfy the condition (16),
apart from two exceptional cases that are of no practical significance. On the other
hand, equation (16) can be well approximated by numeric optimization methods.
Optimal FIR QMF banks
In literature, filter design techniques are given which, for a predefined filter length
N − 1, simultaneously maximize the stop-band attenuation and minimize the
reconstruction errors When using numeric optimising techniques, an objective
function with an error of
E = Er + αEs
which consists of a reconstruction error of
Er = 2
π
Z
Ω=0
(

86. H(ejΩ
)

88. 2
+

90. H(ej(Ω−π)
)

92. 2
− 1)dΩ
a stop - band error of
Er = 2
π
Z
Ω=Ωs
(

94. H(ejΩ
)

96. 2
)dΩ, Ωs = (
1
4
+ ∆)2π
The weighting factor α allows us to vary the relative importance of the two sources of
error.
Example

97. AKUH· AMIN DAVAR
, Fall-1390, WSP 51
0 10 20 30 40 50 60 70 80 90 100
−0.2
−0.1
0
0.1
0.2
0.3
0.4
0.5
n
Figure 29: Impulse response for H0(z).
A QMF that has N = 100 coefficients, a weighting factor α = 1 and
The coefficients are shown in Fig. (29), Fig. (30) shows the
frequency responses of the low - pass filter H0(z) = H(z)
and of the high - pass filter H1(z) = H(−z). It can be seen
that these are mirror images of each other
about the value Ω = π/2.
In MATLAB FIR perfect reconstruction two-channel filter bank as described above.
FIRPR2CHFB designs the four FIR filters for the analysis (H0 and H1) and synthesis
(G0 and G1) sections of a two-channel perfect reconstruction filter bank.
Let’s design a filter bank with filters of order 99 and passband edges of the lowpass and
highpass filters of 0.45 and 0.55, respectively:
N = 99;
[H0,H1,G0,G1] = firpr2chfb(N,.45);
% Analysis filters (decimators).
Hlp = mfilt.firdecim(2,H0);
Hhp = mfilt.firdecim(2,H1);
% Synthesis filters (interpolators).
Glp = mfilt.firinterp(2,G0);
Ghp = mfilt.firinterp(2,G1)
Looking at the first lowpass filter we can see that it meets our 0.45 cutoff specification.

98. AKUH· AMIN DAVAR
, Fall-1390, WSP 52
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
0
0.2
0.4
0.6
0.8
1
1.2
1.4
Ω/2π
Figure 30: Frequency response for H0(z), H1(z).
hfv = fvtool(Hlp);
legend(hfv,’Hlp Lowpass Decimator’);
set(hfv, ’Color’, [1 1 1])
Let’s look at all four filters.
hfv=fvtool([Hlp,Hhp,Glp,Ghp]);
legend(hfv,’Hlp Lowpass Decimator’,’Hhp Highpass Decimator’,...
’Glp Lowpass Interpolator’,’Ghp Highpass Interpolator’);
% Load scaling filter associated with an orthogonal wavelet.
load db10;
subplot(321); stem(db10); title(’db10 low-pass filter’);
% Compute the quadrature mirror filter.
qmfdb10 = qmf(db10);
subplot(322); stem(qmfdb10); title(’QMF db10 filter’);
% Check for frequency condition (necessary for orthogonality):
% abs(fft(filter))^2 + abs(fft(qmf(filter))^2 = 1 at each
% frequency.
m = fft(db10);
mt = fft(qmfdb10);
freq = [1:length(db10)]/length(db10);

99. AKUH· AMIN DAVAR
, Fall-1390, WSP 53
subplot(323); plot(freq,abs(m));
title(’Transfer modulus of db10’)
subplot(324); plot(freq,abs(mt));
title(’Transfer modulus of QMF db10’)
subplot(325); plot(freq,abs(m).^2 + abs(mt).^2);
title(’Check QMF condition for db10 and QMF db10’)
xlabel(’ abs(fft(db10))^2 + abs(fft(qmf(db10))^2 = 1’)
Using
X̂(z) = [G0(z) G1(z)]
1
2
H0(z) H0(−z)
H1(z) H1(−z)
X(z)
X(−z)
We have
X̂(−z) =
1
2
[G0(−z) G1(−z)]
H0( − z) H0(z)
H1( − z) H1(z)
X( − z)
X(z)
=
1
2
[G0(−z) G1(−z)]
H0(z) H0( − z)
H1(z) H1( − z)
X(z)
X(−z)
X̂(z)
X̂(−z)
=
1
2
G0(z) G1(z)
G0( − z) G1( − z)
H0(z) H0( − z)
H1(z) H1( − z)
X(z)
X(−z)
in matrix form
X̂(m)
(z) =
1
2
G(m)
(z)[H(m)
(z)]T
X(m)
(z)
with modulation matrices
G(m)
(z) =
G0(z) G1(z)
G0( − z) G1( − z)
H(m)
(z) =
H0(z) H1(z)
H0( − z) H1( − z)
and the modulation vectors
X̂(m)
(z) =
X̂(z) X̂( − z)
T
,X(m)
(z) =
X(z) X( − z)
T
The required PR conditions are written as
X̂(m)
(z) =
x̂(z)
x̂(−z)
=
z−ℓ
0
0 (−z)−ℓ
X(z)
X(−z)

100. AKUH· AMIN DAVAR
, Fall-1390, WSP 54
synthesis filters:
G(m)
(z) = 2z−ℓ
1 0
0 (−1)−ℓ
([H(m)
(z)]T
)−1
(17)
=
2z−ℓ
det H(m)(z)
H1(−z) −H0(−z)
H1(z) −H0(z)
(18)
det H(m)
(z) = H0(z)H1(−z) − H0(−z)H1(z) (19)
G0(z) =
2z−k
det H(m)(z)
H1(−z),
G1(z) = −
2z−k
det H(m)(z)
H0(−z).
For FIR filters:
detH(m)
(z) = cz−ℓ
, ℓ ∈ Z
Conjugate Quadrature Filters
A solution to the above problem was given by smith and barnwell’1984 in the form of
conjugate quadrature filters.
Starting with a low-pass prototype H(z), whose properties will be discussed later, the
analysis filters are chosen as follows:
H0(z) = H(z), (20)
H1(z) = z−(N−1)
H(−z−1
), (21)
(22)
assuming the FIR filter H(z) to have an even number N of coefficients.
Three operations are needed to derive the analysis filter
H1(z) from prototype H(z) :
1. Changing the sign of the variable z transforms the
low - pass filter H(z) into the high - pass filter H(−z).
2. The substitution z → z−1
causes the causal impulse
response h(n) to be reflected about the line n = 0.

101. AKUH· AMIN DAVAR
, Fall-1390, WSP 55
3. Finally, the factor z−(N−1)
make the filter H1(z) causal again. If we sub-
stitute the choice of analysis filters described by (20) and (21) into the
determinant (19), we get
detH(m)
(z) = − z−(N−1)
(H(z)H(z−1
) + H(−z)H(−z−1
))
Assuming the prototype filter satisfies
H(z)H(z−1
) + H(−z)H(−z−1
) = 1
the determinant is
detH(m)
(z) = − z−ℓ
, ℓ = odd
H0(z) : h0(n) = h(n)
H1(z) : h1(n) = (−1)(N−1−n)
h(N − 1 − n),
G0(z) : g0(n) = 2h(N − 1−n),
G1(z) : g1(n) = 2(−1)n
h(n).
Polyphase Signal Decomposition
X(z) =
M−1
X
λ=0
z−λ
Xλ(zM
)

102. AKUH· AMIN DAVAR
, Fall-1390, WSP 56
X(z) X0(z)
↓ M
z−1
z−1
z−1
XM−1(z)
↓ M
z−1
z−1
X2(z)
↓ M
.
.
.
.
.
.
↓ M z−1
X1(z)
Figure 31: Splitting a signal into its polyphase components.
Y (z) =
M−1
X
λ=0
z−λ
Yλ(zM
)
If we cascade the two structures:
Yλ(z) = z−1
Xλ+1(z), λ = 0, 1, 2.....M − 2
YM−1(z) = X0(z),
Y (z) =
M−1
X
λ=0
z−λ
Yλ(zM
)
=
M−2
X
λ=0
z−λ
z−M
Xλ+1(zM
) + z−(M−1)
X0(zM
)
= z−(M−1)
M−1
X
µ=0
z−µ
Xµ(zM
)
= z−(M−1)
X(z).

103. AKUH· AMIN DAVAR
, Fall-1390, WSP 57
YM−1(z)
YM−2(z)
Y1(z)
Y0(z)
↑ M
z−1
z−1
↑ M
z−1
↑ M
.
.
.
↑ M Y (z)
Figure 32: Constructing a signal from its polyphase components.
Polyphase System Decomposition
A transfer function H(z) of an LTI system can be decomposed into its polyphase
components, just like a signal X(z). For FIR systems:
H(z) =
M−1
X
λ=0
z−λ
Hλ(zM
)
We start with the two-channel analysis filter bank and with the equations H0(z) = H(z)
and H1(z) = H(−z) for the filter transfer functions. Representing the low-pass and
high-pass transfer functions in polyphase form:
Because of the QMF-property:
H0(z) = H00(z2
) + z−1
H01(z2
), H00 = Heven, H01 = Hodd
H1(z) = H00(z2
) − z−1
H01(z2
).

104. AKUH· AMIN DAVAR
, Fall-1390, WSP 58
X(z)
H0(z)
H1(z)
↓ 2
↓ 2
Low frequency component, X0(z)
High frequency component, X1(z)
Figure 33: Two-channel analysis filter bank.
Since
H1(z) = H0(−z).
Modulation and Polyphase Representations: Noble Identities; Block Toeplitz
Matrices and Block z-transforms
Modulation Matrix:
Matrix form of PR conditions:
[F0(z) F1(z)]
H0(z) H0(−z)
H1(z) H1(−z)
| {z }
modulation matrix, Hm(z)
=
2z−ℓ
0
(23)
H−1
m (z) =
1
∆
H1(−z) −H0(−z)
−H1(z) H0(z)
∆ = H0(z)H1(−z) − H1(z)H0(−z), must be non-zero.
F0(z) = 2z−ℓH1(−z)
∆
F1(z) = −2z−ℓH0(−z)
∆
)
Required for FIR (24)
If ∆ = 2z−ℓ
⇒
F0(z) = H1(−z)
F1(z) = −H0(−z)
(25)

105. AKUH· AMIN DAVAR
, Fall-1390, WSP 59
Complete the second row of matrix PR conditions by replacing z with −z:
F0(z) F1(z)
F0(−z) F1(−z)
| {z }
Synthesis
modulation
matrix,
Fm(z)
H0(z) H0(−z)
H1(z) H1(−z)
= 2
z−ℓ
0
0 (−z)−ℓ
(26)

106. AKUH· AMIN DAVAR
, Fall-1390, WSP 60
x(n)
H(z2)
u(n)
2
y(n)
U(z) = H(z2
)X(z)
Y (z) = 0.5
U(z1/2
) + U(−z1/2
)
=
H(z)
1
2
X(z1/2
) + X(−z1/2
⇒
can downsample first
First Noble identity:

107. AKUH· AMIN DAVAR
, Fall-1390, WSP 61
x(n)
2 H(z)
y(n)
≡
x(n)
H(z2) 2
y(n)
Figure 34: First Noble identity.

108. AKUH· AMIN DAVAR
, Fall-1390, WSP 62
x(n)
H(z)
u(n)
2
y(n)
Second Noble Identity:
U(z) = H(z)X(z)
Y (z) = U(z2
)
= H(z2
)X(z2
), can upsample first

109. AKUH· AMIN DAVAR
, Fall-1390, WSP 63
x(n)
2 H(z2)
y(n)
≡
x(n)
H(z) 2
y(n)
Figure 35: Second Noble identity.

110. AKUH· AMIN DAVAR
, Fall-1390, WSP 64
Derivation of Polyphase Form:
1. Filtering and downsampling:
x(n)
H(z) 2
y(n)
heven(n) = h(2n), hodd(n) = h(2n + 1)
Heven(z) =
X
n
h(2n)z−n
, Hodd(z) =
X
n
h(2n + 1)z−n
Heven(z) =
X
m=2n
h(m)z−m/2
, Hodd(z) =
X
m=2n+1
h(m)z−(m−1)/2
Heven(z2
) =
X
m=2n
h(m)z−m
, Hodd(z2
) = z
X
m=2n+1
h(m)z−m
H(z) = Heven(z2
) + z−1
Hodd(z2
)
Heven(z2)
Hodd(z2)
z−1
+ 2 y(n)
x(n)
Heven(z2)
Hodd(z2)
z−1
2
2
+ y(n)
x(n)

111. AKUH· AMIN DAVAR
, Fall-1390, WSP 65
2
2
z−1
Heven(z)
Hodd(z)
xeven(n)
xodd(n − 1)
+ y(n)
x(n)
2. Upsampling and filtering:
x(n)
2 F(z)
y(n)
F(z) = Feven(z2
) + z−1
Fodd(z2
)
feven(n) = f(2n), fodd(n) = f(2n + 1)
2
Feven(z2)
Fodd(z2)
+
z−1
y(n)
x(n)

112. AKUH· AMIN DAVAR
, Fall-1390, WSP 66
2
2
z−1
Feven(z2)
Fodd(z2)
+ y(n)
x(n)
Feven(z)
Fodd(z)
yeven(n)
yodd(n)
2
2
+
z−1
y(n)
x(n)
Polyphase Matrix
Consider the matrix corresponding to the analysis filter bank in interleaved
form. This is a block Toeplitz matrix:
4-tap example:
Hb =









.
.
.
· · · h0(3) h0(2) h0(1) h0(0) 0 0 · · ·
· · · h1(3) h1(2) h1(1) h1(0) 0 0 · · ·
· · · 0 0 h0(3) h0(2) h0(1) h0(0) · · ·
· · · 0 0 h1(3) h1(2) h1(1) h1(0) · · ·
.
.
.









Taking block z-transform:
Hp(z) =
h0(0) h0(1)
h1(0) h1(1]
+ z−1
h0(2) h0(3)
h1(2) h1(3)
=
h0(0) + z−1
h0(2) h0(1) + z−1
h0(3]
h1(0) + z−1
h1(2) h1(1) + z−1
h1(3)
=
H0,even(z) H0,odd(z)
H1,even(z) H1,odd(z)
This is the polyphase matrix for a 2-channel filter bank.

113. AKUH· AMIN DAVAR
, Fall-1390, WSP 67
Similarly, for the synthesis filter bank:
Fb =













.
.
.
.
.
.
.
.
.
.
.
.
f0(0) f1(0) 0 0
f0(1) f1(1) 0 0
· · · f0(2) f1(2) f0(0) f1(0) · · ·
· · · f0(3) f1(3) f0(1) f1(1)
· · · 0 0 f0(2) f1(2) · · ·
· · · 0 0 f0(3) f1(3)
.
.
.
.
.
.
.
.
.
.
.
.













Fp(z) =
f0(0) f1(0)
f0(1) f1(1)
+ z−1
f0(2) f1(2)
f0(3) f1(3)
=
F0,even(z) F1,even(z)
F0,odd(z) F1,odd(z)
Perfect reconstruction condition in polyphase domain:
F0,even(z) F1,even(z)
F0,odd(z) F1,odd(z)
| {z }
Fp(z)
H0,even(z) H0,odd(z)
H1,even(z) H1,odd(z)
| {z }
Hp(z)
=
1 0
0 1
Fp(z)Hp(z) = I, (centered form)
This means that Hp(z) must be invertible for all z on the unit circle, i.e.
det Hp(eiω
) 6= 0 ∀ω.
Given that the analysis filters are FIR, the requirement for the synthesis
filters to be also FIR is:
det Hp(eiω
) = z−ℓ
(simple delay)
because H−1
p (z) must be a polynomial.
Condition for orthogonality: Fp(z) is the transpose of Hp(z), i.e.
HT
p (z−1
)Hp(z) = I ⇒
Hp(z) should be paraunitary.
Relationship between Modulation and Polyphase Matrices:

114. AKUH· AMIN DAVAR
, Fall-1390, WSP 68
H0(z) = H0,even(z2
) + z−1
H0,odd(z2
);
h0,even(n) = h0(2n)
h0,odd(n) = h0(2n + 1)
H1(z) = H1,even(z2
) + z−1
H1,odd(z2
);
h1,even(n) = h1(2n)
h1,odd(n) = h1(2n + 1)
H0(−z) = H0,even(−z2
) − z−1
H0,odd(−z2
);
h0,even(n) = h0(2n)
h0,odd(n) = h0(2n + 1)
H1(−z) = H1,even(−z2
) − z−1
H1,odd(−z2
);
h1,even(n) = h1(2n)
h1,odd(n) = h1(2n + 1)
in matrix form:
H0(z) H0(−z)
H1(z) H1(−z)
| {z }
Hm(z)
Modulation matrix
=
H0,even(z2
) H0,odd(z2
)
H1,even(z2
) H1,odd(z2
)
| {z }
Hp(z2
)
Polyphase matrix
1 1
z−1
−z−1
1 1
z−1
−z−1
=
1 0
0 z−1
| {z }
D2(z)
Delay matrix
1 1
1 −1
| {z }
2-point
DFT matrix
FN =







1 1 1 · 1
1 w w2
· · · wN−1
1 w2
w4
· · · w2(N−1)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
1 wN−1
w2(N−1)
· · · w(N−1)2







, w = ei 2π
N , N-point matrix
F−1
N =
1
N
FN
in general
Hm(z)F−1
N = Hp(zN
)DN (z)
N = # of channels in filterbank
N = 2, in our case so far

115. AKUH· AMIN DAVAR
, Fall-1390, WSP 69
Polyphase Matrix Example:
Daubechies 4-tap filter
h0(0) = 1+
√
3
4
√
2
, h0(1) = 3+
√
3
4
√
2
h0(2) = 3−
√
3
4
√
2
h0(3) = 1−
√
3
4
√
2
,
H0(z) =
1+
√
3
4
√
2
+
3+
√
3
4
√
2
z−1
+
3−
√
3
4
√
2
z−2
+
1−
√
3
4
√
2
z−3
H1(z) =
1−
√
3
4
√
2
+
3−
√
3
4
√
2
z−1
+
3+
√
3
4
√
2
z−2
+
1+
√
3
4
√
2
z−3
Time domain:
h0(0)2
+ h0(1)2
+ h0(2)2
+ h0(3)2
=
1
32
[(4 + 2
√
3) + (12 + 6
√
3)
+(12 − 6
√
3) + (4 − 2
√
3)]
= 1
h0(0)h0(2) + h0(1)h0(3) =
1
32
[(2
√
3) + (−2
√
3)] = 0
i.e. filter is orthogonal to its double shifts
Polyphase Domain:
H0,even(z) =
1+
√
3
4
√
2
+
3−
√
3
4
√
2
z−1
H0,odd(z) =
3+
√
3
4
√
2
+
1−
√
3
4
√
2
z−1
H1,even(z) =
1−
√
3
4
√
2
+
3+
√
3
4
√
2
z−1
H1,odd(z) =
−(3−
√
3)
4
√
2
+
−(1+
√
3)
4
√
2
z−1
Hp(z) =
1
4
√
2
1 +
√
3 3 +
√
3
1 −
√
3 −(3 +
√
3)
+
1
4
√
2
3 −
√
3 1 −
√
3
3 +
√
3 −(1 +
√
3)
z−1
= A + Bz−1
HT
p (z−1
)Hp(z) = (AT
+ BT
z−1
)(A + Bz−1
)
= I ⇒
Hp(z) is a Paraunitary Matrix
Modulation domain:

116. AKUH· AMIN DAVAR
, Fall-1390, WSP 70
H0(z)H0(z−1
) =
1
16
−z3
+ 9z + 16 + 9z−1
− z−3
H0(−z)H0(−z−1
) =
1
16
+z3
− 9z + 16 − 9z−1
+ z−3
H0(z)H0(z−1
) + H0(−z)H0(−z−1
) = 2 ⇒
|H0(eiω
)| + |H0(ei(ω+π)
)|2
= 2
[H_0,H_1,F_0,F_1]=wfilters(’db2’); omega=linspace(-pi,pi,360);
w=exp(i*omega); G=polyval(H_0,w); subplot(211)
plot(omega/pi,abs(G));figure(gcf) xlabel(’$omega / pi$’)
title(’Magnitude Response of Daubechies 4-tap filter.’)
subplot(212) plot(omega/pi,angle(G));figure(gcf) title(’Phase
Response of Daubechies 4-tap filter.’) xlabel(’$omega / pi$’)
plot(omega/pi,abs(G).^2+...
abs(polyval(H_0,exp(i*(omega+pi)))).^2); figure(gcf)

117. AKUH· AMIN DAVAR
, Fall-1390, WSP 71
−1 −0.5 0 0.5 1
0
0.5
1
1.5
ω
/π
Magnitude Response of Daubechies 4−tap filter.
−1 −0.5 0 0.5 1
−4
−2
0
2
4
Phase Response of Daubechies 4−tap filter.
ω
/π
Figure 36: Daubechies 4-tap filter.
Run biorthwavf.m
% 1-D signal analysis
% biorwavf generates symmetric biorthogonal wavelet filters.
% The argument has the form biorNr.Nd, where
% Nr = number of zeros at pi in the synthesis lowpass filter, s[n].
% Nd = number of zeros at pi in the analysis lowpass filter, a[n].
% We find the famous Daubechies 9/7 pair, which have Nr = Nd = 4.
% below,) the vectors s and a are zero-padded to make their lengths equal:
% a[-4] a[-3] a[-2] a[-1] a[0] a[1] a[2] a[3] a[4]
% 0 s[-3] s[-2] s[-1] s[0] s[1] s[2] s[3] 0
[s,a]= biorwavf(’bior4.4’);
% Find the zeros and plot them.
close all clf fprintf(1,’Zeros of H0(z)’) roots(a) subplot(1,2,1)
zplane(a) title(’Zeros of H0(z)’)
fprintf(1,’Zeros of F0(z)’) roots(s) subplot(1,2,2)
zplane(s) % Note: there are actually 4 zeros clustered at z = -1.
title(’Zeros of F0(z)’) pause
% Determine the complete set of filters, with proper alignment.
% Note: Matlab uses the convention that a[n] is the flip of h0[n].
%h0[n] = flip of a[n], with the sum normalized to sqrt(2).
%f0[n] = s[n], with the sum normalized to sqrt(2).
%h1[n] = f0[n], with alternating signs reversed (starting with the first.)
%f1[n] = h0[n], with alternating signs reversed (starting with the second.)
[h0,h1,f0,f1] = biorfilt(a, s);

118. AKUH· AMIN DAVAR
, Fall-1390, WSP 72
clf subplot(2,2,1) stem(0:8,h0(2:10)) ylabel(’h0[n]’) xlabel(’n’)
subplot(2,2,2) stem(0:6,f0(2:8)) ylabel(’f0[n]’) xlabel(’n’) v =
axis; axis([v(1) 8 v(3) v(4)]) subplot(2,2,3) stem(0:6,h1(2:8))
ylabel(’h1[n]’) xlabel(’n’) v = axis; axis([v(1) 8 v(3) v(4)])
subplot(2,2,4) stem(0:8,f1(2:10)) ylabel(’f1[n]’) xlabel(’n’)
pause
% Examine the Frequency response of the filters.
N = 512; W = 2/N*(-N/2:N/2-1); H0 = fftshift(fft(h0,N)); H1 =
fftshift(fft(h1,N)); F0 = fftshift(fft(f0,N)); F1 =
fftshift(fft(f1,N)); clf plot(W, abs(H0), ’-’, W, abs(H1), ’--’,
W, abs(F0), ’-.’, W, abs(F1), ’:’) title(’Frequency responses of
Daubechies 9/7 filters’) xlabel(’Angular frequency (normalized by
pi)’) ylabel(’Frequency response magnitude’) legend(’H0’, ’H1’,
’F0’, ’F1’, 0) pause
% Load a test signal.
load noisdopp x = noisdopp; L = length(x); clear noisdopp
% Compute the lowpass and highpass coefficients using convolution and
% downsampling.
y0 = dyaddown(conv(x,h0)); y1 = dyaddown(conv(x,h1));
% The function dwt provides a direct way to get the same result.
[yy0,yy1] = dwt(x,’bior4.4’);
% Now, reconstruct the signal using upsampling and convolution. We only
% keep the middle L coefficients of the reconstructed signal i.e. the ones
% that correspond to the original signal.
xhat = conv(dyadup(y0),f0) + conv(dyadup(y1),f1); xhat =
wkeep(xhat,L);
% The function idwt provides a direct way to get the same result.
xxhat = idwt(y0,y1,’bior4.4’);
% Plot the results.
subplot(4,1,1); plot(x) axis([0 1024 -12 12]) title(’Single stage
wavelet decomposition’) ylabel(’x’) subplot(4,1,2); plot(y0)
axis([0 1024 -12 12]) ylabel(’y0’) subplot(4,1,3); plot(y1)
axis([0 1024 -12 12]) ylabel(’y1’) subplot(4,1,4); plot(xhat)
axis([0 1024 -12 12]) ylabel(’xhat’) pause

119. AKUH· AMIN DAVAR
, Fall-1390, WSP 73
% Next, we perform a three level decomposition. The following
% code draws the structure of the iterated analysis filter bank.
clf
t = wtree(x,3,’bior4.4’);
plot(t)
pause
close(2)
% For a multilevel decomposition, we use wavedec instead of dwt.
% Here we do 3 levels. wc is the vector of wavelet transform
% coefficients. l is a vector of lengths that describes the
% structure of wc.
[wc,l] = wavedec(x,3,’bior4.4’);
% We now need to extract the lowpass coefficients and the various
% highpass coefficients from wc.
a3 = appcoef(wc,l,’bior4.4’,3); d3 = detcoef(wc,l,3); d2 =
detcoef(wc,l,2); d1 = detcoef(wc,l,1);
clf subplot(5,1,1) plot(x) axis([0 1024 -22 22]) ylabel(’x’)
title(’Three stage wavelet decomposition’) subplot(5,1,2) plot(a3)
axis([0 1024 -22 22]) ylabel(’a3’) subplot(5,1,3) plot(d3) axis([0
1024 -22 22]) ylabel(’d3’) subplot(5,1,4) plot(d2) axis([0 1024
-22 22]) ylabel(’d2’) subplot(5,1,5) plot(d1) axis([0 1024 -22
22]) ylabel(’d1’) pause
% We can reconstruct each branch of the tree separately from the individual
% vectors of transform coefficients using upcoef.
ra3 =upcoef(’a’,a3,’bior4.4’,3,1024);
rd3 =upcoef(’d’,d3,’bior4.4’,3,1024);
rd2 =upcoef(’d’,d2,’bior4.4’,2,1024);
rd1 =upcoef(’d’,d1,’bior4.4’,1,1024);
% The sum of these reconstructed branches gives the full recontructed
% signal.
xhat = ra3 + rd3 + rd2 + rd1;
clf subplot(5,1,1) plot(x) axis([0 1024 -10 10]) ylabel(’x’)
title(’Individually reconstructed branches’) subplot(5,1,2)
plot(ra3) axis([0 1024 -10 10]) ylabel(’ra3’) subplot(5,1,3)
plot(rd3) axis([0 1024 -10 10]) ylabel(’rd3’) subplot(5,1,4)
plot(rd2) axis([0 1024 -10 10]) ylabel(’rd2’) subplot(5,1,5)
plot(rd1) axis([0 1024 -10 10]) ylabel(’rd1’) pause

120. AKUH· AMIN DAVAR
, Fall-1390, WSP 74
clf plot(xhat-x) title(’Reconstruction error (using upcoef)’) axis
tight pause
% We can also reconstruct individual branches from the full vector of
% transform coefficients, wc.
rra3 = wrcoef(’a’,wc,l,’bior4.4’,3); rrd3 =
wrcoef(’d’,wc,l,’bior4.4’,3); rrd2 = wrcoef(’d’,wc,l,’bior4.4’,2);
rrd1 = wrcoef(’d’,wc,l,’bior4.4’,1); xxhat = rra3 + rrd3 + rrd2 +
rrd1;
clf plot(xxhat-x) title(’Reconstruction error (using wrcoef)’)
axis tight pause
% To reconstruct all branches at once, use waverec.
xxxhat = waverec(wc,l,’bior4.4’);
clf plot(xxxhat-x) axis tight title(’Reconstruction error (using
waverec)’) pause
% 2D image analysis.
% Load a test image. Matlab test images consist of a matrix, X,
% color palette, map, which maps each value of the matrix to a
% color. Here, we will apply the Discrete Wavelet Transform to X.
load woman2
%load detfingr; X = X(1:200,51:250);
close all clf image(X) colormap(map) axis image;
set(gca,’XTick’,[],’YTick’,[]); title(’Original’) pause
% We will use the 9/7 filters with symmetric extension at the
% boundaries.
dwtmode(’sym’) wname = ’bior4.4’
% Plot the structure of a two stage filter bank.
t = wtree(X,2,’bior4.4’); plot(t) pause close(2)
% Compute a 2-level decomposition of the image using the 9/7 filters.
[wc,s] = wavedec2(X,2,wname);
% Extract the level 1 coefficients.
a1 = appcoef2(wc,s,wname,1); h1 = detcoef2(’h’,wc,s,1); v1 =
detcoef2(’v’,wc,s,1); d1 = detcoef2(’d’,wc,s,1);

121. AKUH· AMIN DAVAR
, Fall-1390, WSP 75
% Extract the level 2 coefficients.
a2 = appcoef2(wc,s,wname,2); h2 = detcoef2(’h’,wc,s,2); v2 =
detcoef2(’v’,wc,s,2); d2 = detcoef2(’d’,wc,s,2);
% Display the decomposition up to level 1 only.
ncolors = size(map,1); % Number of colors.
sz = size(X); cod_a1 = wcodemat(a1,ncolors); cod_a1 =
wkeep(cod_a1, sz/2); cod_h1 = wcodemat(h1,ncolors); cod_h1 =
wkeep(cod_h1, sz/2); cod_v1 = wcodemat(v1,ncolors); cod_v1 =
wkeep(cod_v1, sz/2); cod_d1 = wcodemat(d1,ncolors); cod_d1 =
wkeep(cod_d1, sz/2); image([cod_a1,cod_h1;cod_v1,cod_d1]); axis
image; set(gca,’XTick’,[],’YTick’,[]); title(’Single stage
decomposition’) colormap(map) pause
% Display the entire decomposition upto level 2.
cod_a2 = wcodemat(a2,ncolors); cod_a2 = wkeep(cod_a2, sz/4);
cod_h2 = wcodemat(h2,ncolors); cod_h2 = wkeep(cod_h2, sz/4);
cod_v2 = wcodemat(v2,ncolors); cod_v2 = wkeep(cod_v2, sz/4);
cod_d2 = wcodemat(d2,ncolors); cod_d2 = wkeep(cod_d2, sz/4);
image([[cod_a2,cod_h2;cod_v2,cod_d2],cod_h1;cod_v1,cod_d1]); axis
image; set(gca,’XTick’,[],’YTick’,[]); title(’Two stage
decomposition’) colormap(map) pause
% Here are the reconstructed branches
ra2 = wrcoef2(’a’,wc,s,wname,2); rh2 = wrcoef2(’h’,wc,s,wname,2);
rv2 = wrcoef2(’v’,wc,s,wname,2); rd2 = wrcoef2(’d’,wc,s,wname,2);
ra1 = wrcoef2(’a’,wc,s,wname,1); rh1 = wrcoef2(’h’,wc,s,wname,1);
rv1 = wrcoef2(’v’,wc,s,wname,1); rd1 = wrcoef2(’d’,wc,s,wname,1);
cod_ra2 = wcodemat(ra2,ncolors); cod_rh2 = wcodemat(rh2,ncolors);
cod_rv2 = wcodemat(rv2,ncolors); cod_rd2 = wcodemat(rd2,ncolors);
cod_ra1 = wcodemat(ra1,ncolors); cod_rh1 = wcodemat(rh1,ncolors);
cod_rv1 = wcodemat(rv1,ncolors); cod_rd1 = wcodemat(rd1,ncolors);
subplot(3,4,1); image(X); axis image;
set(gca,’XTick’,[],’YTick’,[]); title(’Original’) subplot(3,4,5);
image(cod_ra1); axis image; set(gca,’XTick’,[],’YTick’,[]);
title(’ra1’) subplot(3,4,6); image(cod_rh1); axis image;
set(gca,’XTick’,[],’YTick’,[]); title(’rh1’) subplot(3,4,7);
image(cod_rv1); axis image; set(gca,’XTick’,[],’YTick’,[]);
title(’rv1’) subplot(3,4,8); image(cod_rd1); axis image;
set(gca,’XTick’,[],’YTick’,[]); title(’rd1’) subplot(3,4,9);
image(cod_ra2); axis image; set(gca,’XTick’,[],’YTick’,[]);
title(’ra2’) subplot(3,4,10); image(cod_rh2); axis image;

122. AKUH· AMIN DAVAR
, Fall-1390, WSP 76
set(gca,’XTick’,[],’YTick’,[]); title(’rh2’) subplot(3,4,11);
image(cod_rv2); axis image; set(gca,’XTick’,[],’YTick’,[]);
title(’rv2’) subplot(3,4,12); image(cod_rd2); axis image;
set(gca,’XTick’,[],’YTick’,[]); title(’rd2’) pause
% Adding together the reconstructed average at level 2 and all of
% the reconstructed details gives the full reconstructed image.
Xhat = ra2 + rh2 + rv2 + rd2 + rh1 + rv1 + rd1;
sprintf(’Reconstruction error (using wrcoef2) = %g’, max(max(abs(X-Xhat))))
% Another way to reconstruct the image.
XXhat = waverec2(wc,s,wname);
sprintf(’Reconstruction error (using waverec2) = %g’, max(max(abs(X-XXhat))))
% Compression can be accomplished by applying a threshold to the
% wavelet coefficients. wdencmp is the function that does this.
% ’h’ means use hard thresholding. Last argument = 1 means do not
% threshold the approximation coefficients.
% perfL2 = energy recovery = 100 * ||wc_comp||^2 / ||wc||^2.
% ||.|| is the L2 vector norm.
% perf0 = compression performance = Percentage of zeros in wc_comp.
thr = 20; [X_comp,wc_comp,s_comp,perf0,perfL2] =
wdencmp(’gbl’,wc,s,wname,2,thr,’h’,1);
clf subplot(1,2,1); image(X); axis image;
set(gca,’XTick’,[],’YTick’,[]); title(’Original’) cod_X_comp =
wcodemat(X_comp,ncolors); subplot(1,2,2); image(cod_X_comp); axis
image; set(gca,’XTick’,[],’YTick’,[]); title(’Compressed using
global hard threshold’)
xlabel(sprintf(’Energy retained = %2.1f%% ...
nNull coefficients = %2.1f%%’,perfL2,perf0))
pause
% Better compression can be often be obtained if different thresholds
% are allowed for different subbands.
thr_h = [21 17]; % horizontal thresholds.
thr_d = [23 19]; % diagonal thresholds.
thr_v = [21 17]; % vertical thresholds.
thr = [thr_h; thr_d; thr_v]; [X_comp,wc_comp,s_comp,perf0,perfL2]
= wdencmp(’lvd’,X,wname,2,thr,’h’);
clf subplot(1,2,1); image(X); axis image;
set(gca,’XTick’,[],’YTick’,[]); title(’Original’) cod_X_comp =
wcodemat(X_comp,ncolors); subplot(1,2,2); image(cod_X_comp); axis

123. AKUH· AMIN DAVAR
, Fall-1390, WSP 77
image; set(gca,’XTick’,[],’YTick’,[]); title(’Compressed using
variable hard thresholds’)
xlabel(sprintf(’Energy retained = %2.1f%% ...
nNull coefficients = %2.1f%%’,perfL2,perf0))
% Return to default settings.
dwtmode(’zpd’)
Orthogonal Filter Banks; Paraunitary Matrices; Orthogonality Condition
(Condition O) in the Time Domain, Modulation Domain and Polyphase
Domain
Unitary Matrices:
The constant complex matrix A is said to be unitary if A†
A = I
example:
A =
1
√
2
1 −i
i −1
⇒ A†
= A−1
Paraunitary Matrices:
The matrix function H(z) is said to be paraunitary if it is unitary for all
values of the parameter z
HT
(z−1
)H(z) = I, ∀z 6= 0
Frequency Domain:
HT
(e−iω
)H(eiω
) = I, ∀ω
H∗T
(e−iω
)H(eiω
) = I
we are assuming that h(n) are real.
Orthogonal Filter Banks:
Centered form (PR with no delay):
h0(n) is causal.
Synthesis bank = transpose of analysis bank
What are the conditions on h0(n), h1(n), in the
• time domain?
• polyphase domain?
• modulation domain?

124. AKUH· AMIN DAVAR
, Fall-1390, WSP 78
h0(n)
h1(n)
↓ 2
↓ 2
y0(n)
y1(n)
↑ 2
↑ 2
f1(n)
f0(n)
+ x̂(n)
x(n)
Figure 37: Orthogonal Filter Banks: f0(n) = h0(−n), f1(n) = h1(−n).
Time Domain Analysis: N = 3 (filter length = 4)
Analysis























.
.
.
y0(0)
y0(1)
y0(2)
y0(3)
.
.
.
.
.
.
y1(0)
y1(1)
y1(2)
y1(3)
.
.
.























=




















· · ·
h0(3) h0(2) h0(1) h0(0)
h0(3) h0(2) h0(1) h0(0)
h0(3) h0(2) h0(1) h0(0)
h0(3) h0(2) h0(1) h0(0)
· · ·
· · ·
h1(3) h1(2) h1(1) h1(0)
h1(3) h1(2) h1(1) h1(0)
h1(3) h1(2) h1(1) h1(0)
h1(3) h1(2) h1(1) h1(0)
· · ·




















| {z }
W






















.
.
.
x(−3)
x(−2))
x(−1)
x(0)
x(1)
x(2)
x(3)
x(4)
x(5)
x(6)
.
.
.























125. AKUH· AMIN DAVAR
, Fall-1390, WSP 79
Synthesis:






















.
.
.
x(−3)
x(−2))
x(−1)
x(0)
x(1)
x(2)
x(3)
x(4)
x(5)
x(6)
.
.
.






















=






















.
.
.
.
.
.
h0(3) h1(3)
h0(2) h1(2)
h0(1) h0(3) h1(1) h1(3)
h0(0) h0(2) h1(0) h1(2)
h0(1) h0(3) h1(1) h1(3)
h0(0) h0(2) h1(0) h1(2)
h0(1) h0(3) h1(1) h1(3)
h0(0) h0(2) h1(0) h1(2)
h0(1) h1(1)
h0(0) h0(0)
.
.
.
.
.
.






















| {z }
W T























.
.
.
y0(0)
y0(1)
y0(2)
y0(3)
.
.
.
.
.
.
y1(0)
y1(1)
y1(2)
y1(3)
.
.
.























Orthogonality condition (Condition O) is:
WT
W = I = WWT
⇒ W orthogonal matrix
Block Form:
W =
L
B
LT
L + BT
B = I
LT
L LBT
BLT
BBT
LT
L = I ⇒
X
n
h0(n)h0(n − 2k) = δ(k) (27)
LBT
= 0 ⇒
X
n
h0(n)h1(n − 2k) = 0 (28)
BBT
= I ⇒
X
n
h1(n)h1(n − 2k) = δ(k) (29)
Good choice for h1(n):
h1(n) = (−1)n
h0(N − n), N odd
Example: N = 3
h1(0) = h0(3)
h1(1) = −h0(2)
h1(2) = h0(1)
h1(3) = −h0(0)

126. AKUH· AMIN DAVAR
, Fall-1390, WSP 80
With this choice, Equation (28) is automatically satisfied:
LBT
= 0 ⇒
X
n
h0(n)h1(n − 2k) = 0
k = −1 : h0(0)h0(1) − h0(1)h0(0) = 0
k = 0 : h0(0)h0(3) − h0(1)h0(2) + h0(2)h0(1) − h0(3)h0(0) = 0
k = 1 : h0(2)h0(3) − h0(3)h0(2) = 0
k = ±2 : no overlap
Also, Equation (29) reduces to Equation (27)
δ(k) =
X
n
h1(n)h1(n − 2k) =
X
n
(−1)n
h0(N − n)(−1)n−2k
h0(N − n + 2k)
=
X
ℓ
h0(ℓ)h0(ℓ + 2k)
So, Condition O on the lowpass filter + alternating flip for highpass filter
lead to orthogonality

127. AKUH· AMIN DAVAR
, Fall-1390, WSP 81
Polyphase Domain:
↓ 2
↓ 2
Hp(z) HT
p (z−1)
↑ 2
↑ 2
z
+ x(n − 1)
x(n)
xeven(n)
xodd(n − 1)
z−1
y0(n)
y1(n)
xeven(n)
xodd(n − 1)
Figure 38: FB analysis and synthesis, polyphase structure.
Hp(z) =
H0,even(z) H0,odd(z)
H1,even(z) H1,odd(z)
, Polyphase matrix
Condition O:
HT
p (z−1
)Hp(z) = I ⇒ Hp(z) is paraunitary
H0,even(z−1
) H1,even(z−1
)
H0,odd(z−1
) H1,odd(z−1
)
H0,even(z) H0,odd(z)
H1,even(z) H1,odd(z)
=
1 0
0 1
Reverse the order of multiplication:
H0,even(z) H0,odd(z)
H1,even(z) H1,odd(z)
H0,even(z−1
) H1,even(z−1
)
H0,odd(z−1
) H1,odd(z−1
)
=
1 0
0 1
Express Condition O as a condition on H0,even(z) and H0,odd(z) :
H0,even(z)H0,even(z−1
) + H0,odd(z)H0,odd(z−1
) = 1
Frequency domain:
|H0,even(eiω
)|2
+ |H0,odd(eiω
)|2
= 1

128. AKUH· AMIN DAVAR
, Fall-1390, WSP 82
The alternating flip construction for H1(z) ensures that the remaining
conditions are satisfied.
H0(z) = H0,even(z2
) + z−1
H0,odd(z2
)
H1(z) =
N
X
n=0
h1(n)z−n
=
N
X
n=0
(−1)n
h0(N − n)z−n
=
N
X
m=0
(−1)N−m
h0(m)zm−N
= (−1)N
z−N
N
X
m=0
(−1)−m
h0(m)zm
, N is odd
H1(z) = −z−N
H0(−z−1
) = −z−N
H0,even(z−2
) + (−z−1
)−1
H0,odd(z−2
)
= − z−N
|{z}
odd
[H0,even(z−2
)
| {z }
even
−z H0,even(z−2
)
| {z }
odd
], N is odd
= − z−N
|{z}
odd
H0,even(z−2
)
| {z }
even
| {z }
z−1H1,odd(z2)
+ z−N+1
| {z }
even
H0,odd(z−2
)
| {z }
even
| {z }
H1,even(z2)
So, after z →
√
z ⇒
H1,even(z) = z(−N+1)/2
H0,odd(z−1
)
H1,odd(z) = −z(−N+1)/2
H0,even(z−1
) ⇒
H0,even(z)H1,even(z−1
) + H0,odd(z)H1,odd(z−1
) = 0
H1,even(z)H1,even(z−1
) + H1,odd(z)H1,odd(z−1
) = 1

129. AKUH· AMIN DAVAR
, Fall-1390, WSP 83
Modulation Domain
H0(z)
H1(z)
↓ 2
↓ 2
y0(n)
y1(n)
↑ 2
↑ 2
H1(z−1)
H0(z−1)
+ x̂(n)
x(n)
PR conditions:
H0(z)H0(z−1
) + H1(z)H1(z−1
) = 2, No distortion (30)
H0(−z)H0(z−1
) + H1(−z)H1(z−1
) = 0 Alias cancellation (31)
H0(z−1
) H1(z−1
)
H0(z) H0(−z)
H1(z) H1(−z)
| {z }
Hm(z)
modulation matrix
=
2 0
We replace z with −z in Equations (30) and (31).
H0(−z)H0(−z−1
) + H1(−z)H1(−z−1
) = 2, No distortion
H0(z)H0(−z−1
) + H1(z)H1(−z−1
) = 0 Alias cancellation
H0(z−1
) H1(z−1
)
H0(−z−1
) H1(−z−1
)
| {z }
HT
m(z−1)
H0(z) H0(−z)
H1(z) H1(−z)
| {z }
Hm(z)
=
2 0
0 2
| {z }
2I
Condition O:
HT
m(z−1
)Hm(z) = 2I, Hm(z) is paraunitary
Reverse the order of multiplication:
H0(z) H0(−z)
H1(z) H1(−z)
| {z }
Hm(z)
H0(z−1
) H1(z−1
)
H0(−z−1
) H1(−z−1
)
| {z }
HT
m(z−1)
=
2 0
0 2
| {z }
2I

130. AKUH· AMIN DAVAR
, Fall-1390, WSP 84
Express Condition O as a condition on H0(z):
H0(z)H0(z−1
) + H0(−z)H0(−z−1
) = 2
Frequency Domain:
|H0(eiω
)|
2
+

132. H0(ei(ω+π)
)