Wavelet Transform and DSP Applications

Overview

• Fourier Transform: its power and limitations.

• Short Time Fourier Transform.

• The Gabor Transform.

• Discrete Time Fourier Transform and filter banks.

• Continuous Wavelet Transform.

• Wavelet Transform Ideal Case.

• The Multi-Resolution Analysis

• Perfect Reconstruction Filter Banks and wavelets.

• Haar Wavelet.

• Daubechies Wavelet.

Fourier Transform

• Fourier Transform is one of several mathematical tools that is

useful in the analysis and design of LTI systems.

• It is a way to convert the signal from time domain into

frequency domain.

Example of the Fourier Transform

Transfer Function

• The Fourier transform of a periodic discrete-time signal x(n) is

defined as:

• 𝑋 𝑤 = 𝑛=−∞∞ 𝑥(𝑛)𝑒−𝑗𝜔𝑛

• Where X(w) is the frequency content of the signal x(n).

• The signal x(n) can be synthesis and reconstructed from its

spectrum as:

• 𝑥 𝑛 =1

2𝜋 −∞∞

𝑋(𝑤)𝑒−𝑗𝜔𝑡𝑑𝑤

• Example: Determine the Fourier transform of the signal shown in figure below.

𝑥 𝑛 = 𝐴. −𝑀 ≤ 𝑛 ≤ 𝑀0. 𝑒𝑙𝑠𝑒𝑤ℎ𝑒𝑟𝑒

, where M=2, A=1.

• Solution: the Fourier transform of the signal is

𝑋 𝑤 = 𝑛=−∞∞ 𝑥(𝑛)𝑒−𝑗𝜔𝑛

𝑋 𝑤 = 𝑋 −2 𝑒2𝑗𝑤 + 𝑋 −1 𝑒𝑗𝑤 + 𝑥 0 𝑒0 + 𝑋 1 𝑒−𝑗𝑤 + 𝑋 2 𝑒−2𝑗𝑤

= 1 + 2 cos𝑤 + 2 cos 2𝑤

Signal for this example

w X(w)

0 5

𝜋/2 -1

𝜋 1

2𝜋 5

-𝜋/2 -1

-𝜋 1

3𝜋/2 -1

Spectral characteristics of the

rectangular pulse.

• It is a powerful mathematical tool that allows to view signals in a different

domain, making many difficult problems easy to analyze.

• i.e. any periodic function g(x) in the domain D=[−π,π] can be written as:

• 𝑔 𝑥 = 𝑘=−∞∞ 𝜏(𝑘)𝑒𝑗𝑘𝑥

• 𝜏(𝑘) = 𝐷=−𝜋𝜋

𝑔(𝑥)𝑒−𝑗𝑘𝑥𝑑𝑥

• where 𝑒𝜃 = cos 𝜃 + jsin(𝜃). This idea that a function could be broken

down into its constituent frequencies (i.e., into sines and cosines of all

frequencies) was a powerful one and forms the backbone of the Fourier

transform.

The power of the Fourier Transform

Fourier Transform Examples

Fourier Transform Examples Contd.

F1(𝑤)

F2(𝑤)

F3(𝑤)

Fourier Analysis Examples Contd.

F4(𝑤)

Can’t provide simultaneous time and frequency localization.

Limitations of the Fourier transform

Fourier Analysis Examples Contd.

F4(𝑤)

Provides excellent

localization in the

frequency domain

but poor localization

in the time domain.


Not very useful for analyzing time-variant, non-stationary

signals.


Stationary vs. non-stationary Signals

• Stationary signals:

time-invariant spectra

• Non-stationary

signals: time-varying

spectra

)(4 tf

)(5 tf

Not appropriate for representing discontinuities or sharp

corners (i.e., requires a large number of Fourier

components to represent discontinuities).


Fourier Not very useful for analyzing time-variant, non-

stationary signals.


Short-Time Fourier Transform

• 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.

• Denis Gabor (1946), developed a technique called windowing.

STFT - Steps

(1) Choose a window function of finite length.

(2) Place the window on top of the signal at t=0.

(3) Truncate the signal using this window.

(4) Compute the FT of the truncated signal, save results.

(5) Incrementally slide the window to the right.

(6) Go to step 3, until window reaches the end of the signal.

• Continuous STFT is mathematically represented as:

• Discrete STFT is mathematically represented as:

• Where W(t) is a window function, commonly a Hann window or Gaussian window centered around zero.

Transfer Function

• Fundamental frequency estimation from spectral peaks.

• Cross synthesis.

• Spectral envelope extraction by linear smoothing.

• Sinusoidal modeling of audio signals.

• Time-scale modification.

• Frequency shifting

• Audio FFT filter banks.

STFT Applications

STFT Example

• Used to analyze audio/music and gain

information about an audio sample.

• Horizontal axis represents the frequency.

• The height of each bar is the amplitude of

the frequency.

• Depth dimension represents the time.

• Special case of the STFT used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time.

• Continuous Gabor transformation is represented as:

• 𝐺𝑥 𝑡, 𝑓 = −∞∞

𝑒−𝜋 𝜏−𝑡 2𝑒−𝑗2𝜋𝑓𝜏𝑥 𝜏 𝑑𝜏

• Discrete Gabor transformation is represented as:

• y 𝑡 = 𝑚=−∞∞ 𝑛=−∞

∞ 𝐶𝑛𝑚. 𝑔𝑛𝑚(𝑡)

• With 𝑔𝑛𝑚(𝑡) = 𝑠 𝑡 − 𝑔𝜏0 . 𝑒𝑗𝛺𝑛𝑡

The Gabor Transform

• The main application is in time frequency analysis.

• 𝑓 𝑥 = cos(2𝜋𝑡) , 𝑓𝑜𝑟 𝑡 < 0

cos 4𝜋𝑡 , 𝑓𝑜𝑟 𝑡 ≥ 0

Application and Example

Example contd.

• The input signal has 1 Hz frequency component when t ≤ 0 and has 2 Hz frequency component when t > 0.

• But if the total bandwidth available is 5 Hz, other frequency bands except x(t) are wasted.

• Through time frequency analysis by applying the Gabor transform, the available bandwidth can be known and those frequency bands can be used for other applications and bandwidth is saved.

What are the drawbacks of STFT?

• Unchanged Window.

• Dilemma of Resolution

Narrow window -> poor frequency resolution

Wide window -> poor time resolution

• Can’t know what frequency exists at what time intervals.

Via Narrow Window Via Wide Window

Discrete Time Fourier Transform and filter banks

• DFT converts a finite list of equally spaced samples of a function

into the list of coefficients of a finite combination of complex

sinusoids, ordered by their frequencies.

• 𝑋(𝑘) ≝ 𝑛=0𝑁−1 𝑥𝑛. 𝑒

−𝑗2𝜋𝑘𝑛

𝑁 , 𝑘 ∈ 𝑍 (𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠)

• Filter Banks means split up signal into different frequency bands.

Then reassemble bands into original signal.

Wavelet Transform

• A wavelet is a small wave.

• It is a mathematical function that represent scaled and translated

(shifted) copies of a finite-length waveform called mother

wavelet.

Fourier Transform Wavelet Transform

Some of the most popular mother wavelets

Wavelet Transform Ideal Case (computing CWT of a signal)

• The continuous wavelet transform is given by:

• XWT τ, s =1

s −∞

+∞x t ѱ∗ t−τ

sdt

• Where:

• τ Is the translation parameter.

• s Is the scale parameter.

• ψ is the Mother Wavelet.

Transfer Function

• The Discrete Wavelet Transform (DWT) is given by:

𝑔 𝑡 =

𝑚∊ℤ

𝑛∊ℤ

𝑥, ѱ𝑚,𝑛 . ѱ𝑚,𝑛[𝑡]

Where: ѱ𝑚,𝑛 𝑡 = 𝑠0−𝑚/2

ѱ(𝑠0−𝑚𝑡 − 𝑛τ0)

Transfer Function

The Multi-Resolution Analysis

• Analyzing a signal at different frequencies with

different resolutions. By applying bank filters.

Filter Bank

In signal processing, a filter bank is an array of band-pass filters

that separates the input signal into multiple components, each

one carrying a single frequency sub-band of the original signal.

In DSP, this term is applied to a bank of receivers, and these

receivers also down-convert the sub-bands to a low center

frequency that can be re-sampled at a reduced rate.

Filter Bank

The Analysis of Filter Banks

• Low-pass filter h0(n) whose output, sub-band flp(n) is called approximation

sub-band of f(n).

• High-pass filter h1(n) whose output sub-band fhp(n) is called high frequency or

detail part of f(n).

General Perfect Reconstructions Filter Banks

• Two channel orthogonal system can’t have linear phase except in

simple structures (sum and difference of 2 delays). So, in order to

obtain filter banks with linear phase, orthogonality must be

sacrificed, which leads to biorthogonal linear phase filter banks.

Thus, by sacrificing orthogonality, we have greater degree of

freedom. So, we can use arbitrary length linear phase filters.

Haar Wavelet Transform

• The Haar wavelet (HWT) is a discontinuous, and resembles a

step function.

• Where L is the decomposition level, a is the approximation

subband and d is the detail subband.

Pyramid decomposition using Haar wavelet filter

Decomposition Level 1 Decomposition Level 2

Haar Wavelet Transform Example

• The signal image is:1 3 22 2 11 3 2

212

2 2 1 1

• The sub-bands of the first decomposition will be as follows:

L=

4

2

4

24

2

2

2

, 𝐻 =2

20

0 0

4

2

4

24

2

2

2

2

20

0 0

Example contd.

• The sub-bands of the second decomposition will be as follows:

𝐿𝐿 =4 34 3

, 𝐿𝐻 =0 −10 −1

𝐻𝐿 =1 01 0

, 𝐻𝐻 =−1 0−1 0

Decomposition Using Haar Wavelet Example

Daubechies Wavelet

• It is a family of orthogonal wavelets defining a discrete

wavelet transform and characterized by a maximal number of

vanishing moments for some given support.

• Like Haar wavelet, Daubechies wavelets consist of basic building

blocks (scaling functions) and wavelets proper.

• Daubechies wavelets don’t have the jump discontinuities of the

Haar wavelets but at the cost of the expensive computation.

Thanks^_^

Wavelet Transform and DSP Applications

Engineering

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