digital signal processing laboratory transfer function and digital filter design

International University School of Computer Science & Engineering VIETNAM NATIONAL UNIVERSITY - HO CHI MINH CITY INTERNATIONAL UNIVERSITY SCHOOL OF COMPUTER SCIENCE AND ENGINNERING... In

International University School of Computer Science & Engineering

VIETNAM NATIONAL UNIVERSITY - HO CHI MINH CITY

INTERNATIONAL UNIVERSITY SCHOOL OF COMPUTER SCIENCE AND ENGINNERING

Digital Signal Processing Laboratory

TRANSFER FUNCTION AND DIGITAL

FILTER DESIGN

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Contents

1 Theoretical Background

1.1 Fourier Transform

1.2 Transfer Function

1.3 Resonator Filter L.A Hamming Window escssessssssssssessssesscnsessssessesseseeseceesteseeseeeestecssteeeesteessteesseeeeateessaeesseraee

2 Experimental Procedure

2.1 Problem 1

2.2 Problem 2

3 Experimental Results,

3.1 Results of Problem 1

3.2 Results of Problem 2

4 Discussion of Results

5 References

List of Figures

List of Tables

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1 Theoretical Background

1.1 Fourier Transform

- What is it?

> The Fourier transform is a mathematical function that decomposes a time-dependent waveform into frequencies The transform's output is a complex-valued frequency function

In simpler words, the Fourier transform can help find the frequencies of a function in the

time domain "!,

- How to doit?

> For doing the Fourier transform, we will use the following formula ”!:

| f(xje dx

- Example

> Applying the Fourier transform to the square wave reveals its hidden frequency components It shows a series of vertical lines on a frequency axis, representing the

different frequencies present in the signal Here is what it may look like:

DFFT of 1Hz 50% Duty-Cycle Square Wave

500 -+-i \ -4 - L - be -4 -4 - ban - fpe -d -

Se iitfiil aaa Hường ae nee === a

300 +]-F - - - iss Trennn reeeee ren

200 -1-

Frequency [Hz]

Figure 1: Example of Fourier transform (taken from” Why Fourier series and transform of a square

wave are different? - Signal Processing Stack Exchange”)

- What are the applications?

> Fourier transform has numerous applications in real-life and here are some of them:

1 Signal processing

2 Communication system

3 Image Processing

4 Audio Processing

1.2 Transfer Function

- What is it?

> A transfer function is defined as the ratio of the output Laplace transform to the input Laplace transform, assuming that all initial conditions are zero “! It consists of seven

different types:

a) Impulse response

b) Difference equation

c) 1/0 Difference Equation

d) Frequency response

e) Block diagram of realization

f) Sample Processing Algorithm

g) Frequency Response and Pole/zero Pattern

- How to doit?

> We have the direct form to do the transfer function which is:

Y(z)

H(z)=

- Example

> Consider this system:

y(n) =O0.25 p(n - 2)+x(07)

Here by taking the z-transform of both sides of the equation we get:

¥(z) =0.252-¥(2)+ X(z)

Y(z)

Solving for * (2) we get:

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H(z) =- - = +

- What are the applications?

> Transfer function can have many applications in real-life Here are some of them:

1 Control system designs

2 Electrical circuits

3 Mechanical systems

4 Robotics

1.3 Resonator Filter

- What is it?

= A resonator filter, also known as a resonant filter or a resonant circuit, is a form of electrical filter that filters or amplifies certain frequencies using the idea of resonance

- How to doit?

> Resonator filters can be implemented using various electronic components and circuit

configurations The most common types of resonator filters are based on LC (inductor-

capacitor) circuits or RLC (resistor-inductor-capacitor) circuits The resonant frequency of

the filter can be adjusted by varying the values of the inductors, capacitors, or resistors in the circuit

Example

> Let's consider an example of a resonator filter with a resonant frequency of 1 kHz The

LC circuit consists of an inductor with an inductance of 100 mH and a capacitor with a

capacitance of 10 nF By using these values, we can calculate the resonant frequency using the formula:

]

Resonant frequency (f) = 27VLC

Substituting the values: f= 1591 Hz

So, in this example, the resonant frequency of the filter would be approximately 1.591 kHz

- What are the applications?

> Here are some applications of resonator filter:

- Audio and music

- Communication system

- Image and Signal Processing

- What is it?

> The Hamming window is a mathematical function used in signal processing and other applications to taper the edges of a signal, most notably in Fourier analysis

- How to doit?

> The mathematical expression for the Hamming Window is given by:

(0) =0.Š4 - 0.40 cos( = -)

(Eq.3) where:

1.) is the value of the window at sample n

Nis the total number of samples

By multiplying each sample of the signal by the corresponding sample by this formula:

- Example

>

- What are the applications?

> Some of the applications of Hamming Window

- Spectral Analysis

- Speech Processing

- Filter Design

- Image Processing

- RADAR and SONAR Processing

2 Experimental Procedure

2.1 Problem 1

a)

International University School of Computer Science & Engineering

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yea) - 16 z1Y(2) - 03x? X3) + Xca) + 072 X(2) + 0.62” X2)

© X‹z)[ƑA- 4§z2+ 0.9233 < X21 424+ 0.62°*]

Ye2) A= AS a £) 3„!

= Ac2) = M2 = — 5

b)

Code:

a= [1,0.7,0.6];

b = [1,-1.5,0.9];

x = zeros(1,100);

x(1) = 1;

h = filter(b,a,x);

h = h'

stem(h)

xlabel('n')

ylabel(’ h[n]')

title('Impulse response of filter')

b)

Output:

1.0000

-2.2000

EEO93IU

Digital Signal Processing Laboratory 9

1.8400

0.0320

-1.1264

0.7693

0.1373

-0.5577

0.3080

0.1190

-0.2681

0.1163

0.0795

-0.1254

0.0401

0.0472

-0.0571

0.0116

0.0261

-0.0253

0.0020

0.0137

-0.0108

-0.0007

0.0070

-0.0045

-0.0010

0.0034

-0.0018

-0.0008

0.0016

-0.0007

International University School of Computer Science & Engineering

-0.0005

0.0008

-0.0002

-0.0003

0.0003

-0.0001

-0.0002

0.0001

-0.0000

-0.0001

0.0001

0.0000

-0.0000

0.0000

0.0000

-0.0000

0.0000

0.0000

-0.0000

0.0000

0.0000

-0.0000

0.0000

0.0000

-0.0000

0.0000

0.0000

-0.0000

0.0000

0.0000

-0.0000

-0.0000

0.0000

-0.0000

-0.0000

0.0000

-0.0000

-0.0000

0.0000

-0.0000

-0.0000

0.0000

-0.0000

-0.0000

0.0000

-0.0000

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0.0000

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-0.0000

0.0000

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-0.0000

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0.0000

International University School of Computer Science & Engineering

0.0000

-0.0000

0.0000

0.0000

-0.0000

0.0000

0.0000

-0.0000

Impulse response of filter

c)

Code:

close all;

a= [1, 0.7, 0.6];

b= [1, -1.5, 0.9];

x = zeros(1, 100);

x(1) = 1;

h = filter(b, a, x);

h=h';

% Plot pole-zero plot

figure;

zplane(b, a);

% Plot frequency response

figure;

q = fftshift(fft(h));

subplot(2, 1, 1);

plot(linspace(-pi, pi, length(q)), abs(q));

grid on;

grid minor;

xlim([-pi, pil);

title('Magnitude of Frequency Response');

ylabel(' |H(e*{j \Omega})|');

xlabel('\Omega');

subptot(2, 1, 2);

plot(linspace(-pi, pi, Length(q)), 189/pi*angLe(q));

grid on;

grid minor;

xlim([-pi, pil);

title('Angle of Frequency Response');

ylabel('\angle H(e*{j] \Omega})');

xlabel('\Omega');

Output:

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0.6 5

0.2 fF

Magnitude of Frequency Response

Angle of Frequency Response

He!) oOo

2.2 Problem 2

2.3 Problem 3

3 Experimental Results

3.1 Results of Problem 1

International University School of Computer Science & Engineering 3.2 Results of Problem 2

3.3 Results of Problem 3

4 Discussion of Results

- What have you learnt?

- Any interesting findings?

- Potential applications?

5 References

[1]: Stanford University (n.d.) Fourier Transform - GRU Retrieved from https://gru.stanford.edu/doku.php/tutorials/fouriertransform

[2]: https://examplemath.com/fourier-transform-examples