(DSP) Digital Signal Processing - YolaDSP) Digital Signal...DIGITAL SIGNAL PROCESSINGDIGITAL SIGNAL PROCESSINGSIGNAL PROCESSING COMPUTER SIMULATION LAB DEPARTMENT OF ELECTRICAL ENGINEERING

DIGITAL DIGITAL DIGITAL DIGITAL SIGNAL PROCESSINGSIGNAL PROCESSINGSIGNAL PROCESSINGSIGNAL PROCESSING

COMPUTER SIMULATION LAB

DEPARTMENT OF ELECTRICAL ENGINEERING

Prepared By:

Engr. Faisal Baig

Lecturer (Lab) Electrical,

FUUAST-Islamabad

Name: ____________________________________________

Registration No: ____________________________________

Roll No: ___________________________________________

Semester: _________________________________________

Batch: ____________________________________________

SECTION NO. 1 (Software)

EXPERIMENT NO. 1:

Signals in Matlab (Continuous time & Discrete time)

EXPERIMENT NO. 2:

Discrete Time Systems and its Properties

1) Linearity

2) Time invariant

EXPERIMENT NO. 3:

Impulse Response of a system

1) IIR

2) FIR

EXPERIMENT NO. 4:

DTFT and Properties

EXPERIMENT NO. 5:

Sampling, A/D Conversion and D/A Conversion

EXPERIMENT NO. 6:

System Frequency Transform

Introduction to Digital Filter

EXPERIMENT NO. 7:

Simple Digital Filter

1) IIR

2) FIR

EXPERIMENT NO. 8:

Digital Filter Design

1) FIR

2) Introduction to Widows method

EXPERIMENT NO. 9:

FIR Filter Design using Rectangular Windows method

1) LPF

2) HPF

3) BPF

4) BSF

EXPERIMENT NO. 10:

DIGITAL FILTER STRUCTURE

EXPERIMENT NO. 11:

Design of FIR filters using Matlab commands. EXPERIMENT NO. 12:

Design of IIR filters using Matlab commands

EXPERIMENT NO. 13:

DFT

EXPERIMENT NO. 14:

Interpolation & Decimation

EXPERIMENT NO. 15:

Filter designing by Matlab tools

EXPERIMENT NO. 16:

Design an IIR filter to suppress frequencies of 5 Hz and 30 Hz from given signal

EXPERIMENT NO. 1

SIGNALS IN MATLAB (CONTINUOUS TIME & DISCRETE TIME)

Signals are broadly classified into two classifications:

a. Continuous Time Signals

b. Discrete Time Signals

A continuous time signal will be denoted by x(t), in which the variable t can represent any physical

quantity.

A discrete time signal will be denoted x[n], in which the variable n is integer value. In this lab we will

learn to represent and operate on signals in MATLAB.

1. Continuous Time Signals

For the following: Run the following lines and explain why the plots are different.

Provide the snapshots of the plots for each step given below.

close all, clear all t = 0:2*pi; plot(t,sin(t)) figure t = 0:0.2:2*pi; plot(t,sin(t)) figuret = 0:0.02:2*pi; plot(t,sin(t))

For the last graph, add a title and axis labels with: title('Continuous time signal plot') xlabel('t (Seconds)') ylabel('y(t)')

Change the axis with:

axis([0 2*pi -1.2 1.2])

Put two plots on the same axis t = 0:0.2:2*pi; plot(t,sin(t),t,sin(2*t))

Produce a plot without connecting the points

t = 0:0.2:2*pi; plot(t,sin(t),'.')

Try the following command

t = 0:0.2:2*pi;

plot(t,sin(t),t,sin(t),'g.')

Question 1: What does ‘g’ do?

2. Discrete Time Signals

Use stem to plot the discrete time signals. Provide the snapshots of the step below.

close all clear all n = -10: 10; f = n >= 0; stem(n,f)

3. Elementary sequences in digital signal processing

For each part below, provide an example using any input and also provide the plots of input and

output sequences using subplot.

a. Unit sample sequence

− = , = , ≠

% Generate a vector from -10 to 20 n = -10:20; n0=0; % Generate the unit sample sequence u =n-n0==0; % Plot the unit sample sequence stem(n,u); xlabel( 'Time index n' ); ylabel( 'Amplitude' ); title( 'Unit Sample Sequence' ); axis([-10 20 0 2]);

b. Unit step sequence

− = , ≥ , <

n=-10:30 n0=0; y=n-n0>=0; figure stem(n,y); title('UNIT STEP FUNCTION');

c. Real Valued Exponential sequence

= , ∈

n = 0:35; a = 1.2; K = 0.2; x = K*a.^+n; stem(n,x); xlabel('Time index n'); ylabel('Amplitude'); d. Complex valued exponential sequence

=

c = -(1/12)+(pi/6)*i; K = 2; n = 0:40; x = K*exp(c*n); subplot(2,1,1); stem(n,real(x)); xlabel('Time index n'); ylabel('Amplitude'); title('Real part'); subplot(2,1,2); stem(n,imag(x)); xlabel('Time index n'); ylabel('Amplitude'); title('Imaginary part');

e. Sinusoidal sequence

n = 0:40; f = 0.1; phase = 0; A = 1.5; arg = 2*pi*f*n - phase; x = A*cos(arg); stem(n,x); % Plot the generated sequence axis([0 40 -2 2]); grid; title('Sinusoidal Sequence'); xlabel('Time index n'); ylabel('Amplitude'); axis;

f. Operations on sequence:

• Shifting

In this operation each sample of x(n) is shifted by an amount k to obtain a shifted

sequence y(n)

yn = xn − k

If we let m = n-k, then n = m+k the above operation is given by

ym + k = xm

For this we can use the following code

s=1:100; subplot(2,1,1) stem(s) title('Sequence'); axis([0 100 0 100]); s_new=[zeros(1,10) s]; subplot(2,1,2) stem(s_new); axis([0 100 0 100]); title('Delayed sequence');

• Folding

In this operation each sample of x(n) is flipped around its axis to obtain a folded sequence

y(n)

!" = #−"

For this the following code is shown.

Exercise:

Generate and plot each of the following sequences over the indicated interval. Provide the

scripts used to generate the plots.

Ex.1:

a. $" = 2&" + 2 − &" − 4,−5 ≤ & ≤ 5

b. #" = "[+" − +" − 10] + 10/^−0.3" − 10[+" − 10 − +" − 20], 0 ≤ " ≤ 20

Ex.2:

Let x(n) = 1,2,3,4,5,6,7,6,5,4,3,2,1, Determine and plot the following sequences.

a. #3" = 2#" − 5 − 3#" + 4 b. #4" = #3 − " + #"#" − 2

EXPERIMENT NO. 2

(Discrete Time Systems)

DSP LAB # 3: Discrete Time Systems

To provide an overview of discrete time signals and systems on MATLAB, to analyze various properties

of discrete time systems and verify them on MATLAB.

Introduction:

Mathematically, a discrete-time system is described as an operator T[.] that takes a sequencex(n) called

excitation and transforms it into another sequence y(n) (called response). Discretetime systems can be

classified into two categories i) LTI systems ii) NON-LTI systems. Adiscrete system T[.] is a linear operator

L[.] if and only if L[.] satisfies the principle ofsuperposition, namely

5[ + 66] = 5[] +5[6]

A discrete system is time-invariant if shifting the input only causes the same shift in the output.

A system is said to be bounded-input bounded-output (BIBO) stable if every bounded input produces a

bounded output.

|| < ∞ → |:| < ∞,∀#, !

An LTI system is BIBO stable if and only if its impulse response is absolutely sum able.

<=<>?@ABCB@: ↔ E|F| < ∞

A system is said to be causal if the output at any instant depends only on the present & past values only.

An LTI system is causal if and only if the impulse response is

F = , < 0

1. Linearity and Non-Linearity: We now investigate the linearity property of a causal system of described by the following

equation.

:[] − . G:[ − ] + . HI:[ − 6] = 6. 6[] + 6. J[ − ] + 6. G[ − 6]

Following program simulates the above mentioned equation.

Question 1: Run above program and compare y[n] obtained with weighted input with y[n] obtained by

combining the two outputs y1[n] and y2[n] with the same weights. Are these two sequences equal? Is

this system linear?

Exercise 1: Consider another system described by :*. − :* − 6. + . HI:* − G. = 6. 6*. + 6. J* − J. + 6. G* − I.

:*. − :* − 6. + :* − G. = *. + * − J. + * − G.

:*. − :* − 6. + . HI:* − G. + J:* − I. = *. + . J* − . + 6. G* − J.

Modify given program to compute the output sequences y1[n], y2[n], and y[n] of the above system.

Compare y[n] with y[n]. Are these two sequences equal? Is this system linear?

2. Time-Invariant and Time-Varying Systems:

We next investigate the time-invariance property. Following program simulates following

difference equation

:*. − . G:* − . + . HI:* − 6. = 6. 6*. + 6. J* − . + 6. G* − 6.

Two input sequence x[n] and x[n-D], are generated and corresponding output sequences

y1[n],y2[n] are plotted.

clear all, close all n = 0:40; a = 2; b = -3; x1 = cos(2*pi*0.1*n); x2 = cos(2*pi*0.4*n); x = a*x1 + b*x2; num = [2.2 2.3 2.4]; den = [1 -0.4 0.75]; ic = [0 0]; % Set zero initial conditions y1 = filter(num,den,x1,ic); % Compute the output y1 [n] y2 = filter(num,den,x2,ic); % Compute the output y2 [n] y = filter(num,den,x,ic); % Compute the output y[n] yt = a*y1 + b*y2; d = y - yt; % Compute the difference output d[n] % Plot the outputs and the difference signal subplot (3,1,1) stem(n ,y); ylabel('Amplitude'); title('Output Due to Weighted Input'); subplot(3,1,2) stem(n,yt); ylabel('Amplitude'); title('Weighted Output'); subplot(3,1,3) stem(n,d); xlabel('Time index n'); ylabel('Amplitude'); title('Difference Signal');

Exercise 2: Consider another system described by:

:*. − :* − . + :* − 6. + . :* − G. = [] + 6. J[ − ] + 6. G[ − I] :[] − :[ − 6]+= [] + . 6[ − 6] + [ − G]

:[] = [] + 6. J[ − J] + 6. G[ − I]

Modify program to simulate the above system and determine whether this system is time-

invariant or not.

3. Linearity and Time-Invariant: (LOOPS)

Using loops to solve the given equation

:[] = [] + [ − ]

close all, clear all n = 0:40; D = 10;a = 3.0;b = -2; x = a*cos(2*pi*0.1*n) + b*cos(2*pi*0.4*n); xd = [zeros(1,D) x]; num = [2.2 2.3 2.4]; den = [1 -0.4 0.75]; ic = [0 0];% Set initial conditions % Compute the output y[n] y = filter(num,den,x,ic); % Compute the output yd[n] yd = filter(num,den,xd,ic); % Compute the difference output d[n] d = y - yd(1+D:41+D); % Plot the outputs subplot(3,1,1) stem(n,y); ylabel('mplitude'); title('Output y[n]');grid; subplot(3,1,2) stem(n,yd(1:41)); ylabel('Amplitude'); title(['Output Due to Delayed Input x[n , num2str(D ),]']);grid; subplot(3,1,3) stem(n,d); xlabel('Time index n'); ylabel('Amplitude'); title('Difference Signal');grid;

x1=[1 1 2]; x2=[2 2 0]; %for y1 for i=1:3 if(i==1) y1(i)=x1(i); else if(i>1) y1(i)=x1(i)+x1(i-1); end end end %for y2 for i=1:3 if(i==1) y2(i)=x2(i); else if(i>1) y2(i)=x2(i)+x2(i-1); end end end x3=x1+x2; %for y3 for i=1:3 if(i==1) y3(i)=x3(i); else if(i>1) y3(i)= x3(i)+x3(i-1);

end

end

end

y3

y1+y2

if(y3==(y1+y2))

display('System is linear')

end

x1=[1 1 2]; %for y1 for i=1:3 if(i==1) y1(i)=x1(i); else if(i>1) y1(i)=x1(i)+x1(i-1); end end end y1 x2=[0 0 x1] %for y2 for i=1:5 if(i==1) y3(i)=x2(i); else if(i>1) y3(i)=x2(i)+x2(i-1); end end end y3 if(y3(3:5)==y1) display('System is Time invariant') end

Exercise 1: Consider another system described by:

:*. + :* − . = [] + [ − J] :[] = [] + [ − ]

Modify program to simulate the above system and determine whether this system is time-

invariant or not.

EXPERIMENT NO. 3

Impulse Response of System

Impulse Response Computation:

Response of a system to an impulse is called impulse response, two types

• FIR( Finite impulse response)

• IIR (infinite impulse response)

FIR:

In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or

response to any finite length input) is of finite duration, because it settles to zero in finite time.

For a discrete-time FIR filter, the output is a weighted sum of the current and a finite number of

previous values of the input. The operation is described by the following equation, which defines

the output sequence y[n] in terms of its input sequence x[n]:

y[n]=b0x(n)+ b1x(n-1)+ b2x(n-2) ……….+ bNx(n-N)

y [n]=∑bi x(n-i)

where:

• is the input signal,

• is the output signal,

• are the filter coefficients, also known as tap weights, that make up the impulse

response,

is the filter order; an th-order filter has terms on the right-hand side. The in

these terms are commonly referred to as taps, based on the structure of a tapped delay line that in

many implementations or block diagrams provides the delayed inputs to the multiplication

operations. One may speak of a 5th order/6-tap filter, for instance.

Properties: An FIR filter has a number of useful properties which sometimes make it preferable to an infinite

impulse response (IIR) filter. FIR filters:

• Require no feedback. This means that any rounding errors are not compounded by summed

iterations. The same relative error occurs in each calculation. This also makes implementation

simpler.

• Are inherently stable. This is due to the fact that, because there is no required feedback, all the

poles are located at the origin and thus are located within the unit circle (the required condition

for stability in a Z transformed system).

• They can easily be designed to be linear phase by making the coefficient sequence symmetric;

linear phase, or phase change proportional to frequency, corresponds to equal delay at all

frequencies. This property is sometimes desired for phase-sensitive applications, for example

data communications, crossover filters, and mastering.

Impulse response:

The impulse response can be calculated if we set in the above relation, where

is impulse. The impulse response for an FIR filter then becomes the set of coefficients , as follows

for to .

FIR filters are clearly bounded-input bounded-output (BIBO) stable, since the output is a sum of a

finite number of finite multiples of the input values, so can be no greater than times the

largest value appearing in the input.

:*. = 6. 6*. * . * 6.

Exercise 1: Compute Impulse Response

:*. 6. 6*. * . * 6. . * K.

IIR:

% Compute the impulse response y

close all, clear all

N = 40;

num = [2.2 1 1];

den = [1];

y = impz(num,den,N);

% Plot the impulse response

stem(y);

xlabel('Time index n'); ylabel('Amplitude');

title('Impulse Response'); grid;

IIR systems have an impulse response function that is non-zero over an infinite length of time. This

is in contrast to finite impulse response (FIR) filters, which have fixed-duration impulse responses.

The simplest analog IIR filter is an RC filter made up of a single resistor (R) feeding into a node

shared with a single capacitor (C).

Digital filters are often described and implemented in terms of the difference equation that defines

how the output signal is related to the input signal:

where:

Is the feed forward filter order

are the feed forward filter coefficients

is the feedback filter order

are the feedback filter coefficients

Is the input signal

Is the output signal.

A more condensed form of the difference equation is:

which, when rearranged, becomes:

To find the transfer function of the filter, we first take the Z-transform of each side of the above

equation, where we use the time-shift property to obtain:

We define the transfer function to be:

Considering that in most IIR filter designs coefficient is 1, the IIR filter transfer function takes the

more traditional form:

Properties:

The main advantage IIR filters have over FIR filters is that through recursion they use fewer taps.

Therefore in digital signal processing applications IIR filters use fewer computing resources than an

equivalent FIR filter. A disadvantage of IIR filters is they can be unstable and subject to limit cycle

behavior.

:*. − 6:* − . = 6. 6*. * . * 6.

Know For N=100

% Compute the impulse response y

close all, clear all

N = 40;

num = [2.2 1 1];

den = [1 2];

y = impz(num,den,N);

% Plot the impulse response

stem(y);

xlabel('Time index n'); ylabel('Amplitude');

title('Impulse Response'); grid;

Following equation computes impulse response of following difference equation

:*. − . G:* . . HI:* 6. 6. 6*. 6. J* . 6. G* 6. Exercise 3: Write MATLAB program to generate and plot the step response of a causal LTI

system.

Exercise 2:

:*. . G:* . . HI:* 6. 6. 6*. 6. J* . 6. G* 6. Using this program compute and plot the first 40 samples of the step response above

mentioned LTI system.

% Compute the impulse response y close all, clear all N = 40; num = [2.2 2.3 2.4]; den = [1 -0.4 0.75]; y = impz(num,den,N); % Plot the impulse response stem(y); xlabel('Time index n'); ylabel('Amplitude'); title('Impulse Response'); grid;

EXPERIMENT NO. 4

Z-Transform

Study of Z-Transform in MATLAB:

THEORY: Z-Transform technique is an important tool in the analysis of characterization of discrete time

signals and LTI systems; Z-Transform gives the response of various signals by its pole zero locations.

Z-Transform is an important tool in DSP that gives the solution of difference equation in one go.

The Z-Transform of a discrete time system x (n) is defined as power series;

And the inverse Z-Transform is denoted by;

Since, Z-Transform is the infinite power series; it exists only for the region for which this series

converges (region of convergence). Inverse Z-Transform is the method for inverting the Z-

Transform of a signal so as to obtain the time domain representation of signal. The features of Z-

Transform which are explained are as fellows;

Z-Transform of a Discrete time function

EXAMPLE:

MATLAB CODE:

% Z transform close all, clear all

b=[1 -1.6 180 1];

a=[1 -1.5 161 0.878];

A=roots(a)

B=roots(b)

zplane(b,a)

A =

0.7527 +12.6666i

0.7527 -12.6666i

-0.0055

B =

0.8028 +13.3927i

0.8028 -13.3927i

-0.0056

Z-TRRANSFORM OF A DISCRETE TIME FUNCTION:

Z-transform is defined as

X(z)

Or

X(z)=Z(x(n))

Let the function equation be

*. = /G^

*. = MB ∗

% Z-transform close all, clear all syms z n u =1/(4^n); U=ztrans(u) U= 4*z/(4*z-1)

% Z-transform close all, clear all syms z n a x=sin(a*n) X=ztrans(x) X= z*sin(a)/(z^2-2*z*cos(a)+1)

Exercise 1:

Consider that the equation is

*. = ^G

Find Z-transform

INVERSE Z-TRANSFORM: The inverse Z-Transform is denoted by,

Let the Z-domain is:

% Z-transform close all, clear all syms z n X=2*z/(2*z-1) x=iztrans(X) x= (1/2)^n

EXPERIMENT NO. 5

Fourier Transform and Properties

Discrete-Time Fourier Transform: The discrete-time Fourier transform (DTFT) X(ejω ) of a sequence x[n] is a continuous function of ω.

Since the data in MATLAB is in vector form, X(ejω ) can only be evaluated at a prescribed set of

discrete frequencies. Moreover, only a class of the DTFT that is expressed as a rational function in

e−jω in the form

OP/QRS = TU + T3/VQR + T4/VQR4 + ⋯ + TX/VQRXYU + Y3/VQR + Y4/VQR4 + ⋯ + YZ/VQRZ

(5.1)

can be evaluated. In the following two projects you will learn how to evaluate and plot the DTFT

and study certain properties of the DTFT using MATLAB.

DTFT Computation

The DTFT X(ejω ) of a sequence x[n] of the form of Eq. (5.1) can be computed easily at a prescribed

set of L discrete frequency points ω = ω₃ using the MATLAB function freqz. Since X(ejω ) is a

continuous function of ω, it is necessary to make L as large as possible so that the plot generated

using the command plot provides a reasonable replica of the actual plot of the DTFT. In MATLAB,

freqz computes the L-point DFT of the sequences p0 p1 . . . PM and d0 d1 . . . dM , and then

forms their ratio to arrive at X(ejωl ), l = 1, 2, . . . , L. For faster computation, L should be chosen as a

power of 2, such as 256 or 512.

Program P5_1 can be used to evaluate and plot the DTFT of the form of Eq. (5.1).

% Program P5_1 % Evaluation of the DTFT clear all; close all; clc % Compute the frequency samples of the DTFT w = -4*pi:8*pi/511:4*pi; num = [2 1];den = [1 -0.6]; h = freqz(num, den, w); % Plot the DTFT subplot(2,1,1) plot(w/pi,real(h));grid title('Real part of H(e^j\omega)') xlabel('\omega /\pi'); ylabel('Amplitude'); subplot(2,1,2) % plot the imaginary part in the same way your self and give it title pause subplot(2,1,1) % In this subplot plot magnitude of FFT your self subplot(2,1,2) % In this subplot plot magnitude of FFT your self

DTFT Properties

Most of the properties of the DTFT can be verified using MATLAB. Since all data in MATLAB have to

be finite-length vectors, the sequences being used to verify the properties are thus restricted to be

of finite length.

Program P5_2 can be used to verify the time-shifting property of the DTFT.

-1 -0.5 0 0.5 10

20

40

60Magnitude Spectrum of Original Sequence

-1 -0.5 0 0.5 10

20

40

60Magnitude Spectrum of Time-Shifted Sequence

-1 -0.5 0 0.5 1-4

-2

0

2

4Phase Spectrum of Original Sequence

-1 -0.5 0 0.5 1-4

-2

0

2

4Phase Spectrum of Time-Shifted Sequence

% Program P5_2 % Time-Shifting Properties of DTFT close all; clear all; clc w = -pi:2*pi/255:pi; wo = 0.4*pi;D=10; num=[1 2 3 4 5 6 7 8 9]; h1 = freqz(num, 1, w); h2 = freqz([zeros(1,D) num], 1, w); subplot(2,2,1) plot(w/pi,abs(h1));grid title('Magnitude Spectrum of Original Sequence') subplot(2,2,2) plot(w/pi,abs(h2));grid title('Magnitude Spectrum of Time-Shifted Sequence' ) subplot(2,2,3) plot(w/pi,angle(h1));grid title('Phase Spectrum of Original Sequence') subplot(2,2,4) plot(w/pi,angle(h2));grid title('Phase Spectrum of Time-Shifted Sequence')

Program P5_3 can be used to verify the frequency-shifting property of the DTFT.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 11.3579

1.3579

1.3579

1.3579

1.3579

1.3579

1.3579

1.3579

1.3579x 10

10 Magnitude Spectrum of Original Sequence

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 11.3579

1.3579

1.3579

1.3579

1.3579

1.3579

1.3579

1.3579

1.3579x 10

10 Magnitude Spectrum of Frequency-Shifted Sequence

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1.5

-1

-0.5

0

0.5

1

1.5x 10

-9 Phase Spectrum of Original Sequence

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1.5

-1

-0.5

0

0.5

1

1.5x 10

-9 Phase Spectrum of Frequency-Shifted Sequence

% Program P5_3 % Frequency-Shifting Properties of DTFT clear all; close all; clc w = -pi:2*pi/255:pi; wo = 0.4*pi; num1=[13579111315 17];

L = lengt h(num1); h1 = freqz(num1, 1, w); n = 0:L-1; num2 = exp(wo*i*n).*num1; h2 = freqz(num2, 1, w); subplot(2,2,1) plot(w/pi,abs(h1));grid title('Magnitude Spectrum of Original Sequence') subplot(2,2,2) plot(w/pi,abs(h2));grid title('Magnitude Spectrum of Frequency-Shifted Sequ ence') subplot(2,2,3) plot(w/pi,angle(h1));grid title('Phase Spectrum of Original Sequence') subplot(2,2,4) plot(w/pi,angle(h2));grid title('Phase Spectrum of Frequency-Shifted Sequence ')

Program P5_4 can be used to verify the convolution property of the DTFT.

Program P5_5 can be used to verify the modulation property of the DTFT.

% Program P5_4

% Convolution Property of DTFT

close all; clear all; clc

w = -pi:2*pi/255:pi;

x1=[1 3 5 7 9 11 13 15 17];

x2=[1 -23 -21];

% convolve x1 and x2 yourself

h1 = freqz(x1, 1, w);

% compute freqz of x2 at w points and save it in h2 yourself

hp = h1.*h2;

h3 = freqz(y,1,w);

subplot(2,2,1)

plot(w/pi,abs(hp));grid

title('Product of Magnitude Spectra')

subplot(2,2,2)

plot(w/pi,abs(h3));grid

title('Magnitude Spectrum of Convolved Sequence')

subplot(2,2,3)

% Program P5_5

% Modulation Property of DTFT

close all; clear all; clc

w = -pi:2*pi/255:pi;

x1=[1 3 5 7 9 11 13 15 17];

x2=[1 -1 1 -1 1 -1 1 -1 1];

y = x1.*x2;

h1 = freqz(x1, 1, w);

h2 = freqz(x2, 1, w);

% similarly compute freqz of y and save it in h3 yourself

subplot(3,1,1)


title('Magnitude Spectrum of First Sequence')

subplot(3,1,2)


Program P5_6 can be used to verify the time-reversal property of the DTFT.

-1 -0.5 0 0.5 12

4

6

8

10Magnitude Spectrum of Original Sequence

-1 -0.5 0 0.5 12

4

6

8

10Magnitude Spectrum of Time-Reversed Sequence

-1 -0.5 0 0.5 1-4

-2

0

2

4Phase Spectrum of Original Sequence

-1 -0.5 0 0.5 1-4

-2

0

2

4Phase Spectrum of Time-Reversed Sequence

% Program P5_6 % Time-Reversal Property of DTFT close all; clear all; clc w = -pi:2*pi/255:pi; num=[1 2 3 4]; L = length(num)-1; h1 = freqz(num, 1, w); h2 = freqz(fliplr(num), 1, w); h3 = exp(w*L*1i).*h2; subplot(2,2,1) plot(w/pi,abs(h1));grid title('Magnitude Spectrum of Original Sequence') subplot(2,2,2) plot(w/pi,abs(h3));grid title('Magnitude Spectrum of Time-Reversed Sequence ') subplot(2,2,3) plot(w/pi,angle(h1));grid title('Phase Spectrum of Original Sequence') subplot(2,2,4) plot(w/pi,angle(h3));grid title('Phase Spectrum of Time-Reversed Sequence')

EXPERIMENT NO. 6

Sampling, A/D Conversion and D/A

Conversion

The Sampling Process in the Time Domain:

The purpose of this section is to study the relation in the time domain between a continuous-time

signal xa (t) and the discrete-time signal x[1] generated by a periodic sampling of xa (t).

1. Sampling of a Sinusoidal Signal

In this project you will investigate the sampling of a continuous-time sinusoidal signal xa (t) at

various sampling rates. Since MATLAB cannot strictly generate a continuous-time signal, you will

generate a sequence xa (nTH ) from xa (t) by sampling it at a very high rate, 1/TH , such that the

samples are very close to each other. A plot of xa (nTH ) using the plot command will then look like

a continuous-time signal.

2. Aliasing Effect in the Time Domain

In this project you will generate a continuous-time equivalent ya (t) of the discrete-time signal x[1]

generated in Program P4_1 to investigate the relation between the frequency of the sinusoidal

signal xa (t) and the sampling period. To generate the reconstructed signal ya (t) from x[1], we pass

x[1] through an ideal low pass filter that in turn can be implemented according to Eq. (4.1). If Eq.

(4.1) is computed at closely spaced values of t, a plot of ya (t) will resemble a continuous-time

signal. In order to implement this equation on MATLAB, the summation in Eq. (4.1) needs to be

replaced with a finite sum, and hence we can generate only an approximation to the desired

reconstructed continuous-time signal ya (t).

% Program 6_1 % Illustration of the Sampling Process % in the Time Domain clear all; close all; clc; t = 0:0.0005:1; f = 13; xa = cos(2*pi*f*t); subplot(2,1,1) plot(t,xa,'LineWidth',1.5); xlabel('Time, msec');ylabel('Amplitude'); title('Continuous-time signal x_a(t)'); axis([0 1 -1.2 1.2]) subplot(2,1,2); T = 0.1; n = 0:T:1; xs = cos(2*pi*f*n); k = 0:length(n)-1; stem(k,xs,'r'); xlabel('Time index n');ylabel('Amplitude'); title('Discrete-time signal x[n]'); axis([0 (length(n)-1) -1.2 1.2])

h\t = 12π _ H\jΩedefdt

h

Vh

= T2π_ edefdΩ = sinΩltΩmt/2

enVen

,− ∞ ≤ t ≤ ∞

(4.1)

3. Effect of Sampling in the Frequency Domain

Aliasing Effect in the Frequency Domain

% Program P6_2 % Illustration of Aliasing Effect in the Time Domai n clear all; close all; clc; T = 0.1; f = 13; n = (0:T:1)'; xs = cos(2*pi*f*n); t = linspace(-0.5,1.5,500)'; ya = sinc((1/T)*t(:,ones(size(n))) - (1/T)*n(:,ones (size(t)))')*xs; plot(n,xs,'bo',t,ya, 'r','Linewidth',1.5);grid; xlabel('Time, msec');ylabel('Amplitude'); title('Reconstructed continuous-time signal y_a(t )'); axis([0 1 -1.2 1.2]);

% Program P6_3 % Illustration of the Aliasing Effect % in the Frequency Domain clear all; close all; clc; t = 0:0.005:10; xa = 2*t.*exp(-t); subplot(2,2,1) plot(t,xa);grid xlabel('Time, msec');ylabel('Amplitude'); title('Continuous-time signal x_a(t)'); subplot(2,2,2) wa = 0:10/511:10; ha = freqs(2,[1 2 1],wa); plot(wa/(2*pi),abs(ha));grid; xlabel('Frequency, kHz');ylabel('Amplitude'); title('|X_a(j\Omega)|'); axis([0 5/pi 0 2]); subplot(2,2,3) T=1 ; n = 0:T:10; xs = 2*n.*exp(-n); k = 0:length(n)-1; stem(k,xs);grid; xlabel('Time index n');ylabel('Amplitude'); title('Discrete-time signal x[n]'); subplot(2,2,4) wd = 0:pi/255:pi; hd = freqz(xs,1,wd); plot(wd/(T*pi), T*abs(hd));grid; xlabel('Frequency, kHz');ylabel('Amplitude'); title('|X(e^j\omega)|'); axis([0 1/T 0 2])

4. Analog Low pass Filters

Analog low pass filters are employed as anti-aliasing filters and as anti-imaging filters in the digital

processing of continuous-time signals.

Design of Analog Low pass Filters

The first step in the design of any of these filters is the determination of the filter order N and the

appropriate cutoff frequency ΩC. These parameters can be determined using the MATLAB

commands buttord for the Butterworth filter, cheb1ord for the Type 1 Chebyshev filter, cheb2ord

for the Type 2 Chebyshev filter, and ellipord for the elliptic filter. ΩC is the 3-dB cutoff frequency

for the Butterworth filter, the passband edge for the Type 1 Chebyshev filter, the stopband edge

for the Type 2 Chebyshev filter, and the pass band edge for the elliptic filter. For the design of

filters MATLAB commands are butter for the Butterworth filter, cheby1 for the Type 1 Chebyshev

filter, cheby2 for the Type 2 Chebyshev filter, and ellip for the elliptic filter.

Program P4 4 can be used for the design of the Butterworth lowpass filter.

% Program P6_4 % Design of Analog Lowpass Filter clear all; close all; clc; Fp = 3500;Fs = 4500; Wp = 2*pi*Fp; Ws = 2*pi*Fs; [N, Wn] = buttord(Wp, Ws, 0.5, 30,'s'); [b,a] = butter(N, Wn, 's'); wa = 0:(3*Ws)/511:3*Ws; h = freqs(b,a,wa); plot(wa/(2*pi), 20*log10(abs(h)),'r', 'LineWidth',1 .5);grid xlabel('Frequency, Hz');ylabel('Gain, dB'); title('Gain response'); axis([0 3*Fs -60 5]);

0 2000 4000 6000 8000 10000 12000-60

-50

-40

-30

-20

-10

0

Frequency, Hz

Gai

n, d

BGain response

EXPERIMENT NO. 7:

System Frequency Transform

Introduction to Digital Filter

Frequency Transform:

Is to represent mathematical function of time in frequency domain.

:*. − 6:* − . = 6. 6*. 6* . 6. 6* 6.

% Program P7_1

% Illustration of Aliasing Effect in the Time Domain

clear all; close all; clc;

num=[1 2 2.2];

den=[1 2];

freqz(num,den)

% Program P7_2



num=[1 2 2.2];

den=[1 2];

Y=freqz(num,den)

Plot(abs(Y))

Can also view the frequency response of the system by defining “w”

% Program P7_3



w=-pi:pi/100:pi

num=[1 2 2.2];

den=[1 2];

freqz(num,den)

Another way to do this is by computing the frequency response of the system then define w and

compute the response

% Program P7_4



w=-pi:pi/100:pi

Y=(1+2*exp(-j*w)+2.2*exp(-2*j*w))./(1-2*exp(-j*w))

plot(w,abs(Y))

Exercise 1:

:*. − 6:* − 6. = 6. 6[] + 6[ − ] + 6. 6[ − 6]

:[] − . :[ − ] + :[ − J] = [] + . I[ − ] + [ − 6]

:[] − . :[ − ] + :[ − J] + :[ − I] = [] + [ − G] Plot the magnitude and phase response of the given difference equation.

Filter:

Filters are usually used to discriminate a frequency or a band of frequency from a given signal

which is normally a mixture of both desired and undesired signals. The undesired portion of the

signal commonly comes from noise sources such as power line hum etc. Or other signals which are

not required for the current application. Analog filters were being used successfully for decades to

serve this purpose. Although analog filters are excellent in some aspects, especially in cost, they do

have some serious demerits. One of the drawbacks of analog filters is there non-linear phase

characteristics. This is not a serious problem in many of the applications, but it become serious in

applications like telecommunication, voice processing etc... Another drawback is the less sharp cut-

off frequency. It is possible to increase roll-off rate by cascading filter stages, but this would

increase system cost and complexity. On the other hand, it is possible to achieve all these

characteristics fairly by using a digital filter

Two types of Filter

• FIR (Finite impulse Response)

• IIR (Infinite impulse Response)

Filters can be designed by using different techniques but the most simple type of filters are simple

digital filters. In case of simple digital filters these can be implemented simply by the difference

equation,

Like FIR filters are

Low Pass:

:[] = 6[] +6[ − ]

High Pass:

:[] = 6[] −6[ − ]

Band Pass:

:[] = 6[] −6[ − 6]

Band Stop:

:[] = 6[] +6[ − 6]

EXPERIMENT NO. 8:

Simple Digital Filter

1) FIR

2) IIR

Low Pass:

:*. = 6 *. +

6 * − . Find the frequency transform of the given difference equation

High Pass:

:*. = 6 *. −

6 * − . Find the frequency transform of the given difference equation

% Program

% Low pass FIR filter


w=-pi:pi/100:pi;

num=[1/2 1/2];

den=[1];

H=freqz(num,den,w);

plot(w,abs(H))

xlabel('w frequency')

ylabel('abs(H)')

Band Pass:

:*. = 6 *. −

6 * − 6. Find the frequency transform of the given difference equation

% Program



w=-pi:pi/100:pi;

num=[1/2 -1/2];

den=[1];

H=freqz(num,den,w);

plot(w,abs(H))


ylabel('abs(H)')

% Program



w=-pi:pi/100:pi;

num=[1/2 0 -1/2];

den=[1];

H=freqz(num,den,w);

plot(w,abs(H))


ylabel('abs(H)')

Band Stop:

:*. = 6 *. +

6 * − 6. Find the frequency transform of the given difference equation

% Program



w=-pi:pi/100:pi;

num=[1/2 0 1/2];

den=[1];

H=freqz(num,den,w);

plot(w,abs(H))


ylabel('abs(H)')

EXPERIMENT NO.9

Design of FIR filters using

Rectangular window

Low Pass Filter:

Compute the impulse response of the Low pass filter. After computing impulse response computes

the frequency response of the response to visualize the frequency response.

Pass the signal of any frequency from the filter and observe the output

% Program

% Low pass FIR filter (Rectangular Window)

clc

clear all

w=-pi:pi/200:pi;

n=0:40;

%Impulse Responce

h=(1/pi)*sinc((1*(n-20))/pi);

subplot(2,3,1)

plot(n,h)

xlabel('Impulse Responce for n=0:40')

%Frequency Responce

y=freqz(h,1,w);

subplot(2,3,2)

plot(w,abs(y))

xlabel('freqz Responce for n=0:40')

n1=0:50;

x=cos(1.5*n1);

subplot(2,3,3)

plot(n1,x)

xlabel('Input Signal')

subplot(2,3,4)

y=conv(x,h);

plot(y)

axis([0 90 -1 1])

xlabel('Output Signal')

subplot(2,3,5)

X=freqz(x,1,w);

plot(w,abs(X))

xlabel('X(e^j^w)')

High Pass Filter:




% Program

% High pass FIR filter (Rectangular Window)

clc

clear all

w=-pi:pi/200:pi;

n=0:40;

%Impulse Responce

h=sinc((n-20))-(1.5/pi*sinc(1.5*(n-20)/pi))

subplot(2,3,1)

plot(n,h)


%Frequency Responce

y=freqz(h,1,w);

subplot(2,3,2)

plot(w,abs(y))


n1=0:50;

x=cos(2*n1)+cos(0.5*n1);

subplot(2,3,3)

plot(n1,x)


subplot(2,3,4)

y=conv(x,h);

plot(y)

axis([0 90 -1 1])


subplot(2,3,5)

X=freqz(x,1,w);

plot(w,abs(X))

xlabel('X(e^j^w)')

Band Pass Filter:




% Program

% Band pass FIR filter (Rectangular Window)

clc

clear all

w=-pi:pi/200:pi;

wL=1.5;wP=2.3;

n=0:40;

%Impulse Responce

h=(wL/pi*sinc(wL*(n-20)/pi))-(wP/pi*sinc(wP*(n-20)/pi))

subplot(2,3,1)

plot(n,h)


%Frequency Responce

y=freqz(h,1,w);

subplot(2,3,2)

plot(w,abs(y))


n1=0:50;

x=cos(2*n1)+cos(0.7*n1);

subplot(2,3,3)

plot(n1,x)


subplot(2,3,4)

y=conv(x,h);

plot(y)

axis([0 90 -1 1])


subplot(2,3,5)

X=freqz(x,1,w);

plot(w,abs(X))

xlabel('X(e^j^w)')

subplot(2,3,6)

Y=freqz(y,1,w);

plot(w,abs(Y))

xlabel('Y(e^j^w)')

Band Pass Filter:

Compute the impulse response of the Band stop filter. After computing impulse response

computes the frequency response of the response to visualize the frequency response.


% Program

% Band Stop FIR filter (Rectangular Window)

clc

clear all

w=-pi:pi/200:pi;

wL=0.5;wP=2.3;

n=0:100;

%Impulse Responce

h=(wL/pi*sinc(wL*(n-50)/pi))-(wP/pi*sinc(wP*(n-50)/pi))+sinc(n-50)

subplot(2,3,1)

plot(n,h)


%Frequency Responce

y=freqz(h,1,w);

subplot(2,3,2)

plot(w,abs(y))


n1=0:50;

x=cos(1*n1)+cos(1.5*n1);

subplot(2,3,3)

plot(n1,x)


subplot(2,3,4)

y=conv(x,h);

plot(y)

axis([0 90 -1 1])


subplot(2,3,5)

X=freqz(x,1,w);

plot(w,abs(X))

xlabel('X(e^j^w)')

subplot(2,3,6)

Y=freqz(y,1,w);

plot(w,abs(Y))

xlabel('Y(e^j^w)')

EXPERIMENT NO. 10

DIGITAL FILTER STRUCTURE

FIR Filter Design:

Conceptually the simplest approach to FIR filter design is to simply truncate to a finite number of

terms the doubly infinite-length impulse response coefficients obtained by computing the inverse

discrete-time Fourier transform of the desired ideal frequency response. However, a simple

truncation results in an oscillatory behavior in the respective magnitude response of the FIR filter,

which is more commonly referred to as the Gibbs phenomenon.

The Gibbs phenomenon can be reduced by windowing the doubly infinite-length impulse response

coefficients by an appropriate finite-length window function. The functions fir1 and fir2 can be

employed to design windowed FIR digital filters in MATLAB. Both functions yield a linear-phase

design.

The function fir1 can be used to design conventional lowpass, highpass, bandpass, and bandstop

linear-phase FIR filters. The command

b = fir1(N,Wn)

returns in vector b the impulse response coefficients, arranged in ascending powers of z—1 , of a

lowpass or a bandpass filter of order N for an assumed sampling frequency of 2 Hz. For lowpass

design, the normalized cutoff frequency is specified by a scalar Wn, a number between 0 and 1. For

bandpass design, Wn is a two-element vector [Wn1, Wn2] containing the specified passband edges

where 0 < Wn1 < Wn2 < 1. The command

b = fir1(N,Wn,’high’)

with N an even integer, is used for designing a highpass filter. The command

b = fir1(N,Wn,’stop’)

with Wn a two-element vector, is employed for designing a bandstop FIR filter. If none is specified,

the Hamming window is employed as a default. The command

b = fir1(N, Wn, taper)

makes use of the specified window coefficients of length N+1 in the vector taper. However, the

window coefficients must be generated a priori using an appropriate MATLAB function such as

blackman, hamming, hanning, chebwin, or kaiser. The commands to use are of the following

forms:

taper = blackman(N) taper = hamming(N) taper = hanning(N) taper = chebwin(N) taper = kaiser(N, beta)

The function fir2 can be used to design linear-phase FIR filters with arbitrarily shaped magnitude

responses. In its basic form, the command is

b = fir2(N, fpts, mval)

which returns in the vector b of length N+1 the impulse response coefficients, arranged in

ascending powers of z—1 . fpts is the vector of specified frequency points, arranged in an

increasing order, in the range 0 to 1 with the first frequency point being 0 and the last frequency

point being 1. As before, the sampling frequency is assumed to be 2 Hz. mval is a vector of specified

magnitude values at the specified frequency points and therefore must also be of the same length

as fpts. The Hamming window is used as a default. To make use of other windows, the command

to use is

b = fir2(N, fpts, mval,taper)

where the vector taper contains the specified window coefficients.

A more widely used linear-phase FIR filter design is based on the Parks–McClellan algorithm, which

results in an optimal FIR filter with an equiripple weighted error ε (w) defined in Eq.

ℰω = qrstuP/QRSt − vrw

It makes use of the Remez optimization algorithm and is available in MATLAB as the function firpm.

This function can be used to design any type of single-band or multiband filter, the differentiator,

and the Hilbert transformer. In its basic form, the command

b = firpm(N,fpts,mval)

returns a vector b of length N+1 containing the impulse response coefficients of the desired FIR filter

in ascending powers of z—1 . fpts is the vector of specified frequency points, arranged in

increasing order, in the range 0 to 1 with the first frequency point being 0 and the last frequency

point being 1. As before, the sampling frequency is assumed to be 2 Hz. The desired magnitudes

of the FIR filter frequency response at the specified band edges are given by the vector mval, with

the elements given in equal-valued pairs. The desired magnitudes between two specified

consecutive frequency points f(k) and f(k+1) are determined according to the following rules. For

k odd, the magnitude is a line segment joining the points mval(k), fpts(k) and mval(k+1),

fpts(k+1), whereas, for k even, it is unspecified with the frequency range [fpts(k), fpts(k+1)]

being a transition or “don’t care” region. The vectors fpts and mval must be of the same length

with the length being even. Figure 7.4 illustrates the relationship between the vectors fpts and

mval given by

fpts = [ 0 0.2 0.4 0.7 0.8 1.0] mval = [0.5 0.5 1.0 1.0 0.3 0.3]

FIGURE 9.1: ILLUSTRATION OF RELATIONSHIP BETWEEN VECTORS FPTS AND MVAL

The desired magnitude responses in the passband(s) and the stopband(s) can be weighted by an

additional vector wgts included as the argument of the function firpm. The function can be used to

design equiripple Types 1, 2, 3, and 4 linear-phase FIR filters. Types 1 and

2 are the default designs for order N even and odd, respectively. Types 3 (N even) and 4 (N odd) are

used for specialized filter designs, the Hilbert transformer and the differentiator. To design these

two types of FIR filters the fiags hilbert and differentiator are used for ftype in the last two

versions of firpm. The command

b = firpm(N,fpts,mval,wgts)

is used to design an FIR filter weighted in each band by the elements of the weight vector wgts

whose length is thus half that of fpts. The elements of the vector wgts can be determined

from the specified passband and stopband ripples by dividing the maximum ripple value by the

ripple values. To design a Hilbert transformer or a differentiator, use the forms

firpm(N,fpts,mval,ftype) firpm(N,fpts,mval,wgts,ftype)

where ftype is the string hilbert or differentiator. In the case of a Hilbert transformer

design, the smallest element in fpts should not be a 0.

The order N of the FIR filter to meet the given specifications can be estimated using either Kaiser’s

formula of Eq. (9.2).

x ≅ −20 log3UP&~&S 1314.6r/2

(9.2)

The MATLAB function kaiord given below implements Kaiser’s formula:

function N = kaiord(Fp, Fs, dp, ds, FT) % Computation of the length of a linear-phase % FIR multiband filter using Kaiser’s formula % dp is the passband ripple % ds is the stopband ripple % Fp is the passband edge in Hz % Fs is the stopband edge in Hz % FT is the sampling frequency in Hz. % If none specified default value is 2 % N is the estimated FIR filter order if nargin == 4, F T=2 ; end if length(Fp) > 1, TBW = min(abs(Fp(1) - Fs(1)), abs(Fp(2) - Fs(2))); else TBW = abs(Fp - Fs); end num = -20*log10(sqrt(dp*ds)) - 13; den = 14.6*TBW/FT; N = ceil(num/den);

The function kaiserord in the Signal Processing Toolbox can also be used for estimating the filter

order using Kaiser’s formula. It can be used in one of the following forms:

[N, Wn, beta, ftype] = kaiserord(fedge, aval, dev) [N, Wn, beta, ftype] = kaiserord(fedge, aval, dev, FT) c = kaiserord(fedge, aval, dev, FT, ’cell’)

where FT is the sampling frequency in Hz whose default value is 2 Hz if not specified; fedge is a

vector of bandedge frequencies in Hz, in increasing order between 0 and FT/2; and aval is a vector

specifying the desired values of the magnitude response at the specified bandedges given by fedge.

The length of fedge is 2 less than twice the length of aval and therefore must be even. dev is a

vector of maximum deviations or ripples in dB allowable for each band. If the deviations specified

are unequal, the smallest one is used for all bands.

The output data are in the desired format for use in fir1, with normalized bandedges Wn and the

parameter beta used for computing the window coefficients as given in Eq. (7.36). The string ftype

specifies the filter type for fir1. It is high for highpass filter design, and stop for bandstop filter

design. The last form of kaiserord specifies a cell array whose elements are parameters to fir1.

The MATLAB function firpmord implements the formula of Eq. (7.8). It can be used in one of the

following forms:

[N,fts,mval,wgts] = firpmord(fedge,aval,dev) [N,fts,mval,wgts] = firpmord(fedge,aval,dev,FT)

where FT is the sampling frequency in Hz whose default value is 2 Hz if not specified, fedge is a

vector of bandedge frequencies in Hz, in increasing order between 0 and FT/2; and aval is a vector

specifying the desired values of the magnitude response at the specified bandedges given by fedge.

The length of fedge is 2 less than twice the length of aval and therefore must be even. dev is a

vector of maximum deviations or ripples in dB allowable for each band. A third form of firpmord is

given by

c = firpmord(fedge,aval,dev,FT, ’cell’)

and specifies a cell array whose elements are the parameters to firpm.

In some cases, the order N determined using either method may not result in an FIR filter meeting

the original specifications. If it does not, the order should either be increased or decreased by 1

gradually until the specifications are met. Moreover, the order estimates may be highly inaccurate

for very narrowband or very wideband FIR filters.

Lab task

1.1..1 Using MATLAB determine the lowest order of a digital IIR lowpass filter of all four types.

The specifications are as follows: sampling rate of 40 kHz, passband edge frequency of 4

kHz, stopband edge frequency of 8 kHz, passband ripple of 0.5 dB, and a minimum

stopband attenuation of 40 dB. Comment on your results.

1.1..2 Using MATLAB determine the lowest order of a digital IIR highpass filter of all four

types. The specifications are as follows: sampling rate of 3,500 Hz, passband edge

frequency of 1,050 Hz, stopband edge frequency of 600 Hz, passband ripple of 1 dB,

and a minimum stopband attenuation of 50 dB. Comment on your results.

1.1..3 Using MATLAB determine the lowest order of a digital IIR bandpass filter of all four types.

The specifications are as follows: sampling rate of 7 kHz, passband edge frequencies at

1.4 kHz and 2.1 kHz, stopband edge frequencies at 1.05 kHz and 2.45 kHz, passband

ripple of 0.4 dB, and a minimum stopband attenuation of 50 dB. Comment on your

results.

1.1..4 Using MATLAB determine the lowest order of a digital IIR bandstop filter of all four

types. The specifications are as follows: sampling rate of 12 kHz, passband edge

frequencies at 2.1 kHz and 4.5 kHz, stopband edge frequencies at 2.7 kHz and 3.9

kHz, passband ripple of 0.6 dB, and a minimum stopband attenuation of 45 dB.

Comment on your results.

1.1..5 Design the Butterworth bandstop filter by running Program P9_1. Write down the

exact expression for the transfer function generated. What are the filter

specifications? Does your design meet the specifications? Using MATLAB, compute

and plot the filter’s unwrapped phase response and the group delay response.

1.1..6 Modify Program P10_1 to design a Type 1 Chebyshev lowpass filter meeting the given

specifications of Question Q 1. Write down the exact expression for the transfer

function generated. Does your design meet the specifications? Using MATLAB,

compute and plot the filter’s unwrapped phase response and the group delay

response.

1.1..7 Modify Program P10_1 to design a Type 2 Chebyshev highpass filter meeting the

specifications given in Question Q 2. Write down the exact expression for the

transfer function generated. Does your design meet the specifications? Using MATLAB,

compute and plot the filter’s unwrapped phase response and the group delay

response.

1.1..8 Modify Program P10_1 to design an elliptic bandpass filter meeting the specifications

given in Question Q 3. Write down the exact expression for the transfer function

generated. Does your design meet the specifications? Using MATLAB, compute and

plot the filter’s unwrapped phase response and the group delay response.

1.1..9 Repeat the above question for the following cases: (a) sampling rate of 20 kHz, (b) δp

= 0.002 and δS = 0.002, and (c) stopband edge = 2.3 kHz. Compare the filter length

obtained in each case with that obtained in the above question. Comment on the

effect of the sampling rate, ripples, and the transition bandwidth on the filter order.

EXPERIMENT NO.11

Design of FIR filters using Matlab

commands.

Description:

Digital filters refers to the hard ware and software implementation of the mathematical Algorithm

which accepts a digital signal as input and produces another digital signal as output whose wave

shape, amplitude and phase response has been modified in a specified manner.

Digital filter play very important role in DSP. Compare with analog filters they are preferred in

number of application due to following advantages.

• Truly linear phase response

• Better frequency response

• Filtered and unfiltered data remains saved for further use.

There are two types of digital filters.

• FIR (finite impulse response) filter

• IIR (infinite impulse response) filter

Description of the Commands Used In FIR Filter Design:

FIR1:

FIR filters design using the window method. B = FIR1(N,Wn) designs an N'th order low

pass FIR digital filter and returns the filter coefficients in length N+1 vector B. The cutoff

frequency Wn must be between 0 < Wn < 1.0, with 1.0 corresponding to half the

sample rate. The filter B is real and has linear phase. The normalized gain of the filter

at Wn is -6 dB.

B = FIR1(N,Wn,'high') designs an N'th order high pass filter. You can also use B =

FIR1 (N,Wn,'low') to design a low pass filter. If Wn is a two-element vector, Wn = [W1

W2], FIR1 returns an order N band pass filter with pass band W1 < W < W2.

B = FIR1 (N,Wn,'stop') is a band stop filter if Wn = [W1 W2]. You can also specify If Wn

is a multi-element vector, Wn = [W1 W2 W3 W4 W5 ... WN], FIR1 returns an order N

Multiband filter with bands 0 < W < W1, W1 < W < W2, ..., WN < W < 1.

B = FIR1 (N,Wn,'DC-1') makes the first band a pass band.

B = FIR1(N,Wn,'DC-0') makes the first band a stop band.

By default FIR1 uses a Hamming window. Other available windows, including Boxcar,

Hann, Bartlett, Blackman, Kaiser and Chebwin can be specified with an optional trailing

argument. For example, B = FIR1(N,Wn,kaiser(N+1,4)) uses a Kaiser window with

beta=4. B = FIR1(N,Wn,'high',chebwin(N+1,R)) uses a Chebyshev window.

For filters with a gain other than zero at Fs/2, e.g., high pass and band stop filters, N must

be even. Otherwise, N will be incremented by one. In this case the window length

should be specified as N+2.

By default, the filter is scaled so the center of the first pass band has magnitude exactly

one after windowing. Use a trailing 'noscale' argument to prevent this scaling, e.g.

B = FIR1(N,Wn,'noscale')

B = FIR1(N,Wn,'high','noscale')

B = FIR1(N,Wn,wind,'noscale').

You can also specify the scaling explicitly, e.g. FIR1(N,Wn,'scale'), etc.

FREQZ Digital Filter Frequency Response.

[H,W] = FREQZ(B,A,N) returns the N-point complex frequency response vector H and

the N-point frequency vector W in radians/sample of the filter: given numerator and

denominator coefficients in vectors B and A. The frequency response is evaluated at N

points equally spaced around the upper half of the unit circle. If N isn't specified, it

defaults to 512.

[H,W] = FREQZ(B,A,N,'whole') uses N points around the whole unit circle.

H = FREQZ(B,A,W) returns the frequency response at frequencies designated in vector

W, in radians/sample (normally between 0 and pi).

[H,F] = FREQZ(B,A,N,Fs) and [H,F] = FREQZ(B,A,N,'whole',Fs) return frequency

vector F (in Hz), where Fs is the sampling frequency (in Hz).

H = FREQZ(B,A,F,Fs) returns the complex frequency response at the frequencies

designated in vector F (in Hz), where Fs is the sampling frequency (in Hz).

[H,W,S] = FREQZ(...) or [H,F,S] = FREQZ(...) returns plotting information to be used

with FREQZPLOT. S is a structure whose fields can be altered to obtain different

frequency response plots. For more information see the help for FREQZPLOT.

FREQZ(B,A,...) with no output arguments plots the magnitude and unwrapped phase of

the filter in the current figure window.

Designing A Low Pass Filter:

Suppose out target is to pass all frequencies below 1200 Hz

fs=8000; % sampling frequency

n=50; % order of the filter

w=1200/ (fs/2);

b=fir1(n,w,'low'); % Zeros of the filter

freqz(b,1,128,8000); % Magnitude and Phase Plot of the filter

figure(2)

[h,w]=freqz(b,1,128,8000);

plot(w,abs(h)); % Normalized Magnitude Plot

grid

figure(3)

zplane(b,1);

Designing High Pass Filter:

Now our target is to pass all frequencies above 1200 Hz

Designing High Pass Filter:

Designing Band Pass Filter:

fs=8000;

n=50;

w=1200/ (fs/2); b=fir1(n,w,'high');

freqz(b,1,128,8000);

figure(2)

[h,w]=freqz(b,1,128,8000);


grid

figure(3)

zplane(b,1);

fs=8000;

n=50;

w=1200/ (fs/2);

b=fir1(n,w,'high');

freqz(b,1,128,8000);

figure(2)

[h,w]=freqz(b,1,128,8000);


grid

figure(3)

zplane(b,1);

fs=8000;

n=40;

b=fir1(n,[1200/4000 1800/4000],’bandpass’);

freqz(b,1,128,8000)

figure(2)

[h,w]=freqz(b,1,128,8000);


grid

figure(3)

zplane(b,1);

Designing Multi Band Filter

Problems:

Design a band pass filter and band stop filter with the help of LPF and HPF

The filter has following specifications.

Band pass = 1200 – 2800 Hz

Band stop =1200-2800 Hz

n=50;

w=[0.2 0.4 0.6];

b=fir1(n,w);

freqz(b,1,128,8000)

figure(2)

[h,w]=freqz(b,1,128,8000);


grid

figure(3)

zplane(b,1);

Design a Multi band filter using HPF and LPF

The filter has following specifications

Pass band=1200 Hz – 1800 Hz

Stop band = 1900 Hz – 2200 Hz

Pass band = 2300 Hz – 2700 Hz

EXPERIMENT NO.12

Design of IIR filters using Matlab

commands.

Description:

Matlab contains various routines for design and analyzing digital filter IIR. Most of these are part of

the signal processing tool box. A selection of these filters is listed below.

• Buttord ( for calculating the order of filter)

• Butter ( creates an IIR filter)

• Ellipord ( for calculating the order of filter)

• Ellip (creates an IIR filter)

• Cheb1ord (for calculating the order of filter)

• Cheyb1 (creates an IIR filter)

Explanation Of The Commands For Filter Design:

Buttord:

Butterworth filter order selection.

[N, Wn] = BUTTORD(Wp, Ws, Rp, Rs) returns the order N of the lowest order digital Butterworth

filter that loses no more than Rp dB in the pass band and has at least Rs dB of attenuation in the

stop band. Wp and Ws are the pass band and stop band edge frequencies, normalized from 0 to 1

(where 1 corresponds to pi radians/sample). For example

Low pass: Wp = .1, Ws = .2

High pass: Wp = .2, Ws = .1

Band pass: Wp = [.2 .7], Ws = [.1 .8]

Band stop: Wp = [.1 .8], Ws = [.2 .7]

BUTTORD also returns Wn, the Butterworth natural frequency (or, the "3 dB frequency") to use

with BUTTER to achieve the specifications. [N, Wn] = BUTTORD(Wp, Ws, Rp, Rs, 's') does the

computation for an analog filter, in which case Wp and Ws are in radians/second. When Rp is

chosen as 3 dB, the Wn in BUTTER is equal to Wp in BUTTORD.

Ellipord:

Elliptic filter order selection.

[N, Wn] = ELLIPORD(Wp, Ws, Rp, Rs) returns the order N of the lowest order digital elliptic filter

that loses no more than Rp dB in the pass band and has at least Rs dB of attenuation in the stop

band Wp and Ws are the pass band and stop band edge frequencies, normalized from 0 to 1

(where 1 corresponds to pi radians/sample). For example,



Band pass: Wp = [.2 .7], Ws = [.1 .8]

Band stop: Wp = [.1 .8], Ws = [.2 .7]

ELLIPORD also returns Wn, the elliptic natural frequency to use with ELLIP to achieve the

specifications. [N, Wn] = ELLIPORD(Wp, Ws, Rp, Rs, 's') does the computation for an analog filter, in

which case Wp and Ws are in radians/second. NOTE: If Rs is much greater than Rp, or Wp and Ws

are very close, the estimated order can be infinite due to limitations of numerical precision.

Cheb1ord:

Chebyshev Type I filter order selection.

[N, Wn] = CHEB1ORD(Wp, Ws, Rp, Rs) returns the order N of the lowest order digital Chebyshev

Type I filter that loses no more than Rp dB in the pass band and has at least Rs dB of attenuation in

the stop band. Wp and Ws are the pass band and stop band edge frequencies, normalized from 0

to 1 (where 1 corresponds to pi radians/sample). For example,



Band pass: Wp = [.2 .7], Ws = [.1 .8]

Band stop: Wp = [.1 .8], Ws = [.2 .7]

CHEB1ORD also returns Wn, the Chebyshev natural frequency to use with CHEBY1 to achieve the

specifications. [N, Wn] = CHEB1ORD(Wp, Ws, Rp, Rs, 's') does the computation for an analog filter,

in which case Wp and Ws are in radians/second.

Butter:

Butterworth digital and analog filter design.

[B,A] = BUTTER(N,Wn) designs an Nth order lowpass digital Butterworth filter and returns the filter

coefficients in length N+1 vectors B (numerator) and A (denominator). The coefficients are listed in

descending powers of z. The cutoff frequency Wn must be 0.0 < Wn < 1.0, with 1.0 corresponding

to half the sample rate. If Wn is a two-element vector, Wn = [W1 W2], BUTTER returns an order 2N

bandpass filter with passband W1 < W < W2.

[B,A] = BUTTER(N,Wn,'high') designs a highpass filter. [B,A] = BUTTER(N,Wn,'stop') is a bandstop

filter if Wn = [W1 W2]. When used with three left-hand arguments, as in [Z,P,K] = BUTTER(...), the

zeros and poles are returned in length N column vectors Z and P, and the gain in scalar K. When

used with four left-hand arguments, as in [A,B,C,D] = BUTTER(...), state-space matrices are

returned. BUTTER(N,Wn,'s'), BUTTER(N,Wn,'high','s') and BUTTER(N,Wn,'stop','s') design analog

Butterworth filters. In this case, Wn is in [rad/s] and it can be greater than 1.0.

Ellip:

Elliptic or Cauer digital and analog filter design.

[B,A] = ELLIP(N,Rp,Rs,Wn) designs an Nth order low pass digital elliptic filter with Rp decibels of

peak-to-peak ripple and a minimum stop band attenuation of Rs decibels. ELLIP returns the filter

coefficients in length N+1 vectors B (numerator) and A (denominator).The cutoff frequency Wn

must be 0.0 < Wn < 1.0, with 1.0 corresponding to half the sample rate. Use Rp = 0.5 and Rs = 20 as

starting points, if you are unsure about choosing them.

If Wn is a two-element vector, Wn = [W1 W2], ELLIP returns an order 2N band pass filter with pass

band W1 < W < W2. [B,A] = ELLIP(N,Rp,Rs,Wn,'high') designs a high pass filter. [B,A] =

ELLIP(N,Rp,Rs,Wn,'stop') is a band stop filter if Wn = [W1 W2]. When used with three left-hand

arguments, as in [Z,P,K] = ELLIP(...), the zeros and poles are returned in length N column vectors Z

and P, and the gain in scalar K. When used with four left-hand arguments, as in [A,B,C,D] =

ELLIP(...), state-space matrices are returned.

ELLIP(N,Rp,Rs,Wn,'s'), ELLIP(N,Rp,Rs,Wn,'high','s') and ELLIP(N,Rp,Rs,Wn,'stop','s')

design analog elliptic filters. In this case, Wn is in [rad/s] and it can be greater than 1.0.

Cheby1:

Chebyshev Type I digital and analog filter design.

[B,A] = CHEBY1(N,R,Wn) designs an Nth order lowpass digital Chebyshev filter with R decibels of

peak-to-peak ripple in the passband. CHEBY1 returns the filter coefficients in length N+1 vectors B

(numerator) and A (denominator). The cutoff frequency Wn must be 0.0 < Wn < 1.0, with 1.0

corresponding to half the sample rate. Use R=0.5 as a starting point, if you are unsure about

choosing R. If Wn is a two-element vector, Wn = [W1 W2], CHEBY1 returns an order 2N bandpass

filter with passband W1 < W < W2. [B,A] = CHEBY1(N,R,Wn,'high') designs a highpass filter. [B,A] =

CHEBY1(N,R,Wn,'stop') is a bandstop filter if Wn = [W1 W2]. When used with three left-hand

arguments, as in [Z,P,K] = CHEBY1(...), the zeros and poles are returned in length N column vectors

Z and P, and the gain in scalar K.

When used with four left-hand arguments, as in [A,B,C,D] = CHEBY1(...), state-space

matrices are returned.

CHEBY1(N,R,Wn,'s'), CHEBY1(N,R,Wn,'high','s') and CHEBY1(N,R,Wn,'stop','s')

design analog Chebyshev Type I filters.In this case, Wn is in [rad/s] and it can be greater

than 1.0.

Buttord and Butter Filter:

Designing IIR Low Pass Filter:

Suppose our target is to design a filter to pass all frequencies below 1200 Hz with pass

band ripples = 1 dB and minimum stop band attenuation of 50 dB at 1500 Hz. The

sampling frequency for the filter is 8000 Hz;

Designing IIR High Pass Filter:

We will consider same filter but our target now is to pass all frequencies above 1200 Hz

Designing IIR Band Pass Filter:

Now we wish to design a filter to pass all frequencies between 1200 Hz and 2800 Hz with pass band ripples = 1 dB and minimum stop band attenuation of 50 dB. The sampling frequency for the filter is 8000 Hz;

fs=8000;

[n,w]=buttord(1200/4000,1500/4000,1,50); % finding the order of the filter

[b,a]=butter(n,w); % finding zeros and poles for filter

figure(1)

freqz(b,a,512,8000);

figure(2)

[h,q] = freqz(b,a,512,8000);

plot(q,abs(h)); % Normalized Magnitude plot

grid

figure(3)

f=1200:2:1500;

freqz(b,a,f,8000) % plotting the Transition band

figure(4)

zplane(b,a) % pole zero constellation diagram

[n,w]=buttord(1200/5000,1500/5000,1,50); [b,a]=butter(n,w,'high'); figure(1) freqz(b,a,512,10000); figure(2) [h,q] = freqz(b,a,512,8000); plot(q,abs(h)); % Normalized Magnitude plot grid figure(3) f=1200:2:1500; freqz(b,a,f,10000) figure(4) zplane(b,a)

Designing IIR Band Stop Filter:

[n,w]=buttord([1200/4000,2800/4000],[400/4000, 3200/4000],1,50); [b,a]=butter(n,w,'bandpass'); figure(1) freqz(b,a,128,8000) figure(2) [h,w]=freqz(b,a,128,8000); plot(w,abs(h)) grid figure(3) f=600:2:1200; freqz(b,a,f,8000); % Transition Band figure(4) f=2800:2:3200; freqz(b,a,f,8000); % Transition Band figure(5) zplane(b,a)

[n,w]=buttord([1200/4000,2800/4000],[400/4000, 3200/4000],1,50); [b,a]=butter(n,w,'stop'); figure(1) freqz(b,a,128,8000) [h,w]=freqz(b,a,128,8000); figure(2) plot(w,abs(h)); grid figure(3) f=600:2:1200; freqz(b,a,f,8000); % Transition Band figure(4) f=2800:2:3200; freqz(b,a,f,8000); % Transition Band figure(5) zplane(b,a);

Problems:

Design all above filter using following commands

• Ellipord( ) • Ellip( ) • Cheb1ord( )

• Cheby1( )

EXPERIMENT NO.13

DFT

The discrete Fourier transform (DFT) X[k] of a finite-length sequence x[n] can be easily computed in

MATLAB using the function fft. There are two versions of this function. fft(x) computes the DFT X[k]

of the sequence x[n] where the length of X[k] is the same as that of x[n]. fft(x,L) computes the L-

point DFT of a sequence x[n] of length N where L ≥ N . If L > N , x[n] is zero-padded with L − N

trailing zero-valued samples before the DFT is computed. The inverse discrete Fourier transform

(IDFT) x[n] of a DFT sequence X[k] can likewise be computed using the function ifft, which also has

two versions.

Project 13.1 DFT Properties

Two important concepts used in the application of the DFT are the circular-shift of a sequence and

the circular convolution of two sequences of the same length. As these operations are needed in

verifying certain properties of the DFT, we implement them as MATLAB functions circshift1 and

circonv as indicated below:

function y = circshift1(x,M) % Develops a sequence y obtained by % circularly shifting a finite-length % sequence x by M samples if abs(M) > length(x) M = rem(M,length(x)); end if M < 0 M = M + length(x); end y = [x(M+1:length(x)) x(1:M)]; end

function y = circonv(x1,x2) L1 = length(x1); L2 = length(x2); if L1 ~= L2, error( 'Sequences of unequal lengths' ), end y = zeros(1,L1); x2tr = [x2(1) x2(L2:-1:2)]; for k = 1:L1 sh = circshift1(x2tr,1-k); h = x1.*sh; y(k) = sum(h); end

Program P13_1 can be used to illustrate the concept of circular shift of a finite-length sequence. It

employs the function circshift1

% Program P13_1 % Illustration of Circular Shift of a Sequence clear all ; close all ; clc M=6; a=[0 1 2 3 4 5 6 7 8 9];

b = circshift1(a,M); L = length(a)-1; n = 0:L; subplot(2,1,1); stem(n,a);axis([0,L,min(a),max(a)]); title( 'Original Sequence' ); subplot(2,1,2); stem(n,b);axis([0,L,min(a),max(a)]); title([ 'Sequence Obtained by Circularly Shifting by ' ,num2str(M), 'Samples' ]);

Program P13_2 can be used to illustrate the circular time-shifting property of the DFT. It employs

the function circshift1.

% Program P13_2 % Circular Time-Shifting Property of DFT close all ; clear all ; clc x=[0 2 4 6 8 10 12 14 16]; N = length(x)-1; n = 0:N; y = circshift1(x,5); XF = fft(x); YF = fft(y); subplot(2,2,1) stem(n,abs(XF)); grid title( 'Magnitude of DFT of Original Sequence' ); subplot(2,2,2) stem(n,abs(YF)); grid title( 'Magnitude of DFT of Circularly Shifted Sequence' ); subplot(2,2,3) stem(n,angle(XF)); grid title( 'Phase of DFT of Original Sequence' ); subplot(2,2,4) stem(n,angle(YF)); grid title( 'Phase of DFT of Circularly Shifted Sequence' );

Program P13_3 can be

used to illustrate the

circular convolution

property of the DFT. It

employs the function

circonv.

0 2 4 6 80

20

40

60

80Magnitude of DFT of Original Sequence

0 2 4 6 80

20

40

60

80Magnitude of DFT of Circularly Shifted Sequence

0 2 4 6 8-4

-2

0

2

4Phase of DFT of Original Sequence

0 2 4 6 8-4

-2

0

2

4Phase of DFT of Circularly Shifted Sequence

% Program P13_3 % Circular Convolution Property of DFT clear all ; close all ; clc g1=[1 2 3 4 5 6]; g2=[1 -2 3 3 -2 1]; ycir = circonv(g1,g2); disp( 'Result of circular convolution = ' ); disp(ycir) G1 = fft(g1); % similarly compute fft of g2 and save in G2 yc = real(ifft(G1.*G2)); disp( 'Result of IDFT of the DFT products = ' ); disp(yc)

Program P13_4 can be used to illustrate the relation between circular and linear convolutions

% Program P13_4 % Linear Convolution via Circular Convolution close all ; clear all ; clc g1=[1 2 3 4 5]; g2 = [2 2 0 1 1]; g1e = [g1 zeros(1,length(g2)-1)]; g2e = [g2 zeros(1,length(g1)-1)]; %Do circular convolution of g1e and g2e and save in ylin yourself disp( 'Linear convolution via circular convolution = ' ); disp(ylin); y = conv(g1, g2); disp( 'Direct linear convolution = ' );disp(y)

Program P6_5 can be used to verify the relation between the DFT of a real sequence, and the DFTs

of its periodic even and the periodic odd parts.

% Program P13_5 % Relations between the DFTs of the Periodic Even % and Odd Parts of a Real Sequence close all ; clear all ; clc x=[1 2 4 2 6 32 6 4 2 zeros(1,247)]; x1 = [x(1) x(256:-1:2)]; xe = 0.5 *(x + x1); XF = fft(x); XEF = fft(xe); k = 0:255; subplot(2,2,1); plot(k/128,real(XF)); grid ylabel( 'Amplitude' ); title( 'Re(DFT\x[n]\)' ); subplot(2,2,2); plot(k/128,imag(XF));

grid ylabel( 'Amplitude' ); title( 'Im(DFT\x[n]\)' ); subplot(2,2,3); plot(k/128,real(XEF)); grid xlabel( 'Time index n' ); ylabel( 'Amplitude' ); title( 'Re(DFT\x_e[n]\ )' ); subplot(2,2,4); plot(k/128,imag(XEF)); grid xlabel( 'Time index n' );ylabel( 'Amplitude' ); title( 'Im(DFT\x_e[n]\)' );

Parseval’s relation can be verified using the following program.

% Program P13_6 % Parseval's Relation x = [(1:128) (128:-1:1)]; XF = fft(x); % Take square of vector x and then add all its % entries and save in “a” . Do yourself b = round(sum(abs(XF).^2)/256)

0 0.5 1 1.5 2-50

0

50

100

Am

plitu

de

Re(DFTx[n])

0 0.5 1 1.5 2-100

-50

0

50

100A

mpl

itude

Im(DFTx[n])

0 0.5 1 1.5 2-50

0

50

100

Time index n

Am

plitu

de

Re(DFTxe[n] )

0 0.5 1 1.5 2

-0.5

0

0.5

x 10-14

Time index n

Am

plitu

de

Im(DFTxe[n])

EXPERIMENT NO. 14

(DECIMATION & INTERPOLATION)

The digital signal processing structures discussed so far belong to the class of single-rate systems as

the sampling rates at the input and the output and all internal nodes are the same. There are

applications where it is necessary and often convenient to have unequal rates of sampling at various

parts of the system including the input and the output. In this laboratory exercise you will investigate

first using MATLAB the properties of the up-sampler and the down-sampler, the two basic

components of a multi-rate system. You will then investigate their use in designing more complex

systems, such as interpolators and decimators, and filter banks.

1.1 Basic Sampling Rate Alteration Devices

The objective of this section is to investigate using MATLAB the operations of the up-

sampler and the down-sampler both in the time domain and in the frequency domain.

Project 10.1 Input-Output Relations in the Time-Domain

Program P10_1 can be used to study the operation of a up-sampler.

%Program14_1 %Illustration of Up-Sampling by an Integer Factor % close all ; clear all ; clc n=0:50; x=sin(2*pi*0.12*n); y=zeros(1,3*length(x)); y([1:3:length(y)])=x; subplot(2,1,1) stem(n,x); title( 'InputSequence' ); xlabel( 'Timeindexn' );ylabel( 'Amplitude' ); subplot(2,1,2) stem(n,y(1:length(x))); title( 'OutputSequence' ); xlabel( 'Timeindexn' );ylabel( 'Amplitude' );

0 5 10 15 20 25 30 35 40 45 50-1

-0.5

0

0.5

1InputSequence

Timeindexn

Am

plitu

de

0 5 10 15 20 25 30 35 40 45 50-1

-0.5

0

0.5

1OutputSequence

Timeindexn

Am

plitu

de

%ProgramP14_2 %Illustration of Down-Sampling by an Integer Factor % close all ; clear all ; clc n = 0:49; m = 0:50*3-1; x = sin(2*pi*0.042*m); y = x([1:3:length(x)]); subplot(2,1,1) stem(n,x(1:50));axis([0 50 -1.2 1.2]); title( 'InputSequence' ); xlabel( 'Timeindexn' ); ylabel( 'Amplitude' ); subplot(2,1,2) stem(n,y);axis([0 50 -1.2 1.2]); title( 'OutputSequence' ); xlabel( 'Timeindexn' ); ylabel( 'Amplitude' );

The Signal Processing Toolbox includes three M-functions which can be employed to design and

implement an interpolator or a decimator. The three M-functions are decimate, interp, and

resample. Each function is available with several options. In this section you will study the

decimation and interpolation operation using these functions.

Project 14.2 Decimator Design and Implementation

Program P10_3 illustrates the use of the M-function decimate in the design and implementation of

a decimator with an integer-valued decimation factor M. In the option utilized in this program,

decimate designs and uses a lowpass decimation filter with a stopband edge.

%ProgramP14_3 %Illustration of Decimation Process % clear all ; close all ; clc M=input( 'Down-samplingfactor=' ); n=0:99; x=sin(2*pi*0.043*n)+sin(2*pi*0.031*n); y=decimate(x,M, 'fir' ); subplot(2,1,1); stem(n,x(1:100)); title( 'InputSequence' ); xlabel( 'Timeindexn' );ylabel( 'Amplitude' ); subplot(2,1,2); m=0:(100/M)-1; stem(m,y(1:100/M)); title( 'OutputSequence' ); xlabel( 'Timeindexn' );ylabel( 'Amplitude' );

1.2 Decimator and Interpolator Design and Implementation

Project 10.3 Interpolator Design and Implementation

Program P10_4 illustrates the use of the M-function interp in the design and implementation of

an interpolator with an integer-valued interpolation factor L. interp designs and uses a lowpass

interpolation filter with a stopband edge satisfying Eq. (10.1).

tuP/QRSt = ,|r| ≤ r/0,/ ≤ |r| ≤

(10.1)

%ProgramP14_4 %Illustration of Interpolation Process % clear all ; close all ; clc L=input( 'Up-samplingfactor=' ); % Generate the input sequence n=0:49; x=sin(2*pi*0.043*n)+sin(2*pi*0.031*n); % Generate the interpolated output sequence y=interp(x,L); % Plot the input and the output sequences

0 10 20 30 40 50 60 70 80 90 100-2

-1

0

1

2InputSequence

Timeindexn

Am

plitu

de

0 5 10 15 20 25 30 35 40 45 50-2

-1

0

1

2OutputSequence

Timeindexn

Am

plitu

de

subplot(2,1,1); stem(n,x(1:50)); title( 'InputSequence' ); xlabel( 'Timeindexn' );ylabel( 'Amplitude' ); subplot(2,1,2); m=0:(50*L)-1; stem(m,y(1:50*L)); title( 'OutputSequence' ); xlabel( 'Timeindexn' );ylabel( 'Amplitude' );

Project 10.4 Fractional-Rate Sampling Rate Alteration

Program P10_5 illustrates the use of the M-function resample in the design and implementation

of an interpolator with a fractional-rate interpolation factor L/M. Resample designs and uses a

lowpass interpolation filter with a stopband edge.

% Program14_5 % Illustration of Sampling Rate Alteration by % a Ratio of Two Integers % close all ; clear all ; clc L=input( 'Up-samplingfactor=' ); M=input( 'Down-samplingfactor=' ); n=0:29; x=sin(2*pi*0.43*n)+sin(2*pi*0.31*n);

0 5 10 15 20 25 30 35 40 45 50-2

-1

0

1

2InputSequence

Timeindexn

Am

plitu

de

0 10 20 30 40 50 60 70 80 90 100-2

-1

0

1

2OutputSequence

Timeindexn

Am

plitu

de

y=resample(x,L,M); subplot(2,1,1); stem(n,x(1:30));axis([0 29 -2.2 2.2]); title( 'InputSequence' ); xlabel( 'Timeindexn' );ylabel( 'Amplitude' ); subplot(2,1,2); m=0:(30*L/M)-1; stem(m,y(1:30*L/M));axis([0 (30*L/M)-1 -2.2 2.2]); title( 'OutputSequence' ); xlabel( 'Timeindexn' );ylabel( 'Amplitude' );

1.3 Lab Task

1. What is the angular frequency in radians of the sinusoidal sequence in Program P10_1? What is its

length? What is the up-sampling factor L?

2. How is the up-sampling operation implemented in Program P10_1?

3. Modify Program P10_1 to study the operation of an up-sampler on a ramp sequence.

4. Program P10_2 can be used to study the operation of a down-sampler

5. What is the angular frequency in radians of the sinusoidal sequence Program P10_2? What is its

length? What is the down-sampling factor M?

6. How is the down-sampling operation implemented in Program P10_2?

7. What are the frequencies of the two sinusoidal sequences forming the input sequence in Program

P10_3? What is the length of the input?

8. What are the type and order of the decimation filter?

9. Run Program P10_3 for M = 4 and comment on the results.

10. Change the frequencies of the two sinusoidal sequences in Program P10_3 in the input signal to

0.045 and 0.029, and the length of the input to 120. Run the modified Program P10 5 for M = 3.

Comment on your results.

EXPERIMENT NO.15

Filter designing by Matlab tools

Description:

There are two tool boxes available for designing, analyzing and for viewing different responses

(Impulse & Step) of FIR and IIR filters.

• fvtool

• fdatool

•

Filter Visualization Tool:

FVTOOL is a Graphical User Interface (GUI) that allows you to analyze digital filters. FVTOOL (B,A)

launches the Filter Visualization Tool and computes the magnitude Response for the filter defined

in B and A. FVTOOL(B,A,B1,A1,...) will perform an analysis on multiple filters. The real advantage of

this visualization tool is that we can view the magnitude response and phase response

simultaneously, the impulse response, step response the coefficients of the filter etc Let us

consider a Low Pass FIR filter of order 30 which passes all frequencies below 2000 Hz with sampling

rate of 8000 Hz.

b=fir1(30,2000/4000,’low’);

fvtool(b,1)

Filter Design & Analysis Tool.

FDATOOL launches the Filter Design & Analysis Tool (FDATool). FDATool is a Graphical User

Interface (GUI) that allows you to design or import, and analyze digital FIR and IIR filters. If the

Filter Design Toolbox is installed, FDATool seamlessly integrates advanced filter design methods

and the ability to quantize filters.

Now we will design a LPF on fdatool, the specifications for the filter are shown in respective

columns of FDA tool

Problems: Design IIR butter worth filter with following specifications

• -50 dB or more for 0 to 1200 Hz ( Stop Band Attenuation ) • -1 dB or less from 2000 Hz to 4000 Hz ( Pass Band Characteristics ) • -50 dB or more above 6000 Hz ( Stop Band Attenuation )

Sampling frequency 16000 Hz

EXPERIMENT No.16

Design an IIR filter to suppress

frequencies of 5 Hz and 30 Hz from

given signal.

Description:

We know from Fourier analysis that signals can be described by a summation of frequency

components. Typically, a filter is used to enhance signals by attenuating unwanted frequency

components and retaining desired frequency components. In this practical we begin by creating a

signals ‘s’ with three sinusoidal components ( at 5,15,30 Hz) and a time vector ‘t’ of 100 samples

with a sampling rate of 100 Hz, and displaying it in the time domain. The Matlab commands are

shown below.

fs=100;

t=(1:100)/fs;

s=sin(2*pi*t*5)+sin(2*pi*t*15)+sin(2*pi*t*30);

plot(t,s)

grid

Now we design a filter to keep the 15 Hz sinusoid and eliminate the 5 and 30 Hz sinusoids. We use

the functions ellipord and ellip to create an infinite impulse response (IIR) filter with a pass band

from 10 to 20 Hz. The ellipord function requires the specification of pass band corner frequencies,

minimum transition band frequencies near the pass band corner frequencies, the maximum pass

band ripple in decibels (dB), and the minimum stop band attenuation in dB. In this example, we

choose a transition frequency to be ±5 Hz near the pass band corners, with a maximum of 0.1 dB

ripple in the pass band, and a minimum of 40 dB attenuation in the stop bands. We start by

determining the minimum order (pass band and stop band frequencies are normalized to the

Nyquist frequency):

wp1 = 10/50;

wp2 = 20/50;

ws1 = 5/50;

ws2 = 25/50;

wp = [Wp1 Wp2];

ws = [Ws1 Ws2];

rp = 0.1;

rs = 40;

[n,wn] = ellipord(wp,ws,rp,rs);

ellipord returns an order of 5, the minimum possible order for a low pass prototype that will meet

the constraints upon transformation to a band pass filter. When we apply this order to the ellip

function, internally we transform the low pass prototype to a band pass filter using the function

lp2bp. This doubles the order, making n = 10. Next we use n, the order, and Wn, the pass band

corner frequencies, to actually design the filter. We also use freqz, a tool for computing and

displaying the frequency response of the descriptive transfer function. When called with no left-

hand-side arguments (i.e., return values), freqz displays the magnitude and phase response of the

filter normalized to the Nyquist frequency.

[b,a] = ellip(n,.1,40,w);

freqz(b,a,128,100)

[h,w]=freqz(b,a,128,100);

plot(w,abs(h));

grid

title(‘Normalized Magnitude Response’);

axis([0 50 0 1.2]);

figure(4)

sf=filter(b,a,s); % Time domain Response of the Filter

plot(t,sf)

grid

xlabel('Time (seconds)');

ylabel('Signal Amplitude');

title('Filtered Signal only 15 Hz frequency');

Problem: Design an IIR filter to remove 100 and 150 frequencies from above signal. x=1+sin (2*pi*50*t) + sin (2*pi*100*t) + 0.5 sin (2*pi*125*t) + 0.25 sin (2*pi*150*t);

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