dsp lab manual

CONTENTS

S.No Exercise Page. No

1 Introduction 1

1 To find DFT of given signal 7

2 To find frequency response of given system 10

3 Impulse response of first order and second order

systems

12

4 Implementation of FFT of given sequence 14

5 Determination of power spectrum of given signal 18

6, 7 Implementation of low pass and High Pass FIR filter 20

8,9 Implementation of Low pass and high pass IIR filter 28

10 Generation of sinusoidal waveform based on recursive

difference equation/IIR filter

35

11 Generation of DTMF signal 38

12,13,

14

Implementation of Interpolation, decimation and I/D

sampling rate Converters

42

DSP Lab Manual

INRODUCTION

MATLAB: MATLAB is a software package for high performance numerical

computation and visualization provides an interactive environment with hundreds of built

in functions for technical computation, graphics and animation. The MATLAB name

stands for MATrix Laboratory

The diagram shows the main features and capabilities of MATLAB.

Dept of ECE VITAE, HYD

MATLAB

ComputationsLinear algebra

Signal processingPolynomials & interpolation

QuadratureSolution of ODEs

External interfaceInterface with C and Fortran programs

Graphics2-D graphics3-D graphicsAnimation

ToolboxSignal processing Image processingStatistics Control systemNeural networks Communications

1

DSP Lab Manual

At its core ,MATLAB is essentially a set (a “toolbox”) of routines (called “m files” or

“mex files”) that sit on your computer and a window that allows you to create new

variables with names (e.g. voltage and time) and process those variables with any of

those routines (e.g. plot voltage against time, find the largest voltage, etc).

It also allows you to put a list of your processing requests together in a file and save that

combined list with a name so that you can run all of those commands in the same order at

some later time. Furthermore, it allows you to run such lists of commands such that you

pass in data and/or get data back out (i.e. the list of commands is like a function in most

programming languages). Once you save a function, it becomes part of your toolbox (i.e.

it now looks to you as if it were part of the basic toolbox that you started with).

For those with computer programming backgrounds: Note that MATLAB runs as an

interpretive language (like the old BASIC). That is, it does not need to be compiled. It

simply reads through each line of the function, executes it, and then goes on to the next

line. (In practice, a form of compilation occurs when you first run a function, so that it

can run faster the next time you run it.)

MATLAB Windows:

MATLAB works with through three basic windows

Command Window : This is the main window .it is characterized by MATLAB

command prompt >> when you launch the application program MATLAB puts you in

this window all commands including those for user-written programs ,are typed in this

window at the MATLAB prompt

Graphics window: the output of all graphics commands typed in the command window

are flushed to the graphics or figure window, a separate gray window with white

background color the user can create as many windows as the system memory will allow

Edit window: This is where you write edit, create and save your own programs in files

called M files.

Dept of ECE VITAE, HYD2

DSP Lab Manual

Input-output:

MATLAB supports interactive computation taking the input from the screen and

flushing, the output to the screen. In addition it can read input files and write output files

Data Type: the fundamental data –type in MATLAB is the array. It encompasses several

distinct data objects- integers, real numbers, matrices, charcter strings, structures and

cells.There is no need to declare variables as real or complex, MATLAB automatically

sets the variable to be real.

Dimensioning: Dimensioning is automatic in MATLAB. No dimension statements are

required for vectors or arrays .we can find the dimensions of an existing matrix or a

vector with the size and length commands.

Basic Instructions in Mat lab

1. T = 0: 1:10

This instruction indicates a vector T which as initial value 0 and final value

10 with an increment of 1

Therefore T = [0 1 2 3 4 5 6 7 8 9 10]

2. F= 20: 1: 100

Therefore F = [20 21 22 23 24 ……… 100]

3. T= 0:1/pi: 1

Therefore T= [0, 0.3183, 0.6366, 0.9549]

4. zeros (1, 3)

The above instruction creates a vector of one row and three columns whose

values are zero

Output= [0 0 0]

5. zeros( 2,4)

Output = 0 0 0 0

0 0 0 0

6. ones (5,2)

The above instruction creates a vector of five rows and two columns


DSP Lab Manual

Output = 1 1

1 1

1 1

1 1

1 1

7. a = [ 1 2 3]

b = [4 5 6]

a.*b = [4 10 18]

Which is multiplication of individual elements?

i.e. [4X1 5X2 6X3]

8 if C= [2 2 2]

b.*C results in [8 10 12]

9. plot (t, x)

Ifx = [6 7 8 9]

t = [1 2 3 4]

This instruction will display a figure window which indicates the plot of x versus t

9

8

7

6

1 2` 3 4

10. stem (t,x)


DSP Lab Manual

This instruction will display a figure window as shown

9

8

7

6

11. Subplot: This function divides the figure window into rows and columns

Subplot (2 2 1) divides the figure window into 2 rows and 2 columns 1

represent number of the figure

Subplot(3, 1, 3)

12 Filter

Syntax: y = filter(b,a,X)

Dept of ECE VITAE, HYD

1 2 (2 2 1) (2,2,2)

3 4 (2 2 3) ( 2 2 4)

1 (3, 1, 1)

2 (3, 1, 2)

3 (3, 1, 3)

5

DSP Lab Manual

Description: y = filter(b,a,X) filters the data in vector X with the filter described by

numerator coefficient vector b and denominator coefficient vector a.If a(1) is not equal to

1, filter normalizes the filter coefficients by a(1). If a(1) equals 0, filter returns an error.

13. Impz

Syntax: [h,t] = impz(b,a,n)

Description: [h,t] = impz(b,a,n) computes the impulse response of the filter with

numerator coefficients b and denominator coefficients a and computes n samples of the

impulse response when n is an integer (t = [0:n-1]'). If n is a vector of integers, impz

computes the impulse response at those integer locations, starting the response

computation from 0 (and t = n or t = [0 n]).If, instead of n, you include the empty vector

[] for the second argument, the number of samples is computed automatically by default.

14. Fliplr

Syntax: B = fliplr(A)

Description: B = fliplr(A) returns A with columns flipped in the left-right direction, that

is, about a vertical axis.If A is a row vector, then fliplr(A) returns a vector of the same

length with the order of its elements reversed. If A is a column vector, then fliplr(A)

simply returns A.

15. Conv

Syntax: w = conv(u,v)

Description: w = conv(u,v) convolves vectors u and v. Algebraically, convolution is the

same operation as multiplying the polynomials whose coefficients are the elements of u

and v.

16.Disp

Syntax: disp(X)

Description: disp(X) displays an array, without printing the array name. If X contains a

text string, the string is displayed.Another way to display an array on the screen is to type

its name, but this prints a leading "X=," which is not always desirable.Note that disp does

not display empty arrays.


DSP Lab Manual

17.xlabel

Syntax: xlabel('string')

Description: xlabel('string') labels the x-axis of the current axes.

18. ylabel

Syntax : ylabel('string')

Description: ylabel('string') labels the y-axis of the current axes.

19.Title

Syntax : title('string')

Description: title('string') outputs the string at the top and in the center of the current

axes.

20.grid on

Syntax : grid on

Description: grid on adds major grid lines to the current axes.


DSP Lab Manual

Experiment 1: To find DFT of given sequence

Aim: To find DFT of given sequence

Theory: Description

Fourier analysis is a family of mathematical techniques, all based on ecomposing signals

into sinusoids. The discrete Fourier transform (DFT) is the family member used with

digitized signals. A signal can be either continuous or discrete, and it can be either

periodic or Aperiodic. The combination of these two features generates the four

categories, described below

Aperiodic-Continuous

This includes, decaying exponentials and the Gaussian curve. These signals extend to

both positive and negative infinity without repeating in a periodic pattern. The Fourier

Transform for this type of signal is simply called the Fourier Transform.

Periodic-Continuous

This includes: sine waves, square waves, and any waveform that repeats itself in a regular

pattern from negative to positive infinity. This version of the Fourier transform is called

the Fourier series.

Aperiodic-Discrete

These signals are only defined at discrete points between positive and negative

infinity,and do not repeat themselves in a periodic fashion. This type of Fourier transform

is called the Discrete Time Fourier Transform.

Periodic-Discrete

These are discrete signals that repeat themselves in a periodic fashion from negative to

positive infinity. This class of Fourier Transform is sometimes called the Discrete Fourier

Series, but is most often called the Discrete Fourier Transform.

Discrete Fourier Transform Computation:

Mathematical Expression to calculate DFT for an input sequence x(n)


DSP Lab Manual

Matlab Code

clf;

a=[1 1 2 2];

x=fft(a,4);

n=0:3;

subplot(2,1,1);

stem(n,abs(x));

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

title('amplitude obtained by dft');

grid;

subplot(2,1,2);stem(n,angle(x));

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

title('Phase obtained by dft');


DSP Lab Manual

Experiment No 2. To find frequency response of LTI system given in

transfer function / difference equation

Aim: To find frequency response of LTI system given in transfer function /

difference equation

% Frequency Response of LTI system using difference equation

clf;

num=input('numerator coefficiuents b = ');

den=input('denominator coefficients a = ');

w = -pi:8*pi/511:pi;

h = freqz(num, den, w);

% Plot the DTFT

subplot(2,1,1)

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

title('Magnitude of H')

xlabel('\omega /\pi');

ylabel('Amplitude');

subplot(2,1,2)

plot(w/pi,phase(h));grid

title('phase of H')

xlabel('\omega /\pi');

ylabel('phase');

Output

freqresp

numerator coefficiuents b = [2 1]

denominator coefficients a = [1 -0.6]


DSP Lab Manual


DSP Lab Manual

EXPERIMENT NO 3: Impulse response of a given system

Aim:

To find the impulse response h(n) of the given LTI system whose response y(n) to an

input x(n) is given.

Theory:

Fig.2.1 A LTI system

A discrete time LTI system (also called digital filters) as shown in Fig.2.1 is

represented by

o A linear constant coefficient difference equation, for example,

y [ n ]+a1 y [n−1 ]−a2 y [ n−2 ]=b0 x [ n ]+b1 x [ n−1]+b2 x [ n−2 ];

o A system function H(z) (obtained by applying Z transform to the

difference equation). H ( z )=

Y ( z )X (z )

=b0+b1 z−1+b2z−2

1+a1 z−1+a2z−2

Given the difference equation or H(z), the impulse response of the LTI system is

found using filter or impz MATLAB functions.

MATLAB Programs:

Program 1

b=input('numerator coefficiuents b = ');

a=input('denominator coefficients a = ');

n=input('enter length n = ');

[h,t] = impz(b,a,n);

disp(h);


DSP Lab Manual

stem(t,h);

xlabel('samples n ' ,'color' , 'm');

ylabel('amplitude h(n)','color','m');

title('impulse response','color','r');

grid on;

Result:

numerator coefficiuents b = [1]

denominator coefficients a = [1 -0.5]

enter length n = 20

1.0000 0.5000 0.2500 0.1250 0.0625 0.0313 0.0156 0.0078

0.0039 0.0020 0.0010 0.0005 0.0002 0.0001 0.0001 0.0000

0.0000 0.0000 0.0000 0.0000


DSP Lab Manual

Experiment No. 4 Implementation of N point FFT

Aim: To Implement FFT algorithm of a given sequence and to plot magnitude and

phase spectra

Theory:

Discrete Fourier Transform (DFT) is used for performing frequency analysis of

discrete time signals. DFT gives a discrete frequency domain representation

whereas the other transforms are continuous in frequency domain.

The N point DFT of discrete time signal x[n] is given by the equation

X ( k )=∑n=0

N-1

x [n ]e− j2 π kn

N ; k=0,1,2,. .. .N-1

Where N is chosen such that N≥L , where L=length of x[n].

The inverse DFT allows us to recover the sequence x[n] from the frequency

samples.

x [ n ]= 1N∑k=0

N−1

x [ n ]ej2 π kn

N ; n=0,1,2,. .. .N-1

X(k) is a complex number (remember ejw=cosw + jsinw). It has both magnitude

and phase which are plotted versus k. These plots are magnitude and phase

spectrum of x[n]. The ‘k’ gives us the frequency information.

Here k=N in the frequency domain corresponds to sampling frequency (fs).

Increasing N, increases the frequency resolution, i.e., it improves the spectral

characteristics of the sequence. For example if fs=8kHz and N=8 point DFT, then

in the resulting spectrum, k=1 corresponds to 1kHz frequency. For the same fs

and x[n], if N=80 point DFT is computed, then in the resulting spectrum, k=1

corresponds to 100Hz frequency. Hence, the resolution in frequency is increased.

Since N≥L , increasing N to 8 from 80 for the same x[n] implies x[n] is still the

same sequence (<8), the rest of x[n] is padded with zeros. This implies that there

is no further information in time domain, but the resulting spectrum has higher

frequency resolution. This spectrum is known as ‘high density spectrum’

(resulting from zero padding x[n]). Instead of zero padding, for higher N, if more


DSP Lab Manual

number of points of x[n] are taken (more data in time domain), then the resulting

spectrum is called a ‘high resolution spectrum’.

Algorithm:

1. Input the sequence for which DFT is to be computed.

2. Input the length of the DFT required (say 4, 8, >length of the sequence).

3. Implement FFT algorithm.

4. Plot the magnitude & phase spectra.

MATLAB Program:

% This programme is for fast fourier transform in matlab.

% Where y is the input argument and N is the legth of FFT . Let

% y = [1 2 3 4 ];

y=input('enter input sequence:');

N= input('size of fft(e.g . 2,4,8,16,32):');

n=length(y);

p=log2(N);

Y=y;

Y=[Y, zeros(1, N - n)];

N2=N/2;

YY = -pi*sqrt(-1)/N2;

WW = exp(YY);

JJ = 0 : N2-1;

W=WW.^JJ;

for L = 1 : p-1

u=Y(:,1:N2);

v=Y(:,N2+1:N);

t=u+v;

S=W.*(u-v);

Y=[t ; S];


DSP Lab Manual

U=W(:,1:2:N2);

W=[U ;U];

N=N2;

N2=N2/2;

end;

u=Y(:,1);

v=Y(:,2);

Y=[u+v;u-v];

Y

subplot(3,1,1)

stem(y);grid

title('input sequence y')

xlabel('n');


subplot(3,1,2)

stem(abs(Y));grid

title('Magnitude of FFT')

xlabel('N');


subplot (3,1,3)

stem(angle(Y));grid

title('phase of FFT')

xlabel('N');

ylabel('phase');

Result:

>> fft1

enter input sequence:[1 2 3 4 5 6 7]

size of fft(e.g . 2,4,8,16,32):8

Y =

28.0000


DSP Lab Manual

-9.6569 + 4.0000i

-4.0000 - 4.0000i

1.6569 - 4.0000i

4.0000

1.6569 + 4.0000i

-4.0000 + 4.0000i

-9.6569 - 4.0000i


DSP Lab Manual

Experiment No. 5 Power Density Spectrum Using MATLAB

Write a MATLAB program to compute Power Density Spectrum of a given signal

Algorithm:

1. Generate a signal

2. find DFT of the signal

3. PSD is given by the formula

y(n) = ∑

k=−∞

∞

x (k )x (k−n )

where n = - (N-1) to (N-1)

PSD= |Y(w)|2

Where Y(w) = fft of y(n)

Matlab program

% computation of psd of signal corrupted with random noise

t = 0:0.001:0.6;

x = sin(2*pi*50*t)+sin(2*pi*120*t);

y = x + 2*randn(size(t));

figure,plot(1000*t(1:50),y(1:50))

title('Signal Corrupted with Zero-Mean Random Noise')

xlabel('time (milliseconds)');

Y = fft(y,512);

%The power spectral density, a measurement of the energy at various frequencies, is:

Pyy = Y.* conj(Y) / 512;

f = 1000*(0:256)/512;

figure,plot(f,Pyy(1:257))

title('power spectrum of y');

xlabel('frequency (Hz)');


DSP Lab Manual

Output


DSP Lab Manual

Experiment No. 6 and 7 Design and implementation of FIR filter to

meet given specifications

Aim: To design and implement a FIR filter for given specifications.

Theory:

There are two types of systems – Digital filters (perform signal filtering in time domain)

and spectrum analyzers (provide signal representation in the frequency domain). The

design of a digital filter is carried out in 3 steps- specifications, approximations and

implementation.

DESIGNING AN FIR FILTER (using window method):

Method I: Given the order N, cutoff frequency fc, sampling frequency fs and the

window.

Step 1: Compute the digital cut-off frequency Wc (in the range -π < Wc < π, with

π corresponding to fs/2) for fc and fs in Hz. For example let fc=400Hz,

fs=8000Hz

Wc = 2*π* fc / fs = 2* π * 400/8000 = 0.1* π radians

For MATLAB the Normalized cut-off frequency is in the range 0 and 1, where 1

corresponds to fs/2 (i.e.,fmax)). Hence to use the MATLAB commands

wc = fc / (fs/2) = 400/(8000/2) = 0.1

Note: if the cut off frequency is in radians then the normalized frequency is

computed as wc = Wc / π

Step 2: Compute the Impulse Response h(n) of the required FIR filter using the

given Window type and the response type (lowpass, bandpass, etc). For example

given a rectangular window, order N=20, and a high pass response, the

coefficients (i.e., h[n] samples) of the filter are computed using the MATLAB

inbuilt command ‘fir1’ as

h =fir1(N, wc , 'high', boxcar(N+1));


DSP Lab Manual

Note: In theory we would have calculated h[n]=hd[n]×w[n], where hd[n] is the

desired impulse response (low pass/ high pass,etc given by the sinc function) and

w[n] is the window coefficients. We can also plot the window shape as

stem(boxcar(N)).

Plot the frequency response of the designed filter h(n) using the freqz function and

observe the type of response (lowpass / highpass /bandpass).

Method 2:

Given the pass band (wp in radians) and Stop band edge (ws in radians) frequencies,

Pass band ripple Rp and stopband attenuation As.

Step 1: Select the window depending on the stopband attenuation required.

Generally if As>40 dB, choose Hamming window. (Refer table )

Step 2: Compute the order N based on the edge frequencies as

Transition bandwidth = tb=ws-wp;

N=ceil (6.6*pi/tb);

Step 3: Compute the digital cut-off frequency Wc as

Wc=(wp+ws)/2

Now compute the normalized frequency in the range 0 to 1 for MATLAB

as

wc=Wc/pi;

Note: In step 2 if frequencies are in Hz, then obtain radian frequencies (for computation

of tb and N) as wp=2*pi*fp/fs, ws=2*pi*fstop/fs, where fp, fstop and fs are the

passband, stop band and sampling frequencies in Hz

Step 4: Compute the Impulse Response h(n) of the required FIR filter using N,

selected window, type of response(low/high, etc) using ‘fir1’ as in step 2 of

method 1.

MATLAB IMPLEMENTATION


DSP Lab Manual

FIR1 Function : B = FIR1(N,Wn) designs an N'th order lowpass FIR digital filter and

returns the filter coefficients in length N+1 vector B. The cut-off 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, i.e., even symmetric coefficients obeying B(k) = B(N+2-k), k =

1,2,...,N+1.

If Wn is a two-element vector, Wn = [W1 W2], FIR1 returns an order N bandpass filter

with passband W1 < W < W2. B = FIR1(N,Wn,'high') designs a highpass filter. B =

FIR1(N,Wn,'stop') is a bandstop filter if Wn = [W1 W2]. 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.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 whose

numerator and denominator coefficients are 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.

For FIR filter enter A=1 and B = h[n] coefficients. Appropriately choose N as 128, 256,

etc

Window Transition Width Δω Min. Stop band Matlab

Name Approximate Exact values Attenuation Command

Rectangular

4 ΠM

1.8 ΠM 21db B= FIR1(N,Wn,boxcar)

Bartlett

8 ΠM

6 .1 ΠM 25db B = FIR1(N,Wn,bartlett)

Hanning

8 ΠM

6 .2 ΠM 44db B = FIR1(N,Wn,hanning)

Hamming

8 ΠM

6 .6 ΠM 53db B= FIR1(N,Wn,hamming)

Blackman

12 ΠM

11 ΠM 74db B = FIR1(N,Wn,blackman)


DSP Lab Manual

Program:

%Method 2: the following program gives only the design of the FIR filter- for

implementation continue with the next program (after h[n])

%input data to be given: Passband & Stopband frequency

% Data given: Passband ripple & stopband attenuation As. If As>40 dB, Choose

hamming

clear

wpa=input('Enter passband edge frequency in Hz');

wsa= input('Enter stopband edge frequency in Hz');

ws1= input('Enter sampling frequency in Hz');

%Calculate transmission BW,Transition band tb,order of the filter

wpd=2*pi*wpa/ws1;

wsd=2*pi*wsa/ws1;

tb=wsd-wpd;

N=ceil(6.6*pi/tb)

wc=(wsd+wpd)/2;

%compute the normalized cut off frequency

wc=wc/pi;

%calculate & plot the window

hw=hamming(N+1);

stem(hw);

title('Fir filter window sequence- hamming window');

%find h(n) using FIR

h=fir1(N,wc,hamming(N+1));

%plot the frequency response

figure(2);

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

mag=20*log10(abs(m));

plot(ws1*w/(2*pi),mag);


DSP Lab Manual

title('Fir filter frequency response');

grid;

%generate simulated input of 50, 300 & 200 Hz, each of 30 points

n=1:30;

f1=50;f2=300;f3=200;fs=1000;

x=[];

x1=sin(2*pi*n*f1/fs);



x=[x1 x2 x3];

subplot(2,1,1);

stem(x);

title('input');

%generate o/p

%y=conv(h,x);

y=filter(h,1,x);

subplot(2,1,2);

stem(y);

title('output');

Result:

Enter passband edge frequency in Hz100

Enter stopband edge frequency in Hz200

Enter sampling frequency in Hz1000

N = 33

Plots are as in Fig.11.1 and 11.2

Inference:

Notice the maximum stopband attenuation of 53 dB from plot 11.2


DSP Lab Manual

%Design and implementation of FIR filter Method 1

%generate filter coefficients for the given %order & cutoff Say N=33, fc=150Hz,

%fs=1000 Hz, Hamming window

h=fir1(33, 150/(1000/2),hamming(34));


n=1:30;

f1=50;f2=300;f3=200;fs=1000;

x=[];




x=[x1 x2 x3];

subplot(2,1,1);

stem(x);

title('input');

%generate o/p

%y=conv(h,x);

y=filter(h,1,x);

subplot(2,1,2);

stem(y);

title('output');

Result:

Plots are in Fig.11.3 & 11.4

Notice that freqs below 150 Hz are passed & above 150 are cutoff


DSP Lab Manual

Fig.11.1 Fig.11.2

Fig.11.3(using filter command) Fig.11.4 (using conv command)

Fig.11.5 Freqs f1=150;f2=300;f3=170;fs=1000;

FIR filter design using Kaiser window:


DSP Lab Manual

disp('FIR filter design using Kaiser window');

M = input ('enter the length of the filter = ');

beta= input ('enter the value of beta = ');

Wc = input ('enter the digital cutoff frequency');

Wn= kaiser(M,beta);

disp('FIR Kaiser window coefficients');

disp(Wn);

hn = fir1(M-1,Wc,Wn);

disp('the unit sample response of FIR filter is hn=');

disp(hn);

freqz(hn,1,512);

grid on;

xlabel('normalized frequency');

ylabel('gain in db');

title('freq response of FIR filter');

Result:

FIR filter design using Kaiser window

enter the length of the filter = 30

enter the value of beta = 1.2

enter the digital cutoff frequency.3

FIR Kaiser window coefficients

0.7175 0.7523 0.7854 0.8166 0.8457 0.8727 0.8973 0.9195

0.9392 0.9563 0.9706 0.9822 0.9909 0.9967 0.9996 0.9996

0.9967 0.9909 0.9822 0.9706 0.9563 0.9392 0.9195 0.8973

0.8727 0.8457 0.8166 0.7854 0.7523 0.7175

the unit sample response of FIR filter is hn=

Columns 1 through 5

0.0141 0.0028 -0.0142 -0.0224 -0.0117

Columns 6 through 10


DSP Lab Manual

0.0133 0.0333 0.0277 -0.0072 -0.0495


-0.0614 -0.0140 0.0895 0.2097 0.2900


0.2900 0.2097 0.0895 -0.0140 -0.0614


-0.0495 -0.0072 0.0277 0.0333 0.0133


-0.0117 -0.0224 -0.0142 0.0028 0.0141

Fig.11.6.Response of FIR filter designed using Kaiser window


DSP Lab Manual

EXPERIMENT NO 8,9: Design and implementation of IIR filter to

meet given specifications

Aim: To design and implement an IIR filter for given specifications.

Theory:

There are two methods of stating the specifications as illustrated in previous program. In

the first program, the given specifications are directly converted to digital form and the

designed filter is also implemented. In the last two programs the butterworth and

chebyshev filters are designed using bilinear transformation (for theory verification).

Method I: Given the order N, cutoff frequency fc, sampling frequency fs and the IIR

filter type (butterworth, cheby1, cheby2).

Step 1: Compute the digital cut-off frequency Wc (in the range -π < Wc < π, with π

corresponding to fs/2) for fc and fs in Hz. For example let fc=400Hz, fs=8000Hz

Wc = 2*π* fc / fs = 2* π * 400/8000 = 0.1* π radians

For MATLAB the Normalized cut-off frequency is in the range 0 and 1, where 1

corresponds to fs/2 (i.e.,fmax)). Hence to use the MATLAB commands

wc = fc / (fs/2) = 400/(8000/2) = 0.1

Note: if the cut off frequency is in radians then the normalized frequency is computed

as wc = Wc / π

Step 2: Compute the Impulse Response [b,a] coefficients of the required IIR filter

and the response type (lowpass, bandpass, etc) using the appropriate butter, cheby1,

cheby2 command. For example given a butterworth filter, order N=2, and a high pass

response, the coefficients [b,a] of the filter are computed using the MATLAB inbuilt

command ‘butter’ as [b,a] =butter(N, wc , 'high');

Method 2:

Given the pass band (Wp in radians) and Stop band edge (Ws in radians) frequencies,

Pass band ripple Rp and stopband attenuation As.

Step 1: Since the frequencies are in radians divide by π to obtain normalized

frequencies to get wp=Wp/pi and ws=Ws/pi


DSP Lab Manual

If the frequencies are in Hz (note: in this case the sampling frequency should be

given), then obtain normalized frequencies as wp=fp/(fs/2), ws=fstop/(fs/2),

where fp, fstop and fs are the passband, stop band and sampling frequencies in Hz

Step 2: Compute the order and cut off frequency as

[N, wc] = BUTTORD(wp, ws, Rp, Rs)

Step 3: Compute the Impulse Response [b,a] coefficients of the required IIR filter

and the response type as [b,a] =butter(N, wc , 'high');

IMPLEMENTATION OF THE IIR FILTER

1. Once the coefficients of the IIR filter [b,a] are obtained, the next step is to

simulate an input sequence x[n], say input of 100, 200 & 400 Hz (with sampling

frequency of fs), each of 20/30 points. Choose the frequencies such that they are

>, < and = to fc.

2. Filter the input sequence x[n] with Impulse Response, to obtain the output of the

filter y[n] using the ‘filter’ command.

3. Infer the working of the filter (low pass/ high pass, etc).

MATLAB IMPLEMENTATION

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 passband and has at least Rs dB of

attenuation in the stopband. Wp and Ws are the passband and stopband edge frequencies,

normalized from 0 to 1 (where 1 corresponds to pi radians/sample). For example,

Lowpass: Wp = .1, Ws = .2 Highpass: Wp = .2, Ws = .1

Bandpass: Wp = [.2 .7], Ws = [.1 .8] Bandstop: 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.

BUTTER Butterworth digital and analog filter design.


DSP Lab Manual

[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].

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.

Program (Design & implementation)

%generate filter coefficients for the given %order & cutoff %Say N=2, fc=150Hz,

%fs=1000 Hz, butterworth filter

[b,a]=butter(2, 150/(1000/2));


n=1:30;

f1=100;f2=300;f3=170;fs=1000;

x=[];




x=[x1 x2 x3];

subplot(2,1,1);

stem(x);

title('input');

%generate o/p

y=filter(b,a,x);

subplot(2,1,2);

stem(y);

title('output');


DSP Lab Manual

Result:

Plot is in Fig. 12.1, which shows that 100 Hz is passed, while 300 is cutoff and 170

has slight attenuation.

Note: If fp,fstp,fs, rp,As are given then use

[N,wc]=buttord(2*fp/fs,2*fstp/fs,rp,As)

[b,a]=butter(N,wc);

If wp & ws are in radians

[N,wc]=buttord(wp/pi,ws/pi,rp,As)

[b,a]=butter(N,wc);

If wc is in radians & N is given

[b,a]=butter(N,wc/pi);

For a bandpass output

wc=[150/(1000/2) 250/(1000/2)];

[b,a]=butter(4, wc);

Plot is in Fig.12.2 where only 170 is passed, while 100 & 300 are cutoff.

Fig.12.1 Low pass IIR filter(butterworth) output


DSP Lab Manual

Fig.12.2 IIR output for bandpass (150-250 Hz)

Programs for designing of IIR filters (for theory practice):

% Butterworth filter: Given data: rp=1, rs=40, w1=800, %w2=1200,ws=3600;

rp=1, rs=40, w1=800, w2=1200,ws=3600;

% Analog frequency

aw1=2*pi*w1/ws;

aw2=2*pi*w2/ws;

% Prewrapped frequency

pw1 = 2*tan(aw1/2);

pw2 = 2*tan(aw2/2);

%Calculate order and cutoff freq

[n,wc]= buttord (pw1,pw2,rp,rs,'s');

% analog filter transfer


DSP Lab Manual

[b,a] = butter(n,wc,'s');

%obtaining the digital filter using bilinear transformation

fs=1;

[num,den]= bilinear(b,a,fs);


[mag,freq1]=freqz(num,den,128);

freq=freq1*ws/(2*pi);

m = 20*log10(abs(mag));

plot(freq,m);

grid;

Result: rp = 1 rs = 40 w1 = 800 w2 = 1200

Fig. 12.2 BUTTERWORTH FILTER


DSP Lab Manual

To design a chebyshev filter for given specifications

%Given data

rp=1,rs=40,w1=800,w2=1200,ws=3600

%Analog frequencies

aw1= 2*pi*w1/ws;

aw2=2*pi*w2/ws;

% Prewrapped frequency assuming T=1/fs

pw1 = 2*tan(aw1/2);

pw2 = 2*tan(aw2/2);

[n,wc]= cheb1ord (pw1,pw2,rp,rs,'s');

[b,a] = cheby1(n,rp,wc,'s');

%obtaining the digital filter using bilinear transformation

fs=1;

[num,den]= bilinear(b,a,fs);


[mag,freq1]=freqz(num,den,128);

freq=freq1*ws/(2*pi);

m = 20*log10(abs(mag));

plot(freq,m);grid;

Result: rp = 1 rs = 40 w1 = 800 w2 = 1200 ws = 360

Fig.12.3 CHEBYSHEV FILTER


DSP Lab Manual

EXPERIMENT NO 10: Generation of sinusoidal wave using recursive difference

equation

Aim: To generate sinusoidal signal using recursive difference equation

Theory:

There exists different techniques to generate a sine wave on a DSP processor.Using a lookup table,

interpolation, polynomials, or pulse width modulation are some of these techniques utilized to generate a

sine wave. When a slightly sufficient processing power is considered, it is possible to utilize another

technique which makes use of an IIR filter.In this technique, a second order IIR filter as shown in Figure 1

is designed to be marginally stable which is an undesirable situation for a normal filtering operation. For

this second order filter to be marginally stable, its poles are located in the unit circle. After choosing the

suitable filter coefficients for the filter to be marginally stable, a unit impulse is applied to the filter as an

input at time t = 0, and then the input is disconnected. Then the filter starts to oscillate with a fixed

frequency.

ÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁÁ

ÁÁÁÁÁÁ Á

The difference equation of the second order filter shown in Figure 1 will be:


DSP Lab Manual

Where F0 is the frequency, Fs is the sampling frequency and A is the amplitude of the

sine wave.

Matlab Program

% Generation of sine wave using recursive difference equation or IIR filter

%fs = sampling rate

% t = time in sec

%f0 = frequency

% A = amplitude

A = 2;

t=0.1;

fs = 5000;

f0=50;

length = fs*t;

y = zeros(1,length);

impulse = zeros(1,length);

impulse(1) = 1;

% frequency coefficients

% difference equation of form y[n]= -a1 y[n-1] -a2 y[n-2] + b0 impulse[n]


DSP Lab Manual

% a1 = -2cos(2*pi*f0/fs), a2= 1, b0 = Asin(2*pi*f0/fs)

a1 = -2*cos(2*pi*f0/fs)

a2 = 1;

b0 = A*sin(2*pi*f0/fs)

y(1)= 0;

y(2)= b0;

for i=3:length

y(i)= b0* impulse(i)- a1*y(i-1) - a2* y(i-2);

end

plot (y);

title('Sinusoidal waveform')

xlabel('samples n');

ylabel('y(n)');

Result :


DSP Lab Manual

Experiment No. 11 Generation of DTMF signals

Therory: Dual-tone Multi-Frequency (DTMF) signaling is the basis for voice

communications control and is widely used worldwide in modern telephony to dial

numbers and configure switchboards. It is also used in systems such as in voice mail,

electronic mail and telephone banking.

A DTMF signal consists of the sum of two sinusoids - or tones - with frequencies taken

from two mutually exclusive groups. These frequencies were chosen to prevent any

harmonics from being incorrectly detected by the receiver as some other DTMF

frequency. Each pair of tones contains one frequency of the low group (697 Hz, 770 Hz,

852 Hz, 941 Hz) and one frequency of the high group (1209 Hz, 1336 Hz, 1477Hz) and

represents a unique symbol. The frequencies allocated to the push-buttons of the

telephone pad are shown below:

We will right the program to generate and visualize the DTMF tones for all 12 Symbols.

Also we will estimate its energy content using Gortzet’s Algorithm

The minimum duration of a DTMF signal defined by the ITU standard is 40 ms.

Therefore, there are at most 0.04 x 8000 = 320 samples available for estimation and

detection. The DTMF decoder needs to estimate the frequencies contained in these short

signals.


1209Hz 1336Hz 1477 Hz

697Hz1

ABC

2

DEF

3

770

Hz

GHI

4

JKL

5

MNO

6

852HzPQRS

7

TUV

8

WXYZ

9

941

Hz *0 #

DSP Lab Manual

One common approach to this estimation problem is to compute the Discrete-Time

Fourier Transform (DFT) samples close to the seven fundamental tones. For a DFT-based

solution, it has been shown that using 205 samples in the frequency domain minimizes

the error between the original frequencies and the points at which the DFT is estimated.

At this point we could use the Fast Fourier Transform (FFT) algorithm to calculate the

DFT. However, the popularity of the Goertzel algorithm in this context lies in the small

number of points at which the DFT is estimated. In this case, the Goertzel algorithm is

more efficient than the FFT algorithm.

Plot Goertzel's DFT magnitude estimate of each tone on a grid corresponding to the

telephone pad.

MATLAB Program

% generating all 12 frequencies

symbol = {'1','2','3','4','5','6','7','8','9','*','0','#'};

lfg = [697 770 852 941]; % Low frequency group

hfg = [1209 1336 1477]; % High frequency group

f = [];

for c=1:4,

for r=1:3,

f = [ f [lfg(c);hfg(r)] ];

end

end

f'

% Generate the DTMF tones

Fs = 8000; % Sampling frequency 8 kHz

N = 800; % Tones of 100 ms

t = (0:N-1)/Fs; % 800 samples at Fs

pit = 2*pi*t;


DSP Lab Manual

tones = zeros(N,size(f,2));

for toneChoice=1:12,

% Generate tone

tones(:,toneChoice) = sum(sin(f(:,toneChoice)*pit))';

% Plot tone

subplot(4,3,toneChoice),plot(t*1e3,tones(:,toneChoice));

title(['Symbol "', symbol{toneChoice},'":

[',num2str(f(1,toneChoice)),',',num2str(f(2,toneChoice)),']'])

set(gca, 'Xlim', [0 25]);


if toneChoice>9, xlabel('Time (ms)'); end

end

set(gcf, 'Color', [1 1 1], 'Position', [1 1 1280 1024])

annotation(gcf,'textbox', 'Position',[0.38 0.96 0.45 0.026],...

'EdgeColor',[1 1 1],...

'String', '\bf Time response of each tone of the telephone pad', ...

'FitHeightToText', 'on');

% estimation of DTMF tone using Gortzel's Algorithm

Nt = 205;

original_f = [lfg(:);hfg(:)] % Original frequencies

k = round(original_f/Fs*Nt); % Indices of the DFT

estim_f = round(k*Fs/Nt) % Frequencies at which the DFT is estimated

tones = tones(1:205,:);

figure,

for toneChoice=1:12,

% Select tone

tone=tones(:,toneChoice);

% Estimate DFT using Goertzel

ydft(:,toneChoice) = goertzel(tone,k+1); % Goertzel use 1-based indexing

% Plot magnitude of the DFT


DSP Lab Manual

subplot(4,3,toneChoice),stem(estim_f,abs(ydft(:,toneChoice)));

title(['Symbol "', symbol{toneChoice},'":

[',num2str(f(1,toneChoice)),',',num2str(f(2,toneChoice)),']'])

set(gca, 'XTick', estim_f, 'XTickLabel', estim_f, 'Xlim', [650 1550]);

ylabel('DFT Magnitude');

if toneChoice>9, xlabel('Frequency (Hz)'); end

end

set(gcf, 'Color', [1 1 1], 'Position', [1 1 1280 1024])

annotation(gcf,'textbox', 'Position',[0.28 0.96 0.45 0.026],...

'EdgeColor',[1 1 1],...

'String', '\bf Estimation of the frequencies contained in each tone of the telephone pad

using Goertzel', ...

'FitHeightToText','on');

Output:


DSP Lab Manual

Experiment No. 12, 13, 14: Implementation of Interpolation, decimation

and I/D sampling rate converters

Theory:

Multirate : Changing the Sampling Rate of the Discrete Time Signal.The process of

converting a signal from a given rate to a different rate is called sampling rate

conversion.

Systems that employ multiple sampling rates in the processing of digital signals are

called multirate digital signal processing systems.

Can be accomplished in two ways:

Adv: New sampling rate can be arbitrarily selected

Disadv: Signal distortion is introduced by

D/A converter in signal reconstruction

Quantization effects in A/D converter

Changing the sampling rate in digital domain --Multirate

Fundamental operations in multirate signal processing are

1) Downsampling (Decimation)

2) Upsampling(Interpolation )

Decimation is the process of decreasing the sampling rate by a factor of M i.e. from Fs

to Fs / M


DSP Lab Manual

Down sampling : by a factor of M is achieved by discarding M-1 samples for every M

samples.

This combined operation of filtering and downsampling is called as decimation.


DSP Lab Manual

The rate compressor reduces the sampling rate from Fs to Fs / M

To prevent aliasing at lower rate, digital filter is used to band limit the i/p signal to less

than Fs/2M. (new folding frequency).

Sampling rate reduction is achieved by discarding M-1 samples for every M samples of

filtered signal W(n).

INTERPOLATION

Process of increasing the sampling rate of the signal by a factor of L i.e. from F s to LFs

Upsampling by a factor L means inserting L-1 zeros between two samples


DSP Lab Manual


DSP Lab Manual

In matlab we can use function resample() to achieve interpolation as well as decimation.

MATLAB program:

% % m-file to illustrate simple interpolation and

% decimation operations

% File name:

%

Fs=input('Enter Sampling Frequency:'); % sampling frequency

I= input('interpolation factor:');

A=1.5; % relative amplitudes

B=1;

f1=50; % signal frequencies

f2=100;

t=0:1/Fs:1; % time vector

x=A*cos(2*pi*f1*t)+B*cos(2*pi*f2*t); % generate signal

y=resample(x,I,1); % interpolate signal by 4

stem(x(1:100)) % plot original signal

xlabel('Discrete time, nT ')

ylabel('Input signal level')

figure

stem(y(1:400)) % plot interpolated signal.


DSP Lab Manual

xlabel('Discrete time 4 X nT')

ylabel('Interpolated output signal level')

D=input('Enter the decimation factor:');

y1=resample(x,1,D);

figure

stem(y1(1:50)) % plot decimated signal.

xlabel('Discrete time nT/2')

ylabel('Decimated output signal level')

Result

>> decimation

Enter Sampling Frequency:1000

interpolation factor:4


DSP Lab Manual

Enter the decimation factor:2


DSP Lab Manual


dsp lab manual

Documents

matlab windows

matlab prompt graphics

main window

capabilities of matlab

edit window

figure window

matlab command prompt

basic windows command