Final Dsp Manual

8/3/2019 Final Dsp Manual

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Experiment No: 1

BASIC OPERATIONS ON MATRICES

AIM: To write a MATLAB program to perform some basic operation on matrices such as addition,

subtraction, multiplication and inverse of a matrix.

SOFTWARE REQURIED:

MATLAB R2009a (7.8.0 Version).

THEORY:

Matlab stands for MATrix LABoratory, and was originally written to perform matrix algebra. It is very

efficient at doing this, as well as having the capabilities of any normal programming language. Inmost instances, the data we will be dealing with are simple matrices, either single values, or one- or

two dimensional arrays. An example of a one dimensional array (vector) is a single row or a single

column of values.

The following matrix operations can be performed with very ease in MATLAB.

Consider the two matrices:

a=[1 2 -9 ; 2 -1 2; 3 -4 3]; and b=[1 2 3; 4 5 6; 7 8 9];

i. Addition>>c=a+b

c =

2 4 -6

6 4 8

10 4 12

ii. Subtraction>>d=a-b

d =

0 0 -12

-2 -6 -4

-4 -12 -6


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iii. Multiplication>>e=a*b

e =

-54 -60 -66

12 15 18

8 10 12

iv. Inverse>>f=inv(a)

f =

0.1000 0.6000 -0.1000

0 0.6000 -0.4000

-0.1000 0.2000 -0.1000

CONCLUSION:

Thus we have studied the computations on matrices using MATLAB.


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Experiment No: 2

REALISATION OF SIGNALS

Aim: To write a MATLAB program to generate discrete time and continuous time signals like unit

impulse, unit step, unit ramp, exponential signal and sinusoidal signals.

SOFTWARE REQURIED:


Theory: In the analysis of communication systems, standard test signals play a vital role. Such

signals are used to check the performance of a system. Mathematically, a signal is described as a

function of one or more independent variables. It is a physical quantity which varies with some

dependent or independent variables. The signals can be basically either continuous time or discrete

time signals. Continuous time signals are analog signals having some value at every instant of time.

Discrete time signals are represented at discrete intervals of time i.e. they have a specific value at

specific time.

Some standard test signals are as follows:

Unit Impulse Signal: A discrete time unit impulse function is denoted by (n). Its amplitude is 1 at

n=0 and for all other values of n; its amplitude is zero.

(n)=1; for n=0

=0; for n0

Discrete Unit Impulse Signal


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A continuous time unit impulse function is denoted by (t). Mathematically it is expressed as:

(t)=0; for t0

Continuous Unit Impulse Signal

Unit Step signal: A discrete time unit step signal is denoted by u(n). Its value is unity for n0 while

for other values of n its value is zero.

u(n)=1; for n0

=0; for n


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Unit ramp Signal: A discrete time unit ramp signal is denoted by u r(n). Its value increases linearly

with sample number n.

ur(n)=n; for n0

=0; for n


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x(n)=A sin wn, where A = amplitude

w=Angular

frequency

A continuous time sinusoidal signal is denoted by:

x(t)=A sin wt, where A = amplitude

w=Angular

frequency

Conclusion: Thus the Generation of discrete time and continuous time signals like unit impulse, unit

step, unit ramp, exponential signal and sinusoidal signals is successfully completed.


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Experiment No: 3

OPERATIONS ON SIGNALS AND SEQUENCES

Aim: To write a MATLAB program to perform various operations on signals such as addition,

multiplication, scaling, shifting and folding.

SOFTWARE REQURIED:


Theory: Many a times it is necessary to modify the original signal. This modification is achieved by

performing different operations on given discrete time signal. These operations can be briefly

explained as follows:

Addition: Consider two sequences:

x1(n) = [1, 1, 0, 1,1] and x2(n) = [2, 2, 0, 2, 2]

Let y(n) = x1(n) + x2(n)

y(n) = [3, 3, 0, 3, 3]

Multiplication:

Scaling:

Shifting:

Folding:

Some standard test signals are as follows:

Unit Impulse Signal: A discrete time unit impulse function is denoted by (n). Its amplitude is 1 at

n=0 and for all other values of n; its amplitude is zero.

(n)=1; for n=0

=0; for n0

Discrete Unit Impulse Signal


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A continuous time unit impulse function is denoted by (t). Mathematically it is expressed as:

(t)=0; for t0

Continuous Unit Impulse Signal

Unit Step signal: A discrete time unit step signal is denoted by u(n). Its value is unity for n0 while

for other values of n its value is zero.

u(n)=1; for n0

=0; for n


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Continuous Unit Step Signal

Unit ramp Signal: A discrete time unit ramp signal is denoted by u r(n). Its value increases linearly

with sample number n.

ur(n)=n; for n0

=0; for n


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A continuous time exponential signal is expressed as:

x(t)=at, where a is some real constant

Sinusoidal Signal: A discrete time sinusoidal signal is denoted by:

x(n)=A sin wn, where A = amplitude

w=Angular

frequency

A continuous time sinusoidal signal is denoted by:

x(t)=A sin wt, where A = amplitude

w=Angular

frequency

Conclusion: Thus the Generation of discrete time and continuous time signals like unit impulse, unit

step, unit ramp, exponential signal and sinusoidal signals is successfully completed.

Experiment No: 2

SAMPLING THEOREM

AIM: To generate a MATLAB Program to verify the sampling theorem.

SOFTWARE REQURIED

MATLAB R2006 (7.3 Version).

THEORY:

Example:

x=cos(2*pi*f1*t)+cos(2*pi*f2*t);f1=3;


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f2=23;

%case 1: (fs2fm)

Output:

CONCLUSION:

Thus we have studied the computations on matrices using MATLAB.

Aim

SOFTWARE REQURIED


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Theory

Example

Functions

Figures

Conclusion


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Experiment No: 1

LINEAR PHASE FIR FILTERS

AIM: To write a MATLAB program:

I. To check the linear phase condition and classify the given filter into type I, II, III and IVbased on the impulse response.

II. To study the magnitude and phase responses of the filter.

SOFTWARE REQURIED:


THEORY:

A filter is supposed to operate with a definite pass band and a stop band. An ideal filter must have

constant gain and a uniform delay for all the frequencies in the pass band. The group delay( p) is the

derivative of the phase angle with respect to the frequency.

The group delay becomes constant with respect to frequency if the phase angle happens to be linear

function of the frequency. Hence linear phase filters can have uniform group delay. Linear phase

property is exhibited only by FIR filters. An FIR filter has linear phase it its impulse response is either

symmetric or anti symmetric. The nature of the pass band depends on filter length and its symmetry

as shown below in the table.

Filter Length and Symmetry Type Nature of Pass Band

Symmetric with odd length I LP,HP,BP and BS

Symmetric with even length II LP and BP

Anti symmetric with odd length III BP

Anti symmetric with even length IV HP and BP

The magnitude and phase responses along with zero locations of such filters are investigated

through a MATLAB program.


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MATLAB FUNCTIONS:

fliplr():FLIPLR(X) returns X with row preserved and columns flipped in the left/right direction.

X = 1 2 3 becomes 3 2 1

4 5 6 6 5 4

freqz():[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 a filter.

zplane():ZPLANE(Z,P) plots the zeros Z and poles P (in column vectors) with the unit circle for

reference.

CONCLUSION:

Thus the linear phase property of the FIR filter was verified. Also the zero locations at z=1 and z=-1

for different types of filters was verified.


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Experiment No: 1

TIME AND FREQUENCY DOMAIN BEHAVIOUR OF

FIR WINDOW FUNCTIONS


i. To study the time and frequency domain behaviour of different window functions usedin FIR filters.

ii. To study the change in the parameter of the window in frequency domain w.r.t. thechange in the window length, using thefdatoolof MATLAB.

SOFTWARE REQURIED:


THEORY:

Windowing technique is one of the methods used for FIR filter design. An ideal filter function in the

frequency domain gives infinitely stretching response in the time domain. This response is made

finite by multiplying it with suitable window function. The response is made causal by shifting the

required number of samples towards right. The different window functions along with their

corresponding frequency responses are studied in this experiment. The shown below table shows

the summary of window functions:

M= Filter length

No. Window Function

1 Rectangular ;1nw n = 0 to M-1

2 Triangular Bartlett ;

1

||21

M

nnw 1|| Mn

3 Hanning

1

**2cos1

2

1

M

npinw ; 1|| Mn

4 Hamming ;

1

**2cos46.054.0

M

npinw 1|| Mn

5 Blackman ;

1

**4cos08.0

1

**2cos5.042.0

M

npi

M

npinw 1|| Mn


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MATLAB FUNCTIONS:

freqz():[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 a filter.

abs():ABS(X) is the absolute value of the elements of X. When X is complex, ABS(X) is the complex

modulus (magnitude) of the elements of X.

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

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

CONCLUSION:

Thus the time and frequency domain behaviour of different window functions were studied, along

with the change in the parameters of the window in frequency domain w.r.t. the change in the

window length, using thefdatoolof MATLAB.


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Experiment No: 1

ANALOG IIR FILTERS


i. To find the order of an analog filter for a given set of specifications using Butterworth,Chebyshev and Elliptical approximation methods.

ii. To study the magnitude and phase responses of the filter.

SOFTWARE REQURIED:


THEORY:The ideal response of an IIR filter may be approximated using different response functions. They are

Butterworth, Chebyshev, Inverse Chebyshev and Elliptic approximations. For a given set of filter

specifications, each of these approximations gives a different order of the filter. A comparative study

of filter behaviour is carried out in this experiment by calculating the order of the individual filter for

the same set of specifications.

Butterworth Approximation:

Order

p

s

ww

d

d

n

log

11

11

log

2

12

1

2

2

Chebyshev and Inverse Chebyshev Approximation:

Order

p

s

ww

de

n1

21

2

2

1

cosh

111

cosh

Where

21

2

1

11

de


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Elliptic Approximation:

Let

p

s

w

wk ;

110

1101.0

1.0

p

s

k

k

M

22211

161

MMC ;

121

k

kD

Order DFCFn ee

Where 92 152ln1 xxxpi

XFe

Alsopkd 1log20 ; skd 2log20

MATLAB FUNCTIONS:

buttord():[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.

cheb1ord():[N, Wp] = 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 passband and has at least Rs dB

of attenuation in the stopband. Wp and Ws are the passband and stopband edge

frequencies.

ellipord():[N, Wp] = 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 passband and has at least Rs dB of attenuation in

the stopband. Wp and Ws are the passband and stopband edge frequencies.

round():ROUND(X) rounds the elements of X to the nearest integers.

ceil():CEIL(X) rounds the elements of X to the nearest integers towards infinity.


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CONCLUSION:

Thus for the given set of specifications, filter order was calculated for Butterworth, Chebyshev and

Elliptic responses. It was also observed that the order of the filter increases with the decrease in

transition width, increase in pass band gain and decrease in stop band gain.

Final Dsp Manual

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