INDEX Exp No. Experiment Name Date of Performan ce Date of Checkin g Signatu re Marks (10) 1. Program for generation of sin, cos, exponential, unit impulse, step function, square wave 2. a) Program to determine convolution of two sequences b) Program to determine correlation of two sequences 3. To calculate DTFT & plot magnitude & phase response of following sequence x(n) = 1 0=<n<=3 = 0 elsewhere 4. An LTI system is specified by the difference equation Y(n)=0.9*y(n-1)+x(n) a) Determine H(exp(jw)) b) Calculate & plot the steady state response Y(n) to X(n) = cos(0.5 n)u(n) ∏
Transcript
INDEX
Exp No.
Experiment Name Date ofPerformance
Date of Checking
Signature Marks(10)
1. Program for generation of sin, cos, exponential, unit impulse, step function, square wave
2. a) Program to determine convolution of two sequences
b) Program to determine correlation of two sequences
3. To calculate DTFT & plot magnitude & phase response of following sequence
x(n) = 1 0=<n<=3 = 0 elsewhere
4. An LTI system is specified by the difference equation
Y(n)=0.9*y(n-1)+x(n)
a) Determine H(exp(jw))b) Calculate & plot the steady state response
Y(n) to X(n) = cos(0.5∏n)u(n)5. Given a causal system
Y(n) = 0.9*y(n-1)+x(n)
a) Find H(2) & sketch its pole zero plotb) Plot H(w) & angle H(w)
6. Let X(n) be a 4–point sequenceX(n) = 1 0<n<3 = 0 elsewherecompute the 4-point, 8-point, 16-point DFT & plot magnitude & phase plot
7. Differentiate b/w high density high resolution spectrum, for given sequence
X(n)=cos(0.48*pi*n)+cos(0.52*pi*n)
a) Determine DFT & also plot it for 0<=n<=10 & 0<=n<=100
8. Program to find the circular convolution of two sequences x(n) & h(n)
9. To design a FIR LPL filter using Rectangular, Barlett, Hamming, Hanning, Blackmann Window & plot the phase response of filter. The specification of the filter are as follows:
Wp=0.5*pi, Ws=0.3*pi, 0.25dB, As=50 dB
10. Transform H(s)=(s+1)/(s2+5s+6) into a digital filter using
a) Impulse Invariant Transformationb) Bilinear Transformation
EXPERIMENT No. – 1
AIM
Program for generation of sin, cos, exponential, unit impulse, step function, square wave.
THEORY
Sine Function:
A function f(x)=sin(x) is a periodic function with a period of 2∏, it will form a periodic wave after 2∏. The f(x) will lie between -1 to +1.
Cosine Function:
A function f(x)=cos(x) is a periodic function with a period of 2∏, it will form a periodic wave after 2∏.
Exponential Function:
A function f(x)=ex is defined as:
0<f(x)≤1; for -∞≤x≤01≤f(x)<∞; for 0≤x≤∞
Unit impulse Function:
Unit impulse is a signal that is 0 everywhere except at n=0 where it is 1. In discrete time domain the unit impulse signal is defined as:
∂(n) =1; n=0 = 0; elsewhere
Unit Step Function:
The integral of the impulse function is a unit step signal. In discrete time unit step signal is defined as:
U(n) =1; n>=0 = 0; n<0
Square Wave Function:
It is a periodic function with a period of T. If f(x) is a square wave function thenF(x) = 1; 0<x<T/2 = -1; T/2<x<T
end; for k=c:d s(k)=-1; end; a=a+10; b=b+10; c=c+10; d=d+10;end;n=0:39;figure;subplot(211);stem(n,s);xlabel('Time');ylabel('Amplitude');title('Square Wave');v=0:5;subplot(212);stem(v,v);xlabel('Time');ylabel('Amplitude');title('Ramp');
OUTPUT
EXPERIMENT No. – 2(a)
AIM
To generate convolved output of two sample sequences.
THEORY
A convolution is an integral that expresses the amount of overlap of one function as it is shifted over another function. It therefore “blends” from one function to another. For example, in synthesis imaging, the measured dirty map is a convolution of the “true” CLEAN map with the dirty beam.
This formula gives response y(n) of the LTI system as a function of the input signal x(n) and the unit sample response h(n) is called CONVOLUTION SUM.
We say that the input x(n) is convolved with the impulse response h(n) to yield the output y(n).
The process of computing the convolution between x(k) and h(k) involves the following steps:
1. Folding: Fold h(k) about k=0 to obtain h(-k).
2. Shifting: Shift h(-k) by n to the right(left) if n is positive(negative) to obtain h(n-k).
3. Multiplication: Multiply x(k) by h(n-k) to obtain the value of the output sequence v(k) = x(k)h(n-k).
4. Summation: Sum all the values of the product sequence v(k) to obtain the value of the output at a given time .
Suppose that we have two signal sequences x(n) and y(n) each of which has finite energy.The cross correlation of x(n) and y(n) is a sequence rxy(l), which is defined as
r xy (l) = x(n) y(n-l)
or equivalently as
r xy (l) = x(n+l) y(n)
the index l is the time shift (or lag) parameter and the subscript xy on the cross correlation sequence r xy (l) indicates the sequence being correlated. The order of the subscript, with x preceding y, indicates the direction in which one sequence is shifted relative to other.
If we reverse the role of x(n) and y(n) in above equation we obtain the correlation sequence
r yx (l) = y(n) x(n-l)
or equivalently as
r yx (l) = y(n+l) x(n)
by comparing the equations
r xy (l) = r yx (-l)
therefore r yx (l) is simply folded version of r xy (l) where the folding is done with respect to l=0.In the special case where y(n)=x(n) , we have AUTOCORRELATION of x(n) which is defined as the sequence
r xx (l) = x(n) x(n-l)
or equivalently as
r xx (l) = x(n+l) x(n)
CONCLUSION: The convolution of two discrete time signals has been found resultant signal is shown wave form.
CODE:
x=[2 -1 3 7 1 2 -3 0 0 0 0 0 0 0];y=[0 0 0 0 0 0 0 0 0 1 -1 2 -2 4 1 -2 5];h=fliplr(y);k1=-4;k2=-3;k3=2;k4=4;ler=k1+k2;rey=k3+k4;for n=ler:rey n1=n+8; r(n1)=0; for k=k1:(n1-k2) a=k+5; b=(n-k)+4; if b<=0 break; end r(n1)=r(n1)+((x(a)).*(h(b))); end end n=ler:rey; stem(n,r); xlabel('time'); ylabel('amplitude'); title('correlated output->y');
(a) Find H(z) & sketch its pole - zero plot(b) Plot magnitude & angle of H(exp(jw)) (c) Comment on the stability of the system
THEORY
The given difference equation is,
y(n)=0.9*y(n-1)+x(n)
Now, taking z-transform we get,
Y(z)-0.9*z-1Y(z)=X(z)Or, Y(z)[1-0.9*z-1]=X(z)
This gives,
H(z)=1/(1-0.9z-1)
The pole of the system is lying at, z=0.9, which is inside the unit circle. Also, the condition for stability is that poles must lie inside the circle for the system to be stable. Here, the pole z=0.9 is lying inside of the unit circle. So, the system is stable.
CODE:
zplane(0,0.09);
k=1;for w=0:0.01:(2*pi); h(k)=1/(1-(0.9*exp(-j*w))); k=k+1;end;w=0:0.01:(2*pi);figure;subplot(211);plot(w,abs(h));title('magnitude of h(e(jw))');subplot(212);plot(w,angle(h));title('angle of h(e(jw))');
OUTPUT
EXPERIMENT NO.6
AIM:
Let x(n) be 4 point sequence
X(n)=1 ; 0=< n =<3
=0 ; elsewhere
Compute the 4-point DFT, 8-point DFT, and 16-point DFT and plot their magnitude and phase plot.
THEORY:
The discrete Fourier transform computes the values of the z transform for evenly spaced points around the unit circle for a given sequence.
If the sequence to be represented is of finite duration, i.e., has only a finite number of non-zero values, the transform is discrete Fourier transform.
The magnitude and phase of 4-point, 8-point, 16-point DFT has been computed and is shown in figure.
EXPERIMENT NO.7(a)
AIM
Differentiate between high density high resolutions spectrums. For given sequence:
X(n)=cos(0.48*pi*n) + cos(0.52*pi*n)
Determine DFT & also plot it for 0<=n<=10
CODE:
n=0:10x=cos(0.48*pi*n)+cos(0.52*pi*n);p=1;N=11;for w=0:pi/100:2*pi; sum=0; for n=0:N-1; sum=sum+(x(n+1)*exp(-j*w*n)); end y(p)=sum; p=p+1;endfigure;magx=abs(y);phasex=angle(y);subplot(311);stem(x);xlabel('x--> ');ylabel('y-->');subplot(312);plot(magx);xlabel('x--->');ylabel('y--->');subplot(313);plot(phasex);xlabel('x--->');ylabel('y--->');
OUTPUT
EXPERIMENT NO.7(b)
AIM
Differentiate between high density high resolutions spectrums. For given sequence:
X(n)=cos(0.48*pi*n) + cos(0.52*pi*n)
Determine DFT & also plot it for 0<=n<=100
CODE:
n=0:100x=cos(0.48*pi*n)+cos(0.52*pi*n);p=1;N=101;for w=0:pi/100:2*pi; sum=0; for n=0:N-1; sum=sum+(x(n+1)*exp(-j*w*n)); end y(p)=sum; p=p+1;endfigure;magx=abs(y);phasex=angle(y);subplot(311);stem(x);xlabel('x--> ');ylabel('y-->');subplot(312);plot(magx);xlabel('x--->');ylabel('y--->');subplot(313);plot(phasex);xlabel('x--->');ylabel('y--->');
OUTPUT
EXPERIMENT No. – 8
AIM
Program to find circular convolution of two sequences x(n) & h(n)
CODE:
x = [1 2 2 0]
h = [1 2 3 4]length_x = length(x);length_h = length(h);if length_x ~= length_h; rest = abs(length_x - length_h); a = zeros(1,rest); if length_x > length_h; h = [h,a] length_h = length(h); else x = [x,a] length_x = length(x); end;end;for n = 1 : length_x; sum = 0; for m = 1 : length_x; if (n-m+1) < 1; t = n-m+length_x+1; else t = n-m+1; end; sum = sum + (x(m) * h(t)); end; result(n) = sum;end;resultfigure(1);subplot(3,1,1);stem(x);xlabel('n ----->');ylabel('x(n) ----->');subplot(3,1,2);stem(h);xlabel('n ----->');ylabel('h(n) ----->');subplot(3,1,3);stem(result);xlabel('n ----->');ylabel('y(n) ----->');
OUTPUT
EXPERIMENT No.-9
AIM
To design a FIR low pass filter using Rectangular, Barlett, Hamming, Hanning & Blackmann window & to plot the frequency response of filter. The specification of the filter are as follows:
Wp=0.2*pi, Ws=0.3*pi, As=50db, Ap=0.25 db
THEORY
The Image Processing Toolbox supports one class of linear filter, the two-dimensionalfinite impulse response (FIR) filter. FIR filters have several characteristics that makethem ideal for image processing in the MATLAB environment:
FIR filters are easy to represent as matrices of coefficients.Two-dimensional FIR filters are natural extensions of one-dimensional FIRfilters.There are several well-known, reliable methods for FIR filter design.FIR filters are easy to implement.FIR filters can be designed to have linear phase, which helps prevent distortion.
Another class of filter, the infinite impulse response (IIR) filter, is not as suitable forimage processing applications. It lacks the inherent stability and ease of design and implementation of the FIR filter. Therefore, this toolbox does not provide IIR filter support.
Windowing Method
The windowing method involves multiplying the ideal impulse response with a windowfunction to generate a corresponding filter. Like the frequency sampling method, thewindowing method produces a filter whose frequency response approximates a desiredfrequency response. The windowing method, however, tends to produce better resultsthan the frequency sampling method.
The toolbox provides two functions for window-based filter design, fwind1 andfwind2. fwind1 designs a two-dimensional filter by using a two-dimensional windowthat it creates from one or two one-dimensional windows that you specify. fwind2designs a two-dimensional filter by using a specified two-dimensional window directly.
fwind1 supports two different methods for making the two-dimensional windows ituses:
Transforming a single one-dimensional window to create a two-dimensionalwindow that is nearly circularly symmetric, by using a process similar to rotation
Creating a rectangular, separable window from two one-dimensional windows,by computing their outer product
