Interpolation Process Implementation:

Theory:

“Upsampling” is defined as the process of inserting zero-valued samples between original samples to increase the sampling rate. (This is called “zero-stuffing”.) . “Interpolation”, is determined as the process of upsampling followed by filtering. The filtering removes the undesired spectral images. The main reason for interpolating is simply used to maximize the sampling rate at the output of a particular system so that another system operating at a better rate can input the signal.

Steps to perform interpolation process:

Step I: Let’s define up sampling factor and input frequencies f1 and f2

Step II: Now represents the input sequence with frequencies f1 and f2

Step III: In order to perform the interpolation on input signal by using the matlab command interp.

Step IV: Next to plot the input and output signal/sequence.

Program:

clc;

clear all;

close all;

L=input(‘enter the upsampling factor’);

N=input(‘enter the length of the input signal’); % its Length should be always greater than 8

f1=input(‘enter the frequency of first sinusoidal’);

f2=input(‘enter the frequency of second sinusoidal’);

n=0:N-1;

x=sin(2*pi*f1*n)+sin(2*pi*f2*n);

y=interp(x,L);

subplot(2,1,1)

stem(n,x(1:N))

title(‘input sequence’);

xlabel(‘time(n)’);

ylabel(‘amplitude’);

subplot(2,1,2)

m=0:N*L-1;

stem(m,y(1:N*L))

title(‘output sequence ‘);

xlabel(‘time(n)’);

ylabel(‘amplitude’);

Output:

enter the upsampling factor4

enter the length of the input signal10

enter the frequency of first sinusodal0.1

enter the frequency of second sinusodal0.3

I/D sampling rate convertor implementation:

Steps to perform the I/D sampling rate conversion:

Step I: Let’s define the upsampling factor, downsampling, and input frequencies f1 and

 f2

Step II: Let’s represent the input sequence with frequencies f1 and f2

Step III:  In order to perform I/D sampling rate conversion on the given input signal by using Matlab resample.

Step IV: To plot the input and output signal/sequence.

Program:

clc;

clear all;

close all;

L=input(‘enter the upsampling factor’);

D=input(‘enter the downsampling factor’);

N=input(‘enter the length of the input signal’);

f1=input(‘enter the frequency of first sinusoidal’);

f2=input(‘enter the frequency of second sinusoidal’);

n=0:N-1;

x=sin(2*pi*f1*n)+sin(2*pi*f2*n);

y=resample(x,L,D);

subplot(2,1,1)

stem(n,x(1:N))

title(‘input sequence’);

xlabel(‘time(n)’);

ylabel(‘amplitude’);

subplot(2,1,2)

m=0:N*L/D-1;

stem(m,y(1:N*L/D));

title(‘output sequence ‘);

xlabel(‘time(n)’);

ylabel(‘amplitude’);

output:

enter the upsampling factor4

enter the downsampling factor3

enter the length of the input signal35

enter the frequency of first sinusodal0.01

enter the frequency of second sinusodal0.03

Decimation process implementation:

Theory:

“Decimation” is that the process of reducing the rate.

“Down sampling” may be a more specific term that refers to only the method of throwing away samples, without the low pass filtering operation. The most important reason to decimate is simply to reduce the sampling rate at the output of 1 system so a system operating at a lower rate can input the signal. But the away more common motivation for decimation is to scale back the value of processing: the calculation and the memory required to implement a DSP system generally is proportional to the rate, therefore the use of a lower rate usually leads to a cheaper implementation.

Steps involved in performing decimation process:

Step I: let’s define down sampling factor and input frequencies f1 and f2

Step II: And Represent input sequence with frequencies f1 and f2

Step III: To perform the decimation on input signal by using MatLab command decimate.

Step IV: Next plot the input and output sequence.

Program:

clc;

clear all;

close all;

D=input(‘enter the down sampling factor’);

L=input(‘enter the length of the input signal’);

f1=input(‘enter the frequency of first sinusoidal’);

f2=input(‘enter the frequency of second sinusoidal’);

n=0:L-1;

x=sin(2*pi*f1*n)+sin(2*pi*f2*n);

y=decimate(x,D,’fir’);

subplot(2,1,1);

stem(n,x(1:L));

title(‘input sequence’);

xlabel(‘time(n)’);

ylabel(‘amplitude’);

subplot(2,1,2)

m=0:(L/D)-1;

stem(m,y(1:L/D));

title(‘Decimated sequence’);

xlabel(‘time(n)’);

ylabel(‘amplitude’);

Output:

enter the down sampling factor6

enter the length of the input signal120

enter the frequency of first sinusodal0.002

enter the frequency of second sinusodal0.01

SINUSOIDAL SIGNAL GENERATION THROUGH FILTERING PROCESS:

One of the most important applications of an LTI discrete-time system is to pass certain frequency components in an input sequence with no distortion and block other frequency components. Such systems are called digital filters. The key to the filtering process is the inverse Discrete Fourier transform, which expresses an arbitrary input sequence as a the linear weighted sum of an infinite number of exponential sequences, or equivalently, as a the linear weighted sum of sinusoidal sequences. As a result, by appropriately choosing the values of magnitude function of the LTI digital filter at frequencies like the frequencies of the sinusoidal components of the input, a number of these sinusoidal sequences can be selectively heavily attenuated or filtered with reference to the others.

Program:

close all;

clear all;

clc;

b=[1];

a=[1,-1,0.9];

 n=[-20:120];

 t=0:0.1:2*pi;

 x=sin(t);

 s=filter(b,a,x);

 stem(t,s);

 title(‘sinusoidal response’);

 xlabel(‘n’);

 ylabel(‘s(n)’);

Steps involved in performing FIR filter operation

Step I: Let’s enter the pass band frequency (fp) and stop band frequency (fq).

Step II: And then get the sampling frequency (fs), length of window (n).

Step III: Next calculate cut off frequency.

Step IV: Use the boxcar, hamming, Blackman Commands for designing window.

Step V: And then design filter by using above parameters.

Step VI: To find frequency response of the filter by using matlab command freqz.

Step VII: Then plot the magnitude response and phase response of the filter.

Program:

clc;

clear all;

close all;

n=20;

fp=300;

fq=200;

fs=1000;

fn=2*fp/fs;

window=blackman(n+1);

b=fir1(n,fn,’high’,window);

[H W]=freqz(b,1,128);

subplot(2,1,1);

plot(W/pi,abs(H));

title(‘mag res of lpf’);

ylabel(‘gain in db——–>’);

xlabel(‘normalized frequency——>’);

subplot(2,1,2);

plot(W/pi,angle(H));

title(‘phase res of lpf’);

ylabel(‘angle——–>’);

xlabel(‘normalized frequency——>’);

Steps involved in performing IIR filter operation

Step I: Now enter the pass band ripple (rp) and stop band ripple (rs).

Step II: while the pass band frequency (wp) and stop band frequency (ws).

Step III: Then Get the sampling frequency (fs).

Step IV: Then calculate the normalized pass band frequency, and normalized stop band frequency w1 and w2 respectively.

Step V : Make using the following function for calculating the Butterworth filter order [n,wn]=buttord(w1,w2,rp,rs ) and Chebyshev filter order [n,wn]=cheb1ord(w1,w2,rp,rs)

 Step VI: Let’s design the nth order digital high pass Chebyshev or Butterworth filter using the following commands. Butterworth filter [b,a]=butter (n, wn,’high’) Chebyshev filter [b,a]=cheby1 (n, 0.5, wn,’high’)

Step VII: In order to find the digital frequency response of the filter by using ‘freqz ( )’ function

Step VIII: Then calculate the magnitude for the frequency response in decibels (dB) mag=20*log10 (abs (H))

Step IX: Then plot the magnitude response [magnitude in dB Vs normalized frequency]

Step X: Next calculate the phase response using angle (H).

Step XI: Then plot the phase response [phase in radians Vs normalized frequency (Hz)].

Program:

clc;

clear all;

close all;

disp(‘enter the IIR filter design specifications’);

rp=input(‘enter the passband ripple’);

rs=input(‘enter the stopband ripple’);

wp=input(‘enter the passband freq’);

ws=input(‘enter the stopband freq’);

fs=input(‘enter the sampling freq’);

w1=2*wp/fs;w2=2*ws/fs;

[n,wn]=buttord(w1,w2,rp,rs,’s’);

disp(‘Frequency response of IIR HPF is:’);

[b,a]=butter(n,wn,’high’,’s’);

w=0:.01:pi;

[h,om]=freqs(b,a,w);

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

an=angle(h);

figure, subplot(2,1,1);plot(om/pi,m);

title(‘magnitude response of IIR filter is:’);

xlabel(‘(a) Normalized freq. –>’);

ylabel(‘Gain in dB–>’);

subplot(2,1,2);plot(om/pi,an);

title(‘phase response of IIR filter is:’);

xlabel(‘(b) Normalized freq. –>’);

ylabel(‘Phase in radians–>’);