Download Experiment # 9

Experiment # 9
Sampling Theorem
Objective:
In this lab, we will explore the sampling theorem and its conditions and the
aliasing problem and the solution for it.
Basic Theoretical Principles
A typical time-dependent signal, for example AC voltage, is continuous with
respect to magnitude and time. Such signals are called analog signals. Using a normal
(analog) oscilloscope we get an analog representation of such a signal.
Today mainly digital equipment is used for electrical measurements. The
original analog signal is converted to a digital signal. A digital signal is discrete with
respect to the magnitude as well as to the time. Therefore, conversion of an analog to
a digital signal means the value of the analog signal function F(t) is measured at
discrete times t0 = ti + i. Δt during a time interval tm = t – t0 and the indicated values
changes with steps ΔF . Mostly the measured value is represented by a binary number
and the step ΔF depends on its number of positions, that means of the number of bits
processed by the analog-to-digital converter (ADC).
The correct use of this measurement technology requires one to know how
often the value of a signal must be measured during a given time interval in order to
reconstruct its shape tm correctly. The answer is given by the sampling theorem of
time functions (Nyquist Criterion):
If the given time function F(t) has a limited bandwidth B and the time interval of
measurement is infinite, a correct interpolation between the sampling points is tm
possible when the function is sampled with a time step.
with a sampling rate
Assuming a simple harmonic function this theorem says that it is necessary to sample
the function more than two times per period. An important example is the digital
processing of audio signals Assuming a bandwidth of B= 20kHz for such signals a
sampling rate f> 40kHz must be used (usually: f= 44.1kHz.)
Experimental Procedures
1. Type the following code in m-file and run it.
clc
close all
fs=10000;
f0=1000;
n=0:1:50;
x=cos(2*pi*f0*n/fs);
figure (1)
stem(n,x)
title('part A')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure (2)
for f0=300:400:1500;
x=cos(2*pi*f0*n/fs);
subplot(2,2,(f0+100)/400)
stem(n,x)
end
title('Part B')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure (3)
for f0=8500:400:9700;
x=cos(2*pi*f0*n/fs);
subplot(2,2,(f0-8100)/400)
stem(n,x)
end
title('Part C')
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
figure (4)
for f0=10300:400:11500;
x=cos(2*pi*f0*n/fs);
subplot(2,2,(f0-9900)/400)
stem(n,x)
end
title('Part D')
% close all