University of Central Oklahoma: ENGR 3324L
Lab 8: Sampling and the Discrete Fourier Transform (DFT)
Lab objective:
1234-
Learn the sampling theorem and the reconstruction of the signal from the samples.
Learn the DFT of a sampled signal and some filtering application of the DFT
Learn about aliasing and how to minimize error due to aliasing.
The relationship between the length of the sampling duration in the time domain and the spectral
resolution (frequency sampling interval).
Prelab:
A continuous signal x(t) defined as
x(t) = cos(2π2t - π/4) + 2cos(2π4t) + 3 0 < t < 1 second
Answer the following:
a) What is the Nyquist sampling rate?
b) What is the fundamental frequency ωo?
c) According to the sampling theorem if the above signal was sampled at a rate of 6 samples/sec, then the
constructed signal from the samples will be xs(t) = cos(2π2t- π/4) + 2cos(2πfat) + 3
What is the apparent frequency fa?
d) What is the fundamental frequency of the signal x1(t) shown below?
x1(t) = cos(2π2.5t - π/4) + 2cos(2π4t) + 3
e) If the signal x1(t) was sampled at a rate of 10 samples/sec for 1 second, what is the spectrum resolution
(frequency sampling interval) of the signal spectrum? Will the discrete spectrum X(rωo) of the sampled signal
xn show the 2.5Hz? If not then explain why and propose a solution for this problem?
Introduction
In most practical applications signals exist in the continuous form. However, with digital technology
advancement most signal processing systems are digital. So, in order for the digital systems to process continuous
signals we need to convert the continuous signals into discrete signals. In order to recover the continuous signal
from the samples, the number of samples per seconds must be equal or greater than twice the bandwidth of the
continuous signal (BW includes the highest frequency present in the continuous signal). If the sampling rate is
below the Nyquist rate (2B) then the recovered continuous signal from the samples will not be the same as the
original signal because aliasing will make high frequency components to appear as low frequency components.
Many signal processors operate on the frequency domain of the signal such as compression. Discrete
Fourier Transform (DFT), which is the same as the Fast Fourier Transform (FFT), is a technique that allows us to
determine the frequency components of a continuous signal from its samples. For the DFT to represent the
fundamental frequency and all harmonic frequencies of a continuous signal then the spectrum resolution
(frequency sampling interval) must equal the fundamental frequency. For example if the spectrum resolution is 1
Hz then we will be missing frequency 1.5 Hz in the continuous signal if it was present. To be able to capture the
1.5Hz then the fundamental frequency should be 0.5 Hz. Fundamental frequency could be considered as the
spectrum resolution of the signal spectrum.
In this lab you will investigate the effect of sampling rate and the spectrum resolution on the ability to
reconstruct the continuous signal from its samples. In addition you will build a digital filter to eliminate specific
frequency component from a continuous signal.
Lab Procedures:
Task 1: Sampling and Desampling (Interpolation using the sinc function)
In this task you will sample the signal x(t) at a rate higher than the Nyquist rate and then reconstruct the
continuous signal x(t) from its samples xn. The continuous signal x(t) is shown below.
x(t) = cos(2πf1t - π/4) + 2cos(2πf2t) + 3 f1 = 2 , f2 = 4, 0 < t< 1 second
1- Computer is a digital machine so all signals processed in the computer are discrete signals. However a discrete
signal of a large number of samples (sampling rate>50*Bandwidth) can be considered as a true representation of
the continuous signal x(t). You can verify this by plotting the sampled signal against t.
>> To=1;
>> t=0:0.01: To;
%100 samples per second
>>x=cos(2*pi*2*t-pi/4)+2*cos(2*pi*4*t)+3;
>>plot(t, x);
2- What is the Bandwidth, B, of the signal x(t)? Sample the signal using Nyquist rate to obtain xn.
>> To=1;
>> fs= ? ; % replace the question mark by the Nyquist rate
>> ts=0:1/fs:To;
>> xn=cos(2*pi*2*ts-pi/4)+2*cos(2*pi*4*ts)+3;
Plot the sampled signal, xn, against the sampling time using Matlab stem function.
>> stem(ts, xn);
Use Matlab plot function to plot the sampled signal, xn, against the sampling time. Does the plot look like the
original signal? Explain your answer.
>>plot(ts, xn);
3- According to the sampling theorem we should be able to reconstruct the original signal x(t) from the sampled
signal, xn, if the sampling rate equal or higher than the Nyquist rate.
N 1


x s (t )   x(nT ) sinc  (t  nT )  = x(t)
T

n 0
Where T is the sampling interval and N is the number of samples. The Matlab M-file
sincInterpolation(xn, fs)shown at the end of the lab will construct the continuous signal, x(t) from the
sampled signal, xn. fs is the sampling rate.
>> [xs, sigMatr] = sincInterpolation(xn, fs);
Are the plots of x(t) of part 1 and xs(t) of part 3 the same? How many samples in the estimated signal xs(t)? From
the code can you explain what the variable sigMatr is? If not then try to plot the fifth row
>>plot(sigMatr(5,:));. Are you amazed of the accuracy of estimating all the points of xs(t) from the
small samples in xn? If you are not amazed then you are missing big concepts!!!
Task 2: Discrete Fourier Transform
What is the fundamental frequency of the signal x(t) shown below?
x(t) = cos(2π2t- π/4) + 2cos(2π4t) + 3
1- Sample the signal x(t) at a rate of fs = 10 samples/sec which is above the Nyquist rate to obtain xn.
>> To=1;
>> fs=10;
>> ts=0:1/fs:( To -1/fs);
>> xn=cos(2*pi*2*ts-pi/4)+2*cos(2*pi*4*ts)+3;
2- Use the fft and the fftshift Matlab functions to find the DFT coefficients of the discrete signal xn.
>>Fx=fftshift((fft(xn)));
Plot the magnitude and the phase of the discrete spectrum X(rωo).
>>stem([-fs/2: 1/To: fs/2-1/To],abs(Fx));
>>figure
>>stem([-fs/2: 1/To: fs/2-1/To],angle(Fx).*(abs(Fx)>0.5));
What is the spectrum resolution ωo (frequency sampling interval) of the discrete spectrum X(rωo)? Can you verify
the frequency components and their corresponding phases that are present in the signal x(t) from X(rωo)? Does the
spectrum magnitude and phase agree with theory? Carry on the calculation to find the DC spectrum X(0) from the
samples xn.
3- Recover the samples xn from its discrete spectrum X(rωo) using the Matlab built-in function ifft.
>>xr=ifft(ifftshift(Fx));
Do the recovered samples xr agree with the original samples xn? Carry on the calculation to find the first sample x0
from the spectrum X(rωo).
Task 3: Spectrum resolution and the time duration of the sampled signal.
Change the frequency of the first cosine term to 2.5 Hz instead of 2 Hz.
x(t) = cos(2π2.5t- π/4) + 2cos(2π4t) + 3
What is the fundamental frequency fo of the above signal?
1- Repeat steps 1 to 3 of task 2. Make sure to change the frequency from 2Hz to 2.5Hz in your code.
2- Why does the spectrum X(rωo) of the sampled signal xn show frequency components that are not present in
the continuous signal x(t), and does not show the 2.5Hz frequency which is present in the x(t)?
3- If We sample the signal x(t) for two seconds, what is the spectrum resolution (frequency sampling interval) of
the spectrum of the sampled signal, xn?
4- Repeat steps 1 to 3 of task 3. Make sure to change the frequency from 2Hz to 2.5Hz and To from 1 second to 2
seconds in your code. Does the spectrum of the 2 second sampled signal, xn, show the 2.5Hz component?
Explain your answer.
function [xs, sigMatr]=sincInterpolation1(x, fs)
%x = samples of the continuous signal
%fs= the sampling rate (frequency)
T = 1/fs;
%sampling interval
[xn, No]=size(x);
%No is the number of samples
To = T*(No-1);
%To Time width of the signal
Nc = 50*No;
%Nc is the number of samples of the constructed signals
t=0:1/Nc:To;
%samples of the time axis at which the signal will be estimated
sigMatr=zeros(No, Nc+1);
for n=0:No-1
Argu = (pi/T)*(t-n*T);
for j=1:Nc
if (Argu(j)~=0)
sigMatr(n+1, j:j) = x(n+1)*sin(Argu(j))./Argu(j);
else
sigMatr(n+1, j:j) = x(n+1);
end
end
end
xs = sum(sigMatr);
plot(t, xs)
axis([0 To min(xs)-0.1 max(xs)+0.1]);
Work Sheet
Task 1: Sampling and Desampling (Interpolation using the sinc function)
1- plot the sampled signal against t.
2- What is the Bandwidth, B, of the signal x(t)?
Plot the sampled signal, xn, against the sampling time using Matlab stem function.
Plot the sampled signal, xn, against the sampling time using Matlab plot function.
Does the plot look like the original signal? Explain your answer.
3- Show the plot of xs(t) produced by the sincInterpolation(xn, fs)
Are the plots of x(t) of part 1 and xs(t) of part 3 the same?
How many samples in the estimated signal xs(t)?
From the code can you explain what the variable sigMatr is?
Task 2: Discrete Fourier Transform
What is the fundamental frequency of the signal x(t) shown below?
2- Plot the magnitude and the phase of the discrete spectrum X(rωo).
What is the spectrum resolution ωo
Can you verify the frequency components and their corresponding phases that are present in the signal x(t) from
X(rωo)?
Does the spectrum magnitude and phase agree with theory?
Carry on the calculation to find the DC spectrum X(0) from the samples xn.
3- Do the recovered samples xr agree with the original samples xn?
Carry on the calculation to find the first sample x0 from the spectrum X(rωo).
Task 3: Spectrum resolution and the time duration of the sampled signal.
What is the fundamental frequency fo of the above signal?
1a- Plot the magnitude and the phase of the discrete spectrum X(rωo).
What is the spectrum resolution ωo
Can you verify the frequency components and their corresponding phases that are present in the signal
x(t) from X(rωo)?
Does the spectrum magnitude and phase agree with theory?
b- Do the recovered samples xr agree with the original samples xn?
2- Why does the spectrum X(rωo) of the sampled signal xn show frequency components that are not present in the
continuous signal x(t), and does not show the 2.5Hz frequency which is present in the x(t)?
3- If We sample the signal x(t) for two seconds, what is the spectrum resolution (frequency sampling interval) of
the spectrum of the sampled signal, xn?
4a- Plot the magnitude and the phase of the discrete spectrum X(rωo).
What is the spectrum resolution ωo
Can you verify the frequency components and their corresponding phases that are present in the signal
x(t) from X(rωo)?
Does the spectrum magnitude and phase agree with theory?
b- Do the recovered samples xr agree with the original samples xn?
Does the spectrum of the 2 second sampled signal, xn, show the 2.5Hz component? Explain your answer.