Description
1 Preliminaries
In this week, we review the ”filtering” subject which was previously covered in Signals & Systems lectures.
In communication systems, filters are widely used to demodulate the received signals in the receivers. Filters
can be examined in four categories which are low pass, high pass, band pass and band stop filters. These
filters pass or reject the signals according to the cut-off frequencies in frequency domain. An ideal filter has
a sharp frequency response which does not pass the frequency components of the signals beyond the desired
cut-off frequency. For example, ideal low pass filters only pass the frequency components between 0 Hz and
the determined cut-off frequency or ideal band pass filters only pass the signals between the desired two
frequencies. Nonideal filters do not have a sharp frequency response and have a transition band around the
cut-off frequency. This transition band can cause to have undesired frequency components. The filtering
operation can be accomplished by multiplication in frequency domain.
For this week’s experiment, it is useful to remember the functions that are employed in Week-1’s experiment.
Furthermore, it is also useful to learn about the Matlab functions find(.), butter(.), freqz(.) and filter(.) by
using Matlab Help before doing the labwork given below.
In your reports, the details of the plots should be visible. Do not take screenshots to copy the figures plotted
in Matlab. Instead, after plotting a Figure in Matlab, apply the following steps to copy the figure: Figure
Window −− > Edit −− > Copy Figure.
2 Labwork
Read the preliminaries given above carefully before doing the experiment given below. You must show
the negative part of the frequency spectrum in your figures.
2.1 Construction of the Signals
a. Construct x2(t) = cos(120πt)+cos(500πt) where the sampling frequency fs = 1000 Hz and its duration
is 2 s. Obtain the Fourier transform of x2(t) where the number of DFT points (N) is the length of the
signal.
b. The magnitude of the frequency spectrum of x(t), i.e., |X(f)|, is given in Fig. 1. Construct |X(f)|
which is a triangular signal between −50 Hz and 50 Hz. Hint: You can use find(.) function in Matlab
to find and change the nonzero components of the signal.
-500 -400 -300 -200 -100 0 100 200 300 400 500
f (Hz)
0
0.2
0.4
0.6
0.8
1
|X(f)|
Figure 1: The plot of |X(f)|.
EE 352 – Lab 1: Signals & Systems Review
c. Generate the signal |Y1(f)| by using the relation |Y1(f)| = |X(f)| + |X2(f)|.
d. Plot |X(f)|, |X2(f)| and |Y1(f)| in the same figure (by using subplot(3,1,.) command) where x-axis
shows the frequency in Hz.
2.2 Filtering
a. As shown in Fig. 2, filter |Y1(f)| with an ideal low pass filter, which has the frequency response H1(f),
to obtain the nonzero frequency components of |X(f)|.
H1 Y (f) 1
(f) Y2
(f)
Figure 2: Block diagram of the filtering operation for H1(f).
b. Plot |H1(f)| and the filtered signal |Y2(f)| in the same figure (by using subplot(2,1,.) command) where
x-axis shows the frequency in Hz.
c. As shown in Fig. 3, filter |Y1(f)| with an ideal band pass filter, which has the frequency response
H2(f), to obtain the frequency components of |X2(f)| stemming from cos(500πt). Hint: Do not forget
to filter the negative frequency components!
H2 Y (f) 1
(f) Y3
(f)
Figure 3: Block diagram of the filtering operation for H2(f).
d. Plot |H2(f)| and the filtered signal |Y3(f)| in the same figure (by using subplot(2,1,.) command) where
x-axis shows the frequency in Hz.
e. Using butter(.) function, design a nonideal band pass filter, which has the frequency response Hbpf (f)
to obtain the frequency components of |X2(f)| stemming from cos(500πt). Filter x2(t) with Hbpf (f)
as shown in Fig. 4.
Hbpf x (f) 2
(t) Ybpf(f)
Figure 4: Block diagram of the filtering operation for Hbpf (f).
f. Plot |Hbpf (f)| and the filtered signal |Ybpf (f)| in the same figure (by using subplot(2,1,.) command)
where x-axis shows the frequency in Hz. You can use freqz(.) function which does not show the negative
part of the frequency spectrum to plot only the filter response.
