Communications II
Experiment IV
COMMUNICATIONS II
EXPERIMENT 4
Group member 1
Name:
Number:
Group member 2
Name:
Number:
1. OBJECTIVES


Fourier Transforms and Convolution Related to Delta Function
Improving SNR of a signal within broadband noise
2. BASIC INFORMATION
2.1 Delta Function
The impulse δ(t) is defined to be equal to zero for all t≠ 0 by the requirement that its integral
be unity.
The discrete time counterpart of δ (t) is the (discrete) delta function
While there are a few subtleties (i.e., differences) between δ(t) and δ[k], for the most part they
act analogously. For example, the program specdelta.m given below and included in your
exp4 folder calculates the spectrum of the (discrete) delta function.
% specdelta.m plot the spectrum of a delta function
time=2;
% length of time
Ts=1/100;
% time interval between samples
t=Ts:Ts:time;
% create time vector
x=zeros(size(t));
% create signal of all zeros
x(1)=1;
% delta function
plotspec(x,Ts)
% draw waveform and spectrum
The output of specdelta.m is shown in Figure 1. As expected from the discussions from the
class, the magnitude spectrum of the delta function is equal to 1 at all frequencies.
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Experiment IV
Figure 1: A (discrete) delta function at time 0 has a magnitude spectrum equal to 1 for all
frequencies.
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amplitude
0.8
0.6
0.4
0.2
0
0
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seconds
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0
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frequency
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magnitude
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2.2 Convolution
Matlab has several functions that simplify the numerical evaluation of convolutions. The most
obvious of these is conv, which is used in convolex.m given below and included in exp4
folder. convolex.m calculates the convolution of an input x (consisting of two delta
functions at times t = 1 and t = 3) and a system with impulse response h that is an exponential
pulse. The convolution gives the output of the system.
% convolex.m: example of numerical convolution
Ts=1/100; time=10;
% sampling interval and total time
t=0:Ts:time;
% create time vector
h=exp(-t);
% define impulse response
x=zeros(size(t));
% input is sum of two delta functions
x(1/Ts)=3; x(3/Ts)=2;
% at times t=1 and t=3
y=conv(h,x);
% do convolution
subplot(3,1,1), plot(t,x)
ylabel('input')
subplot(3,1,2), plot(t,h)
ylabel('impulse response')
subplot(3,1,3), plot(t,y(1:length(t)))
ylabel('output')
xlabel('time in seconds')
Figure 2 shows the input to the system in the top plot, the impulse response in the middle plot,
and the output of the system in the bottom plot. Nothing happens before time t = 1, and the
output is zero. When the first spike occurs, the system responds by jumping to 3 and then
decaying slowly at a rate dictated by the shape of h(t). The decay continues smoothly until
time t = 3, when the second spike enters. At this point, the output jumps up by 2, and is the
sum of the response to the second spike, plus the remainder of the response to the first spike.
Since there are no more inputs, the output slowly dies away.
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Communications II
Experiment IV
Figure 2: The convolution of the input (the top plot) with the impulse response of the system
(the middle plot) gives the output in the bottom plot.
.
input
3
2
1
impulse response
0
0
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0
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time in seconds
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output
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1
0
2.2 Improving SNR
We have described several kinds of corruption that a signal may encounter as it travels from
the transmitter to the receiver. In most of the cases linear filters can help. Perhaps the simplest
way a linear bandpass filter can be used is to remove broadband noise from a signal.
A common way to quantify noise is the signal-to-noise ratio (SNR) which is the ratio of the
power of the signal to the power of the noise at a given point in the system. If the SNR at one
point is larger than the SNR at another point, then the performance is better because there is
more signal in comparison to the amount of noise. For example, consider the SNR at the input
and the output of a BPF as shown in Figure 3. The signal at the input (r(t) in part (a)) is
composed of the message signal x(t) and the noise signal n(t), and the SNR at the input is
therefore
Similarly, the output y(t) is composed of a filtered version of the message (yx(t) in part (b))
and a filtered version of the noise (yn(t) in part (b)). The SNR at the output can therefore be
calculated as
Observe that the SNR at the output cannot be calculated directly from y(t) (since the two
components are scrambled together). But, since the filter is linear,
which effectively shows the equivalence of parts (a) and (b) of Figure 3.
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Experiment IV
Figure 3: Two equivalent ways to draw the same system. In part (a) it is easy to calculate the
SNR at the input, while the alternative form (b) allows easy calculation of the SNR at the
output of the BPF.
The Matlab program improvesnr.m given below and included in your exp4 folder explores
this scenario concretely. The signal x is a bandlimited signal, containing only frequencies
between 3000 and 4000 Hz. This is corrupted by a broadband noise n (perhaps caused by an
internally generated thermal noise) to form the received signal. The SNR of this input snrinp
is calculated as the ratio of the power of the signal x to the power of the noise n. The output of
the BPF at the receiver is y, which is calculated as a BPF version of x+n. The BPF is created
using the firpm command. To calculate the SNR of y, however, the code also implements the
system in the alternative form of part (b) of Figure 3. Thus, yx and yn represent the signal x
filtered through the BPF and the noise n passed through the same BPF. The SNR at the output
is then the ratio of the power in yx to the power in yn, which is calculated using the function
pow.m, available in your exp4 folder.
% improvesnr.m: using linear filters to improve SNR
close all
clear all
time=3; Ts=1/20000;
% length of time and sampling interval
freqs=[0 0.29 0.3 0.4 0.41 1];
amps=[0 0 1 1 0 0];
b=firpm(100,freqs,amps);
% BP filter
n=0.25*randn(1,time/Ts);
% generate white noise signal
x=filter(b,1,2*randn(1,time/Ts));
% bandlimited signal between 3K and 4K
y=filter(b,1,x+n);
% (a) filter the received signal+noise
yx=filter(b,1,x); yn=filter(b,1,n); % (b) filter signal and noise
separately
z=yx+yn;
% add them
diffzy=max(abs(z-y))
% and make sure y and z are equal
snrinp=pow(x)/pow(n)
% SNR at input
snrout=pow(yx)/pow(yn)
% SNR at output
% check spectra
figure,plotspec(x+n,Ts)
figure,plotspec(y,Ts)
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Experiment IV
Since the data generated in improvesnr.m is random, the numbers are slightly different each
time the program is run. Using the default values, the SNR at the input is about 7.8, while the
SNR at the output is about 61. This is certainly a noticeable improvement. The variable diffzy
shows the largest difference between the two ways of calculating the output (that is, between
parts (a) and (b) of Figure 3). This is on the order of 10-15, which is effectively the numerical
resolution of Matlab calculations, indicating that the two are (effectively) the same.
Figure 4 and 5 plot the spectra of the input signal plus noise and output of the BPF
respectively. Observe the large noise floor in Figure 4, and how this is reduced
by passage through the BPF. Observe also that the signal is still changed by the noise in the
pass band between 3000 and 4000 Hz, since the BPF has no effect there.
The program improvesnr.m can be thought of as a simulation of the effect of having a BPF at
the receiver for the purposes of improving the SNR when the signal is corrupted by broadband
noise, as was described
Figure 4: The spectrum of the input to the BPF
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x 10
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Experiment IV
Figure 5: The spectrum of the output of the BPF
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amplitude
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x 10
3. EXPERIMENT

Copy exp4 folder to your computer, start Matlab and change the directory of
Matlab to exp4.
3.1 Fourier Transforms Related to Delta Function





Type “specdelta” in Matlab command window and make sure you can observe
Figure 1.
Find the spectrum of the discrete delta function where the delta does not occur
at start of x where x[10]=1. (That means x=zeros(size(t)); x(10)=1). copy your
result into Fig 5.
Repeat for x[100]=1. Copy your result into Fig 6.
Repeat for x[110]=1. Copy your result into Fig 7.
Does the magnitude of the spectrum change when there is a shift in time
domain
Answer:




You will change the magnitude and observe the effect in the spectrum. Change
x[1]=10 (That means x=zeros(size(t)); x(1)=10). Find the spectrum. Copy your
result into Fig 8.
Repeat for x[10]=3. Copy your result into Fig 9.
Repeat for x[110]=0.1. Copy your result into Fig 10.
How does the spectrum change when magnitude of x changes.
Answer:

You will now observe the spectrum of a signal containing two delta functions.
Create a signal containing two delta functions at the start and in the end, x[1]=1
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Experiment IV

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
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and x[end] =1. (That means x=zeros(size(t)); x(1)=1; x(end)=1). Find the
spectrum. Copy your result into Fig 11.
Repeat for x[90]=1 and x[end-90]=1. Copy your result into Fig 12.
Repeat for x[33]=1 and x[120]=1. Copy your result into Fig 13.
Train of impulses in time domain corresponds to train of impulses in frequency
domain. x=zeros(size(t)); x(1:20:end)=1 produces train of impulses 20 samples
apart. Find the spectrum in this case and copy your result into Fig 14.
Repeat when impulses are 25 samples apart. copy your result into Fig 15.
How does the spectrum change when the train of impulses are separated more
in time
Answer:
3.2 Convolution





Type “convolex” in Matlab command window and make sure you can observe
Figure 2.
Find the output when input is a delta function at t=4 given by x=zeros(size(t));
x(4/Ts)=2 and the impulse response is a square wave with duration T=1sec
given by h=zeros(size(t)); h(1:1/Ts)=1; Copy your result into Fig 16.
Find the output when input is two delta functions at t=1 with magnitude 3 and
t=5 with magnitude 5 and the impulse response is a square wave with duration
T=2sec and magnitude is 4. Copy your result into Fig 17.
Find the output when input is three delta functions at t=2 with magnitude 1 and
t=5 with magnitude 2 and t=7 with magnitude 3 and the impulse response is a
square wave with duration T=1sec and magnitude is 1. Copy your result into
Fig 18.
. Find the output when input is square wave with duration T=1sec and
magnitude is 1 and the impulse response is a square wave with duration T=1sec
and magnitude is 1. Copy your result into Fig 19.
3.3 Improving SNR
 Type “improvesnr” in Matlab command window and make sure you can
observe Figure 3-4 and in command window diffzy is close to zero, snrinp is
around 8, and snrout is around 60. Also find snrout/snrinp and copy the results
below.
Answer: diffzy=
snrinp=
snrout=
snrout/ snrinp=
 Change improvesnr so that input is a signal between 2K and 4K. Copy your
result into Fig 20. Also copy the result of diffzy, snrinp, snrout, snrout/snrinp
below.
Answer: diffzy=
snrinp=
snrout=
snrout/ snrinp=
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Communications II
Experiment IV

Change improvesnr so that input is a signal between 2K and 6K. Copy your
result into Fig 21. Also copy the result of diffzy, snrinp, , and snrout below.
Answer: diffzy=
snrinp=
snrout=
snrout/ snrinp=
 How does snrout/snrinp change when bandwidth of the signal increases.
Explain why
Answer:
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Communications II
Experiment IV
4. RECORDS
Figure 5
Figure 6
Figure 7
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Experiment IV
Figure 8
Figure 9
Figure 10
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Experiment IV
Figure 11
Figure 12
Figure 13
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Experiment IV
Figure 14
Figure 15
Figure 16
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Experiment IV
Figure 17
Figure 18
Figure 19
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Experiment IV
Figure 20
Figure 21
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