# FALL 2015 ```LABORATORY 1:
SIGNAL BANDWIDTH AND DISTORTION
EEE 186: COMMUNICATIONS SYSTEMS LABORATORY
Department of Electrical &amp; Electronic Engineering
College of Engineering &amp; Computer Science
California State University, Sacramento
FALL 2015
Signal Bandwidth and Distortion
The bandwidth of a communication system determines the rate at which bits of
information can be sent using it. There are, however, several definitions of bandwidth:
Two different criteria might be 90 or 99% bandwidth, the latter being the bandwidth that
would pass 99% of the power or of the energy of the signal. Sometimes, the inadequacies
of a system are stated in terms of rise time and other distortion of waveforms with square
corners. Sometimes, if one knows what distortions a signal has undergone, compensating
circuits can be added in the receiver to repair some of the damage. A number of these
phenomena are explored in this experiment.
A key piece of apparatus in this experiment is the Dynamic Signal Analyzer. To
build up our confidence in the equipment, we will feed some reproducible waveforms
into the signal analyzer and compare the results to those predicted by Fourier theory.
Let’s first look at the simplest possible way to characterize a waveform, using the theory
of the Fourier series.
1. Theory of Fourier series
Consider the waveform below, which consists of pulses of width , that occur with a
fundamental period of T.
Let us choose t = 0 to be right down the center of the pulse, as shown below. The
coefficients of the cosine terms are given by:
2
2 nt
f (t ) cos(
)dt

T T /2
T
T /2
an 
4

T
T /2

0
A cos(
2 nt
2A
n
)dt 
sin(
)
T
n
T
For a pure square wave, t = T/2, and the coefficients are an 
2A
n
v(t)
 0 
t, sec
T
2. Measurement of waveform harmonics

Connect the HP 3324A Signal Generator to the HP35665 Dynamic Signal Analyzer.
Set the output of the signal generator to be a square wave, with a frequency of 1 KHz,
peak-to-peak amplitude of 5 V and dc offset of zero.

Measure the amplitudes of all Fourier components out to 10 KHz. Compare your
measured results to the theoretical results obtained from Section 1.

Determine the percent of total ac power that is represented by the fundamental,
fundamental plus third harmonic, fundamental plus third and fifth harmonics,
fundamental plus third, fifth and seventh harmonics, and fundamental plus third, fifth,
seventh and ninth harmonics. The percent total harmonic distortion is :
%THD  100

V22  V32  V42  ..
V1
Set your signal generator for a 1 KHz sine wave with peak-to-peak amplitude of 2 V,
and again measure all the spectral components out to 10 KHz. Calculate the THD for
a sinusoidal signal out of this generator.
3. Pulse Distortion and its Cure
There are a number of ways in which a signal may be distorted. In FM broadcasting and
pulse code modulation, a signal may be distorted deliberately according to some plan
before it is transmitted. At the receiver, since one knows the form of the predistortion,
one may compensate for it rather precisely, and in so doing, greatly reduce the noise

Low pass filter
We will look at a very simple type of distortion resulting from a transmission
bandwidth that is not adequate to produce square corners on a signal. However, since we
know the characteristics of the circuit that did the distorting, we can build an equalizer
that can compensate for the characteristics of the distorter and greatly square up the pulse.
The circuit diagram shown below shows the simplest realization of both a low pass filter
and an equalizer designed to compensate for the filter. The left-most Rp and C form the
low pass filter. Choose a value of C of about 0.01 F and Rp to give a 3 dB cutoff
frequency of 2 KHz, where fc (cutoff frequency) = 1/(2 Rp C).

Equalizer
The opamp shown in the diagram is seen to be connected simply as a voltage follower to
isolate the LPF from the other circuitry, which is the equalizer. It can be shown that the
transfer function of the equalizer has a pole and a zero. The zero is given 1/( Rp C)
where C is suggested in the circuit diagram as 0.01 F. The pole has a similar form,
except that the resistance in the expression would be the equivalent parallel resistance of
Rp and Rs .
Hence the design strategy for the equalizer is to make the zero correspond to a
break frequency of 2 KHz to compensate for the pole of the LPF used above. It is
recommended that you make Rs = 0.1 Rp , so that the resulting new pole is 11 times as
large as the pole of the LPF. Thus, if the LPF and equalizer are cascaded, the result on
the pulse waveform ought to correspond to the higher frequency pole.

Measurement
Using the signal generator, obtain a square wave with a fundamental frequency of 1 KHz.
Feed the signal into the circuit shown above, and sketch the waveform across the first
capacitor. Determine the 10 to 90 % rise time from the time display of the waveform on
the signal analyzer. Also, use the signal analyzer to determine the amplitudes of all the
Fourier components up to 10 KHz.
Similarly, determine the 10-90 % rise time at the output across Rs, and again
measure the Fourier components up to 10 KHz
```