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LABORATORY 1: SIGNAL BANDWIDTH AND DISTORTION EEE 186: COMMUNICATIONS SYSTEMS LABORATORY Department of Electrical & Electronic Engineering College of Engineering & 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 which has been received. 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