# HW10_Soln - Escuti/Brickley ECE200 Fall 2008 Homework 10...

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Escuti/Brickley ECE200 – Fall 2008 – Homework 10 Transmission and Reception of Radio Frequency Signals Page 1 of 10 Problem 1: [12 pts] Consider the following signal: a. Find the carrier frequency in Hz. The carrier is the high frequency part of the plot. We need to determine the period of this signal and then invert it to find the frequency. Within the first 1 ms, there are 2.5 periods. Therefore, the period is T c = 1 ms 2.5 = 0.4 ms and the frequency is f c = 1 T c = 1 0.4 ms = 2.5 kHz b. Find the modulating signal frequency in Hz. The modulating signal can be observed by examining the envelope since it is an AM signal. We can measure the period of the modulating signal as T m = 4 ms and the frequency f m = 1 T m = 1 4 ms = 250 Hz . c. Find the modulation index m. m = V max V min V max + V min × 100% We can measure the maximum voltage of the upper envelope as 5 V and the minimum voltage of the upper envelope as 3 V. Therefore the index modulation is

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Escuti/Brickley ECE200 – Fall 2008 – Homework 10 Transmission and Reception of Radio Frequency Signals Page 2 of 10 m = 5 3 5 + 3 × 100% = 25% d. Find the DC offset V dc used in the amplitude modulation processes. (Note: This is indicated in Fig. 9.10 of the textbook.) The book indicates that you can find the DC offset by averaging the maximum and minimum of the upper envelope. If you do this, then you get V dc = (5V + 3V)/2 = 4 V . While this answer will get full credit for this problem, you should know that in fact the average of the max and min is really = A c * V dc , where A c is the voltage gain of the carrier signal. Note that in order to find the real DC offset from this plot, you would need to know this voltage gain. Problem 2: [10 pts] Consider the AM signal, , where 0 < m < 1. a) Derive an equation for the percentage of the total signal power needed to transmit the carrier sinusoid. Your equation will be a function of A and m . b) Assume A = 1 V. Plot the equation derived in part (a) as a function of m . c) What is the optimum modulation index based on your result in part (b)? Briefly explain why. d) Briefly explain what happens when m > 1? Is this a desirable situation? Explain why or why not. SOLUTIONS: (a) v AM t ( ) = A cos 2 π f c t ( ) 1 + m cos 2 π f m t ( ) [ ] = A cos 2 π f c t ( ) + mA cos 2 π f m t ( ) cos 2 π f c t ( ) = A cos 2 π f c t ( ) + mA 2 cos 2 π f c + f m ( ) t [ ] + mA 2 cos 2 π f c f m ( ) t [ ] Power used to transmit the carrier: P carrier = A 2 2 P Total = A 2 2 + mA 2 2 2 + mA 2 2 2 = A 2 1 + m 2 4 + m 2 4 2 = A 2 2 1 + m 2 2
Escuti/Brickley ECE200 – Fall 2008 – Homework 10 Transmission and Reception of Radio Frequency Signals Page 3 of 10 Percentage power used for the carrier: % Power = P

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