hw1_sol_501_Au11 - ECE-501 Introduction to Analog and...

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ECE-501 Introduction to Analog and Digital Communications Autumn 2011 Homework #1 Sep. 30, 2011 HOMEWORK SOLUTIONS #1 1. First, some elementary trigonometric identities useful for understanding quadrature modulation: (a) sin( α ) sin( β ) = 1 2 { cos( α - β ) - cos( α + β ) } (b) cos( α ) cos( β ) = 1 2 { cos( α - β ) + cos( α + β ) } (c) sin( α ) cos( β ) = 1 2 { sin( α - β ) + sin( α + β ) } 2. This exercise reviews the basic structure of digital modulation via visualization using Matlab. In the example, the modulation is binary phase shift keying (BPSK), meaning the symbols are from the set {- 1 , 1 } . (a) From the frequency plot, we can observe a narrow-band signal centered at f c = 1000 Hz. The magnitude spectrum is symmetric about the origin (as expected since the transmit waveform is real-valued) and the positive portion of the magnitude spectrum is symmetric about f c (because m ( t ) is real-valued). In the time-domain, we can observe the 1000 Hz oscillation within the slowly varying envelope given by the base-band message signal. (b) I hear a very brief “chirp” noise that sounds like a single tone at a high frequency (over an octave above concert A = 440 Hz). It appears that the message duration is too short to audibly discern any of the signal envelope. (c) Each symbol has N = 51 samples, and the samples are reconstructed (digital-to-analog con- version) at 8000 samples per second. Therefore, symbol interval = T symb = 51 samples · 1 8000 sec samples 1000 msec 1 sec = 6 . 3750 msec ECE501, Au2011 1
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%set parameters fs = 8000; %8000 samples per second for DAC and ADC fc = 1000;%carrier frequency at 1000 Hz
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