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# HW1 - ECEN 661 Modulation Theory Homework#1 Assignment Date...

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ECEN 661 - Modulation Theory Homework #1 Assignment Date: 1/16/08 Due Date: 1/23/08 1. An audio signal is FM modulated to produce , where is the carrier frequency and is the frequency deviation constant. (a) Find the I and Q components of . (b) Find the real envelope and excess phase of . (c) Find the complex envelope of . 2. An RLC bandpass filter has a transfer function given by , where is the resonant frequency and is the damping factor. Assuming that so that the filter is narowband: (a) Find the lowpass equivalent transfer function, . (b) Find the complex equivalent impulse response, . Hint: It works out easiest if you take the “center” of the passband to be which is close to the resonant frequency of the circuit. 3. The PSD of a narrowband Gaussian noise pro- cess, , is as shown in the figure at right. (a) Find and sketch the PSD of the I and Q compo- nents of the narrowband noise process. (b) Find and sketch the cross spectral density of the I and Q components.

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