# Signal Processing

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1. Find and sketch the spectrum of the following signal.

x(t) = cos(t) + cos (2t)*cos(3t)

(Express the trigonometric functions in terms of complex exponentials using one of Euler's formulae, then perform the arithmetic (multiplication) and then express x(t) in terms of separate complex exponential terms. Finally sketch the result.)

2. We sample a 5 Hz sinewave at 9 samples per second. Sketch the spectrum of the unsampled sinewave, and sketch the spectrum of the sampled sinewave. Assume an ideal impulse sampler.

3. Find the z-transform transfer function

H (z) = Y(z) / X(z)

for a system whose input-output relationship is as follows:

y[n] = 3y[n-1] -2y [n-2] + x[n] + x[n-1]

(Take the z-transform of the discrete-time equation.)

4. The input to a digital filter is

x [n] = for 1 where (0<n<2) and 0 for elsewhere

The impulse response is

h [n] = (1/3)^n for (0<n<2) and 0 for elsewhere

Find the output of the digital filter. First, use convolution x[n]*h[n], and then use z-transforms.

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#### Solution Summary

This in-depth answer addresses the questions on signal processing by using Euler's formula, using a diagram for the spectrum, z-transforming, discrete-time signal processing, and time-domain approach. All steps are shown with brief explanations.