The input to an LTI system is the periodic pulse wave x(t)depicted in Fig. 1. This input c...

The input to an LTI system is the periodic pulse wave x(t)depicted in Fig. 1. This input can be represented by the Fourier series

(a) Determine ω0 in the Fonrier series representation of x(t ). Also, write down the integral that must be evaluated to obtain the Fourier coefficients ak.

(b) Plot the spectrum of the signal x(t);i.e., make a plot showing the ak's plotted at the frequencies for kω0 for

-4ω0 ≤ ω ≤ 4ω0

(c) If the frequency response of the system is the ideal high pass filter

plot the output of the system, y(t), when the input is x(t)as plotted above.

Hint: First determine which frequency is removed by the filter, and then determine what effect this will have on the waveform.

(d) If the frequency response of the system is an ideal low pass filter

where ωco is the cutoff frequency, for what value s of ωco will the output of the system have the form

where A and B are nonzero ?

(e) If the frequency response of the LTI system is H (jω) = I – e-j2ω, plot the output of the system, y(t), when the input is x(t)as plotted above.

Hint: In this case it will be easiest to determine the impulse response h (t)corresponding to H (jω) and from h(t)you can easily find an equation that relates y(t)to x(t ). This will allow you to plot y(t)

Figure 1: