1. Given ??(??) = ??(?? + ??) and

show that:

(a) If ??(??) = ?? (?? + ?? 2 ), then ???? = 0 for ?? odd.

(b) If ??(??) = -?? (?? + ?? 2 ), then ???? = 0 for ?? even.

2. Given a periodic signal with main period equal to

(a) Compute the Fourier coefficients ???? of ??(??) and then write down the amplitude and phase of ????.

(b) If ??(??)is approximated by the sum

compute the mean-square error ??₂ ² .

3. Fourier Series and Fourier Transforms

(a) Write down the Fourier Series of sin² (??), -∞ < ?? < ∞.

(b) Compute the Fourier Transforms of ???? ^-????(??) and sin ?? ??(??). If you cannot compute the Fourier Transform of any of the signals, please explain why.

4. [Problem 4.18] Using properties to find Fourier Series

Use the Fourier series of a square train of pulses (done in this chapter) to compute the Fourier series of the triangular signal x(t) with a period,

x1 (t) = r(t) - 2r(t - 1) + r(t - 2)

(a) Find the derivative of x(r) or y(r) = dx(t)/dt and carefully plot it. Plot also z(t) = y(t) + 1. Use the Fourier series of the square train of pulses to compute the Fourier series coefficients of y(t) and z(t).

(b) Obtain the trigonometric Fourier series of y(t) and z(t) and explain why they are represented by sines and why z(t) has a nonzero mean.

(c) Obtain the Fourier series coefficients of x(r) from those of y(t).

(d)Obtain the sinusoidal form of x(t) and explain why the cosine representation is more appropriate for this signal than a sine representation.

5. Find the Fourier transform F(??) of the following given function ??(??) and make a drawing of it. Clearly indicate the ranges, axes and amplitudes of all your drawings

. ??(??) = (????????(2??) ∗ ???????? ( ??/2 )) · (????????(??) ∗ ????????(2??))

Recall, ????????(??) = sin(????) /????.