Signals and Systems - Electrical Engineering

296 C H A P T E R 4: Frequency Analysis: The Fourier Series

(a) Given that an integral ofp(t)is the area under the curve, find and plot the function

s(t)=

∫t

−∞

p(t)dt t≤ 1

Indicate the values ofs(t)fort= 0 ,0.25,0.5,0.75, and 1.

(b) Find the Fourier series ofp(t)ands(t)and relate their Fourier series coefficients.

(c) Suppose you want to compute the integral

T∫ 0 / 2

−T 0 / 2

p(t)dt

using the Fourier series ofp(t). What is the integral equal to?

(d) You can also compute the integral from the plot ofp(t):

T∫ 0 / 2

−T 0 / 2

p(t)dt

What is it? Does it coincide with the result obtained using the Fourier series? Explain.

4.23. Full-wave rectifying and DC sources

Letx(t)=sin^2 ( 2 πt), a periodic signal of periodT 0 = 1 , andy(t)=|sin( 2 πt)|, which is also periodic of

periodT 1 =0.5.

(a) A trigonometric identity gives that

x(t)=

1

2

[ 1 −cos( 4 πt)]

Use this result to find its complex exponential Fourier series.

(b) Use the Laplace transform to find the Fourier series ofy(t).

(c) Arex(t)andy(t)identical? Explain.

(d) Indicate how you would use an ideal low-pass filter to get a DC source of unit value fromx(t)and

y(t). Indicate the bandwidth and the magnitude of the filters. Compare these two signals in terms of

advantages or disadvantages in generating the desired DC source.

4.24. Windowing and music sounds—MATLAB

In the computer generation of musical sounds, pure tones need to be windowed to make them more

interesting. Windowing mimics the way a musician would approach the generation of a certain sound.

Increasing the richness of the harmonic frequencies is the result of the windowing, as we will see in this

problem. Consider the generation of a musical note with frequencies aroundfA= 880 Hz. Assume our

“musician” while playing this note uses three strokes corresponding to a windoww 1 (t)=r(t)−r(t−

T 1 )−r(t−T 2 )+r(t−T 0 ), so that the resulting sound would be the multiplication, or windowing, of a

pure sinusoidcos( 2 πfAt)by a periodic signalw(t), withw 1 (t)a period that repeats everyT 0 = 5 Twhere

Tis the period of the sinusoid. LetT 1 =T 0 / 4 andT 2 = 3 T 0 / 4.

(a) Analytically determine the Fourier series of the windoww(t)and plot its line spectrum using MATLAB.

Indicate how you would choose the number of harmonics needed to obtain a good approximation to

w(t).

(b) Use the modulation or the convolution properties of the Fourier series to obtain the coefficients of the

products(t)=cos( 2 πfAt)w(t). Use MATLAB to plot the line spectrum of this periodic signal and again

determine how many harmonic frequencies you would need to obtain a good approximation tos(t).