Chapter 7: Fourier Series and Transforms

Q2MP

Page 387

The symbol [x]means the greatest integer less than or equal to x(for example,[3]=3,[2.1]=2,[4.5]=5Expand x[x]12in an exponential Fourier series of period 1.

Q2P

Page 347

For each of the following combinations of a fundamental musical tone and some of its overtones, make a computer plot of individual harmonics (all on the same axes) and then a plot of the sum. Note that the sum has the period of the fundamental (Problem 5).

2cost+cos2t

Q2P

Page 349

(a) Prove that 0π/2sin2xdx=0π/2cos2xdxby making the change of variablex=12π-t in one of the integrals.

(b) Use the same method to prove that the averages ofsin2(nπx/l) andcos2(nπx/l) are the same over a period.

Q2P

Page 377

Prove that if f(x)=-cneinx,then the average value of|f(x)|2is-cnc-n.Show by problem7.12 that for real f(x)this becomes (11.5).

Q2P

Page 370

The functions in Problems 1 to 3 are neither even nor odd. Write each of them as the sum of an even function and an odd function.

(a) In|1-x| (b) (1+x)(sinx+cosx)

Q2P

Page 354

Sketch several periods of the corresponding periodic function of period 2π. Expand the periodic function in a sine-cosine Fourier series.

f(x)=0,-π<x<01,0<x<π2,0,π2<x<π.

Q2P

Page 373

In problem 1to 3, the graphs sketched represent one period of the excess pressure p(t)in a sound wave. Find the important harmonics and their relative intensities. Use a computer to play individual terms or a sum of several terms of the series.

Q2P

Page 384

Do Example 1 above by using a cosine transform (12.15)Obtain (12.17); for x>0, the 0to integral represents the function

f(x)={1,0<x<10,x>1

Represent this function also by a Fourier sine integral (see the paragraph just before Parseval's theorem).

Q30P

Page 386

Represent each of the following functions (a) by a Fourier cosine integral; (b) by a Fourier sine integral. Hint: See the discussion just before Parseval’s theorem.

30.f(x)={1-x2,0<x<20,x>2

Q31P

Page 386

Verify Parseval’s theorem (12.24) for the special cases in Problems 31 to 33.

31. f(x)as in figure 12.1. Hint: Integrate by parts and use (12.18) to evaluate-|g(α)|2dα.

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