Chapter 7: Fourier Series and Transforms
Q2MP
The symbol means the greatest integer less than or equal to x(for example,Expand in an exponential Fourier series of period 1.
Q2P
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).
Q2P
(a) Prove that by making the change of variable in one of the integrals.
(b) Use the same method to prove that the averages of and are the same over a period.
Q2P
Prove that if ,then the average value ofis.Show by problem7.12 that for real f(x)this becomes (11.5).
Q2P
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) (b)
Q2P
Sketch several periods of the corresponding periodic function of period . Expand the periodic function in a sine-cosine Fourier series.
Q2P
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
Do Example 1 above by using a cosine transform (12.15)Obtain (12.17); for , the 0to integral represents the function
Represent this function also by a Fourier sine integral (see the paragraph just before Parseval's theorem).
Q30P
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.
Q31P
Verify Parseval’s theorem (12.24) for the special cases in Problems 31 to 33.
31. as in figure 12.1. Hint: Integrate by parts and use (12.18) to evaluate.