91Ó°ÊÓ

The symbol $\mathbf{\left[}\mathbf{x}\mathbf{\right]}$means the greatest integer less than or equal to x(for example,$\mathbf{\left[}\mathbf{3}\mathbf{\right]}\mathbf{=}\mathbf{3}\mathbf{,}\mathbf{[}\mathbf{2}\mathbf{.1}\mathbf{]}\mathbf{=}\mathbf{2}\mathbf{,}\mathbf{[}\mathbf{−}\mathbf{4}\mathbf{.5}\mathbf{]}\mathbf{=}\mathbf{−}\mathbf{5}$Expand $\mathit{x}\mathbf{−}\mathbf{\left[}\mathbf{x}\mathbf{\right]}\mathbf{−}\frac{\mathbf{1}}{\mathbf{2}}$in an exponential Fourier series of period 1.

Sketch several periods of the corresponding periodic function of period $\mathbf{2}\mathit{Ï€}$. Expand the periodic function in a sine-cosine Fourier series.

$f\left(x\right)=\left\{\begin{array}{llllll}0,& -\mathrm{Ï€}& <& x& <& 0\\ 1,& 0& <& x& <& \frac{\mathrm{Ï€}}{2},\\ 0,& \frac{\mathrm{Ï€}}{2}& <& x& <& \mathrm{Ï€}.\end{array}\right.$

Prove that if $\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}{\mathbf{∑}}_{\mathbf{-}\mathbf{∞}}^{\mathbf{∞}}{\mathbf{c}}_{\mathbf{n}}{\mathbf{e}}^{\mathbf{i}\mathbf{n}\mathbf{x}}$,then the average value of${\left|f\left(x\right)\right|}^{\mathbf{2}}$is${\mathbf{∑}}_{\mathbf{-}\mathbf{∞}}^{\mathbf{∞}}{\mathit{c}}_{\mathbf{n}}{\mathit{c}}_{\mathbf{-}\mathbf{n}}$.Show by problem7.12 that for real f(x)this becomes (11.5).

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) $\mathit{I}\mathit{n}\left|1-x\right|$ (b) $\left(1+x\right)\left(sinx+cosx\right)$

Do Example 1 above by using a cosine transform (12.15)Obtain (12.17); for $\mathit{x}\mathbf{>}\mathbf{0}$, the 0to $\mathit{∞}$integral represents the function

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\{\begin{array}{l}\begin{array}{ll}1,& 0<x<1\end{array}\\ \begin{array}{ll}0,& x>1\end{array}\end{array}$

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

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.

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).

$\mathbf{2}\mathit{c}\mathit{o}\mathit{s}\mathit{t}\mathbf{+}\mathit{c}\mathit{o}\mathit{s}\mathbf{2}\mathit{t}$

(a) Prove that ${\mathbf{∫}}_{\mathbf{0}}^{\mathbf{π}\mathbf{/}\mathbf{2}}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{x}\mathit{d}\mathit{x}\mathbf{=}{\mathbf{∫}}_{\mathbf{0}}^{\mathbf{π}\mathbf{/}\mathbf{2}}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{x}\mathit{d}\mathit{x}$by making the change of variable$\mathit{x}\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{}\mathit{π}\mathbf{-}\mathit{t}$ in one of the integrals.

(b) Use the same method to prove that the averages of$\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathbf{(}\mathit{n}\mathit{Ï€}\mathit{x}\mathbf{/}\mathit{l}\mathbf{)}$ and$\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathbf{(}\mathit{n}\mathit{Ï€}\mathit{x}\mathbf{/}\mathit{l}\mathbf{)}$ are the same over a period.

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.$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{\{}\begin{array}{ll}\mathbf{1}\mathbf{-}\frac{\mathbf{x}}{\mathbf{2}}\mathbf{,}& \mathbf{0}\mathbf{<}\mathbf{x}\mathbf{<}\mathbf{2}\\ \mathbf{0}\mathbf{,}& \mathbf{x}\mathbf{>}\mathbf{2}\end{array}$

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

31. $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}$as in figure 12.1. Hint: Integrate by parts and use (12.18) to evaluate$\underset{\mathbf{-}\mathbf{∞}}{\overset{\mathbf{∞}}{\mathbf{∫}}}{\mathbf{\left|}\mathbf{g}\mathbf{\left(}\mathbf{Î\pm }\mathbf{\right)}\mathbf{\right|}}^{\mathbf{2}}\mathit{d}\mathit{Î\pm }$.