Chapter 7: Fourier Series and Transforms

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

$\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{\{}\begin{array}{llllll}\mathbf{-}\mathbf{1}\mathbf{,}\mathbf{}\mathbf{}& \mathbf{-}\mathbf{\pi }& \mathbf{<}& \mathbf{x}& \mathbf{<}& \frac{\mathbf{\pi }}{\mathbf{2}}\\ \mathbf{1}\mathbf{,}& \frac{\mathbf{\pi }}{\mathbf{2}}& \mathbf{<}& \mathbf{x}& \mathbf{<}& \mathit{\pi }\mathbf{,}\end{array}$

Consider one arch of$\mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{sin}\mathit{x}$. Show that the average value of role="math" localid="1664260742465" $\mathit{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}$ over the middle third of the arch is twice the average value over the end thirds.

In Problemsto, the sketches show several practical examples of electrical signals (voltages or currents). In each case we want to know the harmonic content of the signal, that is, what frequencies it contains and in what proportions. To find this, expand each function in an appropriate Fourier series. Assume in each case that the part of the graph shown is repeated sixty times per second.

. Rectified half-wave; the curve is a sine function for half the cycle and zero for the other half. Let the maximum current beampere. Hint: Be careful! The value of l here is $\mathit{t}\mathbf{=}\frac{\mathbf{1}}{\mathbf{120}}$but I(t) =sint only from t=0 to $\mathit{t}\mathbf{=}\frac{\mathbf{1}}{\mathbf{120}}$

Use Parseval’s theorem and the results of the indicated problems to find the sum of the series in Probllems 5 to 9. The series $\mathbf{1}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{3}}^{\mathbf{2}}}\mathbf{+}\frac{\mathbf{1}}{{\mathbf{5}}^{\mathbf{2}}}\mathbf{+}\mathbf{\cdots }\mathbf{,}$ using problem 9.6.

Using the definition of a periodic function, show that a sum of terms corresponding to a fundamental musical tone and its overtones has the period of the fundamental.

In Problems 3 to 12, find the average value of the function on the given interval. Use equation (4.8) if it applies. If an average value is zero, you may be able to decide this from a quick sketch which shows you that the areas above and below the x axis are the same.

${\mathbf{\text{cos}}}^{\mathbf{2}}\frac{\mathbf{x}}{\mathbf{2}}\mathbf{on}\left(0,\frac{\pi }{2}\right)$

Find the amplitude, period, frequency, and velocity amplitude for the motion of a particle whose distance from the origin is the given function.

$\mathit{s}\mathbf{=}\mathbf{2}\mathit{s}\mathit{i}\mathit{n}\mathbf{3}\mathit{t}\mathit{c}\mathit{o}\mathit{s}\mathbf{3}\mathit{t}$

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

role="math" localid="1659236419546" $\mathbf{f}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{=}\mathbf{\{}\begin{array}{cc}\mathbf{0}\mathbf{,}& \mathbf{-}\mathbf{\pi }\mathbf{<}\mathbf{x}\mathbf{<}\mathbf{0}\\ \mathbf{-}\mathbf{1}\mathbf{,}& \mathbf{0}\mathbf{<}\mathbf{x}\mathbf{<}\frac{\mathbf{\pi }}{\mathbf{2}}\\ \mathbf{1}\mathbf{,}& \frac{\mathbf{\pi }}{\mathbf{2}}\mathbf{<}\mathbf{x}\mathbf{<}\mathbf{\pi }\end{array}$

Find the exponential Fourier transform of the given f(x) and write f(x) as a Fourier integral [that is, find $\mathbf{\text{g}}\mathbf{\left(}\mathbf{\text{α}}\mathbf{\right)}$in equation (12.2) and substitute your result into the first integral in equation (12.2)].

role="math" localid="1664338587279" $\mathit{f}\mathbf{\left(}\mathit{x}\mathbf{\right)}\mathbf{=}\mathbf{\{}\begin{array}{l}\begin{array}{ll}\mathbf{1}\mathbf{,}& \mathbf{0}\mathbf{<}\mathbf{x}\mathbf{<}\mathbf{1}\end{array}\\ \begin{array}{ll}\mathbf{0}\mathbf{,}& \mathbf{otherwise}\end{array}\end{array}$

For each of the periodic functions in Problems 5.1to 5.11.use Dirichlet's theorem to find the value to which the Fourier series converges at$\mathit{x}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{}\mathbf{\pm }\mathit{\pi }\mathbf{/}\mathbf{2}\mathbf{,}\mathbf{}\mathbf{\pm }\mathit{\pi }\mathbf{,}\mathbf{}\mathbf{\pm }\mathbf{2}\mathit{\pi }$.