4.10 Frequency domain problems

Simple fourier series

Find the complex Fourier series representations of the following signals without explicitly calculating Fourier integrals. What is the signal's period in each case?

1. $s(t)=\sin t$
2. $s(t)=\sin t^2$
3. $s(t)=\cos t+2\cos (2t)$
4. $s(t)=\cos (2t)\cos t$
5. $s(t)=\cos (10\pi t+\frac{\pi }{6})(1+\cos (2\pi t))$
6. $s(t)$ given by the depicted waveform .

   

Fourier series

Find the Fourier series representation for the following periodic signals . For the third signal, find the complex Fourier series for the triangle wave without performing the usual Fourier integrals. Hint: How is this signal related toone for which you already have the series?

Phase distortion

We can learn about phase distortion by returning to circuits and investigate the following circuit .

1. Find this filter's transfer function.
2. Find the magnitude and phase of this transfer function. How would you characterize thiscircuit?
3. Let $v_{\mathrm{in}}(t)$ be a square-wave of period $T$ . What is the Fourier series for the output voltage?
4. Use Matlab to find the output's waveform for the cases $T=0.01$ and $T=2$ . What value of $T$ delineates the two kinds of results you found? The software in fourier2.m might be useful.
5. Instead of the depicted circuit, the square wave is passed through a system that delays its input, whichapplies a linear phase shift to the signal's spectrum. Let the delay $\tau$ be $\frac{T}{4}$ . Use the transfer function of a delay to compute usingMatlab the Fourier series of the output. Show that the square wave is indeed delayed.

Approximating periodic signals

Often, we want to approximate a reference signal by a somewhat simpler signal. To assess the quality of anapproximation, the most frequently used error measure is the mean-squared error. For a periodic signal $s(t)$ , $$\epsilon ^2=\frac{1}{T}\int_0^T (s(t)-\tilde{s}(t))^2\,d t$$ where $s(t)$ is the reference signal and $\tilde{s}(t)$ its approximation. One convenient way of findingapproximations for periodic signals is to truncate their Fourier series. $$\tilde{s}(t)=\sum_{k=-K}^K c_{k}e^{i\frac{2\pi k}{T}t}$$ The point of this problem is to analyze whether thisapproach is the best ( i.e. , always minimizes the mean-squared error).

1. Find a frequency-domain expression for the approximation error when we use the truncatedFourier series as the approximation.
2. Instead of truncating the series, let's generalize the nature of the approximation to including any setof $2K+1$ terms: We'll always include the $c_0$ and the negative indexed term corresponding to $c_k$ . What selection of terms minimizes the mean-squarederror? Find an expression for the mean-squared error resulting from your choice.
3. Find the Fourier series for the depicted signal . Use Matlab to find the truncated approximation and bestapproximation involving two terms. Plot the mean-squared error as a function of $K$ for both approximations.

Long, hot days

The daily temperature is a consequence of several effects, one of them being the sun's heating. If thiswere the dominant effect, then daily temperatures would be proportional to the number of daylight hours. The plot shows that the average daily high temperature does not behave that way.