0.2 The six elements  (Page 3/10)

Use specsquare.m to investigate the relationship between the time behavior of the square wave and its spectrum.The M atlab command zoom on is often helpful for viewing details of the plots.

1. Try square waves with different frequencies: f=20, 40, 100, 300 Hz. How do the time plots change? How do the spectra change?
2. Try square waves of different lengths, time=1, 10, 100 seconds. How does the spectrum change in each case?
3. Try different sampling times, Ts=1/100, 1/10000 . seconds. How does the spectrum change in each case?

In your Signals and Systems course, you probably calculated (analytically) the spectrum of a square wave by using the Fourier series.How does this calculation compare with the discrete data version found by specsquare.m ? Mimic the code in specsquare.m to find the spectrum of

1. an exponential pulse $s\left(t\right)=e^{-t}$ for $0<t<10$ ,
2. a scaled exponential pulse $s\left(t\right)=5e^{-t}$ for $0<t<10$ ,
3. a Gaussian pulse $s\left(t\right)=e^{-t^2}$ for $-2<t<2$ ,
4. a Gaussian pulse $s\left(t\right)=e^{-t^2}$ for $-20<t<20$ ,
5. the sinusoids $s\left(t\right)=sin(2\pi ft+\Phi )$ for $f=20$ , 100, 1000, with $\Phi =0$ , $\pi /4$ , $\pi /2$ , and $0<t<10$ .

M atlab has several commands that create random numbers:

1. Use rand to create a signal that is uniformly distributed on $[-1,1]$ . Find the spectrum of the signal by mimicking the code in specnoise.m .
2. Use rand and the sign function to create a signal that is $+1$ with probability $1/2$ and $-1$ with probability $1/2$ . Find the spectrum of the signal.
3. Use randn to create a signal that is normally distributed with mean 0 and variance 3. Find the spectrum of the signal.

Modify the code in plotspec.m to also plot the phase spectrum.

1. Plot the phase spectrum of a sine wave and a cosine wave, both of the same frequency. How do they differ?
2. Plot the phase spectrum of the random signal created in [link] (a).