2.4. Exercises¶

Continous Time Convolution

On http://pages.jh.edu/~signals/convolve/index.html you can find a Java applet that might help you understand the convolution of continous time signals.
Continous Time Convolution

Consider the function:

\[\begin{split}p_1(t) = \begin{cases} 1 &:-1 \leq t \leq 1\\ 0 &: \mbox{elsewhere} \end{cases}\end{split}\]
1. Calculate the convolution \(p_1 \ast p_1\). (Hint: this can be done ‘graphically’).
2. Calculate \(p_1 \ast p_1 \ast p_1\).
3. Let \(x(t)=\sin(t)\). Calculate \(x \ast p_1\).
Discrete Convolution

Show that any discrete time linear time invariant system is completely characterized by its impulse response \(h[n]\) and that the system response \(y[n]\) on input \(x[n]\) is given by:

\[\begin{split}y[n] &= x[n] \ast h[n] \\ &= \sum_{k=-\infty}^{\infty} x[k]\, h[n - k]\end{split}\]
Discrete Convolution

Consider a part from a discrete signal \(x[n]\):

\[\matrix{\cdots& 0& 0& 0& 1& 2& 1& 3& 2& 3& 1& 2& \underline{3}& 8& 7& 8& 9& 9& 7& 8& 8& 8& \cdots}\]

The underlined value indicates the origin (i.e. \(x[0]\)). Calculate the convolution \(x[n]\ast h[n]\) with:
1. \(h[n] = \matrix{1& \underline{1}& 1}\)
2. \(h[n] = \matrix{\underline{1}& 1& 1}\)
3. \(h[n] = \matrix{\underline{0}& 0& 1}\)
4. \(h[n] = \matrix{1& \underline{0}& -1}\)
In all above functions \(h\) the origin is indicated with the underlining and we have used the convention that all values in the (in principle) infinite function (from \(-\infty\) to \(+\infty\)) that are not given are equal to zero.
Eulers Formula
1. Prove that:
  
  \[\sin(\phi) = \frac{e^{j\phi}-e^{-j\phi}}{2j}\]
2. Prove that:
  
  \[\cos(\phi) = \frac{e^{j\phi}+e^{-j\phi}}{2}\]
Eigenfunctions
1. What is the response of an LTI system given an input \(x(t)=\sin(\omega t)\)?
2. What is the response of an LTI system given an input \(x(t)=\cos(\omega t)\)?