2.5. Exercises

  1. 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 \(x \ast p_1\).

  2. 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:

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

  3. 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.

  4. Eigenfunctions

    What is the response of an LTI system characterized with impulse response function \(h\) given an input \(x(t)=\sin(\omega t)\)?

  5. Difference Equation

    In this section it is stated that a system described with the difference equation in Eq. (2.1) is an LTI system. Give a proof of this.