This Jupyter notebook is part of a collection of notebooks in the bachelors module Signals and Systems, Comunications Engineering, Universität Rostock. Please direct questions and suggestions to Sascha.Spors@uni-rostock.de.
The convolution $y(t) = x(t) * h(t)$ of a periodic signal $x(t)$ with an aperiodic signal $h(t)$ results in a periodic signal. This implies for instance that the output signal $y(t)$ of a linear time-invariant (LTI) system with aperiodic impulse response $h(t)$ is periodic for a periodic input signal $x(t)$. This is shown in the following.
The spectrum $X(j \omega)$ of a periodic signal $x(t)$ is given as
\begin{equation} X(j \omega) = X_0(j \omega) \cdot {\bot \!\! \bot \!\! \bot} \left( \frac{\omega T_\text{p}}{2 \pi} \right) \end{equation}where $X_0(j \omega) = \mathcal{F} \{ x_0(t) \}$ denotes the Fourier transform of one period $x_0(t)$ of the periodic signal. The spectrum $Y(j \omega) = \mathcal{F} \{ x(t) * h(t) \} = X(j \omega) \cdot H(j \omega)$ is computed by introducing the spectrum $X(j \omega)$ of the periodic signal and $H(j \omega)$ of the aperiodic signal (e.g. impulse response)
\begin{align} Y(j \omega) &= X_0(j \omega) \cdot {\bot \!\! \bot \!\! \bot} \left( \frac{\omega T_\text{p}}{2 \pi} \right) \cdot H(j \omega) \\ &= \frac{2 \pi}{T_\text{p}} \sum_{\mu = - \infty}^{\infty} X_0 \left( j \, \mu \frac{2 \pi}{T_\text{p}} \right) \cdot H \left( j \, \mu \frac{2 \pi}{T_\text{p}} \right) \cdot \delta \left( \omega - \mu \frac{2 \pi}{T_\text{p}} \right) \end{align}where the definition of the Dirac comb and the multiplication property of the Dirac impulse was used to derive the last equality. The last equality shows that the spectrum $Y(j \omega)$ is a line spectrum. From this it can be concluded that the signal $y(t)$ has to be periodic. Hence, the convolution of a periodic with an aperiodic signal results in a periodic signal.
The Fourier series expansion of the convolution $y(t) = x(t) * h(t)$ can be deduced by considering the relation between the spectrum of a periodic signal and the Fourier series
\begin{equation} y(t) = \frac{1}{T_\text{p}} \sum_{n = - \infty}^{\infty} X_0 \left( j \, n \frac{2 \pi}{T_\text{p}} \right) \cdot H \left( j \, n \frac{2 \pi}{T_\text{p}} \right) e^{j n \frac{2 \pi}{T_\text{p}} t} \end{equation}Copyright
This notebook is provided as Open Educational Resource. Feel free to use the notebook for your own purposes. The text is licensed under Creative Commons Attribution 4.0, the code of the IPython examples under the MIT license. Please attribute the work as follows: Sascha Spors, Continuous- and Discrete-Time Signals and Systems - Theory and Computational Examples.