# Difference between revisions of "Assignment 8, Part 0: convolution practice"

20.309: Biological Instrumentation and Measurement

You may find the Fourier transform Tables 8.0.1 and 8.0.2 useful. Note that there are a few functions that you may not have seen before including:

• u(t) is the unit step function $u(t) = \begin{cases} 0, & \text{if }t<0 \\ 1, & \text{if }t\geq0 \end{cases}$
• sinc(ax) is defined as: $\text{sinc}(ax) = \frac{\sin(ax)}{ax}$
• rect(ax) is the box function: $\text{rect}(ax) = \begin{cases} 0, & \text{if } |ax|> 1/2 \\ 1, & \text{if } |ax| \leq 1/2 \end{cases}$
Table 8.0.1: Short table of Fourier transform properties
Table 8.0.2: Short table of Fourier transform pairs

1. In class we found the Fourier transform of $\cos^2(\omega_0 t)$. Use graphical convolution to determine the transform of $\cos^4(\omega_0 t)$.

2. Using the transform pairs in table 8.0.2, sketch the fourier transform of $e^{-\alpha t} u(t) \times \cos(\omega_0 t)$. Assume that $\alpha\ll\omega_0$.

3. Table 8.0.3 shows plots of eight time-domain signals A-H. The table on the right includes magnitude plots of the Fourier transform of ten signals numbered 1-10. For each time domain signal A-H, write the number 1-10 in the empty column of the matching frequency-domain signal. You may use a numbered plot more than once.
Some of the frequency plots are shown on log-log axes and some are linear, as indicated by the plot title.
Table 8.0.3