Practice Problems

Problem 1

Given the following system, choose the best filter and cut-off frequency.

\[ v_{\text{in}}(t) = 8 + 3\cos(360^\circ \cdot 4k\,t) + 4\cos(360^\circ \cdot 6k\,t) + 5\cos(360^\circ \cdot 8k\,t)\ \text{V} \]

\[ v_{\text{out}}(t) = 8 + 3\cos(360^\circ \cdot 4k\,t)\ \text{V} \]

Problem 2

T/F These two systems perform the same function.

Problem 3

Given that

\[ v(t) = 3 + 2\cos(360^\circ \cdot 1k\,t) + 4\cos(360^\circ \cdot 2k\,t) + 4\cos(360^\circ \cdot 6k\,t)\ \text{mV} \]

and the Band Reject Filter below, what will \(v_{\text{out}}(t)\) be?

Problem 4

Plot the output of the following filters given that

\[ v(t) = 6 + 10\cos(360^\circ \cdot 1k\,t) + 8\cos(360^\circ \cdot 2k\,t) + 7\cos(360^\circ \cdot 4k\,t) + 5\cos(360^\circ \cdot 8k\,t) + \cos(360^\circ \cdot 10k\,t)\ \text{V} \]

a. LPF → \(f_{\text{c/o}} = 5\ \text{kHz}\)

b. HPF → \(f_{\text{c/o}} = 6\ \text{kHz}\)

c. BPF → \(f_{\text{c/o,1}} = 3\ \text{kHz}\) and \(f_{\text{c/o,2}} = 9\ \text{kHz}\)

d. BRF → \(f_{\text{c/o,1}} = 3\ \text{kHz}\) and \(f_{\text{c/o,2}} = 6\ \text{kHz}\)

Problem 5

Graph the following signal in the frequency domain:

\[ v(t) = 6 + 4\cos(360^\circ \cdot 1k\,t) + 8\cos(360^\circ \cdot 2k\,t) + 5\cos(360^\circ \cdot 4k\,t) + 2\cos(360^\circ \cdot 7k\,t)\ \text{mV} \]

Problem 6

The signal given in Practice Problem 5 above is input to each of the following filters. What is the output of each filter? Provide the frequency domain plot and the equation.

Problem 7

What is the bandwidth for each of the output signals in Practice Problem 6?

Problem 8

You just recorded your voice using a voice recorder in a room full of equipment. The equipment emits a loud \(60\ \text{Hz}\) hum. You decide you want to eliminate the hum from your recording. Your voice ranges from \(300\) to \(3400\ \text{Hz}\). Design a filter to eliminate the hum.