# 2.7: Signals and Systems Problems

Learning Objectives

- Signals and Systems practice problems.

### Complex Number Arithmetic

Find the real part, imaginary part, the magnitude and angle of the complex numbers given by the following expressions.

- -1
- \[\frac{1+\sqrt{3}i}{2}\]
- \[1+i+e^{i\tfrac{\pi }{2}}\]
- \[e^{i\tfrac{\pi }{3}}+e^{i\pi }+e^-({i\tfrac{\pi }{3}})\]

### Discovering Roots

Complex numbers expose all the roots of real (and complex) numbers. For example, there should be two square-roots, three cube-roots, etc. of any number. Find the following roots.

- What are the cube-roots of 27? In other words, what is
- What are the fifth roots of 3(3
^{1/5})? - What are the fourth roots of one?

### Cool Exponentials

- \[i^{i}\]
- \[i^{2i}\]
- \[i^{i^{-1}}\]

### Complex-valued Signals

Complex numbers and phasors play a very important role in electrical engineering. Solving systems for complex exponentials is much easier than for sinusoids, and linear systems analysis is particularly easy.

- Find the phasor representation for each, and re-express each as the real and imaginary parts of a complex exponential. What is the frequency (in Hz) of each? In general, are your answers unique? If so, prove it; if not, find an alternative answer for the complex exponential representation.
- \[3\sin (24t)\]
- \[\sqrt{2} \cos \left ( 2\pi 60t + \frac{\pi }{4} \right )\]
- \[2 \cos \left (t + \frac{\pi }{6} \right ) + 4 \sin \left ( t - \frac{\pi }{3} \right )\]

- Show that for linear systems having real-valued outputs for real inputs, that when the input is the real part of a complex exponential, the output is the real part of the system's output to the complex exponential (see figure below).

\[S\left ( \Re (Ae^{i2\pi ft}) \right ) = \Re \left (S (Ae^{i2\pi ft}) \right )\]

*Fig. 2.7.1 Complex system*

- For each of the indicated voltages, write it as the real part of a complex exponential \[v(t) = \Re (Ve^{st})\] Explicitly indicate the value of the complex amplitude
**V**and the complex frequency Represent each complex amplitude as a vector in the**V-plane**, and indicate the location of the frequencies in the complex- \[v(t) = \cos (5t)\]
- \[v(t) = \sin \left ( 8t+\frac{\pi }{4} \right )\]
- \[v(t) = e^{-t}\]
- \[v(t) = e^{-(3t)}\sin \left ( 4t+\frac{3\pi }{4} \right )\]
- \[v(t) = 5e^{(2t)}\sin (8t + 2\pi )\]
- \[v(t) = -2\]
- \[v(t) = 4\sin (2t) + 3\cos (2t)\]
- \[v(t) = 2\cos \left ( 100\pi t + \frac{\pi }{6} \right )- \sqrt{3}\sin \left ( 100\pi t + \frac{\pi }{2} \right )\]

- Express each of the following signals as a linear combination of delayed and weighted step functions and ramps (the integral of a step).

### Linear, Time-Invariant Systems

When the input to a linear, time-invariant system is the signal **x(t), **the output is the signal **y(t)**,

*Fig. 2.7.2 Time -invariant system*

- Find and sketch this system's output when the input is the depicted signal:
- Find and sketch this system's output when the input is a unit step.

### Linear Systems

The depicted input **x(t)** to a linear, time-invariant system yields the output

*Fig. 2.7.3 Linear system*

- What is the system's output to a unit step input
- What will the output be when the input is the depicted sqaure wave:

### Communication Channel

A particularly interesting communication channel can be modeled as a linear, time-invariant system. When the transmitted signal **x(t)** is a pulse, the received signal

*Fig. 2.7.4 Communication channel* *and pulse sequence*

- What will be the received signal when the transmitter sends the pulse sequence
**x**?_{1}(t) - What will be the received signal when the transmitter sends the pulse sequence
**x**that has half the duration as the original?_{2}(t)

### Analog Computers

So-called **analog computers** use circuits to solve mathematical problems, particularly when they involve differential equations. Suppose we are given the following differential equation to solve.

\[\frac{\mathrm{d\: y(t)} }{\mathrm{d} t} + ay(t) = x(t)\]

In this equation, **a** is a constant.

- When the input is a unit step \[(x(t) = u(t))\] the output is given by \[y(t) = (1-e^{-(at)})u(t)\] What is the total energy expended by the input?
- Instead of a unit step, suppose the input is a unit pulse (unit-amplitude, unit-duration) delivered to the circuit at time
**t =10**, what is the output voltage in this case? Sketch the waveform.

### Contributor

- ContribEEOpenStax