**Problem 2.1**: Complex Number Arithmetic

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

**Problem 2.2**: 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
- What are the fourth roots of one?

**Problem 2.3**: Cool Exponentials

Simplify the following (cool) expressions.

**Problem 2.4**: 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.
- 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 the below figure).

**Problem 2.5:**

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

*s*. Represent each complex amplitude as a vector in the

*V*-plane, and indicate the location of the frequencies in the complex

*s*-plane.

**Problem 2.6:**

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

**Problem 2.7**: 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)* (The following figure).

- Find and sketch this system's output when the input is the depicted signal (Figure 2.19).
- Find and sketch this system's output when the input is a unit step.

**Problem 2.8**: Linear Systems

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

**.**

*y (t)*↵

- What is the system's output to a unit step input
*u (t)*? - What will the output be when the input is the depicted square wave in the following figure?

**Problem 2.9**: 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

*r (t)*is as shown (in the following Figure).

- What will be the received signal when the transmitter sends the pulse sequence (Figure 2.23)
*x*?_{1}(t) - What will be the received signal when the transmitter sends the pulse signal (the figure below)
*x*that has half the duration as the original?_{2}(t)

**Problem 2.10:** 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.

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 − eu (t
^{-(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.

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