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Signals and Systems Problems

Module by: Don Johnson. E-mail the author

Summary: (Blank Abstract)

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Problem Set

Problem 1

Complex Number Arithmetic

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

1. -1-1
2. 1+3i2 1 3 2
3. 1+i+eiπ2 1 2
4. eiπ3+eiπ+e(iπ3) 3 3

Problem 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.

1. What are the cube-roots of 27? In other words, what is 2713 27 1 3 ?
2. What are the fifth roots of 3 ( 315 3 1 5 )?
3. What are the fourth roots of one?

Problem 3

Cool Exponentials

Simplify the following (cool) expressions.

1. ii
2. i2i 2
3. iii

Problem 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.

1. 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.
1. 3sin24t 3 24 t
2. 2cos2π60t+π4 2 2 60 t 4
3. 2cost+π6+4sintπ3 2 t 6 4 t 3
2. 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 1). SAei2πft=SAei2πft S A 2 f t S A 2 f t

Problem 5

For each of the indicated voltages, write it as the real part of a complex exponential ( vt=Vest v t V s t ). Explicitly indicate the value of the complex amplitude VV and the complex frequency ss. Represent each complex amplitude as a vector in the VV-plane, and indicate the location of the frequencies in the complex ss-plane.

1. vt=cos5t v t 5 t
2. vt=sin8t+π4 v t 8 t 4
3. vt=et v t t
4. vt=e(3t)sin4t+3π4 v t 3t 4 t 3 4
5. vt=5e(2t)sin8t+2π v t 5 2 t 8 t 2
6. vt=-2 v t -2
7. vt=4sin2t+3cos2t v t 4 2 t 3 2 t
8. vt=2cos100πt+π63sin100πt+π2 v t 2 100 t 6 3 100 t 2

Problem 6

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

Problem 7

Linear, Time-Invariant Systems

When the input to a linear, time-invariant system is the signal xt xt , the output is the signal yt y t (Figure 3).

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

Problem 8

Linear Systems

The depicted input xt x t to a linear, time-invariant system yields the output yt y t .

1. What is the system's output to a unit step input ut u t ?
2. What will the output be when the input is the depicted square wave?

Problem 9

Predicting the Output

In many situations, we know the response of a system to simple inputs, but want to know how it reacts to more complicated ones. When systems are linear and time-invariant, performing such calculations can be very easy.

1. A circuit's response yt y t to a step input xt=ut x t u t is shown in the top row of Figure 7(a). What is the system's output to the input shown in the second row? Sketch the waveform.
2. For the same system, what input produces the output shown in Figure 7(b)? Sketch your answer.
3. For a second system, the sinusoidal input (1+i)ei2πtut 1 2t ut yields the output ((1i)ei2πt+et)ut 1 2t t ut . What is the output when this input is multiplied by a unit-amplitude pulse of duration 1 second that starts at t=0 t0?
Note:
The provided input and output are zero for t<0 t0.

Problem 10

Communication Channel

A particularly interesting communication channel can be modeled as a linear, time-invariant system. When the transmitted signal xt xt is a pulse, the received signal rt rt is as shown.

1. What will be the received signal when the transmitter sends the pulse sequence x 1 t x 1 t ?
2. What will be the received signal when the transmitter sends the pulse signal x 2 t x 2 t that has half the duration as the original?

Problem 11

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. dytdt+ayt=xt t yt a yt xt In this equation, aa is a constant.

1. When the input is a unit step ( xt=ut xt ut ), the output is given by yt=(1e(at))ut yt 1 at ut . What is the total energy expended by the input?
2. Instead of a unit step, suppose the input is a unit pulse (unit-amplitude, unit-duration) delivered to the circuit at time t=10 t 10 . What is the output voltage in this case? Sketch the waveform.

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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