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Problems on Signals and Systems

Module by: Don Johnson. E-mail the author

Summary: (Blank Abstract)

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Problem Set

Problem 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.

  1. Find the phasor representation for each, and re-express each as the real and imaginary parts of a phasor. What is the frequency (in Hz) of each? In general are you answers unique? If so, prove it; if not, find an alternative answer for the phasor representation.
    1. 3sin24t 3 24 t
    2. 2cos2×π×60t+π4 2 2 π 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 phasor, the output is the real part of the system's output to the phasor. SAej2πft=SAej2πft S A j 2 f t S A j 2 f t

Figure 1
Figure 1 (sys28.png)

Problem 2

For each of the indicated voltages, write it as the real part of a complex exponential ( vt=ReVest v t Re V s t ). 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.

  1. vt=cos5t v t 5 t
  2. vt=sin8t+π4 v t 8 t 4
  3. vt=e-t v t -t
  4. vt=e-3tsin4t+3π4 v t -3 t 4 t 3 4
  5. vt=5e2tsin8t+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 3

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

Figure 2
Figure 2(a) (sig1.png)
Figure 2(b) (sig2.png)
Figure 2(c) (sig3.png)
Figure 2(d) (sig4.png)
Figure 2(e) (sig5.png)

Problem 4

Linear, Time-Invariant Systems

When the input to a linear, time-invariant system is the signal xt x t , the output is the signal yt y t

Figure 3
Figure 3 (sig34a.png)

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

Figure 4
Figure 4 (sig34b.png)

Problem 5

Linear Systems

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

Figure 5
Figure 5 (sig39.png)

  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?

Figure 6
Figure 6 (sig40.png)

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