Problems 2.22 2001/08/22 2003/05/21 00:00:00.022 GMT-5 Don Johnson dhj@rice.edu Erin Snavely ernsnave@alumni.rice.edu Don Johnson dhj@rice.edu Benjamin Fite bfite@rice.edu Brent Michael Hendricks brentmh@rice.edu (Blank Abstract)

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 24 t 2 2 60 t 4 2 t 6 4 t 3 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 ). S A 2 f t S A 2 f t

For each of the indicated voltages, write it as the real part of a complex exponential ( v t 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. v t 5 t v t 8 t 4 v t t v t 3t 4 t 3 4 v t 5 2 t 8 t 2 v t -2 v t 4 2 t 3 2 t v t 2 100 t 6 3 100 t 2

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 xt , the output is the signal y t ().

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 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?

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

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 signal x 2 t that has half the duration as the original?