Find the real part, imaginary part, the magnitude and angle of the complex numbers given by the following expressions.
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 ()?
What are the fourth roots of one?
Cool Exponentials
Simplify the following (cool) expressions.
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 [link]).
For each of the indicated voltages, write it as the real
part of a complex exponential
().
Explicitly indicate the value of the complex amplitude
and the complex frequency
.
Represent each complex amplitude as a vector in the
-plane, and indicate the
location of the frequencies in the complex
-plane.
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
,
the output is the signal
([link]).
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
to a linear, time-invariant system yields the output
.
What is the system's output to a unit step input
?
A particularly interesting communication channel can be
modeled as a linear, time-invariant system. When the
transmitted signal
is a pulse, the received signal
is as shown.
What will be the received signal when the transmitter
sends the pulse sequence
?
What will be the received signal when the transmitter
sends the pulse signal
that has half the duration as the original?
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,
is a constant.
When the input is a unit step
(), the output is given by
.
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
.
What is the output voltage in this case?
Sketch the waveform.