Introduction
In this project, we will ultimately be refreshing our previous knowledge of MATLAB
functions, coding styles, and processes. By going through a few introductory problems, we are
able to fall back into our routine of coding in MATLAB and we are
ECE 301 - Spring 2012
1
Test 1 - Sec.1 w/Answers
1. Let a function g() be dened by the following piecewise math form:
g() =
2 , 2 2
0,
else
Let the time-domain function p(t) be given by p(t) = g(3 2 t). (a) Find the simplest piecewise
math form for p(t).
Introduction
In this project, we will be using MATLAB to explain, compute, and plot continuous
convolution functions. We began by working through a few introductory problems found within
the note sets. Below are the plots of two continuous convolution pro
ECE 301
Detailed Solutions - Set 8 : Signal Spectrum
1
8-1. Several signals st are given below. For each signal use the Fourier transform to compute the mathematical form for the spectrum S f . Also, plot each st and S f . (a) st 2 ut 001 ut 001 , (b) st
1
Solutions - Set 5: Signal Spectrum
ECE 301 - Fa2013
5-1. Several signals s(t) are given below. For each signal use the Fourier transform to compute
the mathematical form for the spectrum S(f ). Also, plot each s(t) and | S(f ) |.
(a) s(t) = 2 u(t + .001
1
Solutions - Set 15: Z-Transforms
ECE 301 - Fa2013
15-1. A signal s[m] has the analytical form s[m] = A m u[m]. Compute the z-transform S(z) if
A has the values given below.
(a) A = 0.45,
DETAILED
(b) A = 0.72,
(c) A = 0.3 e j 0.1 ,
(d) A = 0.5 + j0.2
SO
Turn-In Assignment 1: DUE - Fri, 30 Aug 2013
ECE 301 - Fa2013
1
Problem 1.
Let a function g() be dened by the following piecewise form:
2 , 0 3
0,
else
g() =
Find the simplest math form for each of the following functions:
(a)
g(t),
(b) g(t),
(c) g(t 4),
Solutions - Set 9: Analog Filter Design
ECE 301 -Fa2013
1
9-1. The poles of the Butterworth factor H(s)H(s) are given by the solutions to the following
complex-valued equation:
s 2N
= 1
jB
Compute the general expression for the 2N s-domain solutions of th
Solutions - Set 4: Transfer Functions
ECE 301 - Fa2013
1
4-1. A system has impulse response h(t) = 10, 000 e 5000 t u(t), input x(t), and output y(t).
(a)
Compute the transfer function H(s).
(b) Using Laplace transforms and partial fraction expansion, com
Solutions - Set 12: Sampling Theorem
ECE 301 - Fa2013
1
12-1. This problem helps you derive the Fourier transform of the innite series of impulses in
the time-domain. Let the periodic impulse-sampling waveform s (t) be given by
(t m Ts )
s (t) =
(p1)
m=
S
10-1.
1
Solutions - Set 10: Communications Systems
ECE 301 - Fa2013
Plot the magnitude spectrum | X(f ) | if x(t) is given by
x(t) =
DETAILED
5 cos(107 t) , |t| 5(10)7 sec
0,
else
SOLUTION:
Note that the waveform x(t) in the problem statement is actually
1
Solutions - Set 14: Digital Convolution
ECE 301 - Fa2013
14-1. A discrete system has the impulse response h[m] = (0.9)m u[m] u[m 4]
receives the input
. The system
x[m] = 2 u[m 1] 2 u[m 2] u[m 4] + u[m 5]
Plot the output y[m] over the range 5 m 10.
DETA
1
Solutions - Set 13: Digital Signal Processing
ECE 301 - Fa2013
13-1. Plot the following digital signals on the m-axis over the range 0 m 8 :
(a) x[m] = u[m 3] ,
(b) x[m] = u[m 1] u[m 6]
(c) x[m] = (m 2) u[m 2] (m 6) u[m 6] ,
(d) x[m] = m u[m 2] m u[m 6]
Solutions - Set 2: Continuous Convolution I
ECE 301 - Fa2013
1
2-1. A system has an impulse response given by h(t) = e 2 t u(t). Let this system now have input
given by x(t) = 5 (t 1) 3 (t 4). Compute the output of the system. Express your answer in
three
Solutions - Set 3: Continuous Convolution II
ECE 301 - Fa2013
1
3-3. Let a system have impulse response h(t) and input x(t) given by
h(t) =
5, 0 t 1
,
0, else
x(t) =
3, 0 t 1
0, else
Compute the analytical form (math form) for the output y(t) for all time
1
Solutions - Set 7: Frequency Domain Output
ECE 301 - Sp2014
7-1. Let x(t) = 10 + cos(2 1000 t) be input to the system having frequency response given by
| H(f ) | e j (f ) . Let | H(f ) | and (f ) be given as shown below:
| H(f) |
10
90
10
o
( f )
5
5
5
Solutions - Set 6: Frequency Response
ECE 301 - Fa2013
1
A
, where
B+jf
A and B are real-valued and non-negative. Selected inputs to the system and the corresponding
outputs are given in the table below:
6-1.
A low pass lter frequency response is known to
ECE 301 - Fa2013
Solutions - Set 1: Impulse Response
1
1-1. Let a function f () be dened by the following piecewise form:
f () =
2, 0
0, else
Find the simplest math form for each of the following functions: (a) f (t); (b) f (t); (c) f (t 4);
(d) f (4 t).
ECE 301 - Fa2013
Notes Set 12: Sampling Theorem
1
INTRODUCTION
In Notes Set 11 we studied reconstructing an original analog signal by using samples of the
signal. We saw that a rectangular approximation was easy to understand from both an
implementation a
ECE 301 - Fa2013
Solutions - Set 11: Sampling and Reconstruction
1
11-1. An analog signal x(t) = e 5 t u(t) is sampled using a sampling interval Ts = 0.1 seconds.
The digital signal which results can be written in the form x[m] = x(mTs ) = A m u[m] where
ECE 301 - Fa2013
Notes Set 8: Poles and Zeros
1
INTRODUCTION
We have seen in earlier notes that once we have the system transfer function H(s), we can
often nd the frequency response H(f ) by a simple substitution of variables. However, the
transfer funct
Turn-In Assignment 2: DUE - Mon, 9 Sept 2013
ECE 301 - Fa2013
1
Problem 1.
Suppose a wireless transmitter is sending data to a receiver in the reection environment described
by the following:
direct path:
reected path:
Ad = 0.80, distance = d = 5.1 m;
A1
ECE 301 - Fa2013
Notes Set 11: Sampling and Reconstruction
1
INTRODUCTION
Many components of modern media systems are implemented using digital signal processing.
Among these applications are cell phones, audio CDs, video DVDs, digital television, digital
ECE 301 - Fa2013
Notes Set 15: Z-Transforms
1
INTRODUCTION
The Z-transform is the frequency-domain analytical tool for digital signal processing
systems. It serves a purpose for DSP systems similar to the purpose served by the Laplace
transform for analog
ECE 301 - Fa2013
Notes Set 5: Signal Spectrum
1
INTRODUCTION
The previous sets of Notes have examined the properties of systems in the time domain.
In that regard, you could think of a time domain signal g(t) as describing how the signals
voltage or curre
ECE 301 - Fa2013
Notes Set 4: Transfer Functions
1
INTRODUCTION
In this set of Notes we will study some very useful properties of the the Laplace Transform
and the s-domain, which will result in determining the Transfer Function of a system. We
will see t
ECE 301 - Fa2013
Notes Set 1: Impulse Response
1
INTRODUCTION
You have probably studied basic circuits in an earlier course like ECE 211. Typical problems
at that level of investigation might have been to nd the output of a circuit in response to
a partic
ECE 301 - Fa2013
Notes Set 9: Analog Filter Design
1
INTRODUCTION
One of the most important applications of the s-plane is Analog Filter Design. Analog
lter design combines the previous concepts of transfer function, frequency response, and
poles/zeros an
ECE 301 - Fa2013
Notes Set 10: Communications Systems
1
INTRODUCTION
One area which uses a great deal of linear systems theory is Communications. The area
of Communications is very broad and is divided among specialities such as Amplitude
Modulation (AM),
ECE 301 - Fa2013
Notes Set 13: Digital Signal Processing
1
INTRODUCTION
In the rst part of Notes Set 11 we examined the problem of sampling an original analog
signal to acquire signal samples. This sequence of digital samples may now be considered
as a Di
ECE 301-002
Fall 2016
1
Test 3 w/Answers
Problem 1. The analog signal x(t) = 10 e 50 t cos(250 t) is multiplied by a sampling waveform
s (t) to give the impulse sampled signal x (t) = x(t) s (t). The sampling frequency is fs = 250
Hz. This sampling wavefo
ECE 301-002 Fall 2016
1
Test 1 w/Answers
1. Let a linear system with input x(t) and output y(t) be described by the differential equation
d2
d
y(t) + 60 y(t) + 500 y(t) = 5 x(t)
2
dt
dt
(a) Compute the simplest math form of the impulse response h(t) for t
ECE 301-001 Fall 2016
1
Test 2 w/Answers
1. An RLC-circuit with R = 2000 , L = 0.001 H, and C = 105 F is shown below:
+
vi (t)
R
+
C
vo (t)
R
L
Vo (s)
N2 s 2 + N1 s + N0
can be written as H(s) =
, where N0 ,
Vi (s)
s 2 + D1 s + D0
N1 , N2 , D0 , and D1 ar
ECE 301-001 Fall 2016
1
Test 3 w/Answers
1. Let A(s) be the transfer function
of an
N = 3 stable lowpass Butterworth filter, with half-power
bandwidth B = 20 and let A( = 0) = 1. The transfer function A(s) can be written as
A(s) =
G
s + K2
s 2 + K1 s +
ECE 301-001 Fall 2016
1
Test 1 w/Answers
1. Let a linear system with input x(t) and output y(t) be described by the ODE below:
d2
d
d
y(t) + 20 y(t) + 15, 890 y(t) = 5 x(t) + 50 x(t)
2
dt
dt
dt
(a) Compute the simplest real-valued mathematical form (funct