332:345 Linear Systems and Signals Fall 2011
Instructor: Zoran Gajic, ELE-222, tel. 4453415, emai: gajic@ece.rutgers.edu
Class Home Page: http:/www.ece.rutgers.edu/gajic/345.html
Ofce Hours: M3, Th3
Textbook: Zoran Gajic, Linear Dynamic Systems and Signal
Linear Systems and Signals -EXAM 3 Part 1summer 2015
THEORETICAL QUESTION (2 points)
Tl) 2 pts. State and prove the initial value theorem of the Laplace transform.
PROBLEMS (6 points)
1) 3 pts. Find the Laplace transform of the signals
f (t ) t 2 sin(t )u
332: 345 Linear Systems and Signals -EXAM I summer 2015
Theoretical Questions (5pts = 5% of the course grade)
Q1) 1.5 pts. State and justify (in terms of the general formula for a continuous-time system) the
principle of time invariance. Summarize the mai
Linear Systems and Signals -EXAM 3 Part 2summer 2015
THEORETICAL QUESTIONS (3 points)
Tl) 1 pt. State and prove the variant of the integral property of the z-transform, that is
derive cfw_ f [k k 0 ]u[k ] .
T2) 2 pts. State and prove the final value theor
ENEE 322
Due: Tue 12/04/12
Homework 08
Problem 1. Discrete time Fourier transform (DTFT) properties:
(a) What does it mean for the DTFT to converge?
(b) If a signal is real-valued, what does this tell you about the DTFT?
(c) Why is the DTFT a periodic fun
ENEE 322
Due: Tue 10/30/12
Homework 06
Problem 1. Determine the Discrete Time Fourier Series (DTFS) coecients of the periodic
discrete time sequence x[n] with one fundamental period dened as
x[n] = 0.5n u[n], 0 k 14.
Plot the signal and its Fourier coecie
ENEE 322
Due: Tue 10/23/12
Homework 05
Problem 1. Determine whether the following functions satisfy the Dirichlet conditionss
(a)
x(t) = tan(t);
(b)
y(t) = sin
0.5
t
for 0 t < 1 and y(t) = y(t + 1).
(c)
g(t) =
1, 22m1 < t 22m
0, 22m2 < t 22m1
for m being
ENEE 322
Due: Tue 11/08/12
Homework 07
Problem 1.
(a) The CTFT of an aperiodic function g(t) is given by G() = 1. Determine the
aperiodic function g(t).
(b) Determine the signal x(t) whose CTFT is a frequency-shifted impulse function
X() = ( 0 ).
(c) Dete
ENEE 322
Due: Tue 10/04/12
Homework 04
Problem 1. Using the properties of the impulse function, (t), simplify the following expressions
(a)
5 jt
(t)
7 + t2
(b)
(t + 5)(t 2)dt
(c)
ej0.5t+2 (t 5)dt
Problem 2. Determine the output response of a CT LTI system
ENEE 322
Due: Tue 09/25/12
Homework 03
Problem 1. Consider the DT LTI systems with the following input-output relationships:
(i) y[k] = x[k 1] + 2x[k 3]
(ii) y[k + 1] 0.4y[k] = x[k].
Calculate the impulse responses for the two DT LTI systems. Also, determ
ECE 345
Homework 01
Fall 2015
Problem 1. Classify the following signals as power or energy signals
(a)
5
for 2 t 2
x(t) =
0
otherwise
(b)
x(t 8k),
z(t) =
k=
where x(t) is dened in part (a).
Hint: z(t) is a periodic signal with the fundamental period 8.
(c
332:345 Linear Systems and Signals -Final Exam -summer 2015
Closed book and notes; no calculators
Part I. Theoretical Questions (5 points)
1. 1pt. State the definitions of position constant k p and velocity constant k v for a closedloop system. (Hint: ple
332:345 Linear Systems and Signals -EXAM 2 summer 2015
PART I: Theoretical Questions (5 points =5% of the course grade)
Q1) 1.5pts. State the definition of magnitude and phase line spectra of a periodic signal.
Q2) 1.5 pts. State and prove the derivative
RUTGERS UNIVERSITY
School of Engineering
Department of Electrical & Computer Engineering
332:345/347 Linear Systems and Signals Fall 2016
Course Description:
This course is an introduction to the basic principles and applications of linear systems. It
cov
RUTGERS UNIVERSITY
School of Engineering
Department of Electrical & Computer Engineering
332:345/347 Linear Systems and Signals Fall 2016
Please work on the following end-of-chapter problems from Ch.1 of the Lathi text. These are for
practice only and are
332:345 Linear Systems & Signals Fall 2016 S. J. Orfanidis
F()=
f (t)e
jt
dt
(FT)
f (t)
F()
(t)
1
(t t0 )
ejt0
1
f (t)=
2
F()ejt d
(IFT)
Properties
Given the pairs f (t) F(),
and g(t) G(), then,
f (t) F(), (reflection)
F(t) 2f (), (duality/symmetry)
2()
1
11.4. PARAMETRIC EQUALIZER FILTERS
581
11. IIR DIGITAL FILTER DESIGN
582
|H( )|
G
2
boost
|H( )|
2
cut
2
G B2
G 02
G 02
G B2
0
0
G
2
0
0
Fig. 11.4.1 Parametric EQ lter with boost or cut.
Fig. 11.3.3 Peaking and complementary notch lters.
The zeros of the
RUTGERS UNIVERSITY
School of Engineering
Department of Electrical & Computer Engineering
332:345/347 Linear Systems and Signals Fall 2016
Please work on the following end-of-chapter problems from Ch.2 of the Lathi text. These are for
practice only and are
332:345 Linear Systems & Signals
Block Diagram Realizations Fall 2016 S. J. Orfanidis
Observer Canonical Form
Consider the second-order transfer function example:
H(s)=
b0 s2 + b1 s + b2
c1 s + c2
r1
r2
= b0 + 2
= b0 +
+
s2 + a1 s + a2
s + a1 s + a2
s p1
Linear Systems & Signals
Applications
communications (wireless, wired) spectral analysis
software radio speech analysis and synthesis
radar, sonar, target tracking geophysical signal processing
automatic control systems image processing, biometrics
elec
Solutions to HW Problems from Chapter 6
Problem 6.2
Use the graphical method to convolve p2 (t) with p2 (t).
SOLUTION: The convolution duration property implies that p2 (t) 3 p2 (t) = 0 for t
and t 2 = 1 + 1.
For
02 t 0, the convolution produces
p2(t)
For
ECE 345
Homework 02
Fall 2015
Problem 1. Using the sifting property of the unit impulse, (t), prove that
(at) =
1
(t),
|a|
a = 0.
Problem 2. Determine whether the following systems are linear.
(a) Dierentiator: y(t) =
dx(t)
dt
(b) Exponential amplier: x(t