EE 110B - Signals and Systems
Lab 3
Task 1: Compute, plot and discuss the discrete-time Fourier transform (DTFT)
X(f )
x[n] exp j 2fn of each of the following sequences. For each
X ( f ) , plot
n
the amplitude spectrum: X ( f ) versus f , and the phase
EE 110B - Signals and Systems
Lab 2
Task 1: Using Matlab to generate a random sequence x[n] for n 0,1,99 and set x[n] 0 for
n 0 and n 99 .
(a) Consider a discrete-time LTI system with the impulse response
h[n] 0.9n 1u[n 1] u[n 100] and the output y[n] h[n
EE 110B - Signals and Systems
Lab 1
Task 1: Use Matlab to plot the following sequences from n 0 to n 50 , discuss and explain
their patterns:
5
1) cos n ; 2) cos n ; 3) cos n ; 4) cos 0.2n ; 5) 0.9n cos n ;
2
5
2
2
6) 1.1n cos n ; 7) cos
1
EE110B Homework 3
1) Determine the discrete-time Fourier transform of:
a) 0.4n1 u[n 1]; b) 0.3|n2| ; c) [n 1] + [n + 2]; d) sin( n + /4).
4
2) Consider a system whose input x[n] and output y[n] are related as
5
1
y[n] y[n 1] + y[n 2] = x[n]
6
6
a) Deter
1
EE110B Homework 1
1) Let x[n] be a signal with x[n] = 0 for n < 3 and n > 6. For each signal given below,
determine the values of n for which it is guaranteed to be zero: a) x[n 3]; b) x[n + 6];
c) x[n]; and d) x[n + 3].
2) For each signal given below,
1
EE110B Homework 2
1) Let x[n] = [n] + 3[n 2] [n 5] and h[n] = 3[n + 3] + 6[n 4]. Compute and plot
each of the following convolutions: a) y1 [n] = x[n] h[n]; b) y2 [n] = x[n + 1] h[n 1];
and c) y3 [n] = x[n 2] h[n 3]
2) Consider a LTI system with the inp
1
EE110B Note 1
Y. Hua
1) Discrete-time signals versus continuous-time signals
Notation: x[n] (discrete) versus x(t) (continuous). n is an integer. t is a real number.
With any x(t), there is x[n] = x(t)|t=T n = x(T n), which is called uniform sampling
wi
1
EE110B Note 3
Y. Hua
1) Sampling Theorem: Consider a continuous-time signal xc (t) and a sequence of its samples
xd [n], i.e., xd [n] = xc (nT ) where T is the sampling interval. The sampling rate is 1/T .
For xc (t), we have the spectrum
Xc (f ) =
xc (
1
EE110B Note 2
Y. Hua
1) Fourier Transforms
a) Continuous-Time Fourier Transform (CTFT): for any signal x(t) satisfying
|x(t)|dt <
, we can write
XCT F T (f ) =
x(t) exp(j2f t)dt
x(t) =
XCT F T (f ) exp(j2f t)df
b) Continuous-Time Fourier Series (CTFS):
EE 110B Signals and Systems
Fourier Analysis of !
Discrete-Time Signals
Ertem Tuncel
Decomposition of periodic signals
Recall that we can decompose any signal into
shifted impulses:
Also remember how this was useful in
understanding the response of an L
EE 110B Signals and Systems
Linear and Time-Invariant (LTI) !
Systems
Ertem Tuncel
Why LTI systems?
Linear and time-invariant systems are
especially easy to analyze and design.
In a lot of cases, they are good enough to do
the "signal processing" job.
EE 110B Signals and Systems
Introduction to !
Discrete-time Signals and Systems
Ertem Tuncel
Discrete-time signals
Motivation: We may have access to only
periodic samples x(nT) of a signal x(t).
x(t)
T 2T 3T
-T
t
x[n]=x(nT)
1
-1
2
3
n
Discrete-time signa
EE 110B Signals and Systems
LTI Systems Defined by
Difference Equations
Ertem Tuncel
Difference equations
The input/output relation of an LTI system can
sometimes be expressed as a constantcoefficient difference equation.
Analogous to constant-coefficie
UNIVERSITY OF CALIFORNIA, RIVERSIDE
Department of Electrical Engineering
WINTER 2012
EE110B-SIGNALS AND SYSTEMS
HOMEWORK 7
Please turn in on Thursday, March 8th, 2012, at the beginning of the class.
Problem 1: Find the transfer function H(z) of the system
UNIVERSITY OF CALIFORNIA, RIVERSIDE
Department of Electrical Engineering
WINTER 2012
EE110B-SIGNALS AND SYSTEMS
HOMEWORK 8
Please turn in on Thursday, March 15th, 2012, at the beginning of the class.
Problem 1: Consider the signal
xc (t) =
sin(t)
t
2
whic
UNIVERSITY OF CALIFORNIA, RIVERSIDE
Department of Electrical Engineering
WINTER 2012
EE110B-SIGNALS AND SYSTEMS
HOMEWORK 1
Please turn in on Thursday, January 19th, 2012, at the beginning of the class.
Problem 1: Determine whether or not each of the follo
UNIVERSITY OF CALIFORNIA, RIVERSIDE
Department of Electrical Engineering
WINTER 2012
EE110B-SIGNALS AND SYSTEMS
HOMEWORK 3
Please turn in on Thursday, February 2nd, 2012, at the beginning of the class.
Problem 1: Consider an LTI system with the input-outp
UNIVERSITY OF CALIFORNIA, RIVERSIDE
Department of Electrical Engineering
WINTER 2012
EE110B-SIGNALS AND SYSTEMS
HOMEWORK 2
Please turn in on Thursday, January 26th, 2012, at the beginning of the class.
Problem 1: Consider an LTI system with the input x[n]
UNIVERSITY OF CALIFORNIA, RIVERSIDE
Department of Electrical Engineering
WINTER 2012
EE110B-SIGNALS AND SYSTEMS
HOMEWORK 5
Please turn in on Thursday, February 23th, 2012, at the beginning of the class.
Problem 1:
Determine the signal x[n] whose DTFT is g
UNIVERSITY OF CALIFORNIA, RIVERSIDE
Department of Electrical Engineering
WINTER 2012
EE110B-SIGNALS AND SYSTEMS
HOMEWORK 6
Please turn in on Thursday, March 1st, 2012, at the beginning of the class.
Problem 1: Determine the z-transform of each of the foll
UNIVERSITY OF CALIFORNIA, RIVERSIDE
Department of Electrical Engineering
WINTER 2012
EE110B-SIGNALS AND SYSTEMS
HOMEWORK 4
Please turn in on Thursday, February 9th, 2012, at the beginning of the class.
Problem:
a) Find the DTFS coecients ak of the periodi