Quiz 1.4 Solutions
Dierence Equation
Convert the block diagram to Direct Form I
The resulting dierence equation is then:
y(n) = x(n) + 8x(n 1) 5y(n 1) 6y(n 2) or
y(n) + 5y(n 1) + 6y(n 2) = x(n) + 8x(n 1) (1)
Homogeneous Solution
To get the homogeneous sol

Linguistics 582
Basics of Digital Signal Processing
Moving Average (Feedforward) Filters
I. Simple digital filters
Suppose that we have a sequence of data points that we think should be characterizable
as a smooth curve, for example, increasing in value a

PROBLEM SET 04: THE z-TRANSFORM
EEE 35: Signals and Systems
ANSWER KEY
Deadline: May 12, 2016, 12:00 NN
INSTRUCTIONS:
Use SHEETS OF BOND PAPER for your answer sheet. Do not forget to write your
COMPLETE NAME, STUDENT NUMBER, and DC SECTION on EACH PAGE.

Inverse Z-transform & computing for the impulse response:
H ( z )=
1+2 z1
13 z14 z2
Method 1:
Method 2:
2
1
2
1
H ( z )=
1+2 z
z
2
1
2
13 z 4 z z
2
H ( z )=
z +2 z
z 3 z4
H ( z )=
1+2 z
z
2
1
2
13 z 4 z z
H ( z )=
z2 +2 z
z 23 z4
2
5 z +4
z 3 z4
H ( z )

Final Exam Review
Week 14
EEE 35: Signals and Systems
1
Causality and stability
Which of the following signals have a Fourier
transform?
A. 2nu[n-1] + 3nu[n]
B. 2nu[n-1] + 3-nu[n]
C. 2-nu[n-1] + 3nu[n]
D. 2-nu[n-1] + 3-nu[n]
Week 14: Review
EEE 35: Signal

EEE 35: Signals & Systems
Lecture 1C
Systems as Functions
Electrical and Electronics Engineering Institute EEE 35
1C-1
EEE 35: Signals & Systems
Component
Type
Process
Some Applications
Signals
Continuous-time,
Continuous-valued
(Analog)
Analysis
Discrete

EEE 35: Signals & Systems
Lecture 2B
LAPLACE TRANSFORM
Properties
Electrical and Electronics Engineering Institute EEE 35
2B-1
Properties of the Laplace Transform
1. Linearity
If L cfw_x1(t) and L cfw_x2(t) exist, then
L cfw_ax1(t) + bx2(t) = aL cfw_x1(t)

Example 2: The switch is closed at t=0, a steady
state having previously existed. Find the current in
resistor R3 using (a) Thvenins theorem and (b)
Nortons theorem.
Source: van Valkenburg, Network Analysis, 2nd edition
Electrical and Electronics Engineer

Systems as functions,
Convolution (CTS)
EEE 35: DC 1.3
Signals and Systems
DC 1.3: Systems as functions, Convolution (CTS)
EEE 35: Signals and Systems
Electrical and Electronics Engineering
Institute - EEE 35
Introduction
Systems are also functions denot

Inverse Laplace Transform
DC 2-2
EEE 35: Signals & Systems
DC 2.2: Inverse Laplace Transform
EEE 35: Signals and Systems
1
Inverse Laplace Transform: Definition
DC 2.2: Inverse Laplace Transform
EEE 35: Signals and Systems
2
Example
DC 2.2: Inverse Laplac

EEE 33 Review, Solutions to
CT Differential Equations
DC 1-1
EEE 35: Signals & Systems
DC 1.1: EEE 33 Review, Solutions to CT Differential Equations
EEE 35: Signals and Systems
1
Signals
Signals are functions of one or more variables that
conveys the beh

Solving LTI Systems using
the Laplace Transform
DC 2-3
EEE 35: Signals & Systems
DC 2.3: Solving LTI Systems using the Laplace Transform
EEE 35: Signals and Systems
1
Concept Map for Continuous-Time Systems
DC 2.3: Solving LTI Systems using the Laplace Tr

Time-Frequency Characterization
of CT Signals and Systems
DC 2-4
EEE 35: Signals & Systems
DC 2.4: Time-Freq. Characterization of CT Signals and Systems
EEE 35: Signals and Systems
1
Bases
DC 2.4: Time-Freq. Characterization of CT Signals and Systems
EEE

EEE 35: Signals & Systems
Lecture 2G
Bode Plot
Electrical and Electronics Engineering Institute EEE 35
2G-1
Bode Plot
Terms encountered:
Km
m
s
p
s
1+
n
p
2
2
1
1+ s + 2 s
n
n
examine the contribution of each term to
Lm G( j ) and
G( j )
Electri

The z-Transform
Deni>on and Proper>es
EEE 35: DC 4.1
Signals and Systems
2S 2013 2014
DC 4.1: The z-Transform: Deni>on and Proper>es
EEE 35: Signals and Systems
1
The z-Transform
Complex plane representa>on of

Quiz 1.1 Solutions
Dierential Equation
KVL around the loop:
Vi (t) = Ri(t) + L
Capacitor equation:
i(t) = C
di(t)
+ Vo (t) (1)
dt
dVo (t)
(2)
dt
Note: We are interested in the voltage across the capacitor.
Substituting (2) to (1):
Vi (t) = R C
Simplifying

Quiz 2.2 Solutions
Dierential Equation
KVL around the outer loop:
vi (t) = L
diL (t)
+ vo (t) (1)
dt
KCL at the RLC common node:
vo (t)
dvo (t)
iL (t) =
+C
(2)
R
dt
Note: We are interested in the voltage across the capacitor.
Substituting (2)to (1):
d
vi

Laplace Transform and Applications
EEE 35: Sample Problems
Signals and Systems
Laplace Transform and Applications
EEE 35: Signals and Systems
Electrical and Electronics Engineering Institute EEE 35
1
Problem 01
Given the two signals:
x1 ( t ) = e-2t u (

Differential Equations
EEE 35: DC Quiz #1
Signals and Systems
DC 2.1: Laplace Transform and Properties
EEE 35: Signals and Systems
1
DC 01 - Quiz
Given the series RLC circuit above, solve for the differential
equation which relates the output voltage to t

Convolution (CTS)
EEE 35: DC Quiz #3
Signals and Systems
DC 1.3: Convolution (CTS)
EEE 35: Signals and Systems
1
DC 03 - Quiz
Let the input x(t) to an LTI system with impulse
response h(t) be given in the figures below. Find the
output y(t).
DC 1.3: Conv

Fourier Series and
Fourier Transform of
Continuous-Time Signals
EEE 35: DC Quiz #9
Signals and Systems
DC 3.1: Fourier Series and Fourier Transform of CTS
EEE 35: Signals and Systems
1
DC 01 - Quiz
Given
Find the Fourier transform, X(), of x(t).
Evalua

Signals as Functions
EEE 35: DC Quiz #2
Signals and Systems
DC 1.2: Signals as Functions
EEE 35: Signals and Systems
1
DC 02 Quiz
DC 1.2: Signals as Functions
EEE 35: Signals and Systems
Determine the
function x(t) in terms
of singularity
functions
Plot t

Difference Equations
EEE 35: DC Quiz #4
Signals and Systems
DC 1.4: Difference Equations
EEE 35: Signals and Systems
1
DC 04 - Quiz
Given the following block diagram:
x(t)
5
z-1
y(t)
8
6
z-1
Solve for the complete response of the system, given
the input

Filters
EEE 35: DC Quiz #12
Signals and Systems
DC 3.4: Filters
EEE 35: Signals and Systems
1
DC 04 - Quiz
For the filter below, determine the following:
Determine H(). What kind of filter is it?
What is the maximum value of |H()|?
At what frequency w

Laplace Transform
and Applications
EEE 35: DC Quiz #6
Signals and Systems
DC 2.1-2.3: Laplace Transform and Applications
EEE 35: Signals and Systems
1
DC Quiz
()
()
Assume the circuit
is relaxed.
Derive the differential equation involving vo(t).
Solve f

EEE 35 Signals and Systems
Electrical Electronics Engineering Institute
University of the Philippines - Diliman
Machine Problem 2: Transfer Function Response
Description: Plot the magnitude response of a given circuit
Credit Points: 100 (before bonus)
Obj