EE2010 Signals and Systems (by Prof. Ma Kai-Kuang)
Example:
Express the following complex numbers in the polar form and the exponential form.
(a) z1 6 j 8, (b) z2 6 j 8, (c) z3 6 j 8, (d ) z4 6 j 8
Solution :
(a) Identify which quadrant of z1 lies in: It
EE2023 Signals & Systems (Chapter 2)
Time and Frequency Domain Representation of
Signals
A signal is a function which represents the time variation of a
physical variable.
A continuous time signal is one that is defined for all time
instants, t, in an int
EE2023 Signals & Systems (Chapter 7)
Use of Laplace Transform in Solving D.E.
Use Laplace Transform to derive the output, Vc(t), if V(t) = V
(constant voltage) and the initial condition is vc(0)>0.
Differential equation describing the RC
circuit is
dVc
EE2023 Signals & Systems (Chapter 6)
General Physical
Systems
Physical systems are interconnections of components, devices or
subsystems
Physical systems have many different properties.
We can write mathematical expressions which approximately
describes t
EE2023 Signals & Systems (Chapter 8)
Parameters of First-order Systems
D.E. of a linear first-order system is generally written as
T
dy (t )
y (t ) Ku (t )
dt
where K is the steady-state/static gain, T is called the timeconstant.
Y (s)
K
G
(
s
)
The tr
Two signal x[n] and y[n] are cross correlation to give a third signal z[k]=Rxy[k]. The first
waveform is x[n]=[2 3 2 -4] for n=0 to 3, and x[n]=0, otherwise. y[n] is x[n-5]
Calculate the following:
a) Energy of x[n]
b) Maximum value of |x[n]|
c) Maximum v
EE2007: ENGINEERING MATHEMATICS II
Complex Analysis
A/P Chua Chin Seng
School of EEE, NTU
Email: ecschua@ntu.edu.sg
Tel: 6790 5412
Acknowledgement:
This set of notes are from A/P Saratchandran/Ling Keck Voon with
modifications
EE2007: ENGINEERING MATHEMAT
EE2008: Data Structures and Algorithms
Recursive Algorithms, Trees and Sorting
Assoc Prof Huang Guangbin
S2.1-B2-06
egbhuang@ntu.edu.sg
Coverage
Weeks 1
to 5
Weeks 6
to 9
Lectures
10 to 13
Topics
Lecturers
Introduction
A/P Low Chor Ping
Principles of Algo
EE2023 Signals & Systems (Chapter 9)
Sinusoidal Response
What is the output when the input is a sinusoidal?
Asin(t)
G(s)
?
Intuitively, the output will also be a sinusoidal signal as the solution
to the system differential equations can be interpreted u
Tutorial 9 (21 Oct 25 Oct)
(1) Find the frequency spectrum of the infinite duration signal x(t)
= cos(0t) where 0 is a constant.
Note:
e jax dx 2 (a )
Let the frequency spectrum of x(t) be X(j). We have,
X ( j )
x(t ) e j t d t
cos(0t ) e j t d t
e j0 t
Tutorial 7 (7 Oct 11 Oct)
(1) Find the Fourier series representation of the following periodic
signal.
2
7
3
t
1
5
9
It can be seen from the waveform that the signal from t = 3 to t =
5 repeats itself indefinitely. Hence the period is
T = 5 (3) = 5 + 3 =
Harry Nyquist
Sampling Theorem
Claude E. Shannon
1
Sampling
A discrete-time signal may be generated by a natural process (say,
daily stock index, hourly-taken temperature recording of a chemical
process, etc.) or produced based on a continuous-time (anal
EE2010 Signals and Systems
g
y
Academic Year 2010/2011
Semester 2
Part 2: Weeks 6 11
6~11
Prof. Ma Kai-Kuang
Office: S2-B2C-83
Tel: 6790 6366
Email: ekkma@ntu edu sg
ekkma@ntu.edu.sg
Topics to be covered in my lectures:
Sinusoids
Fourier Series
Fourier Tr
EE2010/IM2004 Signals and Systems
Academic Year 2010/2011
Semester 2
Part I: Weeks 1-5
Assoc/Prof Teh Kah Chan
Oce: S2-B2a-20
Tel: 6790-5365
Email: ekcteh@ntu.edu.sg
1
Subject Outline
Signals and Systems
Classication of Signals
Elementary and Singulari
EE2010/IM2004 Signals and Systems
Academic Year 2010/2011
Semester 2
Part IV: Weeks 12-13
Assoc/Prof Teh Kah Chan
Oce: S2-B2a-20
Tel: 6790-5365
Email: ekcteh@ntu.edu.sg
1
Subject Outline
Modulation
Types of Modulations
Amplitude Modulation
Time Domain
Fourier Transform
(Part I)
1
Introduction
Fourier series (FS) is applicable only for periodic signals. For
aperiodic (or non-periodic) signals, we need to exploit the Fourier
transform (FT) to conduct spectral analysis.
Fourier transform is, in fact, a
Fourier Series
Jean Baptiste Joseph Fourier (1768 - 1830)
1
Fourier analysis is about a way to represent or analyze a given
signal (be it periodic or non-periodic) to reveal its frequency
content.
It has two cases:
Fourier Series (FS) The Fourier series o
Tutorial 8 (14 Oct 18 Oct. e-Learning. No class)
(1) Find the Fourier series expansion for the periodic function v (t) with period T and
1 for |t | / 2
v(t )
0 elsewhere within a period
v(t)
1
T
2
2
2
T
2
t
The function v(t) is an even function. Hence,
Tutorial 10 (29 Oct 2 Nov)
(1) A 3 kHz sinusoid is sampled at a rate of 10,000 samples per
second. Determine the frequency spectrum of the sampled
signal.
Solution: The frequency spectrum of the sampled signal is
(10,000m 3,000) Hz where m is an integer.