Signal Processing and Linear System Analysis

SIGNAL
PROCESSING AND
LINEARSYSTEM
ANALYSIS
Shieh-Kung Huang
黃謝恭
1

Shieh-Kung
Huang
Copyright © 2018 by McGraw-Hill Education. All rights reserved.
SIGNAL PROCESSING AND LINEAR SYSTEM ANALYSIS
Basic Information about This Course
5
• Instructor
Shieh-Kung Huang (黃謝恭)
Assistant Professor, Civil Engineering, National Chung Hsing University
Office: 508 Concrete Technology Building (混凝土科技研究中心)
E-mail: skhuang@nchu.edu.tw
Tel: (04) 2287-2221 ext. 508
• Course Hour
Tuesday 14:10 – 17:00
• Classroom
201 Civil & Environmental Engineering Building (土木環工大樓)
• Office Hour
Wednesday 2:00 – 4:00, or by appointment
• Teaching Assistant
TBD (To be determined)
• Prerequisites
None

Shieh-Kung
Huang
Course Objectives and Text Book
6
• Course Description
This course focuses on the introduction of signal processing and linear system analysis related to the
civil engineering field. The scopes of this course include continuous-time and discrete-time signals and
systems. Moreover, the lectures range from frequency-domain analysis, time-domain analysis, and
time-frequency-domain analysis. In the end of the semester, the contents cover the related application
in the field of civil engineering, especially focusing on structural monitoring and structural control.
• Course Objectives
TBD
• Competency Indicators
TBD

Shieh-Kung
Huang
Course Objectives and Text Book
7
1. Lathi, B. P., & Green, R. A. (1998). Signal processing and linear systems (Vol. 2). Oxford: Oxford University Press.
2. Stephane, M. (1999). A wavelet tour of signal processing. Elsevier.
3. Golyandina, N., Nekrutkin, V., & Zhigljavsky, A. A. (2001). Analysis of time series structure: SSA and related techniques.
CRC press.
4. Mandal, M. K., & Asif, A. (2007). Continuous and discrete time signals and systems. Cambridge Univeresity Press.
5. Bendat, J. S., & Piersol, A. G. (2011). Random data: analysis and measurement procedures. John Wiley & Sons.
6. Huang, N. E. (2014). Hilbert-Huang transform and its applications (Vol. 16). World Scientific.
7. Addison, P. S. (2017). The illustrated wavelet transform handbook: introductory theory and applications in science,
engineering, medicine and finance. CRC press.
8. https://en.wikipedia.org/wiki/Wiki

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Course Map
8
Ch1
Introduction of
Signals and Systems
Ch2
Fourier Transform
Ch3
Sampling of Signals
Ch4
Filters
Ch5
Short-time Fourier
Transform
Ch6
Wavelet Transform
Ch7
Hilbert–Huang
Transform
Ch8
Singular Spectrum
Analysis
Ch9
Linear System
Analysis

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STRUCTURAL THEORY I
Course Topics and Schedule
9
Chapter 1 Introduction of Signals and Systems
Chapter 2 Fourier Transform
Chapter 3 Sampling of Signals
Chapter 4 Filters
Chapter 5 Short–time Fourier Transform
Mid-term
Chapter 6 Wavelet Transform
Chapter 7 Hilbert–Huang Transform
Chapter 8 Singular Spectrum Analysis
Chapter 9 Linear System Analysis
Final

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STRUCTURAL THEORY I
Grading
10
• Attendance and Homework: 30%
• Midterm Project and Report: 30%
• Final Project and Report: 40%

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Software
11
• Matlab
https://www.mathworks.com/products/matlab.html
• 中興大學計算機及資訊網路中心（需要登入）
http://softservice.nchu.edu.tw/html/

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12
• Are there any questions before we get started?

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INTRODUCTION OF SIGNALS AND SYSTEMS
Chapter Outline
13
1.1 Introduction of Signal Processing
1.2 Introduction of Signals
1.3 Classification of Signals
1.4 Elementary Signals
1.5 Introduction of Systems
1.6 Classification of Systems
1.7 Elementary Systems
1.8 Time-domain Analysis of Systems
CHAPTER 1

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INTRODUCTION OF SIGNAL PROCESSING
14

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Examples of image signal processing
15
• Google Used a 64-Camera, 331-Light Array to Train Its Portrait Lighting AI
https://petapixel.com/2020/12/14/google-used-a-64-camera-331-light-array-to-train-its-portrait-lighting-ai/ December 2020

Shieh-Kung
Huang
16
• OPPO Unveils 6nm Cutting-edge Imaging NPU (neural processing unit) – MariSilicon X
https://www.oppo.com/en/newsroom/press/oppo-imaging-npu-marisilicon-x/ December 2021

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17
• AI ‘Photos’ of What Cartoon Characters Would Look Like in Real Life
https://magnumx.me/ June, 2023

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Examples of signal processing
18
• Audio Equalizer
• Step Tracker

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Applications of signal processing techniques
19
• Consumer Electronics
HDTV, cell phones, cameras,…
• Transportation
GPS, engine control, airplane tracking,…
• Medical
Imaging, monitoring (EEG, ECG),…
• Military
Target tracking, surveillance, UAV,…
• Remote Sensing
Astronomy, climate monitoring, weather forecasting,…
• And so on …

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INTRODUCTION OF SIGNALS
General description of signals and signal processing
20
• Signal
A signal describes how some physical quantity varies over time and/or space.
• Signal Processing
Signal processing is an electrical engineering subfield that focuses on analysing, modifying, and
synthesizing signals such as sound, images, and scientific measurements. Signal processing
techniques can be used to improve transmission, storage efficiency and subjective quality and to also
emphasize or detect components of interest in a measured signal.
• Signal Processing
Signal processing is an electrical
engineering subfield that focuses on
analyzing, modifying, and synthesizing
signals such as sound, images, and scientific
measurements.
Signal processing techniques can be used to
improve transmission, storage efficiency and
subjective quality and to also emphasize or
detect components of interest in a measured
signal.
• Signal processing is everything that deals
with a signal except creating or generating
one.

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INTRODUCTION OF SIGNALS
General description of signals
21
Signals are represented mathematically as functions of one or more independent variables. For
example, a speech signal can be represented mathematically by acoustic pressure as a function of time,
and a picture can be represented by brightness as a function of two spatial variables.
In this course, we focus our attention on signals involving a single independent variable. For
convenience, we will generally refer to the independent variable as time, t, although it may not in fact
represent time in specific applications.

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CLASSIFICATION OF SIGNALS
Continuous-time and discrete-time signals
22
Throughout this course we will be considering two basic types of signals: continuous-time signals and
discrete-time signals. In the case of continuous-time signals the independent variable is continuous, and
thus these signals are defined for a continuum of values of the independent variable. On the other hand,
discrete-time signals are defined only at discrete times, and consequently, for these signals, the
independent variable takes on only a discrete set of values.
To distinguish between continuous-time and discrete-time signals, we will use the symbol t to denote
the continuous-time independent variable and k to denote the discrete-time independent variable. In
addition, for continuous-time signals we will enclose the independent variable in parentheses ( · ),
whereas for discrete-time signals we will use brackets [ · ] to enclose the independent variable. We will
also have frequent occasions when it will be useful to represent signals graphically. Illustrations of a
continuous-time signal x(t) and a discrete-time signal x[k] are shown in the following figure.

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Continuous-time and discrete-time signals
23
A discrete-time signal x[k] may represent a phenomenon for which the independent variable is
inherently discrete. Signals such as demographic data are examples of this. On the other hand, a very
important class of discrete-time signals arises from the sampling of continuous-time signals. In this case,
the discrete-time signal x[k] represents successive samples of an underlying phenomenon for which the
independent variable is continuous. The relationship under a constant sampling rate can be denoted by
x[k] := x(kΔt), where n denotes discrete-time instant and is an integer ranging from −∞ to +∞; Δt denotes
sampling period or sampling interval.
It is important to note that the discrete-time signal x[k] is defined only for integer values of the
independent variable, and for further emphasis we will on occasion refer to x[k] as a discrete-time
sequence.

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Analog and digital signals
24
The concept of continuous-time is often confused with that of analog. The two are not the same. The
same is true of the concepts of discrete-time and digital. A signal whose amplitude can take on any value
in a continuous range is an analog signal. This means that an analog signal amplitude can take on an
infinite number of values. Digital signal, on the other hand, is one whose amplitude can take on only a
finite number of values. Signals associated with a digital computer are digital because they take on only
two values (binary signals), The terms continuous-time and discrete-time qualify the nature of a signal
along the time (horizontal) axis. The terms analog and digital, on the other hand, qualify the nature of the
signal amplitude (vertical axis).
discrete-time
continuous-time
analog digital

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Periodic and aperiodic signals
25
A continuous-time signal x(t) is said to be periodic if it satisfies the following property:
at all time t and for some positive constant T0. The smallest positive value of T0 that satisfies the
periodicity condition is referred to as the fundamental period of x(t).
Likewise, a discrete-time signal x[k] is said to be periodic if it satisfies
at all time k and for some positive constant K0. The smallest positive value of K0 that satisfies the
periodicity condition is referred to as the fundamental period of x[k]. Moreover, the reciprocal of the
fundamental period of a signal is called the fundamental frequency. If radians per second is used as a unit
of frequency, the frequency is referred to as the angular frequency. A signal that is not periodic is called an
aperiodic or non-periodic signal.
0
( ) ( )
x t x t T
= +
0
[ ] [ ]
x k x k K
= +

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Odd and even signals
26
A continuous-time signal xe(t) is said to be an even signal if
Conversely, a continuous-time signal xo(t) is said to be an odd signal if
A discrete-time signal xe[k] is said to be an even signal if
Conversely, a discrete-time signal xo[k] is said to be an odd signal if
The even signal property, both the equations for continuous-time signals or discrete-time signals, implies
that an even signal is symmetric about the vertical axis (usually t = 0). Likewise, the odd signal property,
both the equations for continuous-time signals or discrete-time signals, implies that an odd signal is
antisymmetric about the vertical axis (usually t = 0). The symmetry characteristics of even and odd signals
are illustrated in the following figure.
Noteworthy, most practical signals are neither odd nor even. Such signals are classified in the
“neither odd nor even” category.
( ) ( )
e e
x t x t
= −
( ) ( )
o o
x t x t
= − −
[ ] [ ]
e e
x k x k
= −
[ ] [ ]
o o
x k x k
= − −

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Deterministic and random signals
27
If the value of a signal can be predicted for all time (t or k) in advance without any error, it is referred
to as a deterministic signal. Conversely, signals whose values cannot be predicted with complete
accuracy for all time are known as random signals.
Deterministic signals can generally be expressed in a mathematical, or graphical, form. Unlike
deterministic signals, random signals cannot be modeled precisely. Random signals are generally
characterized by statistical measures such as means, standard deviations, and mean squared values.

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Almost-periodic and transient signals
28
It is noted that periodic signals can generally be reduced to a series of sine waves with
commensurately related frequencies. Conversely, the signals formed by summing two or more
commensurately related sine waves will be periodic. However, the signals formed by summing two or
more sine waves with arbitrary frequencies generally will not be periodic. Specifically, the sum of two or
more sine waves will be periodic only when the ratios of all possible pairs of frequencies form rational
numbers. This indicates that a fundamental period exists that will satisfy the equation of periodic signals.
Transient signals are defined as all nonperiodic signals other than the almost-periodic signals
discussed above. In other words, transient signals include all signals not previously discussed that can be
described by some suitable time-varying function.
Periodic Signals Almost-periodic Signal

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Random signals
29
A single time history representing a random phenomenon is called a sample function (or a sample
record when observed over a finite time interval). The collection of all possible sample functions that the
random phenomenon might have produced is called a random process or a stochastic process. Hence, a
signal for a random physical phenomenon may be thought of as one physical realization of a random
process (signal).
For example, consider the collection of sample
functions (also called the ensemble) that forms the
random process (signals). The mean value (first
moment) can be computed by taking the instantaneous
value of each sample function of the ensemble at time,
summing the values, and dividing by the number of
sample functions. In a similar manner, a correlation (joint
moment) between the values of the random process
(signals) at two different times (called the
autocorrelation function) can be computed by taking the
ensemble average of the product of instantaneous
values at two times variables.

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Stationary and ergodic signals
30
We first consider the mean value and a correlation as the statistical measures. For the general case
statistical measures vary as time varies, the random process (signals) is said to be non-stationary. For the
special case statistical measures do not vary as time varies, the random process (signals) is said to be
weakly stationary or stationary in the wide sense.
An infinite collection of higher order moments and joint moments of the random process (signals)
could also be computed to establish a complete family of probability distribution functions describing the
process. For the special case where all possible moments and joint moments are time invariant, the
random process (signals) is said to be strongly stationary or stationary in the strict sense. For many
practical applications, verification of weak stationarity will justify an assumption of strong stationarity.
In most cases, however, it is also possible to describe the properties of a stationary signals by
computing time averages over specific sample functions in the ensemble. If the random process (signals)
is stationary, and statistical measures do not differ when computed over different sample functions, the
random process (signals) is said to be ergodic. Otherwise, it is non-ergodic signals.

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ELEMENTARY SIGNALS
Functions of elementary signals
31
• Unit step function
The unit step function is defined as follows:
• Rectangular pulse function
The rectangular pulse function is defined as follows:
1 0
( )
0 0
t
u t
t


= 


1 | | 2
rect( )
0 | | 2
t
t
t





= 



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ELEMENTARY SIGNALS
32
• Signum (or sign) function
The signum (or sign) function is defined as follows:
• Ramp function
The ramp function is defined as follows:
1 0
sgn( ) 0 0
1 0
t
t t
t



= =

− 

0
( ) ( )
0 0
t t
r t tu t
t


= = 



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ELEMENTARY SIGNALS
33
• Sinusoidal function
The sinusoidal function is defined as follows:
• Sinc function
The sinc function is defined as follows:
0 0
( ) sin( ) sin(2 )
x t t f t
   
= + = +
0
0
0
sin( )
( ) sinc( )
t
x t t
t



= =

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ELEMENTARY SIGNALS
34
• Exponential function
The exponential function is defined as follows:
0
( )
0 0
( ) (cos sin )
i t
st t
x t e e e t i t
  
 
+
= = = +

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ELEMENTARY SIGNALS
35
• (Unit) impulse function
The (unit) impulse function, also known as the Dirac delta function or simply the delta function, is
defined as follows:
( ) 0 0
( ) 1
t t
t dt



−
= 
=


Shieh-Kung
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INTRODUCTION OF SYSTEMS
General description of systems
36
Another important component of signal processing is a system that usually abstracts a physical
process. The systems studied here is assumed to have some input terminals and output terminals as
shown in the following figure. We assume that if an excitation or input is applied to the input terminals, a
unique response or output signal can be measured at the output terminals. This unique relationship
between the excitation and response, input and output, or cause and effect is essential in defining a
system.

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CLASSIFICATION OF SYSTEMS
SISO, SIMO, and MIMO systems
37
A system with only one input terminal and only one output terminal is called a single-variable system
or a single-input single-output (SISO) system. A system with two or more input terminals and/or two or
more output terminals is called a multivariable system. More specifically, we can call a system a multi-
input multi-output (MIMO) system if it has two or more input terminals and output terminals, a single-input
multi-output (SIMO) system if it has one input terminal and two or more output terminals.

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Continuous-time and discrete-time systems
38
A system is called a continuous-time system if it accepts continuous-time signals as its input and
generates continuous-time signals as its output. The input will be denoted by lowercase italic x(t) for single
input or by boldface x(t) for multiple inputs. Similarly, the output will be denoted by y(t) or y(t). The time t is
assumed to range from −∞ to +∞.
A system is called a discrete-time system if it accepts discrete-time signals as its input and generates
discrete-time signals as its output. All discrete-time signals in a system will be assumed to have the same
sampling period Δt. The input and output will be denoted by x[k] := x(kΔt) and y[k] := y(kΔt), where k
denotes discrete-time instant and is an integer ranging from −∞ to +∞. They become boldface for multiple
inputs and multiple outputs.

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Linear and non-linear systems
39
A system is called a linear system if the additivity and homogeneity properties can be applied for any
time instant.
Additivity (or super-position) property:
Homogeneity property:
Otherwise, it is non-linear systems. The systems to be studied here are limited to linear systems.
A system's output for t > 0 is the result of two independent causes: the initial conditions of the system
(or the system state) at t = 0 and the input x(t) for t > 0. If a system is to be linear, the output must be the
sum of the two components resulting from these two causes: first, the zero-input response component
that results only from the initial conditions at t = 0 with no input for t > 0, and then the zero-state response
component that results only from the input x(t) for t > 0 when the initial conditions (at t = 0) are assumed to
be zero.
When all the appropriate initial conditions are zero, the system is said to be in zero state. The system
output is zero when the input is zero only if the system is in zero state. In other words, if the input x(t) to a
linear system is zero, then the output y(t) must also be zero for all time t. This property is referred to as the
zero-input, zero-output property.
system system
( ) ( ) ( ) ( ) and
x t y t x t y t
  
⎯⎯⎯
→  ⎯⎯⎯
→ 
system system system
1 1 2 2 1 2 1 2
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
x t y t x t y t x t x t y t y t
⎯⎯⎯
→ ⎯⎯⎯
→  + ⎯⎯⎯
→ +

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Time-varying and time-invariant systems
40
A system is said to be time invariant if the time shifting property can be applied for any time instant.
Time Shifting Property:
In other words, if the initial state and the input are the same, no matter at what time they are applied, the
output waveform will always be the same. Therefore, for time-invariant systems, we can always assume,
without loss of generality, that t0 = 0. If a system is not time invariant, it is said to be time invariant (time-
varying).
system system
0 0 0
( ) ( ) ( ) ( ) and
x t y t x t T y t T T
⎯⎯⎯
→  + ⎯⎯⎯
→ + 

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Systems with and without memory
41
A continuous-time system is said to be without memory (memoryless or instantaneous) if its output y(t)
at time t = t0 depends only on the values of the applied input x(t) at the same time t = t0. On the other hand,
if the response of a system at t = t0 depends on the values of the input x(t) in the past or in the future of
time t = t0, it is called a dynamic system, or a system with memory. Likewise, a discrete-time system is
said to be memoryless if its output depends only on the value of its input at the same instant. Otherwise,
the discrete-time system is said to have memory.

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Other classifications
42
• Analog and digital systems
A system whose input and output signals are analog is an analog system; a system whose input
and output signals are digital is a digital system. A digital computer is an example of a digital (binary)
system Observe that a digital computer is an example of a system that is digital as well as discrete-
time.
• Causal and non-causal systems
A system is causal (also known as a physical or non-anticipative) if the output at time t0 depends
only on the input x(t) for t ≤ t0. A system that violates the causality condition is called a non-causal (or
anticipative) system. Note that all memoryless systems are causal systems because the output at any
time instant depends only on the input at that time instant. Systems with memory can either be causal
or non-causal.
• Invertible and non-invertible systems
A system is invertible if the input signal x(t) can be uniquely determined from the output y(t)
produced in response to x(t) for all time t ∈ (−∞,∞). To be invertible, two different inputs cannot produce
the same output since, in such cases, the input signal cannot be uniquely determined from the output
signal. At the same time, the system is said to be non-invertible.

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ELEMENTARY SYSTEMS
Structural systems
43
The following figure is the free-body diagram at time t with the mass replaced by its inertia force. The
forces acting on the mass at some instant of time are balanced according to D’Alember’s principle of
dynamic equilibrium. These include the external force p, the elastic (or inelastic) resisting force fS, the
damping resisting force fD, and the inertial force fI.
( ) ( ) or ( ) ( ) ( ) and ( ) or ( ( ), ( ))
S D D S D S S
p t f f mu t mu t f f p t f cu t f ku t f f u t u t
− − = + + =  = = =

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ELEMENTARY SYSTEMS
Mass–spring–damper systems
44
We have introduced the SDOF system by idealizing a one-story structure, an approach that should
appeal to structural engineering students. However, the classic SDOF system is the mass–spring–damper
system of the following figure.
( ) ( ) ( ) ( ) or ( ) ( ) ( ) and ( ) or ( ( ), ( ))
D S D S S
mu t cu t ku t p t mu t f f p t f cu t f ku t f f u t u t
+ + = + + =  = = =

Shieh-Kung
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ELEMENTARY SYSTEMS
RLC (resistor–inductor–capacitor) circuit systems
45
Another example is an electrical circuit consisting of three passive components: resistor R, inductor L,
and capacitor C. Applying Kirchhoff’s voltage law, the relationship between the input voltage x(t) and the
loop current q(t) is given by.
1 1
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
t
Lq t Rq t q t dt x t Lq t Rq t q t x t
C C
−
+ + =  + + =


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TIME-DOMAIN ANALYSIS OF SYSTEMS
Impulse responses of a system
46
The impulse response h(t) of an linear time-invariant (LTI) system is the output of the system when a
unit impulse (t) is applied at the input. Following the notation introduced the impulse response function
can be expressed as
with zero initial conditions. Because the system is LTI, it satisfies the linearity and the time-shifting
properties. If the input is a scaled and time-shifted impulse function a(t − t0), the output of the system is
also scaled by the factor of a and is time-shifted by T0, i.e.
for any arbitrary constants a and T0. Hence, the zero-input response that results only from the initial
conditions at t = 0 can be computed by the impulse response.
system
( ) ( )
t h t
 ⎯⎯⎯
→
system
0 0
( ) ( )
a t T ah t T
 − ⎯⎯⎯
→ −

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Convolution integral
47
An arbitrary continuous-time signal x(t) can be approximated by the staircase approximation
illustrated in the following figure. The approximated function is given by
As Δ → 0, the summations on all the staircase approximation become integrations. Substituting kΔt by
 and Δt by d , we obtain the following relationship:
where  is the dummy variable that disappears as the integration with limits is computed. The integral on
the left-hand side of the equation is referred to as the convolution integral and is denoted by x(t) ∗ h(t).
( ) ( ) ( )
k
x t x k t t k t t



=−
=  −  

( ) ( ) ( ) ( ) ( )
x t x t d x h t d
      
 
− −
= − = −
 

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Properties of convolution integral
48
Mathematically, the convolution of two functions x(t) and h(t) is defined as follows:
Some important properties of the convolution integral includes
• Commutative Property
• Distributive Property
• Associative Property
• Shift Property
( ) ( ) ( ) ( )
x t h t x h t d
  

−
 = −

1 2 2 1
( ) ( ) ( ) ( )
x t x t x t x t
 = 
 
1 2 3 1 2 1 3
( ) ( ) ( ) ( ) ( ) ( ) ( )
x t x t x t x t x t x t x t
 + =  + 
   
1 2 3 1 2 1
( ) ( ) ( ) ( ) ( ) ( )
x t x t x t x t x t x t
  =  
1 2 1 1 2 2 1 2
( ) ( ) ( ) ( ) ( ) ( )
x t x t g t x t T x t T g t T T
 =  −  − = − −

Shieh-Kung
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Graphical understanding of convolution
49
Given input x(t) and impulse response h(t) of the system, convolution
integral can be performed graphically by following figures
( ) ( ) ( ) ( )
x t h t x h t d
  

−
 = −


Shieh-Kung
Huang
50
( ) ( ) ( ) ( )
x t h t x h t d
  

−
 = −


Shieh-Kung
Huang
51
( ) ( ) ( ) ( )
x t h t x h t d
  

−
 = −


Shieh-Kung
Huang
52
0
[ ] [ ] [ ] [ ]
N
m
x k h k x m h k m
=
 = −


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Huang
FOURIER TRANSFORM
Chapter Outline
53
2.1 Fourier Series
2.2 Fourier Transform
2.3 Discrete Fourier Transform
2.4 Fast Fourier Transform
2.5 Frequency-Domain Analysis
CHAPTER 2

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Huang
FOURIER SERIES
Trigonometric Fourier series
54
An arbitrary periodic function x(t) with fundamental period T0 can be expressed as follows:
where is the fundamental frequency of x(t) and coefficients a0, an, and bn are referred to as the
trigonometric continuous-time Fourier series coefficients. The coefficients are calculated as follows:
 
0 0 0
1
( ) cos( ) sin( )
n n
n
x t a a n t b n t
 

=
= + +

0 0
2 T
 
=
0
0
0
0
0
0
0
0
0
1
( )
2
( )cos( )
2
( )sin( )
T
n T
n T
a x t dt
T
a x t n t dt
T
b x t n t dt
T


=
=
=



Sawtooth Wave
Square Wave
Triangle Wave

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Huang
FOURIER SERIES
Exponential Fourier series
55
An arbitrary periodic function x(t) with a fundamental period T0 can be expressed as follows:
where the exponential continuous-time Fourier series coefficients Dn are calculated as
0 being the fundamental frequency given by .
0
0
( ) in t
n
n
x t D e 

=
= 
0
0
0
1
( ) in t
n T
D x t e dt
T

−
= 
0 0
2 T
 
=

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Huang
FOURIER TRANSFORM
Fourier transform
56
Consider the aperiodic signal x(t). In order to extend the Fourier framework of the continuous-time
Fourier series coefficients to aperiodic signals, let us define a continuous function X() (with the
independent variable ) as
where F[ ·] is Fourier transform operator. The inverse Fourier transform can be written as
 
( ) ( ) ( ) i t
X F x t x t e dt



−
−
= = 
 
1 1
( ) ( ) ( )
2
i t
x t F X X e d

  


−
−
= = 

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Huang
FOURIER TRANSFORM
Fourier transform pairs
57

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Huang
FOURIER TRANSFORM
58

Shieh-Kung
Huang
FOURIER TRANSFORM
59

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Huang
FOURIER TRANSFORM
60

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Huang
FOURIER TRANSFORM
61

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FOURIER TRANSFORM
Properties of Fourier transform
62
Given the Fourier transform of a continuous-time function x(t), we are interested in calculating the
Fourier transform of a function produced by a linear operation on x(t) in the time domain. The linear
operations being considered include superposition, time shifting, scaling, differentiation and integration.
We also consider some basic non-linear operations like multiplication of two signals, convolution in the
time and frequency domain, and Parseval’s relationship.
• Symmetry
The symmetry property states that
FT FT
IFT IFT
( ) ( ) ( ) 2 ( )
x t X X t x
  
⎯⎯→ ⎯⎯→ −
⎯⎯ ⎯⎯

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FOURIER TRANSFORM
63
• Linearity
The linearity property states that
FT
1 1 2 2 1 1 2 2
IFT
( ) ( ) ( ) ( )
a x t a x t a X a X
 
⎯⎯→
+ +
⎯⎯

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Huang
FOURIER TRANSFORM
64
• Time shifting
The time shifting property states that
0
FT
0 IFT
( ) ( )
i t
x t t e X


−
⎯⎯→
− ⎯⎯

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Huang
FOURIER TRANSFORM
65
• Frequency shifting
The linearity property states that
0
FT
0
IFT
( ) ( )
i t
e x t X

 
⎯⎯→ −
⎯⎯

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Huang
FOURIER TRANSFORM
66
• Time scaling
The time scaling property states that
FT
IFT
1
( ) ( )
| |
x at X
a a

⎯⎯→
⎯⎯

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Huang
FOURIER TRANSFORM
67
• Frequency scaling
The frequency scaling property states that
FT
IFT
1
( ) ( )
| |
t
x X a
a a

⎯⎯→
⎯⎯

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Huang
FOURIER TRANSFORM
68
• Even or odd function
The Fourier transform of an even or odd function is

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Huang
FOURIER TRANSFORM
69
• Time Convolution
The time convolution property states that
1 2 1 2
( ) ( ) ( ) ( )
x t x t X X
 
 =

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Huang
FOURIER TRANSFORM
70
• Time Convolution
The time convolution property states that
1 2 1 2
( ) ( ) ( ) ( )
x t x t X X
 
 =

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Huang
FOURIER TRANSFORM
71
• Frequency Convolution
The frequency convolution property
states that
 
1 2 1 2
1
( ) ( ) ( ) ( )
2
x t x t X X
 

 = 

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Huang
FOURIER TRANSFORM
72

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Huang
FOURIER TRANSFORM
73

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Huang
FOURIER TRANSFORM
Time convolution of structural system
74
Given input x(t) and impulse response h(t) of the system, convolution integral can be performed
simply in the frequency domain
( ) ( ) ( ) ( ) ( ) ( )
x t h t x h t d X H
    

−
 = − =


Shieh-Kung
Huang
FOURIER TRANSFORM
Animated interpretation of Fourier transform
75
https://www.youtube.com/watch?v=spUNpyF58BY

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DISCRETE FOURIER TRANSFORM
Discrete-time Fourier transform
76
• Discrete-time Fourier series
A discrete-time periodic function x[k] with period N can be expressed as a superposition of
discrete-time complex exponentials as follows:
where W0 is the fundamental frequency, given by W0 = 2/N, and the discrete-time Fourier series
coefficients Dn for 0 ≤ n ≤ N − 1 are given by
In the equations, the limit of 0 ≤ k ≤ N − 1 implies that the sum can be taken over any N consecutive
samples of x[k].
• Discrete-time Fourier Transform
The discrete-time Fourier transform pair for an aperiodic sequence x[k] is given by discrete-time
Fourier transform shown as
and inverse discrete-time Fourier transform shown as
0
1
0
1
[ ]
N
in k
n
k
D x k e
N
−
− W
=
= 
0
1
0
[ ]
N
in k
n
n
x k D e
−
W
=
= 
( ) [ ] i k
k
X x k e

− W
=−
W = 
2
1
[ ] ( )
2
i t
x k X e d


W
= W W


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Huang
Discrete-time Fourier transform pairs
77

Shieh-Kung
Huang
Discrete-time Fourier transform pairs
78

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Huang
Properties of discrete-time Fourier transform
79

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Huang
Properties of discrete-time Fourier transform
80

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Huang
Discrete Fourier transform
81
The M-point discrete Fourier transform and inverse discrete Fourier transform for a time-limited
sequence x[k], which is non-zero within the limits 0 ≤ k ≤ N − 1, is given by
The above two equations are also, respectively, known as discrete Fourier transform pair. In the equations,
the length M of the discrete Fourier transform is typically set to be greater or equal to the length N of the
aperiodic sequence x[k]. Unless otherwise stated, we assume M = N in the discussion that follows.
Collectively, the DFT pair is denoted as
• Properties of Discrete Fourier Transform
The discrete Fourier transform generally holds the same properties as the discrete-time Fourier
transform.
2
1 ( )
0
2
1 ( )
0
[ ] [ ] for 0 1
1
[ ] [ ] for 0 1
r
N i k
M
k
k
M ir
M
r
X r x k e r M
x k X r e k N
M


− −
=
− −
=
=   −
=   −


DFT
IDFT
[ ] [ ]
x k X r
⎯⎯⎯
→
⎯⎯
⎯

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Huang
82
• Matrix Formulation
An alternative representation for computing the discrete Fourier transform is obtained by
expanding the equations in terms of the time and frequency indices (k,r). For N = M, the resulting
equations are expressed as follows:
and
(2 ) (4 ) (2( 1) )
(4 ) (8 ) (4( 1) )
(2( 1) ) (4( 1) ) (2( 1)( 1) )
[ ] [ ]
[0] 1 1 1 1 [0]
[1] 1 [1]
[2] 1 [2]
[ 1] 1 [ 1]
i N i N i N N
i N i N i N N
i N N i N N i N N N
r k
X x
X e e e x
X e e e x
X N e e e x N
  
  
  
− − − −
− − − −
− − − − − − −
=
    
   
   
   
=
   
   
   
− −
   
X Fx

 
 
 
 
 
 
 
1
(2 ) (4 ) (2( 1) )
(4 ) (8 ) (4( 1) )
(2( 1) ) (4( 1) ) (2( 1)( 1) )
[ ] [ ] [ ]
[0] 1 1 1 1 [0]
[1] 1 [1]
1
[2] 1 [2]
[ 1] 1 [ 1]
i N i N i N N
i N i N i N N
i N N i N N i N N N
k r r
x X
x e e e X
x e e e X
M
x N e e e X N
  
  
  
−
−
−
− − − −
= =
   
   
   
   
=
   
   
   
− −
   
x GX F X
 
 
 
 
 
 
 
 

Shieh-Kung
Huang
83
Or, we can use the following matrix formulation by assuming
and
because and
2 1
2 4 2
1 2
[ ] [ ]
[0] 1 1 1 1 [0]
[1] 1 [1]
[2] 1 [2]
[ 1] 1 [ 1]
N
N
N N
r k
X x
X W W W x
X W W W x
X N W W W x N
−
−
− −
=
     
     
     
     
=
     
     
     
− −
     
X Fx
1
[ ] [ ] [ ]
k r r
−
= =
x GX F X
(2 )
i N
W e 
−
=
(2 ( 1) ) (2 )
i N N i N
e e W
 
− + −
= =
(2 )
1
i N N
e 
−
=

Shieh-Kung
Huang
84
• Frequency Resolution
From the equations, we can see the data points in the frequency domain are the same as the
ones in the time domain. It possesses a constant frequency range and, therefore, the resolution of the
frequency axis r can be determined by equally distributing the data points in the constant frequency
range.
• Zero Padding
To improve the resolution of the frequency axis r in the discrete frequency domain, a commonly
used approach is to append the discrete-time sequences with additional zero-valued samples. This
process is called zero padding, and for an aperiodic sequence x[k] of length N is defined as follows:
The zero-padded sequence xzp[k] has an increased length of M. The frequency resolution Δr of the
zero-padded sequence is improved from 2/N to 2/M.
zp
[ ] 0 1
[ ]
0 1
x k k N
x k
N k M
  −

= 
  −


Shieh-Kung
Huang
Illustration of discrete Fourier transform
85

Shieh-Kung
Huang
86
• A band-limited periodic waveform: truncation interval equal to period.

Shieh-Kung
Huang
87
• A band-limited periodic waveform: truncation interval not equal to period.

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Huang
FAST FOURIER TRANSFORM
Fast Fourier transform
88
There are several well known techniques including the radix-2, radix-4, split radix, Winograd, and
prime factor algorithms that are used for computing the discrete Fourier transform . These algorithms are
referred to as the fast Fourier transform (FFT) algorithms. In this course, we explain the radix-2
decimation-in-time FFT algorithm.
• Intuitive Observation
2 3 2 3
2 4 6 2 2
3 6 9 3 2
[0] 1 1 1 1 [0] 1 1 1 1 [0]
[1] 1 [1] 1 [1]
[2] 1 [2] 1 1 [2]
[3] 1 [3] 1 [3]
X x x
X W W W x W W W x
X W W W x W W x
X W W W x W W W x
         
         
         
= =
         
         
         
0 0
2 0
2
3 2
[0] [0]
1 0 0 1 0 0
[2] [1]
1 0 0 0 1 0
[1] [2]
0 0 1 1 0 0
[3] [3]
0 0 1 0 1 0
X x
W W
X x
W W
X x
W W
X x
W W
   
   
   
   
   
   
=
   
   
   
   
   
   

Shieh-Kung
Huang
Fast Fourier transform
89

Shieh-Kung
Huang
Fast algorithms
90
• Dual Nodes
Inspection of the figure reveals
that in every array we can always
find two nodes whose input
transmission paths stem from the
same pair of nodes in the previous
array. For example, nodes xl[0] and
xl[8] are computed in terms of
nodes x0[0] and x0[8]. Note that
nodes x0[0] and x0[8] do not enter
into the computation of any other
node. We define two such nodes as
a dual node pair.

Shieh-Kung
Huang
Fast algorithms
91
• Alternative Ways

Shieh-Kung
Huang
Fast algorithms
92
• fft Function
The MATLAB functions fft and ifft can compute the fast Fourier transform and its inverse of a
discrete-time signal.

Shieh-Kung
Huang
FREQUENCY-DOMAIN ANALYSIS
Parametric study
93
• Effect of Data Length (truncation interval equal to period)

Shieh-Kung
Huang
Parametric study
94
• Effect of Data Length (truncation interval equal to period)
The data length has a direct effect on the resolution in frequency domain.

Shieh-Kung
Huang
Parametric study
95
• Effect of Data Length (truncation interval not equal to period)
3.3 Hz Sine Waves

Shieh-Kung
Huang
Parametric study
96
3.3 Hz Sine Waves

Shieh-Kung
Huang
Parametric study
97
The data length has a direct effect on the resolution in frequency domain, and the spectral
leakage occurred.
Leakage

Shieh-Kung
Huang
Parametric study
98
• Effect of Sampling Rate (truncation interval equal to period)

Shieh-Kung
Huang
Parametric study
99
• Effect of Sampling Rate (truncation interval equal to period)
The sampling rate has a direct effect on the range in frequency domain.

Shieh-Kung
Huang
Parametric study
100
• Effect of Sampling Rate (truncation interval not equal to period)
3.3 Hz Sine Waves

Shieh-Kung
Huang
Parametric study
101
3.3 Hz Sine Waves

Shieh-Kung
Huang
Parametric study
102
The sampling rate has a direct effect on the range in frequency domain, and the spectral leakage
occurred.
Leakage
Leakage

Shieh-Kung
Huang
Spectral leakage
103
Leakage, more explicitly called spectral leakage, is a smearing of power across a frequency
spectrum that occurs when the signal being measured is not periodic in the sample interval. It occurs
because discrete sampling results in the effective computation of a Fourier series of a waveform having
discontinuities, which result in additional frequency components. Leakage is the most common error
encountered in digital signal processing, and while its effects cannot be entirely eliminated, they may
sometimes be reduced with the aid of a suitable window functions (or apodization functions).

Shieh-Kung
Huang
Spectral leakage
104
• Window Functions
To reduce spectral leakage, a mathematical function called a window function is applied to the
data. Window functions are designed to reduce the sharp transient in the re-created signal as much as
possible. They are typically shaped as functions that start at a value of zero, move to a value of one,
and then return to a value of zero over one frame.

Shieh-Kung
Huang
Spectral leakage
105

Shieh-Kung
Huang
Spectral leakage
106
• rectwin
• hamming
• hann
• gausswin

Shieh-Kung
Huang
Spectral leakage
107
• Sidelobes (or Ripples)
The sidelobes (or ripples) from a strong source, or the combined sidelobes from an extended
source, make it impossible to analyze subtle features in the raw signals (the raw signals being that
formed by simply transforming the Fourier data).

Shieh-Kung
Huang
Spectral leakage
108
• Effect of Window Functions
3.3 Hz Sine Waves

Shieh-Kung
Huang
Spectral leakage
109
3.3 Hz Sine Waves

Shieh-Kung
Huang
Spectral leakage
110
The window functions can effectively reduce the spectral leakage in the discrete frequency
domain.

Shieh-Kung
Huang
Spectral leakage
111
• Effect of Zero Padding
3.3 Hz Sine Waves

Shieh-Kung
Huang
Spectral leakage
112
3.3 Hz Sine Waves

Shieh-Kung
Huang
Spectral leakage
113
The zero padding can not only improve the resolution of the frequency axis, but also reduce the
spectral leakage in the discrete frequency domain.

Shieh-Kung
Huang
FFT of free decay responses
114
• Free Decay Responses (Zero-input Responses)

Shieh-Kung
Huang
115
• FFT of Free Decay Responses

Shieh-Kung
Huang
116
• FFT of Free Decay Responses 0
( )
0 0
2
( ) (cos sin )
(cos sin ) where 1
n
i t
st t
t
D D D n
x t e e e t i t
e t i t
  

 
    
+
−
= = = +
= + = −

Shieh-Kung
Huang
SAMPLING OF SIGNALS
Chapter Outline
117
3.1 Sampling of Signals
3.2 Aliasing
3.3 Quantization
CHAPTER 3

Shieh-Kung
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Continuous-time and
Analog Signals
Discrete-time and
Digital Signals
SAMPLING OF SIGNALS
Measurement process
118
• Typical Process
• Data Acquisition
The process of representing an analog signal as a series of
digital values is a basic requirement of modern digital signal
processing analyzers. This process is called data acquisition. In
practice, the goal of the analog to digital conversion process is to
obtain the conversion while maintaining sufficient accuracy in terms
of frequency, magnitude, and phase.
Physical
Response
Sensor Sensor Gain
Analog-to-digital
Converter (ADC)
Digital
Computer
Sampling & Quantization

Shieh-Kung
Huang
SAMPLING OF SIGNALS
Sampling theorem
119
• Sampling
In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time
signal. A sampler is a subsystem or operation that extracts samples from a continuous signal. A
theoretical ideal sampler produces samples equivalent to the instantaneous value of the continuous
signal at the desired points.
• Nyquist Frequency
In signal processing, the Nyquist frequency (or folding frequency), named after Harry Nyquist, is a
characteristic of a sampler, which converts a continuous function or signal into a discrete sequence.
For a given sampling rate (samples per second), the Nyquist frequency (cycles per second) is the
frequency whose cycle-length (or period) is twice the interval between samples, thus 0.5 cycle/sample.
2
s
n
f
f =

Shieh-Kung
Huang
SAMPLING OF SIGNALS
Sampling theorem
120
• General Description of Sampling Theorem
A baseband signal x(t), band-limited to 2f radians/s, can be reconstructed accurately from its
samples x[k] := x(kΔt) if the sampling rate s, in radians/s, satisfies the following condition:
Alternatively, the sampling theorem may be expressed in terms of the sampling rate s = 2fs in
samples/s, or the sampling interval Δt = Ts. To prevent aliasing
sampling rate (samples/s)
or
sampling interval
The minimum sampling rate fs (Hz) required for perfect reconstruction of the original band-limited
signal is referred to as the Nyquist rate.
4
s f
 

1
2
s
T t
f
=  
2
s
f f


Shieh-Kung
Huang
SAMPLING OF SIGNALS
Sampling theorem
121
• Illustration of Sampling Theorem

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Huang
ALIASING EFFECT
Aliasing
122
In signal processing and related disciplines, aliasing is the overlapping of frequency components
resulting from a sample rate below the Nyquist rate. This overlap results in distortion or artifacts when the
signal is reconstructed from samples which causes the reconstructed signal to differ from the original
continuous signal.

Shieh-Kung
Huang
ALIASING EFFECT
Aliasing effect
123
• Illustration of Aliasing in Frequency Domain

Shieh-Kung
Huang
ALIASING EFFECT
Aliasing effect
124
• Illustration of Aliasing in Frequency Domain

Shieh-Kung
Huang
• Examples of Aliasing
Aliasing that occurs in signals sampled in time, for
instance in digital audio or the stroboscopic effect, is
referred to as temporal aliasing. Aliasing in spatially
sampled signals (e.g., moiré patterns in digital images) is
referred to as spatial aliasing.
Aliasing is generally avoided by applying low-pass
filters or anti-aliasing filters (AAF) to the input signal before
sampling and when converting a signal from a higher to a
lower sampling rate. Suitable reconstruction filtering should
then be used when restoring the sampled signal to the
continuous domain or converting a signal from a lower to a
higher sampling rate. For spatial anti-aliasing, the types of
anti-aliasing include fast approximate anti-aliasing (FXAA),
multisample anti-aliasing, and supersampling.
ALIASING EFFECT
Aliasing effect
125
lower sampling rate. Suitable
reconstruction filtering should then
be used when restoring the sampled
signal to the continuous domain or
converting a signal from a lower to a
higher sampling rate.

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QUANTIZATION
Quantization
126
• Uniform and Non-uniform Quantization
Quantization, in mathematics and digital signal processing, is the process of mapping input
values from a large set (often a continuous set) to output values in a (countable) smaller set, often with
a finite number of elements. Rounding and truncation are typical examples of quantization processes.
The following figure illustrates the input–output relationship for an L-level uniform and non-uniform
quantizer. The quantization levels from d0 to dL are referred to as the decision levels, while the output
levels from r0 to rL−1 are referred to as the reconstruction levels.

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QUANTIZATION
Quantization
127
• Analog-to-digital Converter (ADC)
An analog-to-digital converter (ADC) can be modeled as two processes: sampling and
quantization. Sampling converts a time-varying voltage signal into a discrete-time signal, a sequence
of real numbers. Quantization replaces each real number with an approximation from a finite set of
discrete values. Most commonly, these discrete values are represented as fixed-point words. Though
any number of quantization levels is possible, common word-lengths are 8-bit (256 levels), 16-bit
(65,536 levels) and 24-bit (16.8 million levels). Quantizing a sequence of numbers produces a
sequence of quantization errors which is sometimes modeled as an additive random signal called
quantization noise because of its stochastic behavior. The more levels a quantizer uses, the lower is
its quantization noise power.

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QUANTIZATION
Examples of quantization
128
• Examples of Sinusoidal Signal

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QUANTIZATION
129
• Examples of Ramp Signal

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QUANTIZATION
130
• Examples of Random Signal

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QUANTIZATION
131
• Examples of Random (Small Noise) Signal

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FILTERS
Chapter Outline
132
4.1 Introduction of Filters
4.2 Classification of digital filter
4.3 Filter realization
4.4 Filter Design
4.5 Filter Implementation
4.6 Applications of Filters
CHAPTER 4

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Chapter 4 Filters
INTRODUCTION OF FILTERS
General description of filters
133
A filter is defined as a system that transforms a signal, applied at the input of the filter, by changing its
frequency characteristics in a predefined manner. A convenient classification of filters is obtained by
specifying the shape of their magnitude and phase spectra in the frequency domain.

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Chapter 4 Filters
CLASSIFICATION OF DIGITAL FILTER
Analog and digital filters
134
In signal processing, a digital filter is a system that performs mathematical operations on a sampled,
discrete-time signal to reduce or enhance certain aspects of that signal. This is in contrast to the other
major type of electronic filter, the analog filter, which is typically an electronic circuit operating on
continuous-time analog signals.

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Chapter 4 Filters
Ideal lowpass, highpass, bandpass, and bandstop filters
135
A filter is often classified on the basis of the magnitude and phase spectra derived from its transfer
function. In the case of ideal filters, the shape of the magnitude spectrum is rectangular with a sharp
transition between the range of frequencies passed and the range of frequencies blocked by the filter. The
range of frequencies passed by the filter is referred to as the passband of the filter, while the range of
blocked frequencies is referred to as the stopband.
Lowpass filter Highpass filter
Bandpass filter Bandstop filter

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Chapter 4 Filters
136
Typically in electronic systems such as filters and communication channels, cut-off frequency applies
to an edge in a lowpass, highpass, bandpass, or band-stop characteristic – a frequency characterizing a
boundary between a passband and a stopband. It is sometimes taken to be the point in the filter response
where a transition band and passband meet, for example, as defined by a half-power point (a frequency
for which the output of the circuit is −3 dB of the nominal passband value).

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Chapter 4 Filters
137
• Decibel
The decibel (symbol: dB) is a relative unit of measurement equal to one tenth of a bel (B). It
expresses the ratio of two values of a power or root-power quantity on a logarithmic scale. Two signals
whose levels differ by one decibel have a power ratio of 101/10 (approximately 1.26) or root-power ratio
of 101⁄20 (approximately 1.12).
1
10
0
2
1 1
10 10
2
0 0
10log ( )
10log ( ) 20log ( )
db
db
P
L
P
A A
L
A A
=
= =

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Chapter 4 Filters
Finite impulse response (FIR) and infinite impulse response (IIR) filters
138
Another classification of filters is made on the length of their impulse response h(t). A finite impulse
response (FIR) filter is defined as a filter whose length is finite. The length (or width) of a filter is limited
beyond which the impulse response h(t) is zero in both directions along the t-axis. On the other hand, if
the length of the filter is infinite, the filter is called an infinite impulse response (IIR) filter.
A IIR Lowpass Filter
A FIR Lowpass Filter

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Chapter 4 Filters
FILTER REALIZATION
Infinite impulse response (IIR) filters
139
In the time domain, the output response (or filtered response) y[k] can be determined from its input
x[k] either by solving a linear, constant-coefficient, difference equation of the following form:
or, alternatively, by calculating the convolution sum between the input x[k] and the impulse response h[k].
The convolution sum is given by
In the frequency domain, the convolution property is used to express the convolution sum in terms of the
transfer function H() and the continuous-time Fourier transform X() of the input as follows:
1 0 1
[ ] [ 1] [ ] [ ] [ 1] [ ]
N M
y k a y k a y k N b x k b x k b x k M
+ − + + − = + − + + −
[ ] [ ] [ ] [ ] [ ]
m
y k x k h k x m h k m

=−
=  = −

( ) ( ) ( )
Y X H
  
=

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Chapter 4 Filters
FILTER REALIZATION
Finite impulse response (FIR) filters
140
The previous slide is a non-causal IIR filter; for a causal FIR filter, the output response (or filtered
response) y[k] can be determined from its input x[k]:
or, alternatively, by calculating the convolution sum between the input x[k] and the impulse response h[k].
The convolution sum is given by
In the frequency domain, the convolution property is also expressed in terms of the transfer function H()
and the continuous-time Fourier transform X() of the input as follows:
0 1
[ ] [ ] [ 1] [ ]
M
y k b x k b x k b x k M
= + − + + −
( ) ( ) ( )
Y X H
  
=
0
[ ] [ ] [ ] [ ] [ ]
M
m
y k x k h k h m x k m
=
=  = −


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Chapter 4 Filters
FILTER REALIZATION
Transfer function of filters
141
• Bode Plots
Sketching frequency response plots is considerably facilitated by the use of logarithmic scales.
The amplitude and phase response plots as a function of  on a logarithmic scale are known as the
Bode plots. By using the asymptotic behavior of the amplitude and the phase response, we can sketch
these plots with remarkable ease, even for higher-order transfer functions.

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Chapter 4 Filters
FILTER REALIZATION
142
• Bode Plots

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Chapter 4 Filters
FILTER REALIZATION
143
• Bode Plots

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Chapter 4 Filters
FILTER REALIZATION
Other mathematical models
144
• Representations of Filters, Systems, or Processes
Typically, engineers have several ways to develop a mathematical description of the filters that
they want. The filters to be realized are also a system with their impulse response.
• Transform Function (TF) Models
A transfer function (TF) relates a particular input/output pair of (possibly vector) signals.
• Zero/Pole/Gain (ZPK) Models
Like the TF format, the zero/pole/gain (ZPK) format relates an input/output pair of (possibly
vector) signals. The difference is that the ZPK numerator and denominator polynomials are
factored.
• State-Space (SS) Models
The state-space format is convenient if your model is a set of (linear time invariant) LTI
differential and algebraic equations.
2
0 1 2
2
0 1 2
( )
( )
( )
M
M
N
N
Y s b b s b s b s
H s
X s a a s a s a s
+ + + +
= =
+ + + +
0 1 2
0 1 2
( ) ( )( )( ) ( )
( )
( ) ( )( )( ) ( )
M
N
Y s s b s b s b s b
H s k
X s s a s a s a s a
− − − −
= =
− − − −
( ) ( ) ( )
( ) ( ) ( )
t t x t
y t t x t
= +
= +
q Aq B
Cq D

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Chapter 4 Filters
FILTER REALIZATION
145
Discrete-time Model Continuous-time Model
Transform
Function (TF)
Model
Zero/Pole/Gain
(ZPK) Model
State-Space (SS)
Model
2
0 1 2
2
0 1 2
[ ]
M
M
N
N
b b z b z b z
H z
a a z a z a z
− −
− −
+ + + +
=
+ + + +
2
0 1 2
2
0 1 2
( )
M
M
N
N
b b s b s b s
H s
a a s a s a s
+ + + +
=
+ + + +
0 1 2
0 1 2
( )( )( ) ( )
( )
( )( )( ) ( )
M
N
s b s b s b s b
H s k
s a s a s a s a
− − − −
=
− − − −
0 1 2
0 1 2
( )( )( ) ( )
[ ]
( )( )( ) ( )
M
N
z b z b z b z b
H z k
z a z a z a z a
− − − −
=
− − − −
[ 1] [ ] [ ]
[ ] [ ] [ ]
d d
d d
k k x k
y k k x k
+ = +
= +
q A q B
C q D
( ) ( ) ( )
( ) ( ) ( )
t t x t
y t t x t
= +
= +
q Aq B
Cq D

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Chapter 4 Filters
FILTER REALIZATION
146
Numeric linear-time-invariant (LTI) models are the basic building blocks that you use to represent
linear systems. Numeric LTI model objects let you store filters (as well as dynamic systems) in
commonly-used representations. For example, tf models represent transfer functions in terms of the
coefficients of their numerator and denominator polynomials, and ss models represent LTI systems in
terms of their state-space matrices.

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Chapter 4 Filters
FILTER REALIZATION
Table list of MATLAB conversion between models
147
Transfer
Function
State-Space
Zero-pole-
gain Form
Partial
Fraction
Expansion
Lattice Filter
Form
Second-
order
Sections
Form
Convolution
Matrix
Transfer
Function
tf2ss
tf2zp
roots
residuez
residue
tf2latc tf2sos convmtx
State-Space ss2tf ss2zp ss2sos
Zero-pole-
gain Form
zp2tf
poly
zp2ss zp2sos
Partial
Fraction
Expansion
residuez
residue
Lattice Filter
Form
latc2tf
Second-
order
Sections
Form
sos2tf sos2ss sos2zp
Convolution
Matrix

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Chapter 4 Filters
FILTER DESIGN
Practical lowpass, highpass, bandpass, and bandstop filters
148
Lowpass filter Highpass filter
Bandpass filter Bandstop filter

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Chapter 4 Filters
FILTER DESIGN
Practical filter design
149
In the design of lowpass and highpass filters, we need to first select the pass-band frequency Wp and
stop-band frequency Ws. Sometimes, the stop-band frequency Ws is determined by the pass-band
frequency Wp and filter order n.

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Chapter 4 Filters
FILTER DESIGN
Practical filter design
150
Similarly, in the design of bandpass and bandstop filters, we need to select the pass-band frequency
pair and stop-band frequency pair. Sometimes, the stop-band frequency pair is determined by the pass-
band frequency pair and filter order n.

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Chapter 4 Filters
FILTER DESIGN
MATLAB filters design toolbox
151
The design of digital filters can be easily done through the function filterDesigner provided by
MATLAB.

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Chapter 4 Filters
FILTER DESIGN
MATLAB filters design toolbox
152
Various design functions are also provided by MATLAB.

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Chapter 4 Filters
FILTER DESIGN
Comparison of different filters
153
Comparison of digital filters provided by MATLAB.

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Chapter 4 Filters
FILTER DESIGN
154
Various design functions are also provided by MATLAB.
2
1
| ( ) |
1 ( ) n
c
H i


=
+

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Chapter 4 Filters
FILTER DESIGN
155
• Characteristics of Butterworth Filter

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Chapter 4 Filters
FILTER DESIGN
156

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Chapter 4 Filters
FILTER DESIGN
157

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Chapter 4 Filters
FILTER DESIGN
Design of Ormsby filter
158
Ormsby filter can be composed of a
common type of synthetic wavelet in
reflection seismology. According to frequency
convolution, the filter can be designed as
where fa is central pass frequency and B is
transition band. Or, alternatively, the filter can
be designed with superposition as
where fp is pass-band frequency and fs is
stop-band frequency.
( ) 2 sinc(2 )sinc( )
a a
h t f f t Bt
=
2
2
2 2
( )
( )
( ) sinc ( ) sinc ( )
p
s
s p
p s p s
f
f
h t f t f t
f f f f


= −
− −

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Chapter 4 Filters
FILTER IMPLEMENTATION
Filter Implementation
159
There are several ways to implement digital filters. The simplest way is recursively to compute
(convolution integral) the filtered value. Fortunately, MATLAB provides filter and filtfilt to
implement the digital filter.
• filter Function: Filters a signal using a digitalFilter

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Chapter 4 Filters
Filter Implementation
160
• filtfilt Function: Performs zero-phase filtering of a signal with a digitalFilter

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Chapter 4 Filters
Phase distortion
161
Filtering a signal introduces a delay. This means that the output signal is shifted in time with respect
to the input. When the shift is constant, you can correct for the delay by shifting the signal back in time.
Sometimes the filter delays some frequency components more than others. This phenomenon is called
phase distortion. MATLAB provide phasedelay and grpdelay to estimate the delay produced by the
filter.

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Chapter 4 Filters
Phase distortion
162
• phasedelay Function: Phase delay of digital filter
( )
( )
p
 
 

=

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Chapter 4 Filters
Phase distortion
163
• grpdelay Function: Average filter delay (group delay)
( )
( )
g
d
d
 
 

=

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Chapter 4 Filters
APPLICATIONS OF FILTERS
Baseline correction
164
• Remove Offset from
Numerical Integration

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Chapter 4 Filters
Baseline correction
165
• Remove Offset from
Numerical Integration
Characteristics of Butterworth Filter

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Chapter 4 Filters
Noise reduction
166
• Enhancement of Experimental Measurement

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Chapter 4 Filters
Noise reduction
167
• Enhancement of Experimental
Measurement
DAQ
Sampling Rate is 200 Hz
Load Cell
Velocimeter
LVDT

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Chapter 4 Filters
Noise reduction
168
• Sine Wave with f = 1.5 Hz and A = +/- 40 mm
• MR Damper with 0 Volt
• Sampling Rate is 200 Hz
The raw data were too noisy for velocity.

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Chapter 4 Filters
Noise reduction
169
• Noisy Corruption
• Undesirable Components

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Chapter 4 Filters
Noise reduction
170
• Enhancement
of Experimental
Measurement
n = 12
Wn = 0.05

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SHORT-TIME FOURIER TRANSFORM
Chapter Outline
171
5.1 Introduction of Time-frequency Analysis
5.2 Short-time Fourier Transform
5.3 Short-time Fourier Transform Implementation
5.4 Examples of Short-time Fourier Transform
5.5 Map of Time-frequency Analysis
CHAPTER 5

Shieh-Kung
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Chapter 5 Short-time Fourier Transform
INTRODUCTION OF TIME-FREQUENCY ANALYSIS
Time-frequency representation of chirp signals
172
• Chirp Signals
A chirp is a signal in which the frequency increases (up-chirp) or decreases (down-chirp) with time.
In some sources, the term chirp is used interchangeably with sweep signal. It is commonly applied to
sonar, radar, and laser systems, and to other applications, such as in spread-spectrum
communications (see chirp spread spectrum). This signal type is biologically inspired and occurs as a
phenomenon due to dispersion (a non-linear dependence between frequency and the propagation
speed of the wave components).
Start from
1 Hz
End with
3.3 Hz

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Time-frequency representation of chirp signals
173
• Spectrogram
A spectrogram is a visual representation of the spectrum of frequencies of a signal as it varies
with time. They are used extensively in the fields of music, linguistics, sonar, radar, speech processing,
seismology, and others. When applied to an audio signal, spectrograms are sometimes called
sonographs, voiceprints, or voicegrams. When the data are represented in a 3D plot they may be
called waterfall displays.
2
( , ) | ( , ) |
x
SP t X t
 
=

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Time-frequency representation of shifted frequency signals
174
• Example for Shifted Frequency Signals
The following signal starts from 1 Hz and is shifted to 5 Hz after 7 seconds. Then, the signal is
shifted again to 3 Hz after 13 seconds. From the Fourier spectra, we cannot observe any time
information about the signal but only see the frequency information. In short, It is hard to observe the
variation of spectrum with time by the Fourier transform.

Shieh-Kung
Huang
Time-frequency representation of shifted frequency signals
175

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Hilbert transform
176
• Analytic Signal
If a real signal, xr(t), with the spectrum, X(), is given, then the complex signal, z(t), whose
spectrum is composed of the positive frequencies of X() only, is given by the inverse transform of
X(), where the integration goes only over the positive frequencies,
The factor of 2 is inserted so that the real part of the analytic signal will be xr(t); otherwise it would be
one half of that. We now obtain the explicit form for z(t) in terms of the real signal xr(t). Since
After some derivation, we obtain
We use denote the analytic signal, z(t), corresponding to the signal xr(t). The reason for the name
analytic is that these types of complex functions satisfy the Cauchy-Riemann conditions for
differentiability and have been traditionally called analytic functions. The second part of the third
equation is the Hilbert transform of the signal and there are two conventions to denote the Hilbert
transform as
1 ( )
( ) ( ) ( ) ( ) r
r i r
x
z t x t ix t x t i d
t


 

−
= + = +
−

 
1 ( )
ˆ
( ) ( ) r
x
H x t x t d
t


 

−
= =
−

0
2
( ) ( )
2
i t
z t X e dt




= 
( ) ( ) i t
r
X x t e dt



−
−
= 

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Huang
Hilbert transform
177
• hilbert Function
The MATLAB function hilbert can compute the Hilbert transform easily.

Shieh-Kung
Huang
Hilbert transform
178
• Instantaneous Frequency
We seek a complex signal, z(t), whose real part is the "real signal", xr(t), and whose imaginary
part, xi(t), is our choice, chosen to achieve a sensible physical and mathematical description,
If we can analytically find the imaginary part we can then unambiguously define the amplitude A(t) and
phase (t) by
which gives
for the instantaneous frequency.
( )
( ) ( ) ( ) ( ) i t
r i
z t x t ix t A t e
= + =
2 2 1 ( )
( ) ( ) ( ) ( ) tan
( )
i
r i
r
x t
A t x t x t t
x t
 −
= + =
( )
( )
i
d t
t
dt

 =

Shieh-Kung
Huang
Hilbert transform
179
• Related Functions
The MATLAB function angle and unwrap can compute the Hilbert transform easily.

Shieh-Kung
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Short-time Fourier transform
180
For a given signal x(t) the Fourier transform X() (with the independent variable ) can be calculated
as
However, in order to estimate the time evolution of the frequency components present in the signal, the
short-time Fourier transform (STFT) parses the signal into smaller segments. Hence, we can fix a non-
zero function w(t) (called the window function or mask function). Then, the STFT of a given signal x(t) with
respect to the window function is defined as
where STFT[ ·] denotes STFT. It is also known as windowed Fourier transform or time-dependent Fourier
transform. The inverse short-time Fourier transform can be written as
 
( ) ( ) ( ) i t
X F x t x t e dt



−
−
= = 
 
( , ) ( ) ( ) ( ) i
X t STFT x t x w t e d

   

−
−
= = −

 
1 1
( ) ( , ) ( , )
2
i t
x t STFT X t X e d d

    

 
−
− −
= =  

Shieh-Kung
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Short-time Fourier transform
181
• STFT in Discrete-time Signals
The discrete-time Fourier transform of each segment is calculated separately and plotted as a
function of time k. The STFT is therefore a function of both frequency W and time k. Mathematically,
the STFT of a discrete-time signal x[k] is defined as follows:
where the subscript s in X(W, j) denotes the STFT and j indicates the amount of shift in the time-
localized window w[k] along the time axis. Typical windows used to calculate the STFT are rectangular,
Hanning, Hamming, Blackman, and Kaiser windows. Compared to the rectangular window, the
tapered windows, such as Hanning and Blackman, reduce the amount of sidelobes (or ripples) and
are generally preferred.
( , ) [ ] [ ] i k
k
X j x k w j k e

− W
=−
W = −


Shieh-Kung
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Properties of short-time Fourier transform
182
Given the short-time Fourier transform of a continuous-time function x(t), the linear operations being
considered include superposition, time shifting, scaling, differentiation and integration have almost the
same properties with the Fourier transform.
• Linearity
• Time & Frequency Shifting
• Time & Frequency Scaling

Shieh-Kung
Huang
Windowed Fourier transform
183
Generally, any non-zero function can be a window function, including those we used to reduce
spectral leakage. They are typically shaped as functions that start at a value of zero, move to a value
of one, and then return to a value of zero over one frame. Usually, the window functions used for the
short-time Fourier transform have the following properties
1 2 2 1
( ) ( ) max( ( )) (0)
( ) ( ) when | | | | ( ) 0 when | |is large
w t w t w t w
w t w t t t w t t
= − =
  

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184
• Rec-STFT
The simplest way is to choose a rectangular window to perform a short-time Fourier transform as
It’s called Rec-STFT for short.
2
2
( , ) ( )
T
t
i
T
t
X t x e d

  
+
−
−
= 

Shieh-Kung
Huang
185
• Rec-STFT
The rec-STFT has an advantage of the least computation time for digital implementation but its
performance is worse than other types of time-frequency analysis.
Sidelobes Sidelobes

Shieh-Kung
Huang
186
• STFT with Hanning window

Shieh-Kung
Huang
187
• STFT with Hanning window
Sidelobes

Shieh-Kung
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Gabor transform
188
• Gabor Transform
Theoretically, the best way is to choose a Gaussian-like function as a window to perform a short-
time Fourier transform as
And, the Gabor transform is written as
Although the range of integration is from −∞ to +∞, due to the fact that
So, the Gabor transform can be simplified as:
The reason why we choose the Gaussian function as a mask is because, among all functions, the
Gaussian function has the advantage that the area in time-frequency distribution is minimal. Moreover,
the properties of the Gabor transform in the time and frequency domains is symmetric because the
Gaussian function is the eigenfunction of the Fourier transform.
2
( )
( , ) ( ) t i
X t x e e d
  
  

− − −
−
= 
2
( ) t
w t 
−
=
2
0.00001 when | | 1.9143
t
e t

−
 
2
1.9143
( )
1.9143
( , ) ( )
t
t i
t
X t x e e d
  
  
+
− − −
−
= 

Shieh-Kung
Huang
Gabor transform
189
• Scaled Gabor Transform
As in short time Fourier transform, the resolution in time and frequency domain can be adjusted
by choosing different window function width. In Gabor transform cases, by adding variance , as
following equation:
And, the scaled Gabor transform is written as
With a large , the window function will be narrow, causing higher resolution in time domain but lower
resolution in frequency domain. Similarly, a small  will lead to a wide window, with higher resolution in
frequency domain but lower resolution in time domain.
2
( )
4
( , ) ( ) t i
X t x e e d
  
   

− − −
−
= 
2
( ) t
w t 
−
=

Shieh-Kung
Huang
Gabor transform
190
• Scaled Gabor Transform
Gaussian function is also an
eigenmode in optics, radar
system, and other
electromagnetic wave systems.
Gabor transform for
Gaussian function exp(−t2)
t-axis
f-axis
-4 -3 -2 -1 0 1 2 3 4
-4
-3
-2
-1
0
1
2
3
4
f-axis
t-axis
-4 -3 -2 -1 0 1 2 3 4
-4
-3
-2
-1
0
1
2
3
4
rec-STFT, T = 0.5 for
-4 -3 -2 -1 0 1 2 3 4
-4
-3
-2
-1
0
1
2
3
4
Gabor transform for
 = 0.2
f-axis
t-axis
Gabor transform for
 = 5
-4 -3 -2 -1 0 1 2 3 4
-4
-3
-2
-1
0
1
2
3
4
f-axis
t-axis

Shieh-Kung
Huang
SHORT-TIME FOURIER TRANSFORM IMPLEMENTATION
Uncertainty principle
191
The time-bandwidth product theorem, or uncertainty principle, is a fundamental statement regarding
Fourier transform pairs. The uncertainty principle has played a prominent role in discussions,
metaphysical and otherwise, of joint time-frequency analysis.
where
It’s a little bit complex to prove the uncertainty principle. However, remember that Δf and Δt in the Fourier
transform need to follow the equation
The equation can be re-written as
The equation shows that Δf and Δt cannot go infinitely small at the same time, which is one kind of
uncertainty principle.
2 2 1
2
t 
  
2 2 2 2
2 2
2 2
2 2
2 2
( ) | ( ) | ( ) | ( ) |
| ( ) | | ( ) |
| ( ) | | ( ) |
.
| ( ) | | ( ) |
t f
t
t f
t x t dt f X f df
x t dt X f df
t x t dt f X f df
x t dt X f df

 
 
 
− −
= =
= =
 
 
 
 
s
f
f
N
 =
1
N
f t
=
 

Shieh-Kung
Huang
Practical Implementation
192
After the window function is selected, we first compute the effective window length, l, as
Noteworthily, the window length is under the constraint as
where Δf and Δt are the sampling rate in the frequency and time axis, respectively. The STFT can be
implemented from the definition of the STFT. By limiting the integration within the effective window
where W and j are the discrete points in the frequency and time axis, respectively. The summation range
of the above formulation can be re-written as
The, we can further simplify the equation as
The zero padding may be applied if 2l is smaller than N.
( , ) [ ] [ ]
j l
i k f t
k j l
X f j t x k w j k e t
+
− W  
= −
W  = − 

when [ ] 0
k l w k
 
1
2 1
N l
f t
=  +
 
( )
2
0
( , ) [ ] [ ] if ( )
q j l
l i
N
q
X f j t x q l j w l q e t q k j l
W + −
−
=
W  = − + −  = − −

 
( )
2
0
( , ) [ , ] [ ] [ ] where [ , ]
[ , ] [ ] where [ ] [ ] [ ]
q l j
l i i
N N
q
X f j t C j x q l j w l q e C j e t
C j FFT x q x q x q l j w l q
W W −
−
=
W  = W − + − W = 
= W = − + −


Shieh-Kung
Huang
Practical Implementation
193
1
2 1
N l
f t
=  +
 
 
[ , ] [ ]
C j FFT x q
W
select l select Δf
compute N (FFT
points)
compute C[W, j]
compute FFT of
windowed signal
pad windowed
signal with zeros
compute STFT by
multiplication
given signal x[k] and other sampling
information
( )
[ , ]
l j
i
N
C j e t
W −
W = 
[ ] [ ] [ ]
x q x q l j w l q
= − + −
 
[ ]
FFT x q

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