lecture 1-6 newppt - imperial college londonee1 and eie1: introduction to signals and communications...
TRANSCRIPT
EE1 and EIE1: Introduction to Signals and Communications
Professor Kin K. Leung
EEE and Computing Departments
Imperial College
Lecture one
2
Course Aims
To introduce:1. How signals can be represented and interpreted in
time and frequency domains2. Basic principles of communication systems3. Methods for modulating and demodulating signals to
carry information from an source to a destination
Recommended text book
B.P Lathi and Z. Ding, Modern Digital and Analog Communication Systems, Oxford University Press
● Highly recommended
● Well balanced book
● It will be useful in the future
● Slides based on this book, most of the figures are taken from this book
3
Handouts
● Copies of the transparencies
● Problem sheets and solutions
● Everything is on the webhttp://www.commsp.ee.ic.ac.uk/~kkleung/Intro_Signals_Comm_2018
4
Syllabus
● Fundamentals of Signals and Systems
– Energy and power
– Trigonometric and Exponential Fourier Series
– Fourier transform
– Linear system and convolution integral
• Modulation
– Amplitude modulation: DSB, Full AM, SSB
– Angle modulation: PM, FM
• Advanced Topics: Digital communications, CDMA
5
6
Another example of Communication Systems...
7
From the movie ’The Blues Brothers’
Communication Systems
A source originates a message, such as a human voice, a television picture, a teletype message.
The message is converted by an input transducer into an electrical waveform (baseband signal).
The transmitter modifies the baseband for efficient transmission.
The channel is a medium such as a coaxial cable, an optical fiber, a radio link.
The receiver processes the signal received to undo modifications made at the transmitter and the channel.
The output transducer converts the signal into the original form.
8
Communications
9
Input
transducerTransmitter Channel Receiver Output
transducer
Distortionand
noise
Inputmessage
Inputsignal
Transmitted signal
Receivedsignal
Outputsignal
Outputmessage
10
Analog and digital messages
● Message are digital or analog.
● Digital messages are constructed with a finite number of symbols. Example: a Morse-coded telegraph message.
● Analog messages are characterized by data whose values vary over a continuous range. For example, the temperature of a certain location.
11
Digital Transmission
Digital signals are more robust to noise.
An analog signal is converted to a digital signal by means of an analog-to-digital (A/D) converter.
12
Filter Sampler Quantizer
v(t) m(t) ms(kT) mq(kT)
A/D conversion
The signal m(t) is first sampled in the time domain.
The amplitude of the signal samples ms(kT) is partitioned into a finite number of intervals (quantization).
13
14
Sampling theorem
The sampling theorem states that
If the highest frequency in the signal spectrum is B, the signal can be reconstructed from its samples taken at a rate not less than 2B sample per second.
15
What did we learn today?
● The main elements of a communication systems
● The importance of the Fourier transforms
● Concept of signal bandwidth
● Analog and digital signals
16
EE1 and EIE1: Introduction to Signals and Communications
Professor Kin K. Leung
EEE and Computing Departments
Imperial College
Lecture two
18
Lecture Aims
● To introduce signals
● Classifications of signals
● Some particular signals
Signals
● A signal is a set of information or data
● Examples
- a telephone or television signal
- monthly sales of a corporation
- the daily closing prices of a stock market
● We deal exclusively with signals that are functions of time
How can we measure a signal? How can we distinguish two different signals?
19
Classifications of Signals
● Continuous-time and discrete-time signals
● Analog and digital signals
● Periodic and aperiodic signals
● Energy and power signals
● Deterministic and probabilistic signals
20
Continuous-time and discrete-time signals
● A signal that is specified for every value of time t is a continuous-time signal
● A signal that is specified only at discrete values of t is a discrete-time signals
21
Continuous-time and discrete-time signals (continued)
● A discrete-time signal can be obtained by sampling a continuous-time signal.
● In some cases, it is possible to ’undo’ the sampling operation. That is, it is possible to get back the continuous-time signal from the discrete-time signal.
Sampling Theorem
The sampling theorem states that if the highest frequency in the signal spectrum is B, the signal can be reconstructed from its samples taken at a rate not less than 2B samples per second.
22
Analog and digital signals
● A signal whose amplitude can take on any value in a continuous range is an analog signal
● The concept of analog and digital signals is different from the concept of continuous-time and discrete-time signals
● For example, we can have a digital and continuous-time signal, or a analog and discrete-time signal
23
Analog and digital signals (continued)
● One can obtain a digital signal from an analog one using a quantizer
● The amplitude of the analog signal is partitioned into L intervals. Each sample is approximated to the midpoint of the interval in which the original value falls
● Quantization is a lossy operation
Notice that: One can obtain a digital discrete-time signal by sampling and quantizing an analog continuous-time signal
24
25
Conversion of continuous-time analog to discrete-time digital signal
Periodic and aperiodic signals
● A signal g(t) is said to be periodic if for some positive constant T0,
g(t) = g(t + T0) for all t
● A signal is aperiodic if it is not periodic
Same famous periodic signals:
sin ω0t, cos ω0t, ejω0t,
where ω0 = 2π/T0 and T0 is the period of the function
(Recall that ejω0t = cos ω0t + j sin ω0t)
26
Periodic Signal
A periodic signal g(t) can be generated by periodic extension of any segment of g(t) of duration T0
27
Energy and power signal
First, define energy
● The signal energy Eg of g(t) is defined (for a real signal) as
● In the case of a complex valued signal g(t), the energy is given by
● A signal g(t) is an energy signal if Eg < ∞
28
Eg g2(t)
dt.
2 *( ) ( ) ( ) . gE g t g t dt g t dt
Power
A necessary condition for the energy to be finite is that the signal amplitude goes to zero as time tends to infinity.In case of signals with infinite energy (e.g., periodic signals), a more meaningful measure is the signal power.
A signal is a power signal if
A signal cannot be an energy and a power signal at the same time
29
2 2
2
1 lim ( )
T
g TTP g t dt
T
2 2
2
10 lim ( )
T
TTg t dt
T
Energy signal example
Signal Energy calculation
30
Eg g2(t)
dt (2)2 dt1
0 4e tdt0
4 4 8.
Power signal example
Assume g(t) = Acos(ω0t + θ), its power is given by
31
0
2 2 202
22
2
2 22 2
02 2
2
1 lim cos ( )
1 lim 1 cos(2 2 )
2
lim lim cos(2 2 )2 2
2
T
g TT
T
TT
T T
T TT T
P A w t dtT
Aw t dt
T
A Adt w t dt
T T
A
Power of Periodic Signals
Show that the power of a periodic signal g(t) with period T0 is
Another important parameter of a signal is the time average:
32
0
0
2 2
20
1 ( )
T
g TP g t dt
T
gaverage limT
1
Tg(t)dt
T 2
T 2 .
Deterministic and probabilistic signals
● A signal whose physical description is known completely is a deterministic signal.
● A signal known only in terms of probabilistic descriptions is a random signal.
33
Summary
● Signal classification
● Power of a periodic signal of period T0
● Time average
● Power of a sinusoid A cos(2πf0t + θ) is
34
0
0
2 2
20
1 ( )
T
g TP g t dt
T
2
2)(
1lim T
TTaverage dttg
Tg
A2
2
EE1 and EIE1: Introduction to Signals and Communications
Professor Kin K. Leung
EEE and Computing Departments
Imperial College
Lecture three
36
Lecture Aims
● To introduce some useful signals
● To present analogies between vectors and signals
- Signal comparison: correlation
- Energy of the sum of orthogonal signals
- Signal representation by orthogonal signal set
Useful Signals: Unit impulse function
The unit impulse function or Dirac function is defined as
Multiplication of a function by an impulse:
37
( ) 0 0
( ) 1
t t
t d t
( ) ( ) ( ) ( )
( ) ( ) ( ).
g t t T g T t T
g t t T dt g T
Area = 1
∆→0
Useful Signals: Unit step function
Another useful signal is the unit step function u(t), defined by
Observe that
Therefore
Use intuition to understand this relationship: The derivative of a ’unit step jump’ is an unit impulse function.
38
1 0( ) =
0 0
t td
t
1 0( )
0 0
tu t
t
du(t)
dt(t).
1
t
Useful Signals: Sinusoids
Consider the sinusoid
x(t) = C cos(2πf0t + θ)
f0 (measured in Hertz) is the frequency of the sinusoid and T0 = 1/f0 is the period.
Sometimes we use ω0 (radiant per second) to express 2πf0.
Important identities
with and
39
1 1cos sin , cos , sin ,
2 2
1cos cos cos( ) cos( )
2cos sin cos( )
jx jx jx jx jxe x j x x e e x e ej
x y x y x y
a x b x C x
C a2 b2 tan1 b
a
Signals and Vectors
● Signals and vectors are closely related. For example,
- A vector has components
- A signal has also its components
● Begin with some basic vector concepts
● Apply those concepts to signals
40
Inner product in vector spaces
x is a certain vector.
It is specified by its magnitude or length |x| and direction.
Consider a second vector y .
We define the inner or scalar product of two vectors as
<y, x> = |x||y| cos θ.
Therefore, |x|2 = <x, x>.
When <y, x> = 0, we say that y and x are orthogonal (geometrically, θ = π/2).
41
x
yθ
Signals as vectors
The same notion of inner product can be applied for signals.What is the useful part of this analogy?We can use some geometrical interpretation of vectors to understand signals!Consider two (energy) signals x(t) and y(t).The inner product is defined by
For complex signals
where y*(t) denotes the complex conjugate of y(t).
Two signals are orthogonal if <x(t), y(t)> = 0.
42
x(t), y(t) x(t)y(t)
dt
x(t), y(t) x(t)y *(t)
dt
Energy of orthogonal signals
If vectors x and y are orthogonal, and if z = x + y
|z|2 = |x|2 + |y|2 (Pythagorean Theorem).
If signals x(t) and y(t) are orthogonal and if z(t) = x(t) + y(t) then
Ez = Ex + Ey .
Proof for real x(t) and y(t) :
since
43
2
2 2
( ( ) ( ))
( ) ( ) 2 ( ) ( )
2 ( ) ( )
z
x y
x y
E x t y t dt
x t dt y t dt x t y t dt
E E x t y t dt
E E
x(t)y(t)
dt 0.
z y
x
Power of orthogonal signals
The same concepts of orthogonality and inner product extend to power signals.
For example, g(t) = x(t) + y(t) = C1 cos(ω1t + θ1) + C2 cos(ω2t + θ2) and ω1 ≠ ω2.
The signal x(t) and y(t) are orthogonal: <x(t), y(t)> = 0. Therefore,
44
Px C1
2
2, Py
C22
2.
Pg Px Py C1
2
2
C22
2.
Signal comparison: Correlation
If vectors x and y are given, we have the correlation measure as
Clearly, −1 ≤ cn ≤ 1.
In the case of energy signals:
again −1 ≤ cn ≤ 1.
45
cn cos
y,x
x y
1( ) ( )n
y x
c y t x t dtE E
Best friends, worst enemies and complete strangers
● cn = 1. Best friends. This happens when g(t) = Kx(t) and K is positive. The signals are aligned, maximum similarity.
● cn = −1. Worst Enemies. This happens when g(t) = Kx(t) and K is negative. The signals are again aligned, but in opposite directions. The signals understand each others, but they do not like each others.
● cn = 0. Complete Strangers The two signals are orthogonal. We may view orthogonal signals as unrelated signals.
46
Correlation
Why do we bother poor undergraduate students with correlation?
Correlation is widely used in engineering.
For instance
● To design receivers in many communication systems
● To identify signals in radar systems
● For classifications
47
Correlation examples
Find the correlation coefficients between:
● x(t) = A0 cos(ω0t) and y(t) = A1 sin(ω1t).
● x(t) = A0 cos(ω0t) and y(t) = A1 cos(ω1t) and ω0 ≠ ω1.
● x(t) = A0 cos(ω0t) and y(t) = A1 cos(ω0t).
● x(t) = A0 sin(ω0t) and y(t) = A1 sin(ω1t) and ω0 ≠ ω1.
● x(t) = A0 sin(ω0t) and y(t) = A1 sin(ω0t).
● x(t) = A0 sin(ω0t) and y(t) = −A1 sin(ω0t).
48
Correlation examples
Find the correlation coefficients between:
● x(t) = A0 cos(ω0t) and y(t) = A1 sin(ω1t) cx,y = 0.
● x(t) = A0 cos(ω0t) and y(t) = A1 cos(ω1t) and ω0 ≠ ω1 cx,y = 0.
● x(t) = A0 cos(ω0t) and y(t) = A1 cos(ω0t) cx,y = 1.
● x(t) = A0 sin(ω0t) and y(t) = A1 sin(ω1t) and ω0 ≠ ω1 cx,y = 0.
● x(t) = A0 sin(ω0t) and y(t) = A1 sin(ω0t) cx,y = 1.
● x(t) = A0 sin(ω0t) and y(t) = −A1 sin(ω0t) cx,y = -1.
49
Signal representation by orthogonal signal sets
● Examine a way of representing a signal as a sum of orthogonal signals
● We know that a vector can be represented as the sum of orthogonal vectors
● The results for signals are parallel to those for vectors
● Review the case of vectors and extend to signals
50
Orthogonal vector space
Consider a three-dimensional Cartesian vector space described by three mutually orthogonal vectors, x1, x2 and x3.
Any three-dimensional vector can be expressed as a linear combination of those three vectors: g = a1x1+ a2x2+ a3x3
where
In this case, we say that this set of vectors is complete.
Such vectors are known as a basis vector.
51
xm, xn 0 m n
| xm |2 m n
aig,x
i
|xi|2
Orthogonal signal space
Same notions of completeness extend to signals. A set of mutually orthogonal signals x1(t), x2(t), ..., xN(t) is complete if it can represent any signal belonging to a certain space. For example:
If the approximation error is zero for any g(t) then the set of signals x1(t), x2(t), ..., xN(t) is complete. In general, the set is complete when N → ∞. Infinite dimensional space (this will be more clear in the next lecture).
52
g(t) ~ a1x
1(t) a
2x
2(t) ... a
Nx
N(t)
Summary
● Analogies between vectors and signals
● Inner product and correlation
● Energy and Power of orthogonal signals
● Signal representation by means of orthogonal signal
53
EE1 and EIE1: Introduction to Signals and Communications
Professor Kin K. Leung
EEE and Computing Departments
Imperial College
Lecture four
55
Lecture Aims
● Trigonometric Fourier series
● Fourier spectrum
● Exponential Fourier series
Trigonometric Fourier series
● Consider a signal set
{1, cos ω0t, cos 2ω0t, ..., cos nω0t, ..., sin ω0t, sin 2ω0t, ..., sin nω0t, ...}
● A sinusoid of frequency nω0t is called the nth harmonic of the sinusoid, where n is an integer.
● The sinusoid of frequency ω0 is called the fundamental harmonic.
● This set is orthogonal over an interval of duration T0 = 2π/ω0, which is the period of the fundamental harmonic.
56
Trigonometric Fourier series
The components of the set {1, cos ω0t, cos 2ω0t, ..., cos nω0t, ..., sin ω0t, sin 2ω0t, ..., sin nω0t, ...} are orthogonal as
for all m and n
means integral over an interval from t = t1 to t = t1 + T0 for any value of t1.
57
0
0
0
0 0 0
0 0 0
0 0
0 cos cos
02
0 sin sin
02
sin cos 0
T
T
T
m nn t m tdt T
m n
m nn t m tdt T
m n
n t m tdt
T0
Trigonometric Fourier series
This set is also complete in T0. That is, any signal in an interval t1 ≤ t ≤ t1 + T0 can be written as the sum of sinusoids. Or
Series coefficients
58
g(t) a0 a1 cos0t a2 cos20t ... b1 sin0t b2 sin20t ...
a0 an cosn0tn1
bn sin n0t
0 0
0 0 0 0
( ),cos ( ),sin
cos ,cos sin ,sinn n
g t n t g t n ta b
n t n t n t n t
Trigonometric Fourier Coe cients
Therefore
As
We get
59
1 0
1
1 0
1
0
20
( ) cos
cos
t T
tn t T
t
g t n tdta
n tdt
1 0
1
1 0
1
1 0
1
0 0
0 0
0 0
1( )
2( )cos 1, 2,3,...
2( )sin 1,2,3,...
t T
t
t T
n t
t T
n t
a g t dtT
a g t n tdt nT
b g t n tdt nT
1 0 1 0
1 1
2 20 0 0 0
cos 2, sin 2.t T t T
t tn tdt T n tdt T
Compact Fourier series
Using the identity
where
The trigonometric Fourier series can be expressed in compact form as
For consistency, we have denoted a0 by C0.
60
an cosn0t bn sinn0t Cn cos(n0t n )
Cn an2 bn
2 n tan1(bn an ).
g(t) C0 Cn cos(n0t n )n1
t1 t t1 T0 .
Periodicity of the Trigonometric series
We have seen that an arbitrary signal g(t) may be expressed as a trigonometric Fourier series over any interval of T0 seconds.
What happens to the Trigonometric Fourier series outside this interval?
Answer: The Fourier series is periodic of period T0 (the period of the fundamental harmonic).
Proof:
for all t
and
for all t
61
(t) C0 Cn cos(n0t n )n1
0 0 0 01
0 01
0 01
( ) cos
cos 2
cos
( )
n nn
n nn
n nn
t T C C n t T
C C n t n
C C n t
t
Properties of trigonometric series
● The trigonometric Fourier series is a periodic function of period T0 = 2π/ω0.
● If the function g(t) is periodic with period T0, then a Fourier series representing g(t) over an interval T0 will also represent g(t) for all t.
62
Example
63
Example
ω0 = 2π / T0 = 2 rad / s.
We can plot
● the amplitude Cn versus ω this gives us the amplitude spectrum
● the phase θn versus ω (phase spectrum).
This two plots together are the frequency spectra of g(t).
64
n 0 1 2 3 4
Cn 0.504 0.244 0.125 0.084 0.063
θn 0 -75.96 -82.87 -85.84 -86.42
g(t) C0 Cn cos(2nt n )n1
Amplitude and phase spectra
65
Exponential Fourier Series
Consider a set of exponentials
The components of this set are orthogonal.
A signal g(t) can be expressed as an exponential series over an interval T0:
66
e jn0t n 0,1,2,...
0 0
00
1( ) ( )jn t jn t
n n Tn
g t D e D g t e dtT
Trigonometric and exponential Fourier series
Trigonometric and exponential Fourier series are related. In fact, a sinusoid in the trigonometric series can be expressed as a sum of two exponentials using Euler’s formula.
67
0 0
0 0
0 0
( ) ( )0cos( )
2
2 2
n n
n n
j n t j n tnn n
j jn t j jn tn n
jn t jn tn n
CC n t e e
C Ce e e e
D e D e
1 1
2 2n nj j
n n n nD C e D C e
Amplitude and phase spectra. Exponential case
68
Parseval’s Theorem
Trigonometric Fourier series representation
The power is given by
Exponential Fourier series representation
Power for the exponential representation
69
Pg C02
1
2Cn
2
n1
.
g(t) C0 Cn cos(n0t n ).n1
g(t) Dnejn 0t .
n
2
g nn
P D
Conclusions
● Trigonometric Fourier series
● Exponential Fourier series
● Amplitude and phase spectra
● Parseval’s theorem
70
EE1 and EIE1: Introduction to Signals and Communications
Professor Kin K. Leung
EEE and Computing Departments
Imperial College
Lecture five
72
Lecture Aims
● To introduce Fourier integral, Fourier transformation
● To present transforms of some useful functions
● To discuss some properties of the Fourier transform
Introduction
● We electrical engineers think of signals in terms of their spectral content.
● We have studied the spectral representation of periodic signals.
● We now extend this spectral representation to the case of aperiodic signals.
73
Aperiodic signal representation
We have an aperiodic signal g(t) and we consider a periodic version gT0(t) of such signal obtained by repeating g(t) every T0 seconds.
74
The periodic signal gT0(t)
The periodic signal gT0(t) can be expressed in terms of g(t) as follows:
Notice that, if we let T0 → ∞, we have
75
0 0( ) ( ).Tn
g t g t nT
00
lim ( ) ( ).TT
g t g t
The Fourier representation of gT0(t)
The signal gT0(t) is periodic, so it can be represented in terms of its Fourier series. The basic intuition here is that the Fourier series of gT0(t) will also represent g(t) in the limit for T0 → ∞. The exponential Fourier series of gT0(t) is
where
and
76
0
0( ) ,jn t
T nn
g t D e
Dn 1
T0
gT0
(t)T0 2
T0 2 e jn0tdt
00
2.
T
The Fourier representation of gT0(t)
Integrating gT0(t) over ( −T0/2, T0/2) is the same as integrating g(t) over ( −∞, ∞). So we can write
If we define a function
then we can write the Fourier coefficients Dn as follows:
77
0
0
1( ) .jn t
nD g t e dtT
( ) ( ) j tG g t e dt
00
1( ).nD G n
T
Computing the lim T0→∞ gT0(t)
Thus gT0(t) can be expressed as:
Assuming (i.e., replace notation by ∆ω), we get
In the limit for T0 → ∞, ∆ω → 0 and gT0(t) → g(t).
We thus get:
78
1
T0
2
gT0
(t) Dne jn0t
n
G(n
0)
T0
e jn0t
n
0
( )( )( ) .
2j n t
Tn
G ng t e
g(t) limT0 g
T0(t) lim0
G(n)2n
e j (n )t
1
2G()e jt d.
0
where 0 2
T0
.
Fourier Transform and Inverse Fourier Transform
What we have just learned is that, from the spectral representation G(ω) of g(t), that is, from
we can obtain g(t) back by computing
Fourier transform of g(t):
Inverse Fourier transform:
Fourier transform relationship:
79
( ) ( ) ,j tG g t e dt
1( ) ( ) .
2j tg t G e d
( ) ( ) .j tG g t e dt
1( ) ( ) .
2j tg t G e d
( ) ( ).g t G
Example
Find the Fourier transform of g(t) = e-atu(t).
Since , we have that Therefore:
80
( ) ( )
00
1( ) ( ) .at j t a j t a j tG e u t e dt e dt e
a j
1
2 2
1 1( ) , ( ) , ( ) tan ( ).gG G
a j aa
1j te limt eate jt .
Some useful functions
The Unit Gate Function:
The unit gate function rect(x) is defined as:
81
rect(x) 0 x 1 2
1 x 1 2
Some useful functions
The function sin(x)/x ‘sine over argument’ function is denoted by sinc(x):
● sinc(x) is an even function of x.
● sinc(x) = 0 when sin(x) = 0 and x ≠ 0.
● Using L’Hospital’s rule, we find that sinc(0) = 1
● sinc(x) is the product of an oscillating signal sin(x) and a monotonically decreasing function 1/x.
82
Example
Find the Fourier transform of g(t) = rect( t/τ ).
Therefore
83
2
2
2 2
( )
1 2sin( 2) ( )
sin( 2) sin ( 2).
( 2)
j t j t
j j
tG rect e dt e dt
e ej
c
( ) sin ( 2)rect t c
Example
Find the Fourier transform of the unit impulse δ(t):
Therefore
Find the inverse Fourier transform of δ(ω):
Therefore
84
0( ) 1.j t j t
tt e dt e
1 1( ) .
2 2j te d
( ) 1t
1 2 ( )
Example
Find the inverse Fourier transform of δ( – 0):
Therefore
and
85
00
1 1( ) .
2 2j tj te d e
00 2 ( )j te
00 2 ( )j te
Example
Find the Fourier transform of the everlasting sinusoid cos(ω0t).
Since
and using the fact that and
we discover that
86
0 00
1cos( )
2j t j tt e e
0 0 0cos( ) .t
002j te 0
02 ,j te
Summary
Fourier transform of g(t):
Inverse Fourier transform:
Fourier transform relationship:
Important Fourier transforms:
87
0 0 0cos( ) .t
( ) ( ) ,j tG g t e dt
1( ) ( ) .
2j tg t G e d
( ) ( ).g t G
( ) sin ( 2)rect t c
( ) 1t
EE1 and EIE1: Introduction to Signals and Communications
Professor Kin K. Leung
EEE and Computing Departments
Imperial College
Lecture six
89
Lecture Aims
● To present some properties of the Fourier transform
90
Topics Covered
● Fourier transform table
● Symmetry of Fourier transformation
● Time and Frequency shifting property
● Convolution
● Time differentiation and time integration
● Please read Lathi & Ding
Some properties of Fourier transform
91
Some properties of Fourier transform
92
Some properties of Fourier transform
93
Some properties of Fourier transform
94
Fourier transform pair
95
Symmetry Property
● Consider the Fourier transform pair
● Then
● Example
96
g(t) G()
G(t) 2g()
Scaling Property
● Consider the Fourier transform pair
● Then
● Example
97
g(t) G()1
( ) ( )g at Ga a
Time-Shifting Property
● Consider the Fourier transform pair
● Time shifting introduces phase shift
● Example
98
g(t) G()
g(t t0) G()e jt0
t0
g(t) g(t-to)
to
Frequency-Shifting Property
● Consider the Fourier transform pair
● Exponential multiplication introduces frequency shift
● Cosine multiplication leads to
99
g(t) G()
g(t)e j0t G( 0) g(t)e j0t G( 0)
0 00
0 0 0
1( ) cos ( ) ( )
21
( ) cos ( ) ( )2
j t j tg t t g t e g t e
g t t G G
Frequency-Shifting Property
100
Frequency-Shifting Property
101
Fourier transform of periodic functions
● Find the Fourier transform of a general periodic signal g(t) of period T0
● A periodic signal g(t) can be expressed as an exponential Fourier series as
102
0
0
00
0
2( )
( )
( ) 2
jn tn
n
jn tn
n
nn
g t D eT
g t F D e
g t D n
Fourier transform of periodic functions
● Consider a periodic waveform given by
● where
103
00
non-zero 2( ) ( ) ( )
0 otherwise
n
n
T tg t w t nT w t
g(t) Dne j n0t
n
n
w(t)W ()
g(t)G() 2 Dn( n
0)
n
n
Dn 1
T0g(t)e jn0t dt
T0 W (n
0)
T0
|D-4| |D-3| |D-2| |D-1| |D0| |D1| |D2| |D3| |D4|
G()
2
-40 -30 -20 -0 0 0 20 30 40
……
……
Fourier transform of periodic functions
● Find the Fourier transform of a unit impulse train δ(t) of period T0
104
w(t) (t) W () F ((t)) 1
Dn
W (n0)
T0
1
T0
g(t) 2T
0
( n0)
n
n
Convolution
The convolution of two functions g(t) and w(t),
● Consider two waveforms
● Convolution in time domain
● Convolution in the frequency domain
105
( )* ( ) ( ) ( )g t w t g w t d
1 1 2 2( ) ( ) ( ) ( )g t G g t G
1 2 1 2( ) * ( ) ( ) ( )g t g t G G
1 2 1 2
1( ) ( ) ( )* ( )
2g t g t G G
Time Differentiation and Time Integration
● Consider the Fourier transform relationship
● The following relationship exists for integration
● The following relationship exists differentiation
106
( ) ( )g t G
( )( ) (0) ( )
t Gg d G
j
dg(t)
dt jG()
d ng(t)
dtn ( j)nG()
Important Fourier Transform Operations
107
An Example of Fourier Transform Properties
108
)(
)(0
)(|)(
)(
)( )(
Gj
dtetgj
detgtge
tdge
dteF
tj
tjtj
tj
tjdt
tdg
Conclusions
● Examined some properties of Fourier transforms
- Scaling property
- Time shifting property
- Frequency shifting property
● Examined Fourier Transform of periodic functions
- General case
- Unit Impulse function
● Examined Convolution
● Examined Fourier transforms for
- Integration
- Differentiation
109