lecture 1-6 newppt - imperial college londonee1 and eie1: introduction to signals and communications...

EE1 and EIE1: Introduction to Signals and Communications

Professor Kin K. Leung

EEE and Computing Departments

Imperial College

[email protected]

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.

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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.

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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).

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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.

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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




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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?

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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

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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

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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.

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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

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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

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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)

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Periodic Signal

A periodic signal g(t) can be generated by periodic extension of any segment of g(t) of duration T0

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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 < ∞

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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

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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

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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




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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.

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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

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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.

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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,

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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.

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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.

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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

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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).

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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.

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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

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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.

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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).

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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




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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.

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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.

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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


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

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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

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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.

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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

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(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.

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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).

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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

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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:

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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.

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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

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Parseval’s Theorem

Trigonometric Fourier series representation

The power is given by

Exponential Fourier series representation

Power for the exponential representation

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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




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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.

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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

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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

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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:

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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:

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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:

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( ) ( ) ,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:

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( ) ( )

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

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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

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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




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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


92


93


94

Fourier transform pair

95

Symmetry Property

● Consider the Fourier transform pair

● Then

● Example

96

g(t) G()

G(t) 2g()

Scaling Property


● Then

● Example

97

g(t) G()1

( ) ( )g at Ga a

Time-Shifting Property


● 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


● Exponential multiplication introduces frequency shift

● Cosine multiplication leads to

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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


100


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


● 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

……

……


● 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

lecture 1-6 newppt - imperial college londonee1 and eie1: introduction to signals and communications...

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