Course Introduction and Review of LinearSystems and Fourier Transform

ELEC3540 - Analog and Digital Communications

Dr. Duy Ngo

School of Electrical Engineering and Computer ScienceUniversity of Newcastle

Feb. 24, 2015

Course Outline, Chapter 1 (1.2, 1.3, 1.4, 1.5, 1.6),Appendices A, B of textbook

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Course Admin Teaching Staff

Lecturer: Dr Duy Ngo Office: EAG19 - EA Building Email: duy.ngo@newcastle.edu.au Phone: (02) 4921 8947 Consultation: Wednesday 10.00 AM - 12.00 NOON

Lab Demonstrator: Mr. Behzad Asadi Office: ES230 - ES Building Email: behzad.asadi@uon.edu.au

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Course Admin Contents

Textbook Martin S. Roden, Analog and Digital CommunicationSystems, 5th edition, 2003. ISBN: 9780964696976.

First half of Semester: Analog Communication Review of Fourier transform, linear systems, random

processes Analog baseband transmission, analog AM, FM, PM

Second half of Semester: Digital Communication Baseband digital transmission, digital AM, FM, PM Multiplexing techniques: TDM, FDM, CDM

References to Chapters/Sections of textbook will be givenat the beginning of lecture slides

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Course Admin Assessments

Assignments (15%) 2 assignment, 7.5% each Due in Weeks 6 and 12 Hard copies; Marked by me

Labs (20%) 5 labs, 4% each Report to be submitted 1 week after your assigned time slot Hard copies; Marked by Lab Demonstrator

Mid-term quiz (15%) To be conducted in lecture time, Week 7; Marked by me Closed-book Review lecture in Week 7

Final exam (50%) Closed-book; Marked by me Review lecture in Week 13

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Course Admin My Advice and Expectations

Lectures: Buy and do read textbook Regular attendance, take notes, no ringing phone, no

private chatting, questions are always welcome (dointerrupt!)

Solutions to example exercises are only given in thelectures

Tutorials: Regular attendance Problem-solving skills needed for assignments, quiz and

exam Solutions to tutorial problems are only given in the tutorials Only solutions to unsolved questions posted on the

Blackboard site

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Course Admin My Advice and Expectations

Labs: Complete online Lab Induction and Access Quiz to get

access Contact Lab Manager (Mr. Peter Turner) if you have any

question regarding access Attend your scheduled sessions

Quiz and Exam: will be based on textbook, lectures notes,tutorial problems

Adverse circumstances: email (from UniMail), see me Check Blackboard site and Email regularly for

announcements Have fun!

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Overview of a Communication System

Transmitter: change raw information into a format that ismatched to channel characteristics

Channel: introduce changes to the signals Linear channels with transfer function: H(f ) = A(f )ej(f ) Amplitude distortion: A(f ) is not a constant Phase distortion: (f ) is not linear in f

Receiver: detect and process signals from channel

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Expanded Block Digram of a Communication System

This course studies the transmission techniques used toreliably and efficiently transmit and receive signals over achannel with uncertainty

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Types of Signals

Analog signals: Waveform takes on a continuum of values for any time

within a range of times E.g., human speech waveform

Analog sampled (or discrete) signals: Sample analog signals at discrete time points Signals can still take an infinite number of values

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Types of Signals

Digital signals: A form of discrete signals Signals can only take a finite number of values E.g., pressing keys at ATM, using computer keyboard, A/D

conversion Advantages of digital over analog transmission:

Improved noise immunity Processing simplicity

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Nyquist Sampling Theorem Suppose a signal s(t) is defined over bandwidth [0, fm]

We sample s(t) at all t = nTs where n is integer to obtainss(t)

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Nyquist Sampling Theorem We can reconstruct s(t) from ss(t) precisely and

accurately if

Ts 2fm (2)

That is, sampling rate must be more than twice the highestfrequency of the signal for a perfect reconstruction

2fm is called Nyquist sampling rate12 / 44

A/D Conversion Sampling an analog signal gives a discrete signal, which

still takes a continuum of values. Analog-to-digital (A/D) conversion: Code the list of

infinite numbers into a set of finite code words. Uniform quantization:

Divide the continuum of functional values into uniform-widthregions

Assign an integer code to each region n-bit quantizer contains 2n regions.

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D/A Conversion

Digital-to-analog (D/A) conversion An integer code is associated with a region Usually, the analog value is taken as the centre point of the

region.

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Communication Channels

Copper wires, coaxial cables, fiber optic cables, air All channels have a maximum frequency beyond which

input signals are almost entirely attenuated. As frequencies increase, parallel capacitance short out

the signal Series inductance open circuits

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Communication Channels

Bandpass channel: Low-frequency cutoff Lowpass channel: No lower-frequency cutoff (i.e., dc

coupled) Narrowband: f < 300Hz Voiceband: 300Hz < f < 4kHz Wideband: f > 4kHz The higher the frequency, the more the transmission takes

on the characteristics of light 16 / 44

Classification of Signals

Signal is any function that carries some information. Continuous-time signal x(t) when t takes real numbers.

E.g., x(t) = A cos(2pif0t + ). Discrete-time signal x [n] when n takes integral values.

E.g., x [n] = A cos(2pif0n + ). Real signals take values in the set of real numbers:x(t) R.

Complex signals take values in the set of complexnumbers: x(t) C.

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Classification of Signals

Deterministic signals: x(t) is a real or complex number atany time instant t .

Random signals: x(t) is random variable at any timeinstant t , defined by a probability density function.

Periodic signals: Continuous: x(t + kT0) = s(t), t ,k Z and period T0 > 0 Discrete: x [n + kN0] = s[n], n, k Z and period N0 Z+.

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Some Important Signals

Sinusoidal: x(t) = A cos(2pif0t + ) with amplitude A,frequency f0 and phase .

Complex exponential: x(t) = Aej(2pif0t+). Impulse (or Delta): Precisely, (t) is not a function, but a

distribution or a generalized function.

(t) ={

0, t 6= 0,, t = 0. (3)

(t)(t)dt = (0). (4)

Sinc:

sinc(t) ={

sin(pit)/(pit), t 6= 0,1, t = 0.

(5)

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Energy and Power

For any signal x(t), the energy content or energy Ex is:

Ex = |x(t)|2dt = lim

T

T/2T/2

|x(t)|2dt . (6)

The power content or power Px is:

Px = limT

1T

T/2T/2

|x(t)|2dt . (7)

For real signals, |x(t)|2 = x2(t). A signal x(t) is an energy-type signal iff Ex

Example Consider signal x(t) = A cos(2pif0t + ) x(t) is not energy-type:

Ex = limT

T/2T/2

A2 cos2(2pif0t + )dt = (8)

The power of x(t):

Px = limT

1T

T/2T/2

A2 cos2(2pif0t + )dt

= limT

1T

T/2T/2

A2

2[1 + cos(4pif0t + 2)]dt

= limT

[A2T2T

+

[A2

8pif0Tsin(4pif0t + 2)

]]=

A2

2

Exercise

Suppose x(t) is periodic with period T0 Show that x(t) is not an energy-type signal.

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Exercise Suppose x(t) is periodic with period T0 Show that the power content of x(t) is equal to the average

power in one period.

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Systems

System is an interconnection of various elements andbehaves as a whole

A set of rules that associate an output time function toevery input time function

Example: y(t) = ax(t) + bx2(t) defines a system Communication viewpoint:

A system is excited by an input signal x(t) and, as a resultof such an excitation, produces an output signal y(t).

Output y(t) must be uniquely defined for any legitimateinput x(t) (i.e., must not be a one-to-many mapping!).

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Classification of Systems

Continuous-time systems: Continuous-time input x(t)and continuous-time output y(t)

E.g., differentiator:

y(t) =ddtx(t) (10)

Discrete-time systems: Discrete-time input x [n] anddiscrete-time output y [n]

E.g., discrete-time differentiator:

y [n] = x [n] x [n 1] (11)

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Classification of Systems

Linear systems: Superposition property is satisfied.

T [x1(t) + x2(t)] = T [x1(t)] + T [x2(t)], (12)

where x1(t), x2(t) are two legitimate inputs and , aretwo scalars.

E.g., y(t) = x2(t) is nonlinear because its response to2x(t) is:

T [2x(t)] = 4x2(t) 6= 2x2(t) = 2T [x(t)] (13)

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Classification of Systems

Time-invariant systems: The response due to an input isindependent upon the actual time of occurrence of theinput.

T [x(t t0)] = y(t t0). (14)

E.g., y(t) = x(t) cos(2pif0t) is a time-varying systembecause the response to x(t t0) is

x(t t0) cos(2pif0t) 6= y(t t0) (15)

Causal systems: For physical realizability of systems

y(t0) = T [x(t) : t t0]. (16)

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LTI Systems

We will consider linear time-invariant systems (LTI). For this class of systems, the input-output relation is

particularly simple and can be expressed in terms of theconvolution integral.

Instead of requiring that we know the response due toevery possible input, we need only know the output h(t) ofone test input (t).

h(t) = T [(t)]. (17)

For an LTI system, the impulse response h(t) completelycharacterizes the system. It is all the information we needto describe the system behavior.

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Analysis of LTI Systems in Time Domain Response to x(t) is the convolution of x(t) and h(t):

y(t) = T [x(t)] = T[

x()(t )d]

=

x()T [(t )]d

=

x()h(t )d

= x(t)Fh(t). (18) Given an LTI system, the time-domain analysis involves:

Find impulse response h(t) by applying an impulse (t) atthe input.

To obtain output y(t) to other input x(t), perform theconvolution integral:

y(t) = x(t)Fh(t) =

x()h(t )d. (19)

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Motivation of Fourier Series

Analyzing LTI systems requires direct application ofconvolution integral. Although straightforward, evaluatingthe integral is not always easy.

Alternative approach: Expand the input as a linear combination of some basic

signals whose output can be obtained easily; Then, employ the linearity properties of the system to

obtain the corresponding output.

Question: Which basic signals? Complex exponentials ej2pif0t are the eigenfunctions of

LTI systems Response to a complex exponential is a complex

exponential with the same frequency and with a change inamplitude and phase.

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Fourier Series of Periodic Signals Question: Which signals can be expanded in terms of

complex exponentials? And how? Fourier series theorem: Let signal x(t) be a periodic

signal with period T0 = 1/f0. Subject to the Dirichletconditions, x(t) can be expanded as:

x(t) =

n=xnej2pinf0t , (20)

where the Fourier series coefficients are

xn =1T0

+T0

x(t)ej2pinf0tdt , (21)

for some arbitrary . To describe x(t), it is sufficient to specify a countable set of

(in general, complex) numbers.31 / 44

Fourier Series of Periodic Signals For real signal x(t):

x(t) =a02

+n=1

an cos(2pinf0t) + bn sin(2pinf0t), (22)

where

an =2T0

+T0

x(t) cos(2pinf0t)dt , (23)

bn =2T0

+T0

x(t) sin(2pinf0t)dt . (24)

Parsevals relation: Result of orthogonality of the basis ofcomplex exponentials

1T0

+T0

|x(t)|2dt =

n=|xn|2. (25)

LHS: power content of x(t); |xn|2: power content ofxnej2pinf0t , the nth harmonic. 32 / 44

Response of LTI Systems to Periodic Signals Given impulse response h(t), response to the exponentialej2pif0t is H(f0)ej2pif0t , where the frequency response H(f ) isdefined as:

H(f ) =

h(t)ej2piftdt . (26)

With Fourier series expansion of x(t), response to x(t) is:

y(t) = T [x(t)] = T[ n=

xnej2pinf0t]

=

n=xnT

[ej2pinf0t

]=

n=

xnH(nf0) yn

ej2pinf0t . (27)

Output y(t) is also periodic with period T0 = 1/f0.33 / 44

From Fourier Series to Fourier Transform Extend the ideas of Fourier series representation for

periodic signal to non-periodic signals. Suppose x(t) is non-periodic, we define truncated signal

xT0(t) ={

x(t), T02 < t T02 ,0, otherwise.

(28)

Define periodic signal xPT0(t) =

n= xT0(t nT0), then

xPT0(t) =

n=xPn e

j2pinf0t

=

n=

[1T0

T0/2T0/2

x()ej2pinf0d

]ej2pinf0t

=

n=

[f T0/2T0/2

x()ej2pifd

]ej2pift , (29)

where f = nf0 and f = (n + 1)f0 nf0 = 1/T0.34 / 44

Fourier Transform for Non-Periodic Signals To obtain the result for non-periodic signal x(t), we letT0 . Then, f0 0 and summation over n becomesintegration.

x(t) =

[

x()ej2pifd]ej2piftdf . (30)

Define the Fourier transform of x(t) as:

X (f ) =

x(t)ej2piftdt . (31)

Then, x(t) can be expressed as:

x(t) =

X (f )ej2piftdf . (32)

Notation: X (f ) = F [x(t)] and x(t) = F1[X (f )].35 / 44

Exercise Fourier Transform

Find the Fourier transform of the rectangular function:

(t) ={

A, T2 < t < T2 ,0, otherwise.

(33)

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Table of Common Fourier TransformsTime Domain Frequency Domain

(t) 1

1 (f )

(t t0) ej2pift0ej2pif0t (f f0)cos(2pif0t) 12(f f0) + 12(f + f0)sin(2pif0t) 12j (f f0) 12j (f + f0)(t) sinc(f )

sinc(t) (f )

(t) sinc2(f )

sinc2(t) (f )n= (t nT0) 1T0

n= (f nf0 )

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Fourier Transform for Periodic Signals Fourier series representation of periodic signal x(t):

x(t) =

n=xnej2pi nT0 t . (34)

Taking Fourier transform of both sides gives:

X (f ) =

n=xn

(f n

T0

). (35)

The spectrum is no longer continuous, but consists ofdiscrete lines at multiples of f0 = 1/T0.

It can be shown that:

X (f ) =1T0

n=

XT0

(nT0

)

(f n

T0

). (36)

A convenient way to compute xn is: xn = 1T0XT0(

nT0

).

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Key Properties of Fourier Transform

Linearity: For any (real or complex) scalars , ,

F [x1(t) + x2(t)] = X1(f ) + X2(f ). (37)

Scaling: For any real a 6= 0,

F [x(at)] = 1|a|X(fa

). (38)

Time shift:

F [x(t t0)] = ej2pift0X (f ). (39)

Modulation:

F [x(t)ej2pif0t ] = X (f f0). (40)

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Key Properties of Fourier Transform

Convolution: If x(t) and h(t) possess Fourier transforms:

F [x(t)Fh(t)] = X (f )H(f ). (41) Much simpler to find response in frequency domain than in

time domain. This result is the basis of frequency domain analysis of LTI

systems.

Parsevals relation: |x(t)|2dt =

|X (f )|2df . (42)

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Fourier Transform of Real, Even and Odd Signals

X (f ) =

x(t)ej2piftdt

=

x(t) cos(2pift)dt j

x(t) sin(2pift)dt

= XR(f ) jXI(f ). (43)

If x(t) is real XR(f ) is real and even; XI(f ) are real and odd. X (f ) = X (f ). |X (f )| = |X (f )|. X (f ) = X (f ).

If x(t) is real and even, XI(f ) = 0. Then X (f ) = XR(f ) isreal and even.

If x(t) is real and odd, XR(f ) = 0. Then X (f ) = jXI(f ) isimaginary and odd.

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Energy and Power Relations The energy content of an energy-type signal x(t):

Ex = |x(t)|2dt =

|X (f )|2df =

Gx(f )df . (44)

Energy spectral density: Gx(f ) = |X (f )|2. We define autocorrelation function Rx() as:

Rx() = x()Fx() =

x(t)x(t )dt . (45)

It can be easily verified thatEx = R(0) (46)

Gx(f ) = F [Rx()]. (47) Input-output relation for y(t) = x(t)Fh(t):

Ry (t) = Rx(t)FRh(t) (48)Gy (f ) = Gx(f )Gh(f ) = |X (f )|2|H(f )|2. (49)

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Energy and Power Relations Time-average autocorrelation of power-type signal x(t):

Rx() = limT

1T

T/2T/2

x(t)x(t )dt . (50)

By setting = 0, we obtain the power content of x(t):

Px = Rx(0) = limT

1T

T/2T/2

|x(t)|2dt . (51)

Define power spectral density: Sx(f ) = F [Rx()], then:

Px = Rx(0) = Sx(f )df . (52)

Input-output relation of y(t) = x(t)Fh(t):Ry (t) = Rx(t)Fh(t)Fh(t) (53)Sy (f ) = Sx(f )|H(f )|2. (54)

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Energy and Power Relations For periodic (i.e., power-type) signal x(t):

Rx() =1T0

T0/2T0/2

x(t)x(t )dt =

n=|xn|2ej2pi

nT0.

Power spectral density of x(t):

Sx(f ) = F [Rx()] =

n=|xn|2

(f n

T0

). (55)

Power content of x(t): Px = Sx(f ) =

n= |xn|2.

Input-output relation for y(t) = x(t)Fh(t):

Sy (f ) = Sx(f )|H(f )|2 =

n=|xn|2

H ( nT0)2 (f nT0

)

Py =

n=|xn|2

H ( nT0)2 . (56)

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