1

ECE 3TR4 Communication Systems (Winter 2004)Dr. T. Kirubarajan (Kiruba)ECE DepartmentCRL-225 kiruba@mcmaster.cawww.ece.mcmaster.ca/~kiruba/3tr4/3tr4.html

2© Jeff Bondy

Course Overview

Communication Systems OverviewFourier Series/Transform ReviewSignals and Systems ReviewIntroduction to NoiseMotivation for ModulationAmplitude ModulationAngle ModulationPulse ModulationMultiplexingTransmitters and Receivers

3© Jeff Bondy

Communication SystemsInformation

Source

Transmitter Channel Receiver Information

Destination

Blackberry Keypad

Speakers Brain

IP Packet

GSM-style RF

Vocal Tract

SONET Router

Wireless RF

Acoustic

Fiber

FM Detector

Ears

Photo Diode

ATM.25 Packet

Brain

Router POTS

Analog Communications (3TR4): Information is encoded in a continuous amplitude, continuous time signal.

Digital Communications (4TK4): Information is encoded into a discrete time sequence with a quantized alphabet.

4© Jeff Bondy

Communication ChannelsChannel: The medium linking the transmitter and receiver. It is ALWAYS analog in nature. That is every communication system is more or less ANALOG.

Channel TypesWireline Channels: use a conductive medium to direct transmitted energy to the receiver:

•Copper wire for telephones, xDSL•Fiber optic cable•Aluminum interconnects for ICs

Wireless Channels: Uses an open propagation medium•RF for cell phones•Underwater acoustic ducts for whales

5© Jeff Bondy

Channel ImpairmentsAs a transmitted signal propagates it loses fidelity in a number of ways. This loss of fidelity makes the received signal look very different from the transmitted signal.

Additive Noise: Thermal noise, multi-transmitter interferenceNoise

Transmitter + Receiver

Multiplicative Noise: Rayleigh FadingNoise

Transmitter x Receiver

Convolution Noise: time-delay multipath, reverberationTransmitter Noise Receiver

6© Jeff Bondy

3TR4 ObjectiveInformation

Source

Transmitter Channel Receiver Information

Destination

1. How to design

2. Taking into account

3. That will provide a system that is:Reliable: information received is what was sentEfficient: Not wasteful of time, power or spectrumSimple: economical for H/W and S/W and usually Robust

7© Jeff Bondy

Tradeoffs in Objectives

Simple H/W

Simple S/W

Spectral Use

Temporal Use Power Use

Accuracy & Robustness

Simple

Efficient

Reliable

8© Jeff Bondy

Digital CommunicationsDigital Information Source

Source

Encoder

Channel

Encoder

Modulator

Digital Information Destination

DACN

Source

Decoder

Channel

Decoder

DeModulatorADC

Channel

The placement of the DAC and ADC is up to the system requirements. They can be anywhere between the Information Sources and Destination and the Modulator and Demodulator, respectively.

9

Fourier Series/Transform Review

10© Jeff Bondy

Fourier ReviewFourier Series and Transforms try to form a signal out of sinusoids. These sinusoids have a specific frequency and go on forever. That is your nice time series which is represented by points in time will now be represented by points in frequency. This is why we use the terms “Fourier domain” and “frequency domain” interchangeably.

Reminder:

)sin()cos()( btjabtaae jbt +=

11© Jeff Bondy

What Transform, When?

I-FTNoContinuousFrequencyI-DTFTYesContinuousFrequencyI-FSNoDiscreteFrequencyI-DTFSYesDiscreteFrequencyFTNoContinuousTimeFSYesContinuousTimeDTFTNoDiscreteTimeDTFSYesDiscreteTime

TransformPeriodicDiscrete or Continuous

Start Domain

12© Jeff Bondy

Discrete Time Fourier Series

∑ >=<Ω−=

Nnnjkenx

NkX 0][1][DTFS:

∑ >=<Ω=

NknjkekX

Nnx 0][1][I-DTFS:

X[k] and x[n] have period N

Ω0 = 2π/N

13© Jeff Bondy

Discrete Time Fourier Transform

∑∞

−∞=

Ω−Ω =n

njj enxeX ][][

∫−

ΩΩ Ω=π

ππdeeXnx njj )(

21][

DTFT:

I-DTFS:

X[k] has period 2π

14© Jeff Bondy

Fourier Series

∫><

−=T

tjk dtetxT

kX 0)(1][ ωFS:

∑∞

−∞=

=k

tjk oekXtx ω][)(I-FS:

X(t) has period T

Ω0 = 2π/T

15© Jeff Bondy

Fourier Transform

∫∞

∞−

= ωωπ

ω dejXtx tj)(21)(

∫∞

∞−

−= dtetxjX tjωω )()(FT:

I-FT:

The Fourier Transform is the general transform, it can handle periodic and non-periodic signals. For a periodic signal it can be thought of as a transformation of the Fourier Series

∑∞

−∞=

−=k

nkXjX )(][2)( 0ωωδπω

16© Jeff Bondy

Fourier Series

∫><

−=T

tjk dtetxT

kX 0)(1][ ω

kk B

To

A

To dtkttx

Tjdtkttx

TkX ∫∫

><><

−= )sin()(1)cos()(1][ ωω

k

k

kk

k

eXeBAkX ke

AB

X

kkθ

θ

=+=

− −1tan22][

17© Jeff Bondy

Fourier Series – Real Signals

kk B

To

A

To dtkttx

Tjdtkttx

TkX ∫∫

><><

−= )sin()(1)cos()(1][ ωω

If x(t) is real valued: Ak = A-k Bk = -B-k

( ) ( ) ( )( )∑∑∑∞

=

−−−

∞

=

−∞

−∞=

++++=−++==11

]0[][][]0[][)(k

tjkkk

tjkkk

k

tjktjk

k

tjk ooooo ejBAejBAXekXekXXekXtx ωωωωω

( ) ( )( ) ( ) ( )( )∑∑∞

=

−−∞

=

− ++++=−+++=11

]0[]0[)(k

tjktjkk

tjktjkk

k

tjkkk

tjkkk

oooooo eejBeeAXejBAejBAXtx ωωωωωω

( ) ( )∑∑∞

=

∞

=

+=++=11

][Re2]0[)sin()cos(2]0[)(k

tjk

kokok

oekXXtkBtkAXtx ωωω

( )=+= ∑∞

=1][Re2]0[)(

k

tjkj ok eekXXtx ωθ ∑∞

=

++1

0 )cos(][2]0[k

ktkkXX θω

18© Jeff Bondy

Fourier Series – Real +Even/Odd( )∑

∞

=

+=1

][Re2]0[)(k

tjkj ok eekXXtx ωθ

( )( ) ∑∞

=

+−+=1

)sin()cos(Re2]0[)(k

ookk tkjtkjBAXtx ωω

( )∑∞

=

++=1

)sin()cos(2]0[)(k

okok tkBtkAXtx ωω

Even: f(t) = f(-t), therefore Bk = 0; Cosine Series

Odd: f(t) = -f(-t), therefore Ak = 0; Sine Series

19© Jeff Bondy

Cosine Fourier Series

Even Function

tjtj eettf 00

21

21)cos()( 0

ωωω −+==

21]1[]1[ =−= XXFS

)()()( 00 ωωπδωωπδω −++=jXFT = 2π(FS)

When is FT the continuous counterpart to 2πFS?How do the Delta’s move as frequency changes?

20© Jeff Bondy

Sine Fourier Transform

Odd Function

tjtj ej

ej

ttf 00

21

21)sin()( 0

ωωω −−==

jXX 21]1[]1[ =−−=FS

)()()( 00 ωωπδωωπδω −−+= jjjXFT = 2π(FS)

The Fourier Transform of an Odd Signal is Odd.Notice the Fourier Domain graph is in jF(ω). It is imaginary.

21© Jeff Bondy

DC Fourier Transform

DC Function0;1)( 0

0 === ωω tjetf

1]0[ =XFS

FT (FS)

)(2)( ωπδω =jXThe FT of a signal with a DC component is separable.The DC component of a time signal is statistically the MEAN.

( )∑∞

−∞=

−=k

kkXjX 0][2)( ωωδπω

FT

22© Jeff Bondy

Delta Fourier Transform

Delta Function)0()( δ=tf

FS - No Fourier Series, Not Periodic

∫∞

∞−

−− === 1)()( )0(22 kjktj edtetjX ππδω

The FT is only congruent with the FS for PERIODIC signals.A delta has an infinitely steep rise time, therefore it has a great deal of high frequencies

FT

23© Jeff Bondy

Pulse Train Fourier Transform

Function with Period T

∑∞

−∞=

−=n

nTttf )()( δ

( )∑∫ ∑∫ ∑∞

−∞=

∞

∞−

∞

−∞=

−∞

∞−

∞

−∞=

− −==−=kn

knTj

n

ktj

Tk

TdtedtenTtjX πωδπδω ππ 22)()( 22

kallforTkX 1][ =FS

What happens in the Frequency Domain when the time between pulses is shortened? When T 0? When T = 0?

24© Jeff Bondy

Time Window Fourier Transform

Not Periodic – No FS

≥

<=

2,02,1

)( τ

τ

t

ttf

( )22sin2)( ωττ

ω

ωτω SincjX ==FT

( )τtrect≡

( ) ( ) ( )xxxSaxSinc sin

≡≡

25© Jeff Bondy

Ideal Filter Fourier Transform

Not Periodic – No FS( )WtSinctx =)(

( )WrectWW

WWjX 2,0,

)( ωπωωπ

ω =

≥

<=FT

Why is this called the “ideal filter”?Notice similarities between this and rectangular time window, and how W here is a counterpart to τ there in controlling width.

26© Jeff Bondy

Triangle Fourier Transform

Not Periodic – No FS

( )ττττ t

ttt

tx Λ≡

≥

<−=,0

,1)(

( )[ ]22)( ωττω SincjX =FT

Sinc squared can never be negative. Why are we introducing these signals? They are the foundation of most analog communication signals.

27© Jeff Bondy

More Complex Example

- 2 - 1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2

0

0 . 5

1

-2 -1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2

0

0 . 5

1

-2 -1 . 5 - 1 - 0 . 5 0 0 . 5 1 1 . 5 2

0

0 . 5

1

An pulse train with period (T) one second is convolved with a time windowing function with timing (τ) of 0.5 seconds, to produce a 50% duty cycle square wave.

28© Jeff Bondy

More Complex Example

( )∑∞

−∞=

−=k

Tk

TjX πωδπω 22)(1

The spectrum of the pulse train is:

The spectrum of the square-wave is:

( )2)(2 ωττω SincjX =

Convolution turns into Multiplication in the Freq Domain

( ) ( )∑∞

−∞=

−=⋅k

TkSincTjXjX πωδωτπτωω 2

22)()( 21

This turns into a line spectra, and how it changes with changing the parameters is very informative

29© Jeff Bondy

5.0=τConstant τ

T = 2

T = 4

T = 8

• Amplitude DECREASES as 1/T• Line spectra resolution INCREASES as T• The envelope is INDEPENDENT of T

30© Jeff Bondy

Constant T T = 2

25.0=τ

• Amplitude INCREASES in proportion to Tau• Line spectra resolution is INDEPENDENT of Tau• The spectrum SPREADS as the window shortens !!! TIME RESOLUTION AND FREQUENCY RESOLUTION ARE

INVERSELY RELATED !!!!!!!!

5.0=τ

1=τ

31© Jeff Bondy

The Sampling TheoremOne of the fundamental concepts in dealing with the representation of analog signals in the digital domain is the Nyquist Rate, or Minimum Time-Bandwidth product. This law states the minimum sample frequency necessary to exactly represent an analog signal as a digital signal.Since one of the main constraints in judging the efficiency of a communication system is spectral efficiency, the Nyquist rate forms a large part of the back-bone of system design.

A real-valued band-limited signal having no spectral components above a frequency of B Hz is determined uniquely by its values at uniform intervals spaced no greater than 1/2B seconds apart

32© Jeff Bondy

Sampling TheoremConsider a signal f(t) sampled with an impulse train p(t)

∑

∑

∑

∑

∞

−∞=

∞

−∞=

∞

−∞=

∞

−∞=

−=

=

=

==

=

ns

ns

n

tjns

n

tjn

s

nFF

nFF

TransformFourieretftf

Tetp

tptftf

)()(

)()()(

,)()(

2,)(

)()()(

0

0

0

0

0

ωωω

ωδωω

πω

ω

ω

33© Jeff Bondy

Sampling Theorem VisualBand limited signal + spectrum

Periodic gating function + spectrum

Size of sampling window controls envelope of spectrum, sample frequency controls spacing of original spectrum replicas

34© Jeff Bondy

Nyquist RateSince the periodic gating function controls the center of the replicas and the replicas are 2W (W = 2πB) wide, then to make sure there is no overlap:

BT

WT

21

22

≤

≤π

If the signal is sampled at a lower rate there will be overlap, and in the final spectrum you won’t know if the overlapped part is from the spectrum that is suppose to be there or from the “ALIASED” part of the spectrum

35

Signals and Systems Review

36© Jeff Bondy

Energy and PowerSignal Energy

∫∞

∞−

= ][,)()( 2* sVUNITSdttftfE f

Signal Power

∫−∞→

=2

2

2* ][,)()(1lim

T

TTf VUNITSdttftf

TP

An energy signal cannot be a power signal, nor vice-versa

To be an energy signal:Amplitude 0

As |Time| inf

37© Jeff Bondy

Energy and Power ExampleFind Ex

( )

( ) ∞=++=

++=+=

+=

∞

−∞=

∞

−∞=

∞

∞−

∞

∞−∫∫

ttx

x

tAtAE

dttAdttAE

tAtx

φωω

φωφω

φω

22sin42

)22cos(12

)(cos

)cos()(

00

22

0

2

022

0

0

( )

( ) ( )( )2

2sin2sin42

)22cos(12

1)(cos1

2

000

22

2/

2/0

22/

2/0

22

lim

limlimATT

TA

TTAP

dttAT

dttAT

P

Tx

T

TT

T

TTx

=+−++=

++=+=

∞→

−∞→−∞→∫∫

φωφωω

φωφω

38© Jeff Bondy

Parseval’s TheoremEnergy calculated in the Time domain is equal to energy calculated in the Frequency domain.

∫∫

∫

∫ ∫∫

∫ ∫∫

∫

∫∫

∞

∞−

∞

∞−

−∞

∞−

∞

∞−

−∞

∞−

∞

∞−

∞

∞−

−∞

∞−

∞

∞−

−∞

∞−

∞

∞−

∞

∞−

=

=

=

=

=

=

ωωωπ

ω

ωωπ

ωωπ

ωωπ

ωωωπ

ω

ω

ω

ω

dFFdttftf

dtetfF

ddtetfFdttftf

dtdeFtfdttftf

deFtf

dFFdttftf

tj

tj

tj

tj

)()(21)()(

)()(

)()(21)()(

)(21)()()(

)(21)(

)()(21)()(

**

**

**

**

**

39© Jeff Bondy

Power Spectral Density

∫

∫∞

∞−

−∞→

≡

=

ωωπ

ωωπ

dSP

dFT

P

ff

T

TTf

)(21

)(21 2/

2/

2lim

TF

Sd

dG

duTuF

duuSG

duuFT

duuSG

dFT

dS

Tf

f

Tff

Tff

Tf

2

2

2

2

)()(

)(2

)()()(2

)(211)(

21)(

)(211)(

21

lim

lim

lim

lim

ωω

ωω

π

ωπ

ππω

ωωπ

ωωπ

ωω

ωω

∞→

∞− ∞→∞−

∞−∞→∞−

∞

∞−∞→

∞

∞−

==

==

==

=

∫∫

∫∫

∫∫

40© Jeff Bondy

PSDSf(ω) is the power spectral density function, it has units of power per Hz.

Gf(ω) is the cumulative spectral power function, it the amount of energy in the signal in those components less then ω.

41© Jeff Bondy

Autocorrelation

)()()(1

)()()(1

21)()(1

)()(211

)()(121

)()(

2/

2/

*

11

2/

2/1

2/

2/

*

1)(

2/

2/1

2/

2/

*

2/

2/11

2/

2/

*

*

2

lim

lim

lim

lim

lim

lim

1

1

ττ

τδ

ωπ

ωπ

ωωωπ

ωω

τω

ωτωω

ωτ

f

T

TTf

T

T

T

TTf

ttjT

T

T

TTf

jT

T

tjT

T

tj

Tf

j

Tf

Tf

RdttftfT

SIFT

dtdttttftfT

SIFT

dtdtdetftfT

SIFT

dedtetfdtetfT

SIFT

deFFT

SIFT

TF

S

=+=

−+=

=

=

=

=

∫

∫∫

∫∫∫

∫ ∫∫

∫

−

−∞→

−

−

−

−∞→

∞

∞−

−+−

−

−

−∞→

∞

∞−

−

−

−−

−∞→

∞

∞− ∞→

∞→

42© Jeff Bondy

AutocorrelationRf(τ) should look familiar in a way. It is equivalent to convolving the function f(t) with f(-t).

The autocorrelation function is often used for signal detection in a background of random noise. When we get into random noise it will become very evident why this is so.

∫

∫∞

∞−

−∞→

+=−∗

+=

dttftfff

dttftfT

RT

TTf

)()()()(

)()(1)(

*

2/

2/

*lim

τττ

ττ

43© Jeff Bondy

Linear Time Invariant SystemsFundamental way of describing many components in a communication system. Models filters, amplifiers and equalizers very well.

Model an LTI system with the impulse response, h(t), of the system, the response of an impulse input to the system. The Fourier Transform of the impulse response is the frequency transfer function.

x(t) h(t) y(t)∫∞

∞−

−=

∗=

τττ dtxh

txthty

)()(

)()()(

44© Jeff Bondy

Time Operatorsf(t)

f(t-a)

g(t)

f(t+b)

g(2t)

g(t/2)

What happens to in the Fourier domain to each of these?

45© Jeff Bondy

InvertibilityLTI systems are invertibleIf you can determine the input given the output then a system is called InvertibleGiven input x and it’s output is y:

y(t) = 2 x(t)Is inverted by z:

z(t) = ½ y(t) = x(t)

Not invertible:y(t) = floorx(t)

!!! A non-invertible system usually maps multiple points from the input space to the same point in the output space.

46© Jeff Bondy

LTI Systems)()()( txthty ∗=x(t) y(t)h(t)

)()()( ωωω XHY =X(ω) Y(ω)H(ω)

In the frequency domain the convolution integral becomes a multiplication, and vice-versa. By assessing the frequency domain magnitude and phase we can see how H can effect specific frequencies differently:

)()()()()()(

)()()( )()()(

ωθωθωθ

ωωω

ωωω ωθωθωθ

xhy

jjj

XHY

eXeHeY xhy

+=

=

=

!!! This is the beginning of the filtering interpretation

47© Jeff Bondy

LTI SystemsThe Law of Superposition:Given inputs a and b to system x, a linear system:

x(a)+x(b) = x(a+b)Given input a and some scalar constant to system x,

x(c a) = c x(a)The Law of Time Invariance:Given some input function g(t) and is input to a system X produces an output f(t)

Xg(t) = f(t)If g(t) is shifted in time by T0 then the output has the same shift

Xg(t-T0) = f(t-T0)The Law of Commutation:Given some function g(t) and f(t)

g(t) * f(t) = f(t) * g(t)

48© Jeff Bondy

Ideal Filter Introduction

-10 -5 0 5 100

0.5

1

10 20 30 40 50 60

00.020.040.060.08

-10 -5 0 5 100

0.5

1

10 20 30 40 50 60

-0.05

0

0.05

-10 -5 0 5 100

0.5

1

10 20 30 40 50 60

-0.1

0

0.1

-10 -5 0 5 100

0.5

1

10 20 30 40 50 60

00.20.40.6

Low Pass Filter(LPF)

High Pass Filter(HPF)

BandPass Filter(BPF)

BandStop Filter(BSF)

Frequency Response Impulse Response

49© Jeff Bondy

Real FiltersIn reality one cannot make the Brick Wall type ideal filters. This is due to the fundamental tradeoff between time and frequency resolution. If you have a jump in the frequency response that is infinitesimally resolved, you’d need infinite time to represent that.

One deals with filter specifications such as bandwidth, roll-off, implementation complexity, passband ripple and so on for most of this course, and for many future courses.

It is of great practical importance to understand the tradeoffs implicit in the time-frequency bandwidth tradeoff.

50© Jeff Bondy

Filters cont’dMost filters bandwidths are defined by the 3 dB point, or where the frequency transfer response is 1/2 less then the maximum point.

51© Jeff Bondy

Filter Truncation - TimeOne can never implement an ideal filter because the infinite frequency resolution requires infinite time. What happens when you just get rid of some of the time window?

-100 -50 0 50 100

0

0.2

0.4

0 50 100 150 2000

0.5

1

-50 0 50

0

0.2

0.4

0 20 40 60 80 1000

0.5

1

-10 -5 0 5 10

0

0.2

0.4

0 5 10 15 200

0.5

1

-6 -4 -2 0 2 4

0

0.2

0.4

0 2 4 6 8 100

0.5

1

W = 100

50

10

5

Ringing = Gibbs effect

Longer Time Window, steeper frequency roll-off