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ECE 3TR4 Communication Systems (Winter 2004)Dr. T. Kirubarajan (Kiruba)ECE DepartmentCRL-225 [email protected]/~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