Transcript Document

ECE 3TR4 Communication
Systems (Winter 2004)
Dr. T. Kirubarajan (Kiruba)
ECE Department
CRL-225
[email protected]
www.ece.mcmaster.ca/~kiruba/3tr4/3tr4.html
1
Course Overview
Communication Systems Overview
Fourier Series/Transform Review
Signals and Systems Review
Introduction to Noise
Motivation for Modulation
Amplitude Modulation
Angle Modulation
Pulse Modulation
Multiplexing
Transmitters and Receivers
© Jeff Bondy
2
Communication Systems
Overview
3
Communication Systems
Information
Transmitter
Channel
Receiver
Source
Information
Destination
Blackberry Keypad
GSM-style RF
Wireless RF
FM Detector
ATM.25 Packet
Speakers Brain
Vocal Tract
Acoustic
Ears
Brain
IP Packet
SONET Router
Fiber
Photo Diode 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.
© Jeff Bondy
4
Communication Channels
Channel: The medium linking the transmitter and receiver.
It is ALWAYS analog in nature. That is every communication
system is more or less ANALOG.
Channel Types
Wireline 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
© Jeff Bondy
5
Channel Impairments
As 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 interference
Noise
Transmitter
+
Receiver
Multiplicative Noise: Rayleigh Fading
Noise
Transmitter
x
Receiver
Convolution Noise: time-delay multipath, reverberation
Transmitter
Noise
Receiver
© Jeff Bondy
6
3TR4 Objective
Information
Transmitter
Source
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 sent
Efficient: Not wasteful of time, power or spectrum
Simple: economical for H/W and S/W and usually Robust
© Jeff Bondy
7
Tradeoffs in Objectives
Spectral Use
Temporal Use
Simple H/W
Simple
Efficient
Power Use
Reliable
Accuracy & Robustness
Simple S/W
© Jeff Bondy
8
Digital Communications
Digital
Information
Source
N Source
Encoder
Channel
Encoder
DAC
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.
Digital
Information
Destination
Source
Channel
Decoder
Decoder
ADC
Modulator
Channel
DeModulator
© Jeff Bondy
9
Fourier Series/Transform
Review
10
Fourier Review
Fourier 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:
ae( jbt )  a cos(bt)  ja sin(bt)
© Jeff Bondy
11
What Transform, When?
Start
Domain
Time
Time
Time
Time
Frequency
Frequency
Frequency
Frequency
Discrete or
Continuous
Discrete
Discrete
Continuous
Continuous
Discrete
Discrete
Continuous
Continuous
Periodic
Transform
Yes
No
Yes
No
Yes
No
Yes
No
DTFS
DTFT
FS
FT
I-DTFS
I-FS
I-DTFT
I-FT
© Jeff Bondy
12
Discrete Time Fourier Series
DTFS:
I-DTFS:
1
X [k ]  n  N  x[n]e  jk0 n
N
1
x[n]  k  N  X [k ]e jk0 n
N
X[k] and x[n] have period N
Ω0 = 2π/N
© Jeff Bondy
13
Discrete Time Fourier Transform
DTFT:

 jn
x
[
n
]
e

X [e j ] 
n  
I-DTFS:
1
x[n] 
2

j
jn
X
(
e
)
e
d


X[k] has period 2π
© Jeff Bondy
14
Fourier Series
FS:
1
X [k ]   x(t )e jk0t dt
T T 
I-FS:
x(t ) 

 X [k ]e
jk ot
k  
X(t) has period T
Ω0 = 2π/T
© Jeff Bondy
15
Fourier Transform

FT:
I-FT:
X ( j )   x(t )e jt dt
1
x(t ) 
2


jt
X
(
j

)
e
d


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
X ( j )  2

 X [k ] (  n0 )
k  
© Jeff Bondy
16
Fourier Series
1
 jk0t
X [k ]   x(t )e
dt
T T 
1
X [k ]   x(t ) cos(o kt)dt 
T T 

1
j  x(t ) sin(o kt)dt
T T 

Ak
Bk
 tan1 

Bk

Ak 
X [k ]  Ak2  Bk2 e  X k e k


k
Xk
e
© Jeff Bondy
17
Fourier Series – Real Signals
1
X [k ]   x(t ) cos(o kt)dt 
T T


1
j  x(t ) sin(o kt)dt
T T


Ak
If x(t) is real valued:
x(t ) 

 X [k ]e
jk ot
k  

k 1



 X [0]   X [k ]e
jk ot
Bk
Ak = A-k
 X [k ]e
 jk ot
Bk = -B-k
  X [0]    A

k 1

k
 jBk e jkot   Ak  jBk e  jkot
 



x(t )  X [0]    Ak  jBk e jkot   Ak  jBk e  jkot  X [0]   Ak e jkot  e  jkot  jBk e jkot  e  jkot
k 1

k 1
k 1

x(t )  X [0]  2  Ak cos(kot )  Bk sin(kot )   X [0]  2 Re X [k ]e jkot




 X [k ] cos(k t  
 X [0]  2
x(t )  X [0]  2 Re X [k ] e j k e jkot 
k 1

k 1

k 1
0
k
)
© Jeff Bondy

18
Fourier Series – Real +Even/Odd


x(t )  X [0]  2 Re X [k ] e j k e jkot

k 1

x(t )  X [0]  2 Re Ak  jBk cos(kot )  j sin(kot ) 
k 1

x(t )  X [0]  2  Ak cos(kot )  Bk sin(kot )
k 1
Even: f(t) = f(-t), therefore Bk = 0; Cosine Series
Odd: f(t) = -f(-t), therefore Ak = 0; Sine Series
© Jeff Bondy
19
Cosine Fourier Series
1 j 0 t 1  j  0 t
f (t )  cos(  0t )  e  e
2
2
Even Function
FS X [1]  X [1]  1
FT = 2π(FS)
2
X ( j )   (  0 )   (  0 )
When is FT the continuous counterpart to 2πFS?
How do the Delta’s move as frequency changes?
© Jeff Bondy
20
Sine Fourier Transform
1 j0t 1  j 0t
f (t )  sin( 0t ) 
e  e
2j
2j
Odd Function
FS
X [1]   X [1]  1
FT = 2π(FS)
2j
X ( j )  j (  0 )  j (  0 )
The Fourier Transform of an Odd Signal is Odd.
Notice the Fourier Domain graph is in jF(ω). It is imaginary.
© Jeff Bondy
21
DC Fourier Transform
f (t )  1  e j0t ;
DC Function
FS
FT
(FS)
FT
0  0
X [0]  1
X ( j )  2

 X [k ]   k 
k  
0
X ( j )  2 ( )
The FT of a signal with a DC component is separable.
The DC component of a time signal is statistically the MEAN.
© Jeff Bondy
22
Delta Fourier Transform
f (t )   (0)
Delta Function
FS
- No Fourier Series, Not Periodic

FT
X ( j )    (t )e j 2kt dt  e j 2k (0)  1

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
© Jeff Bondy
23
Pulse Train Fourier Transform
f (t ) 

  (t  nT )
n  
Function with Period T
X [k ]  1
FS
T
X ( j ) 


  (t  nT)e
n 
 j 2kt
dt 
for all k


 e
n  
 j 2knT
dt  2



 2k 

T
T

k 
What happens in the Frequency Domain when the time
between pulses is shortened? When T  0? When T = 0?
© Jeff Bondy
24
Time Window Fourier Transform
 


1, t  2
t
f (t )  

rect

0
,
t



2

Not Periodic – No FS
2 sin 
2   Sinc 
FT X ( j ) 
2

sin  x 
Sinc  x   Sa  x  
x
 
© Jeff Bondy
25
Ideal Filter Fourier Transform
x(t )  SincWt 
Not Periodic – No FS
FT

 W ,   W 
X ( j )  
 rect 
2W
 W W

 0,


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
 t

x(t )  1   , t     t

0
,
t




Not Periodic – No FS
2
FT
X ( j )   Sinc 
2
Sinc squared can never be negative. Why are we introducing
these signals? They are the foundation of most analog
communication signals.
 
  
© Jeff Bondy
27
More Complex Example
1
0.5
0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-1.5
-1
-0.5
0
0.5
1
1.5
2
1
0.5
0
-2
1
0.5
0
-2
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.
© Jeff Bondy
28
More Complex Example
The spectrum of the pulse train is:
X 1 ( j )  2
    2k T 

T k 
The spectrum of the square-wave is:
 2
X 2 ( j )   Sinc
Convolution turns into Multiplication in the Freq Domain
X 1 ( j )  X 2 ( j )  2
T
 2     2k T 
Sinc

k  
This turns into a line spectra, and how it changes with
changing the parameters is very informative
© Jeff Bondy
29
Constant τ
  0.5
T=2
T=4
T=8
•
•
•
Amplitude DECREASES as 1/T
Line spectra resolution INCREASES as T
The envelope is INDEPENDENT of T
© Jeff Bondy
30
Constant T
T=2
  0.25
  0.5
 1
•
•
•
!!!
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 !!!!!!!!
© Jeff Bondy
31
The Sampling Theorem
One 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
© Jeff Bondy
32
Sampling Theorem
Consider a signal f(t) sampled with an impulse train p(t)
f s (t )  f (t ) p(t )
p(t ) 

 e jn  t ,
0
n  
 0  2 T

f s (t )  f (t )  e jn  0t , FourierTransform
n  
Fs ( ) 
Fs ( ) 

 F ( ) (n 0 )
n  

 F (  n 0 )
n  
© Jeff Bondy
33
Sampling Theorem Visual
Band limited signal +
spectrum
Periodic gating
function + spectrum
Size of sampling
window controls
envelope of
spectrum, sample
frequency controls
spacing of original
spectrum replicas
© Jeff Bondy
34
Nyquist Rate
Since 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:
2
 2W
T
1
T
2B
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
© Jeff Bondy
35
Signals and Systems Review
36
Energy and Power
Signal Energy
Ef 


f (t ) f * (t )dt, UNITS [V 2 s ]

Signal Power
1
Pf  lim
T  T
T

2
T
f (t ) f * (t )dt, UNITS [V 2 ]
2
An energy signal cannot be a power signal, nor vice-versa
To be an energy signal:
Amplitude  0
As
|Time|  inf
© Jeff Bondy
37
Energy and Power Example
Find Ex
x(t )  A cos( 0 t   )

2
A
E x   A 2 cos2 ( 0 t   )dt 
2

Ex 
2
At
2


t  
2

 1  cos(2 0t  2 ) dt


A
sin2 0 t  2 

4 0
t  
0
2
1 T /2 2
1
A
2
Px  lim
A
cos
( 0t   )dt  lim

T  T T / 2
T  T 2
Px  lim
T 
T /2
 1  cos(2 0t  2 )dt
T / 2
A2T
A2
A2
sin 0T  2   sin 0T  2  

2T 4T 0
2
© Jeff Bondy
38
Parseval’s Theorem
Energy calculated in the Time domain is equal to energy
calculated in the Frequency domain.

1
f
(
t
)
f
(
t
)
dt


2

*
1
f (t ) 
2
*

F
*

*
F
(

)
F
( ) d


( )e  jt d


*
 j t
F
(

)
e
d

 dt










1  *
*
 j t
f
(
t
)
f
(
t
)
dt

F
(

)
f
(
t
)
e
dt

 d


2  

 


f (t ) f (t ) dt 

*
F ( ) 


 1
f (t ) 
 2

f (t )e  jt dt


1
f
(
t
)
f
(
t
)
dt


2

*

*
F
(

)
F
( ) d


© Jeff Bondy
39
Power Spectral Density
Pf  lim
T 
1
Pf 
2
1
2
1
2T

F ( ) d
2
T / 2

 S f ( )d


 S f ( )d  lim
T 

1
G f ( ) 
2
2G f ( ) 
dG f ( )
d
1 1
T 2


 S f (u )du  lim
T 


F ( ) d
2


1 1
T 2

 S f (u )du   lim
T 

2
T /2
T 

F ( )
T
2
F (u ) du

F (u )

 S f ( )  lim

2
du
2
T
© Jeff Bondy
40
PSD
Sf(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.
© Jeff Bondy
41
Autocorrelation
S f ( ) 
lim
T 
 
1

2
IFT S f
F ( )
2
T

 lim
T 

   lim
1 1
T 2
   lim
1
T
   lim
1
T
   lim
1
T
IFT S f
IFT S f
IFT S f
IFT S f
T 
T 
T 
T 
1 *
F ( ) F ( )e j d
T
 T / 2
 
f * (t )e jt dt
  T / 2
T / 2
T / 2

f (t1 )e  jt dt1e j d
1

e
j (  t  t 1 )


d  dt1 dt

T / 2
*
f (t )
T / 2

T / 2
 1
*
 f (t )  f (t1 )  2
T / 2
T / 2

T / 2
T / 2

f (t1 ) (  t t 1) dt1dt
T / 2
T / 2

f * (t ) f (t   ) dt  R f ( )
T / 2
© Jeff Bondy
42
Autocorrelation
Rf(τ) should look familiar in a way. It is equivalent to
convolving the function f(t) with f(-t).
1 T /2 *
R f ( )  lim
f (t ) f (t   )dt

T   T T / 2
f * ( )  f ( ) 

 f (t ) f (t   )dt

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.
© Jeff Bondy
43
Linear Time Invariant Systems
Fundamental 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.
y (t )  h(t )  x(t )
x(t)
y(t)
h(t)

  h( ) x(t   )d

© Jeff Bondy
44
Time Operators
f(t) 
g(t) 
f(t-a) 
g(2t) 
f(t+b) 
g(t/2) 
What happens to in the Fourier domain to each of these?
© Jeff Bondy
45
Invertibility
LTI systems are invertible
If you can determine the input given the output then
a system is called Invertible
Given 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) = floor{x(t)}
!!! A non-invertible system usually maps multiple
points from the input space to the same point in the
output space.
© Jeff Bondy
46
LTI Systems
x(t)
h(t)
y(t)
y(t )  h(t )  x(t )
X(ω)
H(ω)
Y(ω)
Y ( )  H ( ) X ( )
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:
Y ( ) e
j y ( )
 H ( ) e j h ( ) X ( ) e j x ( )
Y ( )  H ( ) X ( )
 y ( )   h ( )   x ( )
!!! This is the
beginning of the
filtering interpretation 47
© Jeff Bondy
LTI Systems
The 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)
X{g(t)} = f(t)
If g(t) is shifted in time by T0 then the output has the same shift
X{g(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)
© Jeff Bondy
48
Ideal Filter Introduction
Frequency Response
Low Pass Filter
(LPF)
High Pass Filter
(HPF)
BandPass Filter
(BPF)
1
0.08
0.06
0.04
0.02
0
0.5
0
-10
Impulse Response
-5
0
5
10
10
20
30
40
50
60
10
20
30
40
50
60
10
20
30
40
50
60
10
20
30
40
50
60
1
0.05
0.5
0
-0.05
0
-10
-5
0
5
10
1
0.1
0.5
0
-0.1
0
-10
-5
0
5
10
1
BandStop Filter
(BSF)
0.6
0.4
0.5
0.2
0
-10
0
-5
0
5
10
© Jeff Bondy
49
Real Filters
In 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, rolloff, 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.
© Jeff Bondy
50
Filters cont’d
Most filters bandwidths are defined by the 3 dB point, or
where the frequency transfer response is 1/2 less then
the maximum point.
© Jeff Bondy
51
Filter Truncation - Time
One 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?
1
0.4
0.2
W = 100
0.5
0
-100
-50
0
50
100
0
0
50
100
150
200
1
0.4
0.2
0.5
0
-50
0
50
0
0
20
40
60
80
100
50
Ringing = Gibbs
effect
1
0.4
0.2
10
0.5
0
-10
-5
0
5
10
0
0
5
10
15
20
1
0.4
0.2
5
0.5
Longer Time
Window, steeper
frequency roll-off
0
-6
-4
-2
0
2
4
0
0
2
4
6
8
10
© Jeff Bondy
52