ECE 6640
Digital Communications
Dr. Bradley J. Bazuin
Assistant Professor
Department of Electrical and Computer Engineering
College of Engineering and Applied Sciences
Chapter 2
Chapter 2:
Deterministic and Random Signal Analysis
17
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
Bandpass and Lowpass Signal Representation
Signal Space Representation of Waveforms
Some Useful Random Variables
Bounds on Tail Probabilities
Limit Theorems for Sums of Random Variables
Complex Random Variables
Random Processes
Series Expansion of Random Processes
Bandpass and Lowpass Random Processes
Bibliographical Notes and References
Problems
18
28
40
56
63
63
66
74
78
82
82
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
2
Signal Definitions

1,

1
rect t    t    ,
2
0,


1
2
1
t 
2
otherwise
t 
t  1,

tri t   t    t    t   1  t ,
0,

1  t  0
0  t 1
otherwise
t 0
1,

sinct    sin   t 
   t ,
ECE 6640
1,

sgn t   0,
 1,

t0
t0
1,
1

u 1 t    ,
2
0,
t 0
t0
t 0
t0
t0
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
3
Table of Fourier Transforms
• Continuous Time
and Frequency
Fourier Transforms
and Properties
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
4
Fourier Transforms Pairs
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
5
Lowpass Signal Representations
• The Fourier Transform of a real signal has complex
conjugate symmetry
– the real part is symmetric about 0
– the imaginary part is anti-symmetric about 0
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
6
Bandpass Signal Representations
• The Fourier Transform of a real signal has complex
conjugate symmetry
– the real part is symmetric about 0
– the imaginary part is anti-symmetric about 0
• For real signals, the positive spectrum is sufficient to
completely describe the spectral response.
*
X  f   X   f   X   f   X   f   X   f 
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
7
Lowpass Equivalent of Bandpass
• Complex Signal Baseband representation
• The “analytic signal” is the complex representation of a
real signal
– Based on the Hilbert Transform
x t  
1
1
1
1  1

 xt   j   xˆ t    xt   j   
 xt 
2
2
2
2   t

xl t   2  x t   exp j  2    f 0  t 
xt   Rexl t   exp j  2    f 0  t 
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
8
Errata
• Equation 2.1-5
X l  f   2  X   f  f 0   2  X  f  f 0   u1  f  f 0 
• Dr. Bazuin would prefer the equation to read
– The complex lowpass equivalent has negative frequency
components! Therefore, the step function is incorrect and should
allow “all positive” frequencies!
X l  f   2  X   f  f 0   2  X  f  f 0   u 1  f 
ECE 6640
9
Complex Signal Representation
• While a complex signal has a real and imaginary part, in
communications we typically say:
– The in-phase component for the real part
– The quadrature-phase component for the imaginary part
xl t   xI t   j  xQ t 
xI t   xt   cos2    f 0  t   xˆ t   sin 2    f 0  t 
xQ t    xt   sin 2    f 0  t   xˆ t   cos2    f 0  t 
– Note also
xt   xI t   cos2    f 0  t   xQ t   sin 2    f 0  t 
xˆ t   xQ t   cos2    f 0  t   xI t   sin 2    f 0  t 
ECE 6640
10
Envelope and Phase Representation
• We are often interested in magnitude (envelope) and phase
rx t   xI t   xQ t  
2
2
x t   j  x t  x t   j  x t 
*
I
Q
I
Q
 xQ t  

 x t   arctan


x
t
 I 
xl t   rx t   exp j   x t 
xt   Rerx t   exp j  2    f 0  t  j   x t 
xt   rx t   cos2    f 0  t   x t 
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
11
Modulator: Frequency Up-conversion
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
12
Demodulators: Frequency
Down-conversion
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
13
Signal Constructs
• The RF signal is considered as a bandpass real signal in
time.
• Both the baseband signal to be modulated (up-converted)
and demodulated (down-converted) baseband signal are
often more useful as complex signals.
– Up- or down-conversion is often a better description, as
modulation and demodulation can be applied to signals at
baseband before or after “frequency translation”.
– By the way, we have so far skipped some very important
structures, FILTERS!!!
ECE 6640
14
Signal Energy Consideration
• The energy of a signal is defined as


EX 

xt   dt 
2

 xt  xt   dt
*

• Based on Rayleigh’s Theorems, the Fourier Transform
equivalence

EX 

xt   dt 
2



X  f   df
2

• Also of note is Parseval’s Theorem
xt , y t  

 xt  yt   dt 

ECE 6640
*

 X  f  Y  f   df
*

15
Signal Correlation
• For two non identical signal, a correlation coefficient exists
 x, y 
xt , y t 
Ex  E y
• If two signal are “orthogonal”
 x, y  0
– Note that orthogonal baseband imply that bandpass signals will
also be orthogonal … but not the other way around (as shown in
Example 2.1.-1).
ECE 6640
16
Filters
• Since this chapter skips a discussion of filters …
• See Dr. Bazuin’s filter notes.
End Lecture 2
Begin Lecture 3
ECE 6640
17
Signal Space Representation
• When transmitting digital symbols,
– The symbols can be described in terms of an N-dimensional space
with each individual symbol having a vector description.
– The “set” of symbols form a “constellation” in the N-dimensional
space.
– In this space, we can describe the “distance” between the
constellation points. This distance and the additive noise will
define the probability that we can correctly detect the symbol.
– Therefore, when describing symbols this way, you need to review
and understand vector mathematics and relationships.
ECE 6640
18
Vector Space Concepts
• Vectors
v  v1
v2  vn 
v1  v11
t
v12  v1n 
t
column
vectors
• Inner Product (dot product)
n
v1 , v2  v1  v2   v1i  v2i  v2i  v1i
*
H
i 1
• Orthogonal
v1 , v2  v1  v2  0
• Normal (Unit Magnitude)
v1
• Orthonormal
2
n
 v1  v1   v1i  v1i  v1i  v1i  1
*
H
i 1
– All the vectors in the set are (1) orthogonal and (2) normal.
– Mathematically, we prefer using a “orthonormal basis set”
ECE 6640
19
Vector Space Concepts
• Linearly independent vector set
– No one vector can be represented as a linear combination of the
other vectors.
– The vectors can be “created” from and orthonormal set of vectors.
• If an orthonormal basis set has been defined … each vector
can be defined based on a linear transformation from the
basis set
v  Q  v'
– for A, a full rank matrix, A can be decomposed into a linear
transformation and an orthonormal basis set. A = QR
– Alternately, letting
A  v1 v2  v N   Q  R
– Note: A is mxn, Q is mxn and R is nxn for rank(A)=n and m>=n
ECE 6640
20
Gram-Schmidt Procedure
• Shown in the textbook as Equ 2.2-10 to 2.2-14
A  v1 v2  v N   Q  R
q1 
v1
v1
q2 '  v2  v2 , q1  q1
q2 
ECE 6640
e2  eK   I
r1  v1  e1
r2  v2 , q1  e1  q2 '  e2
q2 '
q2 '
q3 '  v3  v3 , q1  q1  v3 , q2  q2
q3 
e1
q3 '
q3 '
r3  v3 , q1  e1  v3 , q2  e2  q3 '  e3
Note: R is upper triangular
QH Q  I
21
Vector Inequalities
• Triangle Inequality
v1  v2  v1  v2
– Equality condition … orthogonal vectors
• Cauchy-Schwarz Inequality
v1 , v2  v1  v2  v1  v2
– more easily considered if the vectors are normalized … the dot
product of two unit magnitude vectors can not be greater than 1.
ECE 6640
22
Signal Space Concepts
• vectors as a set of complex valued time samples …
x1 t , x2 t  

 x t  x t  dt
1
2

• signal norm (square root of energy)
xt  
2

 xt   xt  dt 
*



xt   dt  E x
2

• Orthogonal signals
x1 t , x2 t  

 x t  x t  dt  0
1
2

ECE 6640
23
Orthogonal Signal Set
• Define and Orthonormal Basis Set

1,
n t , m t    n t   m t   dt  
0,

• Estimated (or defined signal)
si ,n  si t , n t  ,
mn
mn
n  1,2,  , K
The image part with relationship ID rId6 was not found in the file.
• Signal error
K
ei t   si t   sˆi t   si t    si ,k  k t 
k 1
ECE 6640
24
Signal Set
• Minimum mean square approximation error
K
ei t   si t   sˆi t   si t    si ,k  k t 
k 1

Emin   ei t   si t   dt
*


Emin
K

 *
   si t    si ,k  k t   si t   dt
k 1

 

mn
mn
1,
0,
n t , m t    n t   m t   dt  


Emin 
ECE 6640


K
K
si t   dt   si ,k  Es   sk ,i
2
k 1
2
i 1
2
25
Orthogonal Basis Set
• The Fourier Series Coefficients are orthonormal …
 2   k  t 
 2   k  t 
s t    ak  cos
  bk  sin 

T
T




k 0

• Set of functions for the interval [0,T]
k t  
ECE 6640
k 
1

 exp j  2     t 
T
T 

26
Time Domain Gram-Schmidt
• symbol waveforms can be decomposed into for description
by an orthogonal -orthonormal basis set ….
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
27
Example 2.2-3
• Three orthonormal
basis for the 4 signals
can be defined.
• The signals can then
be described as
vectors
 2 ,0,0
 0, 2 ,0 
  2 ,0,1
  2 ,0,1
s1 
s2
s3
s3
ECE 6640
t
t
t
t
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
28
Signal Generation and Vector Analysis
• Vectors and basis
generators to form
symbols
• Orthogonal
integrations to
determine received
symbol vectors
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
29
Example 2.2-4
• Vector representation of the signal space
 2 ,0,0
 0, 2 ,0 
  2 ,0,1
  2 ,0,1
s1 
• The vector magnitudes can be computed
– and the energy derived
Ek  sk
2
• The Euclidean distances between symbols
can also be computed
s2
s3
s3
t
t
t
t
s1  2
s2  2
s3  3
s4  3
– defines how easy they are to detect …
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
30
Orthonormal Basis
• In general, low pass signal can be represented by an Ndimensional complex vector, and the corresponding
bandpass signal can be represented by 2N-dimensional real
t
vectors.
sml  sml1 , sml 2 ,  , smlN 
Baseband
N
sml t    ss ln  nl t , m  1,2,  , M
n 1
 N


sm t   Re   ss ln  nl t   exp j  2    f 0  t ,

 n 1

m  1,2,  , M
Bandpass


sm t   Re   ss ln  nl t   exp j  2    f 0  t ,
m  1,2,  , M

 n 1
1
r 
i 
r 
i 
r 
i  t
sm 
sml1 , sml1 , sml 2 , sml 2 ,  , smlN , smlN
2
N

ECE 6640

Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
31
Example 2.2-6
• Describing M bandpass signal
sm t   Re Am  g t   exp j  2    f 0  t 
sml t   Am  g t 
n t   2  Renl t   exp j  2    f 0  t 
 t  
g t 

Eg
2
 g t   cos2    f 0  t 
Eg
ˆt  
Bandpass
2
gˆ t 

 g t   sin 2    f 0  t 
E
Eg
g
sml t   Am  E g   t 
sm t   Am  g t   cos2    f 0  t   Am  g t   sin 2    f 0  t 
r 
i 
Bandpass symbols
ECE 6640
32
Random Variables
• Probability and Statistics Review
• See Review Notes from ECE3800 – Cooper and McGillem
Textbook
– ECE3800_Review_1to8
– ECE3800_Review_9
ECE 6640
33
Communication RV: Bernoulli
• Bernoulli Random Variable: Example flipping a coin
S X  0,1
p1  p
p0  1  p  q
m X  EX   p
 X 2  VARX   p  1  p   p  q
ECE 6640
Alberto Leon-Garcia, “Probability, Statistics, and Random Processes For Electrical Engineering,
3rd ed.”, Pearson Prentice Hall, 2008
34
Communication RV: Binomial (1)
• Binomial Random Variable: Coin flip sequence
S X  0,1,2, , n
n k
nk
p k     p  1  p 
k 
m X  EX   n  p
 X 2  VARX   n  p  1  p 
ECE 6640
Alberto Leon-Garcia, “Probability, Statistics, and Random Processes For Electrical Engineering,
3rd ed.”, Pearson Prentice Hall, 2008
35
Communication RV: Binomial (2)
• Binomial Random Variable: Example bit/symbol errors
S X  0,1,2, , n
n k
nk
p k     p  1  p 
k 
m X  EX   n  p
 X 2  VARX   n  p  1  p 
ECE 6640
Alberto Leon-Garcia, “Probability, Statistics, and Random Processes For Electrical Engineering,
3rd ed.”, Pearson Prentice Hall, 2008
36
Communication RV: Uniform
• Uniform Random Variable: Received signal phase
 1
, a xb

fU x    b  a
0,
x  0 and x  b
ba
E U  
2
ECE 6640
x0
0,
x  a

, a xb
FU  x   
b

a

xb
1,
2

b  a
VARU  
12
Alberto Leon-Garcia, “Probability, Statistics, and Random Processes For Electrical Engineering,
3rd ed.”, Pearson Prentice Hall, 2008
37
Communication RV: Gaussian/Normal
• Gaussian Random Variable: Noise, Law of large number
f X x  
 x  m 
 exp 
2

2

2  

1
2




 xm
FX  x    




EX   m
1
2
xm




 t2
exp 
 2

  dt

VARX    2
• Normal Random Variable: zero mean and unit variance
x
 t2 
1
x  
  exp    dt
2   2 
 x2 
1
 exp  
 x  
2
 2
 xm
FX  x    

  
ECE 6640
1
2
xm




 t2
exp 
 2

  dt

Alberto Leon-Garcia, “Probability, Statistics, and Random Processes For Electrical Engineering,
3rd ed.”, Pearson Prentice Hall, 2008
38
Communication RV: Gaussian/Normal
• The Q-function is derived from the Gaussian
– a single sided Gaussian tail integration
Q x   1   x  

 t2
  exp 
2 x
 2
1

  dt

x  
 t2
  exp 
2   2
1
x

  dt

 xm
1  F  x   Q

  
ECE 6640
Alberto Leon-Garcia, “Probability, Statistics, and Random Processes For Electrical Engineering,
3rd ed.”, Pearson Prentice Hall, 2008
39
Matlab Functions
• The error and complementary error functions are built-in
to MATLAB.
– see Q_fn.m and Q_fninv.m
erf  x  
2

Q x  
  
x

exp  u 2  du
u 0
1 
 x 
 1  erf 

2 
 2 
 x X
1 
FX x   1   1  erf 
2 
 2 
ECE 6640
erfc x   1  erf  x 
Q x  
1
 x 
 erfc

2
2


 1 1
 x X
    erf 

 2 
 2 2

 x X
1
FX x   1   erfc
2
 2 








40
Communication RV: Rayleigh
• Rayleigh Random Variable: Two dimensional Gaussian
R
f R r 
  r2 
,

 exp
2
2
 2  



 0,
r
E R  
ECE 6640
E  X   E Y   0
X 2 Y2

2

for 0  r
for r  0
FR r 
VARX   VARY    2
  r2 
,
 1  exp
2
 2  


 0,
for 0  r
for r  0


VARR    2     2
2

Alberto Leon-Garcia, “Probability, Statistics, and Random Processes For Electrical Engineering,
3rd ed.”, Pearson Prentice Hall, 2008
41
More
•
Exponential
– the modeling of the time between occurrence of events
•
Chi-Square
– sum of squared Gaussian R.V.
•
Rician
– two dimensional Gaussian, non-zero mean
•
Maxwell
– three dimensional Gaussian, zero mean
•
Geometric
– number of failures before the first success (coins or bits)
•
Poisson
– in counting the number of occurrences of an event in a certain time period
or in a certain region in space
ECE 6640
42
Properties of RV: A Summary
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
43
Jointly Gaussian Random Variables
• Multi-dimensional Gaussian R.V. where the vector values
are not independent.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
44
Jointly Gaussian, n=2
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
45
Probability Bounds
• Often in communications, the exact computation for
performance analysis is very difficult or even
“unreasonable”.
• For these cases we seek to bound the performance,
particularly for bit/symbol error rates.
P X     
• Markov Inequality
• Chernov Bound
ECE 6640
46
Markov Inequality
• The Markov inequality gives an upper bound on the tail
probability of nonnegative random variables. Let us
assume that X is a nonnegative random variable, i.e., p(x)
= 0 for all x < 0, and assume α > 0 is an arbitrary positive
real number. The Markov inequality states that:
P X    
ECE 6640
EX 

Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
47
Chernov Bound
•
•
The Chernov bound is a very tight and useful bound that is obtained
from the Markov inequality. Unlike the Markov inequality that is
applicable only to nonnegative random variables, the Chernov bound
can be applied to all random variables.
Let X be an arbitrary random variable, and let δ and ν be arbitrary real
numbers (ν <>0). Define random variable Y by Y = eνX and constant α
by α = eνδ. Obviously, Y is a nonnegative random variable and α is a
positive real number. Applying the Markov inequality to Y and α
yields

Pe
ECE 6640
vX
e
v
   
E e vX
 v  E e v  X  
e

48
Limit Theorems
Theorems related to the running average of random variables:
• Law of Large Numbers:
 
1 n
 Xi  E X j ,
n i 1
 
for E X j   and X j i.i.d .
• Central Limit Theorem:
 
 
for X j i.i.d . and  2  VAR X j and E X j  
1 n
 Xi  m
n i 1

 N 0,1
Gaussian with zero
mean and unit variance
n
ECE 6640
49
MATLAB Signals and SNR
• Review Matlab for analog signal performance
– Computing signal power for SNR
– Generating and Applying Filters
– Interpolation and decimation primitives used
• interp function
• decimation using (1:M:end)
– Saving and Loading simulation data
ECE 6640
50
Complex Random Variables
• Complex numbers and variables can be expressed as a
vector or matrix.
Z  X  i Y
~ X 
Z  
Y 
~
Z  1 i  Z
• Means and covariance can be described.

E Z   E X   i  E Y 
VARZ   E Z  E Z   Z  E Z 
H
  VARX   VARY 
• If and when we need this stuff we will come back to it.
ECE 6640
51
Wide Sense Stationary
• The RV mean is note dependent on time.
• And, the autocorrelation function is not dependent in time.
E Z t   Const.
RZZ t1 , t 2   E Z t1   Z t 2   RZZ  , for   t 2  t1
RZZ    E Z t   Z t   .
• Numerous simplification in computations result, including:
RZZ     conj RZZ  
• Signal Power


PZ  E Z t   RZZ 0  
2

 S  f  df
ZZ

ECE 6640
52
WSS More Properties
• For two RV that are WSS
RXY     conj RYX  
• For Z=aX+bY
RZZ    a  RXX    a  b*  RXY    a *  b  RYX    b  RYY  
2
2
S ZZ  f   a  S XX    a  b*  S XY    a *  b  SYX    b  SYY  
2
2
S XY  f   conj SYX  f 


S ZZ  f   a  S XX    2  Re a  b*  S XY    b  SYY  
2
ECE 6640
2
53
White Noise Process
• For a noise process … from probability notes.
RNN   
N0
   
2
N
k T
S NN  f   0 
2
2
– where k is Boltzmann’s constant, k=1.38 x 10-23 J/K (text error)
and T is temperature in Kelvin.
ECE 6640
54
Cyclostationary Random Process
• The mean and autocorrelation are periodic functions.
– example, a cosine wave
• The autocorrelation based on the signal periodicity T can
be described as
RXX t  T , t    T   RXX t , t   
– an average autocorrealtion is defined as
T0
RXX    1   RXX t   , t   dt
T0
0
– with the power spectral density


S XX  f    RXX  
ECE 6640
55
Example 2.7-1: Random Symbols
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
56
2.7-3 p. 71 Errata
• Text
• Errata
• Comment: when performing a covariance computation, the
mean must be subtracted. The original would be OK if Z is
a zero-mean process.
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
57
Markov Chains
•
Markov chains are discrete-time, discrete-valued random processes in
which the current value depends on the entire past values only through
the most recent values. In a j th order Markov chain, the current value
depends on the past values only through the most recent j values, i.e.,
•
There are numerous “differential encoding” schemes for
communications where the “difference” between successive bits define
the symbol transmitted. The encoding processes is a 1st order Markov
chain.
AS we study encoding and decoding of signals we will be studying
trellis codes and trellis based signal detection. These are Morkov
chains of higher orders. Therefore some background is required …
•
ECE 6640
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
58
Markov Chains (Cont)
• In this book, the finite-state Markov chain is required.
• The symbols received transition from one state to the next
based on the symbol history and new symbols transmitted
and received.
• We will come back to this material at a alater date.
ECE 6640
59
Sampling Theorem/Digital Processing
• Continuous-time, continuous frequency

X F  
 xt   exp j  2  F  t   dt
xt  


 X F   exp j  2  F  t   dF

• Discrete-time, continuous frequency
X w  

 xn   exp j  w  n 
n  

xn  
1
 X w   exp j  w  n   dw
2 
xn  
1 N 1
k n

  X k   exp j  2 

N 
N k 0

• Discrete-time, discrete frequency
N 1
k n

X k    xn   exp  j  2 

N 

n0
ECE 6640
60
Sampling
•
Continuous Time Signal Relationships for aperiodic signal xa(t)
X a F  

x a t  
 xa t   exp j  2  F  t   dt

•
•
For
Discrete Time Sampling
X F  
•


 X F   expi 2  F  t   df
a

x a n  T  

 x t    t  n  t 
n  
a
xn  xa n  T , for    n  

  x t    t  n  T   exp j  2  F  t   dt
a
 n  
Note: as spectral replication has now occurred, due to sampling, we
use X(F) instead of Xa(F)
X F  
x t    t  n  T   exp j  2  F  t   dt



a
n   
•
But the integral involving a delta function is just the value of the
function at the time …
X F  

 x n  T   exp j  2  F  n  T 
n  
ECE 6640
X F  
a

 xn  exp j  2  F  n  T 
n  
61
Sampling (cont)
•
•
The value of the function at the time …
T
Letting
X F  
•
•
X F  

 xn  exp j  2  F  n  T 
n  
1
Fs


F
n  

s
 xn   exp  j  2  F

 n 

2  F
F
w
f 
allowing
or
F
F
Results in a relationship between the sampling rate and the DTCF
Fourier transform, which can then be written as
s
s
X w  

 xn  exp j  w  n
n  
Xf 
or

 xn  exp j  2  f  n
n  
ECE 6640
62
Reconstruction
x a t  

 X F   exp j  2  F  t   dF
a

• the band limiting can be accounted for as
Fs
x a t  
2
 X F   exp j  2  F  t   dF
a

Fs
2
• using the relationship for a sampled time transform
Fs
x a t  
2
 X a F   exp j  2  F  t   dF

• we have
Fs
X F   Fs 
2
Fs
x a t  
2


Fs
2

 X F  k  F 
k  
a
s
1
 X F   exp j  2  F  t   dF
Fs
• and
Fs
x a t  
2


Fs
2

ECE 6640

1  
F 
   xn   exp  j  2   n   exp j  2  F  t   dF
Fs n  
Fs 

1
x a t    xn   
Fs
n  
Fs
 

 
  j  2  F   t  n    dF
exp


 F 

F 
s  



 s 
2
2


n 
sin   Fs   t   

 Fs  
x a t    xn   

n 
n  
  Fs   t  
 Fs 
63
Reconstruction (cont.)


n 
sin   Fs   t   

 Fs  
x a t    xn   

n 
n  
  Fs   t  
 Fs 
• From
• or using the sampled period T


sin   t  n  T 

T
 t  n T 

x a t    xn   
xn  sinc  



T 

n  
n  
 t  n  T 
T

• Therefore, for band-limited sampled signals, perfect
reconstruction is performed as the sum of sinc functions
centered on each of the sampled data points.
x a t  
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


 xn sinc T  t  n  T 
n  
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Reconstruction (cont.)
x a t  
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


 xn sinc T  t  n  T 
n  
Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
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2.9 Bandpass and Lowpass
Random Processes
•
In general, bandpass and lowpass random processes can be defined as
WSS processes X(t) for which the autocorrelation function RX(τ ) is
either a bandpass or a lowpass signal.
– Recall that the autocorrelation function is an ordinary deterministic
function with a Fourier transform which represents the power spectral
density of the random process X(t).
•
•
Therefore, for a bandpass process the power spectral density is located
around frequencies± f0, and for lowpass processes the power spectral
density is located around zero frequency.
To be more specific, we define a bandpass (or narrowband) process as
a real, zero mean, and WSS random process whose autocorrelation
function is a bandpass signal.
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Example 2.9-1 White Gaussian Noise
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Notes and figures are based on or taken from materials in the course textbook:
J.G. Proakis and M.Salehi, Digital Communications, 5th ed., McGraw-Hill, 2008.
67