Baseband Receiver
Unipolar vs. Polar Signaling
Signal Space Representation
Unipolar vs. Polar
Signaling
Error Probability of Binary Signals

Unipolar Signaling
s1 (t )  A,
0  t  T,
for binary1
s0 (t )  0,
0  t  T,
for binary 0
Ed 
 s
T
0
1
(t )  s0 (t ) d t
2
0 s0 (t ) d t  20 s1 (t ) s0 (t )d t
T
 2 Eb  2  s1 (t ) s0 (t )d t
0
 Eb  0  0
 A2T
 Ed 
 Q
Pb  Q

2
2
N
0 

 A T 00
 2 N0

0 s1 (t )
T
2
dt 
T
2
T
  Eb 

 Q 2N 
0


Polar Signaling (antipodal)
s1 (t )  A,
0  t  T,
for binary1
s0 (t )   A, 0  t  T ,
for binary 0
 Ed
Pb  Q
 2N0
 4 A2T 

 2Eb


Q
 

 2N0  Q N0








Orthogonal

Pb  Q 

Eb
N0
Bipolar





Bipolar signals require a factor of 2
increase in energy compared to
Orthagonal

Since 10log102 = 3 dB, we say that
bipolar signaling offers a 3 dB better
performance than Orthagonal
( Antipodal)
 2 Eb
Pb  Q
 N0




Error Performance Degradation
Two Types of Error-Performance Degradation
Power degradation
Signal distortion
6
(a) Loss in Eb/No. (b) Irreducible PB caused by distortion.
Signal Space Analysis
8 Level PAM
S0(t)
S1(t)
S2(t)
S3(t)
S4(t)
S5(t)
S6(t)
S7(t)
8 Level PAM
Then we can represent
S0 t  as  7 t 
S1 t  as  5 t 
S2 t  as  3 t 
S3 t  as 1 t 
S 4 t  as
 t 
S5 t  as
3 t 
S6 t  as
S7 t  as
5 t 
7 t 
(t)
8 Level PAM

We can have a single unit height window (t) as the receive filter

And do the decisions based on the value of z(T)

We can have 7 different threshold values for our decision (we have one
threshold value for PCM detection)

In this way we can cluster more & more bits together and transmit them as
single pulse

But if we want to maintain the error rate then the

transmitted power = f (clustered no of bitsn)
(t) as a unit vector

One dimensional vector space (a Line)

If we assume (t) as a unit vector i, then we can represent signals s0(t) to s7(t)
as points on a line (one dimensional vector space)
0
(t)
S0(t)
S1(t)
S2(t)
S3(t)
S4(t)
S5(t)
S6(t)
S7(t)
A Complete Orthonormal Basis
12
Signal space

What is a signal space?


Why do we need a signal space?



Vector representations of signals in an N-dimensional orthogonal space
It is a means to convert signals to vectors and vice versa.
It is a means to calculate signals energy and Euclidean distances between
signals.
Why are we interested in Euclidean distances between signals?

For detection purposes: The received signal is transformed to a received
vectors. The signal which has the minimum distance to the received signal
is estimated as the transmitted signal.
Signal space

To form a signal space, first we need to know the inner
product between two signals (functions):

Inner (scalar) product:

 x(t ), y(t )   x(t ) y* (t )dt


Properties of inner product:
= cross-correlation between x(t) and y(t)
 ax(t ), y(t )  a  x(t ), y(t ) 
 x(t ), ay(t )  a*  x(t ), y(t ) 
 x(t )  y(t ), z(t )  x(t ), z (t )    y(t ), z(t ) 
Signal space – cont’d


The distance in signal space is measure by calculating the norm.
What is norm?

Norm of a signal:
x(t )   x(t ), x(t )  
= “length” of x(t)



x(t ) dt  Ex
2
ax(t )  a x(t )

Norm between two signals:
d x, y  x(t )  y(t )

We refer to the norm between two signals as the Euclidean
distance between two signals.
Signal space - cont’d

N-dimensional orthogonal signal space is characterized by N linearly
independent functions
 j (t )Nj1 called basis functions. The
basis functions must satisfy the orthogonality condition
T
  i (t ), j (t )   i (t ) (t )dt  K i ji
*
j
0
where   1  i  j

ij
0  i  j

If all Ki  1 , the signal space is orthonormal.
Orthonormal basis
 Gram-Schmidt procedure
0t T
j, i  1,..., N
Signal space – cont’d

M
Any arbitrary finite set of waveforms
si (t )i1
where each member of the set is of duration T, can be expressed as
a linear combination of N orthonogal waveforms
 j (t )Nj1
where N  M.
N
si (t )   aij j (t )
i  1,...,M
NM
j 1
where
T
1
1
*
aij 
 si (t ), j (t ) 
s
(
t
)

i
j (t ) dt

Kj
Kj 0
si  (ai1 , ai 2 ,...,aiN )
Vector representation of waveform
j  1,..., N
0t T
i  1,...,M
N
Ei   K j aij
2
j 1
Waveform energy
Signal space - cont’d
N
si  (ai1 , ai 2 ,...,aiN )
si (t )   aij j (t )
 1 (t )
si (t )
j 1

T
ai1
0
 N (t )

T
0
aiN
 1 (t )
 a i1 
   s
m
 
 aiN 
Waveform to vector conversion
sm
 a i1 
 
 
 aiN 
ai1
 N (t )
si (t )
aiN
Vector to waveform conversion
Schematic example of a signal space
 2 (t )
s1  (a11 , a12 )
 1 (t )
z  ( z1 , z2 )
s3  (a31 , a32 )
s 2  (a21 , a22 )
Transmitted signal
alternatives
Received signal at
matched filter output
s1 (t )  a11 1 (t )  a12 2 (t )  s1  (a11 , a12 )
s2 (t )  a21 1 (t )  a22 2 (t )  s 2  (a21 , a22 )
s3 (t )  a31 1 (t )  a32 2 (t )  s 3  (a31 , a32 )
z (t )  z1 1 (t )  z2 2 (t )  z  ( z1 , z2 )
Example of distances in signal space
 2 (t )
s1  (a11 , a12 )
E1
ds1 , z
E3
s3  (a31 , a32 )
ds3 , z
E2
 1 (t )
z  ( z1 , z2 )
d s2 , z
s 2  (a21 , a22 )
The Euclidean distance between signals z(t) and s(t):
d si , z  si (t )  z (t )  (ai1  z1 ) 2  (ai 2  z 2 ) 2
i  1,2,3
Example of an ortho-normal basis
functions

Example: 2-dimensional orthonormal signal
space

2

(
t
)

cos(2t / T )
 1
T


 (t )  2 sin(2t / T )
2

T

0t T
 2 (t )
0t T
0
 1 (t )
T
  1 (t ), 2 (t )   1 (t ) 2 (t )dt  0
0
 1 (t )   2 (t )  1

Example: 1-dimensional orthonornal signal
space  (t )
1
1 (t )  1
1
T
0
T
t
0
 1 (t )
Example of projecting signals to an orthonormal
signal space
 2 (t )
s1  (a11 , a12 )
 1 (t )
s3  (a31 , a32 )
s 2  (a21 , a22 )
Transmitted signal
alternatives
s1 (t )  a11 1 (t )  a12 2 (t )  s1  (a11 , a12 )
s2 (t )  a21 1 (t )  a22 2 (t )  s 2  (a21 , a22 )
s3 (t )  a31 1 (t )  a32 2 (t )  s3  (a31 , a32 )
T
aij   si (t ) j (t )dt
0
j  1,..., N
i  1,...,M
0t T
Implementation of matched filter
receiver
Bank of N matched filters
 (T  t )

1
z1
r (t )
  N (T  t )
zN
 z1 
 
 
 z N 
Observation
vector
z
z
N
si (t )   aij j (t )
j 1
i  1,...,M
NM
z  ( z1 , z2 ,...,zN )
z j  r (t )  j (T  t )
j  1,..., N
Implementation of correlator receiver
Bank of N correlators
 1 (t )

T
z1
0
r (t )
 N (t )

T
0
N
si (t )   aij j (t )
zN
 r1 

 
rN 
z
z
i  1,...,M
j 1
z  ( z1 , z2 ,...,zN )
NM
T
z j   r (t ) j (t ) dt
0
j  1,..., N
Observation
vector
Example of matched filter receivers using basic functions
s1 (t )
 1 (t )
s2 (t )
A
T
1
T
0
T
T
0
A
T
t
t
0
T
t
1 matched filter
 1 (t )
r (t )
1
T
0
z1
z1   z
z
T t

Number of matched filters (or correlators) is reduced by 1 compared to using
matched filters (correlators) to the transmitted signal.

Reduced number of filters (or correlators)
Example 1.
26
(continued)
27
(continued)
28
Signal Constellation for Ex.1
29
Notes on Signal Space




Two entirely different signal sets can have the same geometric
representation.
The underlying geometry will determine the performance and the
receiver structure for a signal set.
In both of these cases we were fortunate enough to guess the correct
basis functions.
Is there a general method to find a complete orthonormal basis for an
arbitrary signal set?
Gram-Schmidt Procedure
30
Gram-Schmidt Procedure


Suppose we are given a signal set:
{s1 (t ),..........., sM (t )}
We would like to find a complete orthonormal basis
for this signal set.
{ f1 (t ),.........., f K (t )}, K  M

The Gram-Schmidt procedure is an iterative procedure for
creating an orthonormal basis.
31
Step 1: Construct the first basis function
32
Step 2: Construct the second basis function
33
Procedure for the successive signals
34
Summary of Gram-Schmidt Prodcedure
35
Example of Gram-Schmidt Procedure
36
Step 1:
37
Step 2:
38
Step 3:
* No new basis function
39
Step 4:
* No new basis function. Procedure is complete
40
Final Step:
41
Signal Constellation Diagram
42
Bandpass Signals
Representation
Representation of Bandpass Signals
Bandpass signals (signals with small bandwidth compared to carrier
frequency) can be represented in any of three standard formats:
1. Quadrature Notation
s(t) = x(t) cos(2πfct) − y(t) sin(2πfct)
where x(t) and y(t) are real-valued baseband signals called the inphase and quadrature components of s(t)
44
(continued)
2.
Complex Envelope Notation
s(t )  Re[( x(t )  jy(t ))e j 2 fct )  Re[sl (t )e j 2 fct ]
where sl (t ) is the complex baseband or envelope of s(t .)
3.
Magnitude and Phase
s(t )  a(t )cos(2 f ct   (t ))
where a (t )  x 2 (t )  y 2 (t ) is the magnitude of s (t ) , and
y (t ) is the phase of s (t ).
 (t )  tan 1[
]
x (t )
45
Key Ideas from I/Q
Representation of Signals
We can represent bandpass signals independent of
carrier frequency.
 The idea of quadrature sets up a coordinate system
for looking at common modulation types.
 The coordinate system is sometimes called
a signal constellation diagram.
 Real part of complex baseband maps to x-axis and
imaginary part of complex baseband maps to the y-axis

46
Constellation Diagrams
47
Interpretation of Signal Constellation
Diagram

Axis are labeled with x(t) and y(t)





In-phase/quadrature or real/imaginary
Possible signals are plotted as points
Symbol amplitude is proportional to distance
from origin
Probability of mistaking one signal for another is
related to the distance between signal points
Decisions are made on the received signal based
on the distance of the received signal (in the I/Q
plane) to the signal points in the constellation
For PAM signals How???
Binary PAM
s2
s1
 Eb
s4
E
6 b
5
0
Eb
4-ary PAM
s3
s2
E
2 b
5
0
 1 (t )
E
2 b
5
s1
E
6 b
5
 1 (t )
1
T
0
T t
 1 (t )