Transcript Signal Space Analysis
Baseband Receiver
Unipolar vs. Polar Signaling Signal Space Representation
Unipolar vs. Polar Signaling
Error Probability of Binary Signals
Unipolar Signaling
s
1 (
t
)
s
0 (
t
)
A
, 0 , 0 0
t t
T T
, ,
for binary
1
for binary
0
E d
T
0
s
1 (
t
)
s
0 (
t
) 2
dt
T
0
s
1 (
t
) 2
dt
T
0
s
0 (
t
) 2
dt
2
E b
2
E b
T
0
s
1 (
t
) 0
s
0 (
t
)
dt
0
A
2
T
0 0 2
T
0
s
1 (
t
)
s
0 (
t
)
dt P b
Q
2
E d N
0
Q
A
2 2
N T
0
Q E b 2N
0
Polar Signaling (antipodal)
s
1 (
t
)
A
, 0
t
T
,
for s
0 (
t
)
A
, 0
t
T
,
for binary binary
1 0
P b
Q E d
2
N
0
Q
4
A
2
T
2
N
0
Q
2
E
b
N
0
P b Orthogonal
Q E b N
0 Bipolar signals require a factor of 2 increase in energy compared to Orthagonal Since 10log
10
2 = 3 dB, we say that bipolar signaling offers a 3 dB better performance than Orthagonal
Bipolar
(
Antipodal
)
P b
Q
2
E b N
0
Error Performance Degradation
Two Types of Error-Performance Degradation Power degradation Signal distortion (a) Loss in Eb/No. (b) Irreducible
P B
caused by distortion.
6
Signal Space Analysis
8 Level PAM
S 0 (t) S 1 (t) S 2 (t) S 3 (t) S 4 (t) S 5 (t) S 6 (t) S 7 (t)
8 Level PAM Then we can represent
S S
1
S as
0 2
as as S S
4
S
5 3
t as as as S
6
S
7
as as
7 5 3 1 3
t
t
5 7 (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 bits n )
(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 s 0 (t) to s 7 (t) as points on a line (one dimensional vector space) 0 S 0 (t) S 1 (t) S 2 (t) S 3 (t) S 4 (t) S 5 (t) S 6 (t) S 7 (t) (t)
A Complete Orthonormal Basis
12
Signal space
What is a signal space?
Vector representations of signals in an N-dimensional orthogonal space Why do we need a signal 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:
ax
(
t
),
y
= cross-correlation between x(t) and y(t) (
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
)
ax
(
t
)
x
(
t
),
x
(
t
) = “ length ” of x(t)
a x
(
t
) Norm between two signals:
x
(
t
) 2
dt
E x
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
j N j
1 called basis functions. The
i
(
t
),
j
(
t
) 0
T
i
(
t
)
j
* (
t
)
dt
K i
ji
where
ij
1 0
i i
j j
0
j
,
i
t
T
1 ,...,
N
Orthonormal basis Gram-Schmidt procedure
Signal space – cont’d
Any arbitrary finite set of waveforms
i i M
1 where each member of the set is of duration T, can be expressed as a linear combination of N orthonogal waveforms
j
(
t
)
N j
1
s i
(
t
)
j N
1
a ij
j
(
t
)
i N
1 ,...,
M M
where
a ij
1
K j
s i
(
t
),
j
(
t
) 1
K j T
0
s i
(
t
)
j
* (
t
)
dt j i
1 ,..., 1 ,...,
N M
0
t
T
s
i
(
a i
1 ,
a i
2 ,...,
a iN
) Vector representation of waveform
E i
j N
1
K j a ij
2 Waveform energy
s i
(
t
)
Signal space - cont’d
s i
(
t
)
j N
1
a ij
1 (
t
)
N
(
t
) 0 0
T T a i
1
a iN j
(
t
)
a a
iN i
1
s
m
s
m
s
i
(
a i
1 ,
a i
2 ,...,
a iN
)
a a
iN i
1
a i
1
a iN
1 (
t
)
N
(
t
)
s i
(
t
)
Waveform to vector conversion Vector to waveform conversion
Schematic example of a signal space
2 (
t
)
s
1 (
a
11 ,
a
12 )
z
(
z
1 , 1 (
t
)
z
2 )
s
3 (
a
31 ,
a
32 ) Transmitted signal alternatives Received signal at matched filter output
s
2 (
a
21 ,
a
22 )
s
1 (
t
)
s
2 (
t
)
s
3 (
t
)
z
(
t
)
a
11
a
21 1 1 (
t
) (
t
)
a
12
a
22 2 2 (
t
) (
t
)
a
31
z
1 1 (
t
) 1 (
t
)
a
32
z
2 2 2 (
t
) (
t
)
z s
1
s
2
s
3 (
a
11 , (
z
1 , (
a
21 , (
a
31 ,
z
2 )
a
12
a
22
a
32 ) ) )
Example of distances in signal space
2 (
t
)
s
1 (
a
11 ,
a
12 )
E
1
E
3
s
3 (
a
31 ,
a
32 )
d s
3 ,
z E
2
d s
1 ,
z
z
(
z
1 , 1 (
t
)
z
2 )
d s
2 ,
z
s
2 The Euclidean distance between signals
z(t)
(
a
21 ,
a
and
s(t)
: 22 )
d s i
,
z
s i
(
t i
1 , 2 , 3 )
z
(
t
) (
a i
1
z
1 ) 2 (
a i
2
z
2 ) 2
Example of an ortho-normal basis functions
Example: 2-dimensional orthonormal signal space 1 (
t
) 2
T
cos( 2
t
/
T
) 0
t
T
2 (
t
) 2 (
t
) 2 sin( 2
t
/
T
)
T
0
t
T
1 (
t
), 2 (
t
) 1 (
t
) 2 (
t
)
T
0 1 (
t
) 2 (
t
)
dt
1 0 0 Example: 1-dimensional orthonornal signal space 1 (
t
) 1
T
1 (
t
) 1 0 0
T t
1 (
t
) 1 (
t
)
Example of projecting signals to an orthonormal signal space 2 (
t
)
s
1 (
a
11 ,
a
12 ) 1 (
t
)
s
3 (
a
31 ,
a
32 )
s
2 (
a
21 ,
a
22 ) Transmitted signal alternatives
a ij s
1 (
t
)
s
2 (
t
)
s
3 (
t
)
T
0
s i
(
t
)
a
11 1 (
t
)
a
21
a
31 1 (
t
) 1 (
t
)
a
12
a
22
a
32 2 2 2 (
t
) (
t
) (
t
)
s
1
s
2
s
3 (
a
11 , (
a
21 , (
a
31 ,
a
12 )
a
22 )
a
32 )
j
(
t
)
dt j
1 ,...,
N i
1 ,...,
M
0
t
T
Implementation of matched filter receiver
Bank of N matched filters
r
(
t
) 1 (
T
t
)
N
(
T
t
)
z
1
z N
z z
1
N
z Observation vector z z
s i
(
t
( )
z
1 ,
j N
1
z
2
a ij
,...,
z N j
(
t
) )
z j
r
(
t
)
j
(
T
t
)
i
1 ,...,
M j
1 ,...,
N N
M
Implementation of correlator receiver
Bank of N correlators
1 (
t
)
r
(
t
)
N
(
t
) 0 0
T T z
1
z N
r r N
1
z z Observation vector
s i
(
t
)
z j
z
j N
1
a ij
j
(
t
) (
z
1 ,
z
2 ,...,
z N T
0
r
(
t
)
j
(
t
)
dt
)
i
1 ,...,
M j
1 ,...,
N N
M
Example of matched filter receivers using basic functions
s
1 (
t
)
A T
0
r
(
t
)
T s
2 (
t
) 1 (
t
) 1
T t
A
0
T T t
1 (
t
) 1
T
1 matched filter
z
1 0 1
z z
0
T t 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:
s t
1
s M
We would like to find a complete orthonormal basis for this signal set.
f t
1
f K
( )},
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(2π
fct
) − y(t ) sin(2π fct) where x(t) and y(t) are real-valued baseband signals called the in phase and quadrature components of s(t) 44
2.
3.
(continued) Complex Envelope Notation
x t
( ))
j
2 ) Re[ ( )
l
is the complex baseband or envelope of .
j
2 ] Magnitude and Phase tan [ ] is the phase of .
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
s
2
s
1
E b 0 E b
1 (
t
)
s
4 6
E b
5 4-ary PAM
s
3
s
2 2
E b
5
0
2
E b
5 1 (
t
) 1
T
0
s
1 6
E b
5 1 (
t
)
T t