Digital Communications I: Modulation
and Coding Course
Spring - 2013
Jeffrey N. Denenberg
Lecture 3b: Detection and Signal Spaces
Last time we talked about:



Receiver structure
Impact of AWGN and ISI on the transmitted
signal
Optimum filter to maximize SNR

Matched filter receiver and Correlator receiver
Lecture 3b
2
Receiver job

Demodulation and sampling:

Waveform recovery and preparing the received
signal for detection:




Improving the signal power to the noise power (SNR)
using matched filter
Reducing ISI using equalizer
Sampling the recovered waveform
Detection:

Estimate the transmitted symbol based on the
received sample
Lecture 4
3
Receiver structure
Digital Receiver
Step 1 – waveform to sample transformation
Step 2 – decision making
Demodulate & Sample
Detect
z (T )
r (t )
Frequency
down-conversion
For bandpass signals
Received waveform
Receiving
filter
Equalizing
filter
Threshold
comparison
Compensation for
channel induced ISI
Baseband pulse
(possibly distored)
Lecture 4
Baseband pulse
4
Sample
(test statistic)
mˆ i
Implementation of matched filter receiver
Bank of M matched filters
s (T  t)
*
1
z1 (T )
r (t )
sM (T t)
*
zM
 z1 
  
 
 z M 
(T )
z
Matched filter output:
z Observation
vector

z

r
(
t
)

s
T

t) i 1,...,M
i(
i
z

(
z
(
T
),
z
(
T
),...,
z
(
T
))

(
z
,
z
,...,
z
)
1
2
M
1
2
M
Lecture 4
5
Implementation of correlator receiver
Bank of M correlators
s 1 (t )

T
z1 (T )
0
r (t )
s

M
(t )

T
0
 z1 
  
 
 z M 
z
Correlators output:
z Observation
vector
z M (T )
z

(
z
(
T
),
z
(
T
),...,
z
(
T
))

(
z
,
z
,...,
z
)
1
2
M
1
2
M
T
zi r(t)si(t)dt
i 1,...,M
0
Lecture 4
6
Today, we are going to talk about:

Detection:


Estimate the transmitted symbol based on the
received sample
Signal space used for detection


Orthogonal N-dimensional space
Signal to waveform transformation and vice versa
Lecture 4
7
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 vector. The signal which has the minimum
Euclidean distance to the received signal is estimated as the
transmitted signal.
Lecture 4
8
Schematic example of a signal space
 2 (t )
s1 (a11,a12)
 1 (t )
z (z1, z2)
s3 (a31,a32)
s2 (a21,a22)
Transmitted signal
alternatives
Received signal at
matched filter output
s
(
t)
a

(
t)
a

(
t)
s
(
a
,a
)
1
11
1
12
2
1
11
12
s
(
t)
a

(
t)
a

(
t)
s
(
a
,a
)
2
21
1
22
2
2
21
22
s
(
t)
a

(
t)
a

(
t)
s
(
a
,a
)
3
31
1
32
2
3
31
32
z
(
t)
z

(
t)
z

(
t)
z

(
z
,z
)
1
1
2
2
1
2
Lecture 4
9
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
 (



Analogous to the “dot” product of discrete n-space vectors
= cross-correlation between x(t) and y(t)

Properties of inner product:

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
)

Lecture 4
10
Signal space …


The distance in signal space is measure by calculating
the norm.
What is norm?

Norm of a signal:
 2
x
(
t
)


x
(
t
),
x
(
t
)


x
(
t
)
dt

E
x



= “length” or amplitude of x(t)
ax
(t) ax(t)

Norm between two signals:
d
(t)y
(t)
x
,y x

We refer to the norm between two signals as the
Euclidean distance between two signals.
Lecture 4
11
Example of distances in signal space
 2 (t )
s1 (a11,a12)
E1
d s1 , z
 1 (t )
z (z1, z2)
E3
s3 (a31,a32)
d s3 , z
E2
d s2 , z
s2 (a21,a22)
The Euclidean distance between signals z(t) and s(t):
2
2
d

s
(
t
)

z
(
t
)

(
a

z
)

(
a

z
)
s
,
z i
i
11
i
22
i
i

1
,
2
,
3
Lecture 4
12
Orthogonal signal space

N-dimensional orthogonal signal space Nis characterized by
N linearly independent functions  j (t)j1 called basis
functions. The basis functions must satisfy the orthogonality
condition

 
 
0  t T

(
t
),
(
t
)

(
t
)(
t
)
dt

K
i
j
i
i ji

j,i 1,...,
N
0
T
*
j
where


1
ij

0
ij

ij
If all K i  1, the signal space is orthonormal.
See my notes on Fourier Series
Lecture 4
13
Example of an orthonormal basis

Example: 2-dimensional orthonormal signal space

2

(
t
)

cos(
2t /T) 0t T
 2 (t )
 1
T


 (t)  2 sin(
2t /T) 0t T
2

T

T
0
1(t),2(t) 1(t)2(t)dt0
 1 (t )
0
1(t)  2(t) 1

Example: 1-dimensional orthonormal signal space
 1 (t )
1(t) 1
1
T
0
 1 (t )
0
T
t
Lecture 4
14
Signal space …

si (t )iM1
Any arbitrary finite set of waveforms
where each member of the set is of duration T, can be
expressed as a linear combination of N orthonogal
waveforms  j (t)N where N  M .
j 1
N
si(t)
a
j(t)
ij
j
1
i 1,...,M
NM
where


1
1 *
j 1,...,N
a


s
(
t
),
(
t
)

s
(
t
)
(
t
)
dt
0  t T
ij
i
j
i
j

i

1
,...,
M
K
K
j
j0
T
s
(a
,a
,...,
a
)
i
i1
i2
iN
Vector representation of waveform
Lecture 4
N
E
Kj aij
i 
2
j
1
Waveform energy
15
Signal space …
N
si(t)
a
j(t)
ij
s
(a
,a
,...,
a
)
i
i1
i2
iN
j
1
Waveform to vector conversion
Vector to waveform conversion
 1 (t )
si (t )
 1 (t )

T
ai1
0
 N (t )

T
0
aiN
 a i1 
  
 
 a iN 
Lecture 4
 sm
sm
 a i1 
  
 
 a iN 
ai1
 N (t )
aiN
16
si (t )
Example of projecting signals to an
orthonormal signal space
 2 (t )
s1 (a11,a12)
 1 (t )
s3 (a31,a32)
s2 (a21,a22)
Transmitted signal
alternatives
 
s
(
t
)

a

(
t
)

a

(
t
)

s
(
a
,
a
)
s
(
t
)

a

(
t
)

a

(
t
)

s
(
a
,
a
)
s
(
t
)

a
(
t
)

a
(
t
)

s

(
a
,
a
)
1
11
1
12
2
1
11
12
2
21
1
22
2
2
21
22
3
31
1
32
2
3
31
32
T
a
ij
si(t)j(t)dt j 1,...,N i 1,...,M
Lecture 4
0
0  t T
17
Signal space – cont’d
To find an orthonormal basis functions for a given
set of signals, the Gram-Schmidt procedure can be
used.
Gram-Schmidt procedure:



N
M
Given a signal set si (t )i 1 , compute an orthonormal basis  j (t)j1

(
t
)

s
(
t
)
/E

s
(
t
)
/s
(
t
) i
1. Define 1
1
1
1
1

1
(
t
)

s
(
t
)


s
(
t
),
(
t
)

(
t
)
2. For i 2,...,M compute d

i
i
i
j
j
j

1
t)
d
t)/d
t)
If di (t)  0 let 
i(
i(
i(
If di (t)  0
, do not assign any basis function.
3. Renumber the basis functions such that basis is




(
t),

(
t),...,

(
t)
1
2
N


This is only necessary if di (t)  0for any i in step 2.
Note that N  M
Lecture 4
18
Example of Gram-Schmidt procedure

Find the basis functions and plot the signal space for the following
transmitted signals:
s1 (t )
s2 (t )
A
T
0
0

T
t
T
t
A
T
Using Gram-Schmidt procedure:
T
s1(t)A
1(t)
 1 (t )
t) dtA
1 E
1 s
1(
0
2
2
1(t)s1(t)/ E
t)/A
1 s
1(
2 s2(t),
1(t)
0 s2(t)1(t)dtA
T
d2(t)s2(t)(A
)
t)0
1(
s2(t)A
1(t)
1
T
s1 (A
) s2 (A
)
0
T t
s2
-A
Lecture 4
s1
0
19
A
 1 (t )
Implementation of the matched filter receiver
Bank of N matched filters
z1
 (T t)

1
r (t )
N(T t)
zN
 z1 
 
 
 z N 
Observation
vector
z
z
N
si(t)
a
j(t)
ij
j
1
i 1,...,M
NM
z(z1,z2,...,
zN)
zj 
r
(t)

T

t) j 1,...,N
j(
Lecture 4
20
Implementation of the correlator receiver
Bank of N correlators
 1 (t )

T
z1
0
r (t )
 N (t )

T
0
N
si(t)
a
j(t)
ij
zN
 r1 

 
 r N 
z
z
i 1,...,M
j
1
z(z1,z2,...,
zN)
NM
T
zj r(t)
j(t)dt j 1,...,N
0
Lecture 4
21
Observation
vector
Example of matched filter receivers using
basic functions
s1 (t )
 1 (t )
s2 (t )
1
T
A
T
T
0
0
T
t
A
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.
Lecture 4
22
White noise in the orthonormal signal space

AWGN, n(t), can be expressed as
~
ˆ(t)n
n
(t)n
(t)
Noise projected on the signal space
which impacts the detection process.
Noise outside of the signal space
N
ˆ(t)
n
nj
t)
j(
Vector representation of
j
1
n
n
(t),

t) j 1,...,N
j 
j(
~
n
(t),
j(t)
0 j 1,...,N
Lecture 4
nˆ (t )
n(n
,n
n
1
2,...,
N)
n 
N
independent zero-mean
Gaussain random variables with
nj)N
variance var(
0/2
j j 1
23