Summary of binary detection with vector ob­ servation in iid Gaussian noise.:
First remove center point from signal and its effect on observation.
Then signal is
± a . and v =
± a + Z .
Find v, a and compare with threshold (0 for
ML case).
This does not depend on the vector basis ­ becomes trivial if a normalized is a basis vector.
Received components orthogonal to signal are irrelevant.
1

QAM DETECTION
Input
�
Signal encoder a
∈A
�
Baseband modulator u ( t
�
Baseband
→
) passband x ( t )
�
+
WGN
Output
�
Detector
� v
Baseband
�
Demodulator v ( t )
Passband
→ baseband
� y ( t )
We have seen how to design a MAP or ML detector from observing v . We generalize to an arbitrary complex signal set
A
We also question what the entire receiver should do from observation of y ( t ) or v ( t ) .
2

The baseband waveform is u ( t ) = ap ( t ) where a = a
1
+ ia
2 with a
1
=
{ a
}
, a
2
=
{ a
}
.
The passband transmitted waveform is x ( t ) = a
1
Ψ
1
( t ) + a
2
Ψ
2
( t ) .
Ψ
1
( t ) =
{
2 p ( t ) e
2 πif c t }
; Ψ
2
( t ) =
{
2 p ( t ) e
2 πif c t }
These two waveforms are orthogonal (in real vector space), each with energy 2. For p ( t ) real, they are just p ( t ) modulated by cosine and sine.
The received waveform is
Y ( t ) = ( a
1
+ Z
1
)Ψ
1 where Z
( t ) + ( a
2
+ Z
2
)Ψ
2
( t ) + Z ( t )
( t ) is real passband WGN in other de­ grees of freedom.
3

Y ( t ) = ( a
1
+ Z
1
)Ψ
1
( t ) + ( a
2
+ Z
2
)Ψ
2
( t ) + Z ( t )
Since Ψ
1 and Ψ
2 are orthogonal and equal en­ ergy, Z
1 and Z
2 are iid Gaussian.
After translation of passband signal and noise to baseband,
V ( t ) = [ a
1
+ Z
1
+ i ( a
2
+ Z
2
)] p ( t ) + Z ”( t )
Z ”( t ) is the noise orthogonal (in complex vec­ tor space) to p ( t ) .
First consider detection in real vector space.
Here ( a
1
, a
2
) represents the hypothesis and Z
1 are iid Gaussian,
N
(0 , N
0
/ 2) .
, Z
2
4

V ( t ) = [ a
1
+ Z
1
+ i ( a
2
+ Z
2
)] p ( t ) + Z ”( t )
Let Y
1
= a
1
+ Z
1
, Y
2
= a
2
+ Z
2
.
Note that V ( t ) p
∗
( τ
− t ) dt is the output from a complex matched filter. Sampling this at τ = 0 yields ( Y
1
+ iY
2
) .
The components in an expansion Z
3
, Z
4
, . . .
, in an orthonormal expansion of Z ”( t ) are also ob­ servable, although we will not need them.
5

Z
3
, Z
4
, . . .
, are real Gaussian rv’s and can also be viewed as passband rv’s. Assume a finite number of these variables. For any two hy­ potheses, a and a , the likelihoods are f f
Y
|
H
Y
( y
| a ) =
1
( πN
0
) k
2 −
( y j
− a j
)
2
+
2 k − z
2 j exp j =1
N
0 j =3
N
0
|
H
( y
|
LLR ( y
1
) =
( πN
0
) =
2 j =1
2
) k y j a exp j
−
N j =1
2
0
2 y j a
− j
( y j
− a j
N
=
0
2 j =1
)
2
+
2 y j
2 k
N
0
− z
( a j j =3
− a j
)
N
0
2 j
As usual, ML decoding is minimum distance decoding.
6

We can rewrite the likelihood as f
Y
|
H
( y
| a ) = f ( y
1 y
2
| a ) f ( z
3
, z
4
, . . .
, )
So long as Z
3
, Z
4
, . . .
are independent of Y
1
, Y
2 and H , they cancel out in the LLR.
This is why it makes sense to use the WGN model - all we need in detection is the inde­ pendence from the relevant rv’s.
In other words, ( Y
1
, Y
2
) is a sufficient statistic and Z
3
, Z
4
, . . .
, are irrelevant (so long as they are independent of Y
1
, Y
2
, H ).
This is true for all pairwise comparisons be­ tween input signals.
7

Now view this detection problem in terms of complex rv’s. Let α = a
− a .
LLR ( y ) =
2 2 y j
( a j
− a j
= j =1
N
0
)
=
2 j =1
2 y
2
{ y
}{
α
}
+
{ y
}{
α
} j
α
N
0 j
=
2
{ yα
∗
N
0
N
0
}
=
2
{ y, α
}
N
0
In real vector space, we project y onto α .
In complex space, 2-vectors become scalars, inner product needs real part to be taken.
The real part is a “further projection” of a complex number to a real number.
8

Now let’s look at the general case in WGN.
Must consider problem of real signals and noise for arbitrary modulation.
The signal set
A
=
{ a
1
, . . .
, a
M
}
, is a set of k tuples. a m
= ( a m, 1
, . . .
, a m,k
)
T
.
a m is then modulated to k b m
( t ) = a m,j
φ j
( t ) j =1 where
{
φ
1
( t ) , . . .
, φ k
( t )
} is a set of k orthonormal waveforms.
Successive signals are independent and mapped arbitrarily, using orthogonal spaces.
9

Let X ( t )
∈ { b
1
( t ) , . . .
, b
M
( t )
}
. Then k
X ( t ) = X j
φ j
( t ) j =1 where, under hypothesis m ,
X j
= a m,j for 1
≤ j
≤ k
Let φ k +1
( t ) , φ k +2
( t ) . . .
be an additional set of orthonormal functions such that the entire set
{
φ j
( t ); j
≥
1
} spans the space of real
L
2 wave­ forms. k
Y ( t ) = Y j
φ j
( t ) = ( X j
+ Z j
) φ j
( t ) + j =1 j =1 j = k +1
Z j
φ j
( t ) .
10

k
Y ( t ) = Y j
φ j
( t ) = ( X j
+ Z j j =1 j =1
) φ j
( t ) + j = k +1
Z j
φ j
( t ) .
Assume Z =
{
Z
1
, . . .
, Z k
} are iid Gauss.
{
Z k +1
, . . .
,
} is independent of Z and of a .
Z f
Y ,Z
|
H
( z
| m ) =
Z
( y
− a m
) f
Z
( z ) .
=
Λ m,m
= f f
Z
Z
( y
− a m
)
( y
− a m
)
.
The MAP detector depends only on Y . The other signals and other noise variables are ir­ relevant.
11

This says that detection problems reduce to finite dimensional vector problems; that is, sig­ nal space and observation space are for all practical purposes finite dimensional.
The assumption of independent noise and in­ dependent other signals is essential here.
With dependence, error probability is lowered; what you don’t know can’t hurt you.
12

Detection for Orthogonal Signal Sets
We are looking at an alphabet of size m , map­ ping letter j into Eφ j
( t ) where
{
φ j
( t )
} is an orthonormal set.
WGN of spectral density N
0
/ 2 is added to the transmitted waveform.
The receiver gets
Y ( t ) = Y j
φ j
( t ) = ( X j
+ Z j
) φ j
( t ) j j
Only
{
Y
1
, . . .
, Y m
} is relevant.
Under hypothesis k , Y k
=
√
E + Z j for j = k . and Y j
= Z j
13

Use ML detection. Choose k for which y, k is smallest.
| y, k
| 2
= y
2 j
+ ( y k j = k
− n k
)
2
= m j =1 y
2 y
+ E
−
2
√
Ey k
ML: choose k for which y k is largest.
By symmetry, the probability of error is the same for all hypotheses so we look at hypoth­ esis 1.
First scale outputs by N
0
/ 2 , i.e., W j
= Y j
2 /N
0
.
Under H
1
, W
1 for j = 1 .
∼ N
E/N
0
, 1) and W
Pr( e ) =
∞
−∞ f
W
1
|
H

( w
1
|
1) Pr
 m j =2 j
∼
(0 ,

1)
( W j
≥ w
1
|
1)
 dw
1
14

MIT OpenCourseWare http://ocw.mit.edu
6.450 Principles of Digital Communication I
Fall 2009
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