Detection for M-ary Orthogonal Case

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Detection for M-ary Orthogonal Case
M-ary frequency shift keying, where one of M sinusoidal sig-
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nals is selected every Ts seconds, with the tone spacing chosen
1
to be some multiple of Ts.
2
M-ary pulse-position modulation (PPM), involving transmis-
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sion of a pulse occupying one of M time slots within Ts sec-
onds.
M-ary signaling using binary signal patterns formed by the
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rows of binary Hadamard matrices; These rows having mutual
orthogonality.
To evaluate the exact performance as function of Es=N0 on
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the AWGN channel, we notice that the signal space is M-
dimensional with each located pEs away from the origin,
and d = p2Es between each other.
By symmetry, each signal has the same conditional error prob-
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ability P((cid:15) Si) which is the same as P((cid:15)).
j
1
Error Probability: M-ary Orthogonal Case
In the correlation receiver, the decision statistics is:
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2
yi = r(t)si(t) dt:
N0 Z
and the decision favors the largest correlation.
Given conditioning on S0, Y0 is distributed as
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2Es 2Es 2Es
Y0 N 0 ; 1; Yi N 00; 1;
(cid:24) N0 N0 (cid:24) N0
@ A @ A
for i = 0.
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{ To show this, we notice that
2
y0 = r(t)s0(t) dt
N0 Z
2
= [s0(t) + N(t)]s0(t) dt
N0 Z
2 2Es
= N(t)s0(t) dt +
N0 Z N0
N 1
2 2Es
(cid:0)
= nks0k + :
N0 kX=0 N0
Hence E[Y0] = 2Es=N0.
2
Error Probability: M-ary Orthogonal Case (Continued)
{ For variance, we have
2 2
N 1
2 2Es 2 (cid:0)
(cid:27) = E 20Y0 1 3 = E 20 nks0k1 3
(cid:0) N0 N0 kX=0
6 7 6 7
4@ A 5 4@ A 5
N 1
4 (cid:0) 2 2 2Es
= 2 E[nk]s0k = :
N0 kX=0 N0
{ For Yi, E[Yi] = 0, and
2
N 1
2
2 2 (cid:0)
(cid:27) = E Yi = E 20 nksik1 3
(cid:20) (cid:21) N0 kX=0
6 7
4@ A 5
N 1
4 (cid:0) 2 2 2Es
= 2 E[nk]sik = :
N0 kX=0 N0
The receiver errs if Yi > Y0 for any i = 0. Thus the proba-
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bility of correct decision is
1
P(C) = P(all Y1; ;YM 1 < Y0 S0)f(y0 S0) dy0:
Z(cid:0)1 (cid:1)(cid:1)(cid:1) (cid:0) j j
By independence,
(cid:15)
P(C) = 1 P((cid:15))
(cid:0) 2 2 2 2 M 1
(y0 (cid:22)) =2(cid:27) y =2(cid:27) (cid:0)
e(cid:0) (cid:0) y0 e(cid:0)
= 1 2 dy3 dy0
Z(cid:0)1 p2(cid:25)(cid:27) 6Z(cid:0)1 p2(cid:25)(cid:27) 7
6 7
2 2 4 5
(y0 (cid:22)) =2(cid:27)
M 1
e(cid:0) (cid:0) y0
(cid:0)
1
= 1 Q dy0:
" !#
Z(cid:0)1 p2(cid:25)(cid:27) (cid:0) (cid:27)
3
Error Probability: M-ary Orthogonal Case (Continued)
The error performance with respect to bit energy-to-noise
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m
density ratio is shown in Figure 3.3.17 for M = 2 .
Because all designs support a certain bit rate with a com-
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mon power constraint (Pr) at the receiver, it is important to
compare the performance in terms of equal energy per bit
received, which is
Eb = PrTb = PrTs=log2 M = Es=log2 M:
Thus Figure 3.3.17 is plotted against Eb=N0.
The union bound can be applied as in Example 3.5. But
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an upper bound can also be obtained for error performance
which has the expression:
1=2
Es
P((cid:15)) < 1 81 Q2 39:
(cid:0) >>< (cid:0) 60N01 7>>=
6@ A 7
>>: 4 5>>;
The above indicates below 1 at low SNR.
(cid:15)
4
Discussions (Continued)
To compare options for modulation and detection, it is often
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important to compare performace in terms of equal energy
per bit receieved even if M di(cid:11)ers, which is
Eb = PrTb = PrTs=log2 M = Es=log2 M
as M-ary symbol conveys log2 M bits in uncoded memoryless
modulation.
This gives the rise of performance results against Eb=N0 where
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Es=N0 = (Eb=N0)log2 M. Such a normalization is shown in
Figure 3.3.17.
As seen, in order to maintain the same error probability, Es
(cid:15)
must increase with M. However the e(cid:14)ciency measured in
Eb=N0 actually improves as in Figure 3.3.17.
If M increases without limit, the probability of symbol error
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becomes arbitrarily small as long as Eb=N0 exceeds loge 2 =
1:6dB, which is the channel capacity limit for AWGN chan-
(cid:0)
nel in the case of in(cid:12)nite-bandwidth.
5
Discussions (Continued)
Although the orthogonal signaling represents a constructive
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scheme that operates near the channel capacity, it is not a
practical solution.
First the complexity, measured by the number of correlations
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needed per bit transmitted, grows as M=log2 M.
Second, the dimensionality of the signal set, per bit transmit-
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ted, grows at the same rate, and thus orthogonal sets imply
very large bandwidth.
m
If the messages are just binary data with M = 2 , then bit
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error probability, Pb, may be of more interest than the symbol
error probability Ps. For orthogonal signaling, these two are
easily related combinatorily.
Given that an error is made, then the M 1 incorrect choices
(cid:15) (cid:0)
are equally likely by the symmetry. Consider any position
(1st) that labels signal. Of the M 1 error possibilities,
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exactly M=2 di(cid:11)er in the (cid:12)rst position, and thus
M
Pb = Ps:
2(M 1)
(cid:0)
For any M, there holds 0:5Ps < Pb Ps.
(cid:15) (cid:20)
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M-ary Biothogonal Case
The M-ary biothogonal set consists of
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M=2 1 M=2 1
si(t) i=0 (cid:0) ; si(t) i=0 (cid:0) :
f g f(cid:0) g
Demodulation is accomplished by correlating with the largest
magnitude.
The advantage is that the number of corrlators needed for
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optimal receiver is only M=2, and the signal dimension is
M=2 as well. Moreover we gain in energy e(cid:14)ciency over the
orthogonal design.
To evaluate performance, we assume that S0 is transmitted.
(cid:15)
If correct, then y0 > 0, and yi < y0 for all i. Hence
j j
M=2 1
y0
(cid:0)
1
P(C) = f(y0 S0) f(yi S0)dyi dy0
0 " y0 #
Z j 2 Z(cid:0)2 j
(y0 m) =2(cid:27)
e(cid:0) (cid:0) M=2 1
= 1 [1 2Q(y0=(cid:27))] (cid:0) dy0:
0
Z p2(cid:25)(cid:27) (cid:0)
Figure 3.3.19 indicates that M increases the energy e(cid:14)ciency
(cid:15)
steadily.
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M-ary Biothogonal Case (Discussion)
Regarding the bit error probability, there are two types of
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error events, conditioned on S0:
{ Case 1: Choosing S0 rather than S0 (one possible
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event),
{ Case 2: Choosing one of the M 2 signals orthogonal
(cid:0)
to S0.
If case 1 occurs, all bits decided incorrectly, and thus it is a
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very rare case.
For case 2, there are M 2 equally likely decision errors, and
(cid:15) (cid:0)
of these, (M 2)=2 have bit discrepancies with S0 in any
(cid:0)
given position. Hence for M large, Pb approaches to Ps=2
again.
It should be mentioned that the ability of distinguishing si(t)
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from si(t) requires phase synchronization for the receiver.
(cid:0)
In some cases (optical communication), it is infeasible to phys-
ically realize the complement.
8
M-ary Simplex Case
A simplex design is obtained from an orthogonal set by trans-
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lating the M-dimensional constellation so that its center of
mass is at the origin, and then by rotating it into the (M 1)
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dimensional coordinating system.
Let ~si denote the othogonal set signal-space coordinates, and
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~s0 as the translated one. Then
M 1
1
(cid:0)
~s0i = si ~sj:
(cid:0) M jX=0
For M = 2, the simplex constellation is the antipodal design;
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For M = 3, the simplex constellation is formed by the vertices
of an equilateral triangle; For M = 4, we have a regular
tetrahedron.
After obtaining the simplex constellation, each new signal has
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energy given by
M 1
1
(cid:0)
Es0 = < s0i;s0i > = < si;si > + 2 < sj;sj >
M jX=0
M 1
2
(cid:0)
< si;sj >
(cid:0)M jX=0
Es 2Es
= Es + = Es(M 1)=M:
M (cid:0) M (cid:0)
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M-ary Simplex Case (Continued)
For M = 2, we can easily see it is antipodal. Thus, Es0 =
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Es=2.
For M = 3, we have Es0 = 2Es=3. See the (cid:12)gure below.
(cid:15)
(Read example on page 184).
E1/2
s
E1/2
2
s
2E1/2
s
E1/2
2
s
E1/2
s 2 E1/2
s
3
The normalized correlation between signals is:
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< s0i;s0j > Es=M 1
(cid:21)ij = = (cid:0) = ;
0 0
Es Es (M 1)Es=M (cid:0)M 1
i j
r (cid:0) (cid:0)
M 1 M 1
1 2
(cid:0) (cid:0)
< s0i;s0j > = 2 < sj;sj > < si;sj >
M jX=0 (cid:0)M jX=0
Es 2Es
= = Es=M:
M (cid:0) M (cid:0)
10