presentation signal space representations of
TRANSCRIPT
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Presentation
Signal space representations of communication signals, optimal detection
and error probability
Jac Romme Februari 2005
imst
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imst
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Outline
• Representation of bandpass signal
• Discrete Time equivalent model
• Signal space representations
• Optimal detection on AWGN Channel
• Symbol Error Probability on AWGN Channel
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bandpass signal
• Bandpass Signal• Real valued signal S(f) S*(-f)
• finite bandwidth B infinite time span
• fc denotes center frequency
• Negative Frequencies contain no Additional Infocfcf− 0
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Bandpass to Baseband
• Two step procedure:
• Characteristics:• Complex valued signal• No information loss, truely equivalent
• Reconstruction:
)()(2)( fSfUfS =+
)2exp()()(~ cfjtsts π−= +
∫∞
∞− −= τ
ττ
πd
t
sts
)(1)(ˆ
[ ])2exp()(~Re2)( cfjtsts π=
)(ˆ)()( 21
21 tsjtsts +=+
)()(~
cffSfS −= +
⇔
⇔
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Graphical impression
)( fU
)( fS
)( fS +
)(~
fS
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Discrete Time equivalent model
• Ideal Sampling process with period T
• Reconstruction Process
• results is spectral copies
• Use ideal filter to get rid of them
[ ] ( )kTsks =̂
( ) [ ] ( ) ( ) ( )∑∑ ==kk
kTtskTksts δδˆ) ( ) ∑ ⎟
⎠⎞
⎜⎝⎛ −=
k T
kfSfS ˆ
)
⎩⎨⎧ <
=otherwise
fffH s
0
1)( 2
1
t
tt
ππ )sin(
)sinc( =
( ) [ ] ( )∑ −=k
s ktfksts sinc
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Graphical Proof of Nyquist
sf sf20sf−sf2−
sf sf20sf−sf2−
sf sf20sf−sf2−
Aliasing
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Time Domain Sampling Procedure
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-2
-1
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Time Domain Sampling Procedure (2)
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-1
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Time Domain Sampling Procedure (3)
100 110 120 130 140 150 160 170 180 190 200-4
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-1
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Time Domain Sampling Procedure (4)
100 110 120 130 140 150 160 170 180 190 200-4
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-1
0
1
2
3
4
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General Communication signal
• General Shape:• bk is value of k-th symbol
• bk element of {1,2,..,Ns}
• Waveform has:• Unit energy for simplicity
• Duration <T, Bandwidth <B
• for every possible bk
∑ −=k
bb kTtwEtskk
)(ˆ)(
)(twkb
T
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Signal Space Representation
• Signal/vector space is a set of vectors together with two operators, addition of vector and multiplication by a scalar
• Define a set of 2BT real-valued orthonormalfunctions f1(t),f2(t),... ,fBT(t) spanning the 2BT-dimensional space
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Signal Space Representation
• Each waveform can be described by a vector containing 2BT elements
• Some properties:
∫∞
∞−
= dtttw )()( fw
[ ]TBT tftftft )()()()( 221 K=f
∫∞
∞−
= 22)( wdttw ∫
∞
∞−
= wv,)()( dttwtv
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Signal Space Representation
• Example
∫
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
T
dt0
=w
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Optimal detection on AWGN Channel
• Received signal vector
• Noise is not bandwidth limited, but a proper receiver „looks“ only in the waveform space
• Noise elements are i.i.d. Gaussian RV• Receiver must make decision on transmitted
waveform based on r• Maximum likelihood (ML) receiver
• Given AWGN case and equal likely symbols,• Maximum Likelihood = Minimum Distance
nwr +=kb
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Optimal detection on AWGN Channel
• Symbol Error Prob. (SEP) is minimized if:
• Distance can be written out to:
• In case of equal energy symbols
( )2Βminargˆ
kk
b wr −=∈
222,2 kkk wwrrwr +−=−
( )kk
b wr,maxargˆΒ∈
=
Output of b-th Matched Filter
The same for all k
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Optimal detection on AWGN Channel
• Signal Space Matched Filter receiver
1,wr
2,wr
sNwr,
r kb̂maxarg
Note: for equal EnergySymbols
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Optimal detection on AWGN Channel
• 2-th order Bi-orthogonal (BPSK)• BPSK is 1 dimensional Modulation
• 4-th order Bi-orthogonal (QPSK)• QPSK is 2 dimensional
• Noise at the two MF outputs is independent
• The waveform space is subspace of 2BT space
• Ns-th order Bi-orthogonal modulation• Biorthogonal modulation is Ns/2 dimensional
• Maximum order is
21 ,, wrwr −=
43 ,, wrwr −=21 ,, wrwr −=
BTM 221 <
0, 4,32,1 =ww
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-3 -2 -1 0 1 2 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
r
bEd 2=
b1 b2
Symbol Error Probability on AWGN channel
• BPSK case)2|,,()1|,,()( 212
1122
1 =>+=>= kk bPbPeP wrwrwrwr
⎟⎟⎠
⎞⎜⎜⎝
⎛==>
0
221
112 2)|,,(
N
dQbbP kwrwr
⎟⎟⎠
⎞⎜⎜⎝
⎛=⎟
⎟⎠
⎞⎜⎜⎝
⎛=
00
221 2
2)(
N
EQ
N
dQeP b
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Symbol Error Probability on AWGN channel
• Closed form derivation of higher order modulation often impossible
• Solution: Union-bound
∑∈
==≤Bb
kk bbPbbePeP )()|()(
∑ ∑∈ ≠∈
⎟⎟⎠
⎞⎜⎜⎝
⎛≤
Bb bdBd
db
N
dQ
MeP
, 0
2
2
1)(
∑≠∈
=>≤=bdBd
kbdk bbPbbeP,
)|,,()|( wrwr⎟⎟⎠
⎞⎜⎜⎝
⎛=
0
2
2N
dQ db
M
1=
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Symbol Error Probability on AWGN channel
• 6-th order bi-orthogonal modulation• Each symbol has the same error probability
• Mirror Symmetry
∑≠∈
⎟⎟⎠
⎞⎜⎜⎝
⎛≤
bdBd
db
N
dQeP
, 0
2
2)(
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sE2=sE2=
Symbol Error Probability on AWGN channel
• 6-th order bi-orthogonal modulation• Each symbol has the same error probability
• mirror Symmetry
∑≠∈
⎟⎟⎠
⎞⎜⎜⎝
⎛≤
bdBd
db
N
dQeP
, 0
2
2)(
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛≤
00 2
4
2
24)(
N
EQ
N
EQeP ss
Question: 6-th order Bi-orthogonal worse than BPSK???
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Symbol Error Probability on AWGN channel
• Answer: yes and no• Yes: with respect to• No: with respect to { }
• Second No: Symbol Error is not same as Bit error• Depends on bits to symbol mapping
• Gray Coding
• See Proakis
⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟⎠
⎞⎜⎜⎝
⎛≤
0
2
0
2
2
)6(log4
2
)6(log24)(
N
EQ
N
EQeP bb
bs EE )6(log2=sE
bE
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SEP of Bi-orthogonal modulation
Note: numerically obtained exact SEP
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Any Questions??