Presentation Signal space representations of

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Presentation

Signal space representations of communication signals, optimal detection

and error probability

Jac Romme Februari 2005

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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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Time Domain Sampling Procedure (2)

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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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• 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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• 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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• 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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• Signal Space Matched Filter receiver

1,wr

2,wr

sNwr,

r kb̂maxarg

Note: for equal EnergySymbols

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• 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

pdf

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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• 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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• 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=


• 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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• 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??

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