Optimum Receivers for the Additive Receivers for the Additive White

154 Pages·2010·4.21 MB·English

OOppttiimmuumm RReecceeiivveerrss ffoorr tthhee AAddddiittiivvee
WWhhiittee GGaauussssiiaann NNooiissee CChhaannnneell
WWiirreelleessss IInnffoorrmmaattiioonn TTrraannssmmiissssiioonn SSyysstteemm LLaabb..
IInnssttiittuuttee ooff CCoommmmuunniiccaattiioonnss EEnnggggiinneeeerriinngggg
NNaattiioonnaall SSuunn YYaatt--sseenn UUnniivveerrssiittyy
TTaabbllee ooff CCoonntteennttss
Optimum Receiver for Signals Corrupted by Additive
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WWhhiittee GGaauussssiiaann NNooiissee
Correlation Demodulator
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MMaattcchheedd-FFiilltteerr DDeemmoodduullaattoorr
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The Optimum Detector
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PPerfformance off tthhe OOpttiimum RReceiiver ffor MMemorylless
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Modulation
Probability of Error for Binary Modulation
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Probability of Error for M-ary Orthogonal Signals
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Probability of Error for M-ary Biorthogonal Signals
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Probability of Error for M-ary PAM
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Probability of Error for M-ary PSK
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2
TTaabbllee ooff CCoonntteennttss
Differential PSK (DPSK) and Its Performance
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PPrroobbaabbiilliittyy ooff EErrrroorr ffoorr QQAAMM
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Comparison of Digital Modulation Methods
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OOppttiimmuumm RReecceeiivveerr ffoorr SSiiggnnaallss wwiitthh RRaannddoomm PPhhaassee iinn
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AWGN Channel
OOpttiimum RReceiiver ffor BBiinary SSiignalls
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Optimum Receiver for M-ary Orthogonal Signals
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PProbbabbiilliitty off EError ffor EEnvellope DDettecttiion off MM-ary OOrtthhogonall
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Signals
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OOppttiimmuumm RReecceeiivveerr ffoorr SSiiggnnaallss CCoorrrruupptteedd bbyy
AAddddiittiivvee WWhhiittee GGaauussssiiaann NNooiissee
We assume that the transmitter sends digital information by use of
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M siggnals waveforms {{s ((t))=1,,2,,…,,M }}. Each waveform is
mm
transmitted within the symbol interval of duration T, i.e. 0≤t≤T.
The channel is assumed to corrupt the signal by the addition of white
◊
Gaussian noise, as shown in the following figure:
r(t) = s (t) + n(t) , 0 ≤ t ≤ T
m
where n(t) denotes a sample function of AWGN process with power
spectral density Φ ( f )=½N W/Hz.
nn 0
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OOppttiimmuumm RReecceeiivveerr ffoorr SSiiggnnaallss CCoorrrruupptteedd bbyy
AAddddiittiivvee WWhhiittee GGaauussssiiaann NNooiissee
Our object is to design a receiver that is optimum in the sense that it
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minimizes the pprobabilityy of makingg an error.
It is convenient to subdivide the receiver into two parts—the signal
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demodulator and the detector.
The function of the signal demodulator is to convert the received waveform r(t)
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iinnttoo aann NN-ddiimmeennssiioonnaall vveeccttoorr rr=[[rr rr ....…rr ]] wwhheerree NN iiss tthhee ddiimmeennssiioonn ooff tthhee
11 22 NN
transmitted signal waveform.
The function of the detector is to decide which of the M possible signal
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wavefforms was ttransmiittttedd bbasedd on tthhe vecttor r.
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OOppttiimmuumm RReecceeiivveerr ffoorr SSiiggnnaallss CCoorrrruupptteedd bbyy
AAddddiittiivvee WWhhiittee GGaauussssiiaann NNooiissee
Two realizations of the signal demodulator are described in the
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followingg section:
One is based on the use of signal correlators.
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The second is based on the use of matched ffilters.
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The optimum detector that follows the signal demodulator is
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desiggned to minimize the pprobabilityy of error.
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CCoorrrreellaattiioonn DDeemmoodduullaattoorr
We describe a correlation demodulation that decomposes the
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receiver siggnal and the noise into N-dimensional vectors.
In other words, the signal and the noise are expanded into a series of
◊
lliinneeaarrllyy wweeiigghhtteedd oorrtthhoonnoorrmmaall bbaassiiss ffuunnccttiioonnss {{ff ((tt))}}.
n
It is assumed that the N basis function {f (t)} span the signal space,
◊
n
ssoo eevveerryy oonnee ooff tthhee ppoossssiibbllee ttrraannssmmiitttteedd ssiiggnnaallss ooff tthhee sseett
{s (t)=1≤m≤M } can be represented as a linear combination of
m
{f (t)}.
n
In case of the noise, the function {f (t)} do not span the noise space.
◊
n
However we show below that the noise terms that fall outside the
signal space are irrelevant to the detection of the signal.
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CCoorrrreellaattiioonn DDeemmoodduullaattoorr
Suppose the receiver signal r(t) is passed through a parallel bank of
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N basis functions {f (t)}, as shown in the following figure:
n
T T
[[ ]]
∫∫ r((tt)) ff ((tt))ddtt = ∫∫ s ((tt)) ++ n((tt)) ff ((tt))ddtt
k m k
0 0
⇒ r = s + n , k = 1, 2,.....N
k mk k
T
s = ∫ s (t) f (t)dt , k =1,2,......N
mk m k
0
TT
n = ∫ n(t) f (t)dt , k =1,2,.......N
k k
0
The signal is now represented by the vector s with components s ,
◊
m mk
k=1,2,…N. Their values depend on which of the M signals was
ttrraannssmmiitttteedd..
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CCoorrrreellaattiioonn DDeemmoodduullaattoorr
In fact, we can express the receiver signal r(t) in the interval 0 ≤ t ≤
◊
T as: N N
r((tt)) = ∑∑ s ff ((tt)) + ∑∑ n ff ((tt)) + n′′((tt))
mk k k k
k=1 k=1
N
= ∑∑ r ff ((t)) + n′′((t))
k k
k=1
TThhee tteerrmm nn'((tt)),, ddeeffiinneedd aass
◊◊
N
n′(t) = n(t) − ∑ n f (t)
k k
kk=11
is a zero-mean Gaussian noise process that represents the difference
between origginal noise pprocess n((t)) and the ppart corresppondingg to the
projection of n(t) onto the basis functions {f (t)}.
k
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CCoorrrreellaattiioonn DDeemmoodduullaattoorr
We shall show below that n'(t) is irrelevant to the decision as to
◊
which siggnal was transmitted. Conseqquentlyy,, the decision mayy be
based entirely on the correlator output signal and noise components
r =s +n , k=1,2,…,N.
k mk k
The noise components {n } are Gaussian and mean values are:
◊
k
T
[[ ]]
E(n ) = ∫∫ E n(t) f (t)dt = 0 for all n.
k k PPoowweerr ssppeeccttrraall ddeennssiittyy
00
is Φ (f)=½N W/Hz
and their covariances are:
nn 0
Autocorrelation
T T
[[ ]]
EE((nn nn )) = ∫∫ ∫∫ EE nn((tt))nn((ττ)) ff ((tt)) ff ((ττ))ddttddττ CCoonncclluussiioonn:: TThhee NN nnooiissee
k m k m
0 0
Components {n } are
k
1
T T zero-mean uncorrelated
= N ∫∫ ∫∫ δ((t −τ)) ff ((t)) ff ((τ))dtdτ
00 kk mm
2 0 0 GGaussiian randdom
variables with a common
1 1
T
= N ∫∫ f (t) f (t)dt = N δ
varianceσ 2=½N .
0 k m 0 mk n 0
22 00 22
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