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Luigi Paura » 5.Digital Transmission over AWGN chanel


Summary

  • Introduction to the problem
  • Synthesis of the optimum receiver (minimum P(e)) for AWGN channel
  • Structures of the optimum receiver

Optimum Receiver in AWGN (next)


Optimum Receiver in AWGN (next)


Optimum Receiver in AWGN (next)

It can be shown that the receiver which:

performs independent decisions over the single symbols forming the whole message ξ utilizing the observed signal associated with the symbol interval namely:

âl=f{r(t), lT≤t≤(l+1)T}

It is optimum if:

  1. the symbol (the blocks of k binary symbols at the input of the modulator) are statistical independent
  2. the modulation is a memoryless modulation
  3. the channel is memoryless (in our case is AWGN)

Optimum Receiver in AWGN (next)


Optimum Receiver in AWGN (next)

It can be shown that the receiver is constituted by two sections in cascade:

  • The demodulator that starting from r(t) determines a vector r = (r1,r2,…rN) where N is the dimensionality of the transmitted signal space
  • The detector decides which of the M possible signals (symbols) was transmitted based on observation of the vector r.

Optimum Receiver in AWGN (next)


Optimum Receiver in AWGN (next)

Let us evaluate f (rlSm), namely the pdf of the N-dimensional received vector r conditioned to the transmission of the signal sm(t)↔Sm

rk=Smk+nk k=1,…,N

E[rk]=E [Smk+nk]=E [Smk ]+E [nk]=Smk

The  random variable nk e ni are jointly gaussian with zero mean  and covariance N0/2 δ[n-i]

The RV rk e ri are jointly gaussian with means  smk e  smi , respectively and  covariance N0/2 δ[n-i]

Optimum Receiver in AWGN (next)

Let us show now that  r’(t) ↔ r is a sufficient statistic, namely n’(t) does not add any further information to take the decision over the transmitted signal (symbol).


Optimum Receiver in AWGN (next)


Optimum Receiver in AWGN (next)


Optimum Receiver in AWGN (next)


Optimum Receiver in AWGN implemented with matched filters


Properties of the matched filter

Given a signal s(t) of duration T, the matched filter to s(t) is a filter with impulse response h(t) = s(T-t) of duration T and ≠ 0 in the interval (0,T)

Ex: Show that the output of the matched filter to s(t), when the input is s(t), is Rs(T-t) and, therefore, accounting from the properties of the autocorelation function is maximum in t =T and equal to Εs

Optimum decision : MAP criterion (next)


Optimum decision : MAP criterion (next)


Optimum decision : MAP criterion


Optimum Receiver in AWGN (next)


Optimum Receiver in AWGN (next)


Optimum Receiver in AWGN (next)


Optimum Receiver in AWGN (next)


Optimum Receiver in AWGN (next)


Optimum Receiver in AWGN (next)


Optimum Receiver in AWGN (next)


Digital transmission over AWGN channel (next)

Example:QAM signalling (quadrature amplitude modulation)

um(t)=AmcgT(t)cos2πfct+AmsgT(t)sin2πfct

m=1,2,…,M

{Amc} and {Ams}    are the sets of the amplitudes  of the signals such that any binary k-sequence maps to  a pair of different amplitudes

gT(t) = rect[(t-T/2)/T]

Digital transmission over AWGN channel


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