# 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)

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)

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)

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).

### 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

### Digital transmission over AWGN channel (next)

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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