### 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:

â_{l}=*f*{*r*(*t*), *lT≤t≤*(*l*+1)T}

**It is optimum if:**

- the symbol (the blocks of k binary symbols at the input of the modulator) are statistical independent
- the modulation is a memoryless modulation
- 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 = (r
_{1},r_{2},…r_{N}) 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* (*r*lS_{m}), *namely* the pdf of the N-dimensional received vector r conditioned to the transmission of the signal *s*_{m}(*t*)↔*S*_{m}

*r*_{k}=S_{mk}+n_{k} k*=1,…,*N

*E*[*r*_{k}]*=**E** *[*S*_{mk}+n_{k}]=*E** *[*S*_{mk} * *]+*E** *[*n*_{k}]=*S*_{mk}

The random variable *n*_{k} e *n*_{i} *are jointly* gaussian with zero mean and covariance N_{0}/2 δ[n-i]

The RV *r*_{k} e *r*_{i} are jointly gaussian with means *s*mk e *s*_{mi} , respectively and covariance N_{0}/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 R_{s}(T-t) and, therefore, accounting from the properties of the autocorelation function is **maximum in t =T and equal to Ε**_{s}

### 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)

*u*_{m}(t)=*A*_{mc}g_{T}(t)cos2*πf*_{c}t+*A*_{ms}g_{T}(*t*)sin2*πf*_{c}t

*m*=1,2,…*,M*

{*A*_{mc}} and {*A*_{ms}} are the sets of the amplitudes of the signals such that any binary k-sequence maps to a pair of different amplitudes

g_{T}(t) = rect[(t-T/2)/T]

### Digital transmission over AWGN channel