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:

â_l=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,…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 (rlS_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 N₀/2 δ[n-i]

The RV r_k e r_i are jointly gaussian with means  smk e  s_mi , respectively and  covariance N₀/2 δ[n-i]

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_mcg_T(t)cos2πf_ct+A_msg_T(t)sin2πf_ct

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