Integrator and the multiplier in the Optimal Binary Detector

Question

The optimal binary detector for asymmetric, equiprobable PAM pulse normally have structure as shown below,

But I cannot understand what is the multiplier and the integrator using for.

• Why the detector need to care the signal difference and multiply with the received signal?
• Why can't I directly put the received signal to the threshold comparsion?

Answer 1

There are several things missing/extra in your diagram.

• What you are using is rectangular PAM pulses of duration $T$ to send data across the channel, and so you really don't need the multiplier. It is necessary only if $s_1(t)$ and $s_2(t)$ are different from rectangular pulses (though they are still of duration $T$, and in that case, the input $s_1(t)-s_2(t)$ to the multiplier does not tell the whole story.

• The integrator that you are using is really what is called an integrate-and-dump correlator. The input is integrated over intervals of time $[(n-1)T, nT)$. Just before the end of the integration period, at time $nT^{--}$, the integrator output is sampled and the sample value is what is used in the threshold device. Following the sampling at $nT^{--}$, the integrator is dumped (meaning that the output is reset to $0$) at time $nT^{-}$, and the integrator is restarted at $t = nT$ for the next data bit from $nT$ to $(n+1)T$. Note that there is what computer engineers call a critical race involved: the circuit designer has to make absolutely sure that the sample signal arrives at the integrator before the dump signal. Getting this wrong can be a fireable offense.

• With the integrate-and-dump correlator working as described above, the input to the multiplier during $[(n-1)T, nT)$ needs to be $s_1(t-(n-1)T)-s_2(t-(n-1)T)$, that is, the signal $s_1(t)-s_2(t)$ (which is nonzero only for $t \in [0,T)$) delayed by $(n-1)T$ seconds.

• What the integrate-and-dump correlator is doing is computing the output of the matched filter for signals $s_1(t)$ and $s_2(t)$ in additive white Gaussian noise at times $nT$. Note that the correlator output is not equal to the matched filter output at times other than $nT$.

• The reason you cannot simply sample the received signal and make a decision from the sample is that the SNR is abysmal. Technically, AWGN has infinite variance (cf. this question), but even otherwise this is a big problem. The matched filter maximizes the SNR at the sampling instants, and the correlator is just calculating what the matched filter will give you at the sampling instants.

For details of all this, see, for example, pp. 85-93 of these ancient lecture notes of mine.