Realization of matched filter

A matched filter is the linear filter that maximises the output peak-signal-to-noise ratio at the sampling instant $T$ for a known pulse $s(t)$ in white noise. Its impulse response is $h(t) = k\,s(T-t)$.

Derivation (brief). Input $s(t) + n(t)$, noise PSD $N_0/2$. At $t = T$:

$$\left(\frac{S}{N}\right)_o = \frac{\left|\int H(f)S(f)e^{j2\pi fT}df\right|^2}{\frac{N_0}{2}\int|H(f)|^2df} \le \frac{\int|H|^2df\int|S|^2df}{\frac{N_0}{2}\int|H|^2df} = \frac{2E}{N_0}$$

by Schwarz's inequality. Equality holds when $H(f) = kS^*(f)e^{-j2\pi fT}$, i.e. $h(t) = k\,s(T-t)$, giving $(S/N)_{max} = 2E/N_0$.

Realization 1: Integrate-and-dump (rectangular pulse). For $s(t) = A$, $0 \le t \le T$: $h(t) = kA$ for $0 \le t \le T$, a rectangular impulse response, which is an integrator over the last $T$ seconds:

$$y(T) = \int_0^{T} r(t)\,dt$$

        R                   S1 (dump)
 r(t)--/\/\--+--(-A+)--+-------+---> y(T)
             |   op-amp|   |   |
             +---||----+   +-/-+  short C
                 C              at end of T

An op-amp integrator (or RC with $RC \gg T$) integrates over one bit; the output is sampled at $T$, then the capacitor is shorted (dumped) so the next bit starts from zero.

Realization 2: Correlator. Since

$$y(T) = \int_0^T r(\tau)\,h(T-\tau)\,d\tau = k\int_0^T r(\tau)\,s(\tau)\,d\tau$$

the matched filter output at $t=T$ equals the correlation of $r(t)$ with a stored replica of $s(t)$:

 r(t) --->(X)--->[ integrate 0..T ]--> sample at T --> y(T)
           ^
          s(t)  (local replica)

Realization 3: Tapped delay line (for general pulses): delays, tap weights equal to samples of $s(T-t)$, and a summer, giving a digital (FIR) matched filter.

Note: correlator and matched filter give the same value only at $t = T$; their outputs differ at other times.