Matched filter approximation using RC low pass filter

Matched filter

A matched filter is a linear filter whose impulse response is matched to (a time-reversed, delayed copy of) a known input pulse $s(t)$, $h(t)=k\,s(T-t)$, so that it gives the maximum possible peak signal-to-noise ratio at the sampling instant $t=T$ when the pulse is received in additive white noise. It is the optimum detector for digital signals in AWGN; in practice it is built as a correlator.

RC low-pass filter as approximate matched filter

The exact matched filter for a rectangular pulse is an integrate-and-dump circuit, which is not always convenient. A simple single-pole RC low-pass filter can approximate it; we find the best bandwidth and the loss.

Input pulse: $s(t)=A$ for $0\le t\le T$ (energy $E=A^2T$), white noise PSD $N_0/2$.

Filter: $H(f)=\dfrac{1}{1+jf/f_c}$, $f_c=\dfrac{1}{2\pi RC}$ (3 dB bandwidth, variable).

Signal output. The step response of the RC filter rises as $A(1-e^{-t/RC})$; it is maximum at the end of the pulse, $t=T$:

$$s_o(T)=A\left(1-e^{-T/RC}\right)=A\left(1-e^{-2\pi f_cT}\right)$$

 s(t)                 s_o(t)
 A +-----+          A |      ..-.  peak at t=T
   |     |            |    .'    '.
   |     |            |  .'        '..
 --+-----+--> t     --+-'-----+-------'--> t
   0     T            0       T

Noise output power:

$$N_o=\frac{N_0}{2}\int_{-\infty}^{\infty}\frac{df}{1+(f/f_c)^2}=\frac{N_0}{2}\cdot\pi f_c$$

Output SNR:

$$\text{SNR}_o=\frac{A^2\left(1-e^{-2\pi f_cT}\right)^2}{\pi N_0f_c/2}=\frac{2A^2T}{N_0}\cdot\frac{2\left(1-e^{-x}\right)^2}{x},\qquad x=2\pi f_cT$$

Compared with the matched filter ($\text{SNR}_{max}=2E/N_0=2A^2T/N_0$):

$$\rho(x)=\frac{\text{SNR}_o}{\text{SNR}_{max}}=\frac{2(1-e^{-x})^2}{x}$$

Effect of variable bandwidth:

• $f_c$ too small: the filter cannot follow the pulse, $s_o(T)$ is small, SNR low.
• $f_c$ too large: signal fully passes but much extra noise enters, SNR low.
• An optimum lies between. Setting $d\rho/dx=0$ gives $(1+2x)e^{-x}=1$, solved numerically: $x\approx1.256$.

$$f_cT=\frac{1.256}{2\pi}\approx0.2,\qquad \rho_{max}=\frac{2(1-e^{-1.256})^2}{1.256}\approx0.815$$

Result: the best RC filter has $f_c\approx\dfrac{0.2}{T}$ (i.e. $RC\approx0.8T$) and gives an SNR only $10\log_{10}0.815\approx-0.89$ dB, i.e. less than 1 dB below the ideal matched filter. Hence a simple RC filter is a good practical substitute for the matched filter of a rectangular pulse.