Optimum detector; matched filter impulse response

Optimum detector

An optimum detector is a receiver that makes decisions with the minimum probability of error for given signals and noise. In AWGN this is achieved by first passing the received signal through a filter that maximises the output SNR at the sampling instant (the matched filter or an equivalent correlator), then comparing the sample with a threshold (maximum-likelihood decision).

Matched filter impulse response

Consider a known pulse $g(t)$, $0\le t\le T$, plus white Gaussian noise $w(t)$ of power spectral density $N_0/2$, applied to a linear time-invariant filter with impulse response $h(t)$ and transfer function $H(f)$.

 x(t) = g(t) + w(t) --> [ h(t) ] --> y(t) = go(t) + n(t)
                                         |
                                     sample at t = T

Signal output at the sampling instant:

$$g_o(T)=\int_{-\infty}^{\infty}H(f)G(f)e^{j2\pi fT}df$$

Noise output power (white noise through $H(f)$):

$$E[n^2(t)]=\frac{N_0}{2}\int_{-\infty}^{\infty}\lvert H(f)\rvert^2df$$

Peak pulse SNR to be maximised:

$$\eta=\frac{\lvert g_o(T)\rvert^2}{E[n^2(t)]}=\frac{\left|\int H(f)G(f)e^{j2\pi fT}df\right|^2}{\frac{N_0}{2}\int\lvert H(f)\rvert^2df}$$

Schwarz inequality: for any $\phi_1(f),\phi_2(f)$,

$$\left|\int\phi_1\phi_2\,df\right|^2\le\int\lvert\phi_1\rvert^2df\int\lvert\phi_2\rvert^2df$$

with equality only when $\phi_1(f)=k\,\phi_2^*(f)$. Put $\phi_1=H(f)$ and $\phi_2=G(f)e^{j2\pi fT}$:

$$\left|\int H(f)G(f)e^{j2\pi fT}df\right|^2\le\int\lvert H(f)\rvert^2df\int\lvert G(f)\rvert^2df$$

Substituting into $\eta$:

$$\eta\le\frac{2}{N_0}\int_{-\infty}^{\infty}\lvert G(f)\rvert^2df=\frac{2E}{N_0}$$

using Rayleigh's energy theorem $E=\int\lvert G(f)\rvert^2df=\int g^2(t)dt$.

Optimum filter: the maximum $\eta_{max}=2E/N_0$ is reached when equality holds:

$$H_{opt}(f)=k\,G^*(f)\,e^{-j2\pi fT}$$

Impulse response: take the inverse Fourier transform:

$$\begin{aligned} h_{opt}(t)&=k\int_{-\infty}^{\infty}G^*(f)e^{-j2\pi f(T-t)}df\\ &=k\left[\int_{-\infty}^{\infty}G(f)e^{j2\pi f(T-t)}df\right]^*\quad(\text{real }g)\\ &=k\,g(T-t) \end{aligned}$$

So the impulse response of the optimum filter is the time-reversed and delayed (by $T$) version of the input signal, scaled by $k$. Because it is "matched" to the signal, it is called the matched filter.

  g(t)                     h(t) = g(T - t)
   |\                          /|
   | \                        / |
   |  \                      /  |
   +---+-----> t        +---+---+----> t
   0   T                0       T
  (flip about t = 0, then shift right by T)

Properties

• Maximum output SNR $=2E/N_0$ depends only on the signal energy, not on its shape.
• $\lvert H_{opt}(f)\rvert=k\lvert G(f)\rvert$: the filter passes strongly where the signal spectrum is strong.
• The output $g_o(t)$ is proportional to the autocorrelation of $g(t)$ shifted by $T$, peaking at $t=T$; hence a matched filter is equivalent to a correlator (multiply by $g(t)$ and integrate over $0$ to $T$).
• The delay $T$ makes $h(t)$ causal (zero for $t<0$).