Optimum detector impulse response; Hilbert transform

Optimum detector

An optimum detector is a receiver (filter followed by sampler and threshold device) that makes the decision with the minimum probability of error. For a known pulse in additive white Gaussian noise, this is achieved by the filter that maximises the output signal-to-noise ratio at the sampling instant; this optimum filter is the matched filter.

Impulse response of the optimum detector

Set-up. The received signal is $x(t)=s(t)+w(t)$, $0\le t\le T$, where $s(t)$ is a known pulse and $w(t)$ is white noise of two-sided PSD $N_0/2$. It passes through an LTI filter $h(t)\leftrightarrow H(f)$ and is sampled at $t=T$:

 s(t)+w(t)    +--------+   y(t)    sample     y(T)
 ------------>|  h(t)  |---------->--/ ---->  decision
              +--------+           at t=T

Output $y(t)=s_o(t)+n_o(t)$ with

$$s_o(T)=\int_{-\infty}^{\infty}H(f)S(f)e^{j2\pi fT}df,\qquad E[n_o^2]=\frac{N_0}{2}\int_{-\infty}^{\infty}\lvert H(f)\rvert^2df$$

We want $H(f)$ that maximises the output peak signal-to-noise ratio

$$\eta=\frac{\lvert s_o(T)\rvert^2}{E[n_o^2]}=\frac{\left\lvert\int H(f)S(f)e^{j2\pi fT}df\right\rvert^2}{\frac{N_0}{2}\int\lvert H(f)\rvert^2df}$$

Schwarz inequality: $\left\lvert\int\phi_1\phi_2\,df\right\rvert^2\le\int\lvert\phi_1\rvert^2df\int\lvert\phi_2\rvert^2df$, with equality only when $\phi_1(f)=k\,\phi_2^*(f)$. Put $\phi_1=H(f)$ and $\phi_2=S(f)e^{j2\pi fT}$:

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

(using Rayleigh's theorem, $E$ = pulse energy). The maximum is reached when

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

Impulse response. Take the inverse transform. For real $s(t)$, $S^*(f)=S(-f)$, which is the transform of $s(-t)$; the factor $e^{-j2\pi fT}$ is a delay of $T$:

$$\begin{aligned} h_{opt}(t) &= k\int_{-\infty}^{\infty}S^*(f)e^{-j2\pi fT}e^{j2\pi ft}df \\ &= k\int_{-\infty}^{\infty}S(-f)e^{-j2\pi f(T-t)}df \\ &= k\,s(T-t) \end{aligned}$$

So the optimum (matched) filter impulse response is the input pulse reversed in time and delayed by $T$. Choosing the sampling instant $T$ equal to the pulse duration makes $h(t)$ causal.

   s(t)                      h(t) = s(T - t)
    |\                          /|
    | \                        / |
    |  \                      /  |
 ---+---+---> t          ---+---+---> t
    0   T                   0   T

Results: maximum SNR $=2E/N_0$ depends only on pulse energy, not on pulse shape; the filter's frequency response is the conjugate of the pulse spectrum, so it passes strongly the frequencies where the signal is strong.

Hilbert transform

Hilbert transform. The Hilbert transform $\hat g(t)$ of $g(t)$ is obtained by shifting the phase of every frequency component by $-90^\circ$ (positive frequencies) and $+90^\circ$ (negative frequencies), without changing amplitudes:

$$\hat g(t)=g(t)*\frac{1}{\pi t}=\frac1\pi\int_{-\infty}^{\infty}\frac{g(\tau)}{t-\tau}d\tau,\qquad \hat G(f)=-j\,\operatorname{sgn}(f)\,G(f)$$

Example: the Hilbert transform of $\cos\omega_0t$ is $\sin\omega_0t$. Properties: $g$ and $\hat g$ have the same amplitude spectrum and energy, and are orthogonal. Uses: SSB generation by the phase-shift method $s(t)=m(t)\cos\omega_ct\mp\hat m(t)\sin\omega_ct$, analytic signals $g_+(t)=g(t)+j\hat g(t)$, and band-pass signal representation.