Hilbert transform; ideal LPF is non-causal

Hilbert transform

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

$$\hat{x}(t) = \frac{1}{\pi}\int_{-\infty}^{\infty} \frac{x(\tau)}{t-\tau}\,d\tau = x(t) * \frac{1}{\pi t}$$ $$\hat{X}(f) = -j\,\text{sgn}(f)\,X(f)$$

Example: the Hilbert transform of $\cos 2\pi f_c t$ is $\sin 2\pi f_c t$. It is used in SSB generation (phase-shift method) and in analytic signal / pre-envelope representation.

Impulse response of an ideal LPF is non-causal

An ideal low-pass filter of bandwidth $B$, gain $K$ and delay $t_d$ has

$$H(f) = \begin{cases} K\,e^{-j2\pi f t_d}, & |f| \le B \\ 0, & |f| > B \end{cases}$$

Its impulse response is the inverse Fourier transform:

$$\begin{aligned} h(t) &= \int_{-B}^{B} K\,e^{-j2\pi f t_d}e^{j2\pi f t}\,df = K\int_{-B}^{B} e^{j2\pi f(t-t_d)}\,df \\ &= K\,\frac{e^{j2\pi B(t-t_d)} - e^{-j2\pi B(t-t_d)}}{j2\pi(t-t_d)} \\ &= K\,\frac{\sin 2\pi B(t-t_d)}{\pi(t-t_d)} = 2BK\,\text{sinc}\left[2B(t-t_d)\right] \end{aligned}$$

where $\text{sinc}(x) = \sin(\pi x)/(\pi x)$.

             h(t)  peak 2BK at t = td
               |       *
               |      * *
     . .  .  . |  .  *   *  .  .   . .
 ----------------+----+-+----+-----------> t
   (non-zero     0    td
    for t < 0)

Causality check: a system is causal only if $h(t) = 0$ for all $t < 0$. The sinc function extends from $-\infty$ to $+\infty$; it is zero only at the discrete points $t = t_d + n/(2B)$, $n \ne 0$. For example at $t = -\frac{1}{4B}$, $h(t) \ne 0$ in general. However large the delay $t_d$ is made, the oscillating tails for $t < 0$ never vanish completely.

So $h(t) \ne 0$ for $t < 0$, i.e. the filter would respond before the impulse is applied. Hence the ideal LPF is non-causal and physically unrealizable. (This also follows from the Paley–Wiener criterion: a causal filter cannot have $|H(f)| = 0$ over a whole band.) Practical filters (Butterworth, Chebyshev) only approximate the ideal response; a large delay $t_d$ makes the truncated sinc a good approximation.