The ideal LPF is non-causal because

Its sinc h(t) extends to t < 0, so h(t) is nonzero before the impulse

The Paley-Wiener criterion rules out the ideal LPF because

|H(f)| = 0 over a band makes ln|H| infinite so the integral diverges

2064 Shrawan (CS I) · 4+4 marks

Classification of systems; ideal LPF is non-causal

Short answer

Systems are classified as linear/non-linear, time-invariant/variant, causal/non-causal, static/dynamic, stable/unstable, and invertible/non-invertible. An ideal LPF with H(f)=rect(f/2B)e^{-j2πft₀} has impulse response h(t)=2B sinc[2B(t-t₀)], which is non-zero for t<0 for any finite t₀, violating causality. The Paley-Wiener criterion also confirms non-causality since |H(f)|=0 over a band makes the integral diverge.

Explain how can you classify systems according to their basic properties. Show that an ideal LPF is non-causal.

A system is any device or process that transforms an input $x(t)$ into an output $y(t)=\mathcal{T}\{x(t)\}$ . Systems are classified by the following basic properties.

Classification of systems

Class	Condition	Example
Linear / non-linear	Linear if superposition holds: $a x_1+b x_2\to a y_1+b y_2$	$y=3x$ linear; $y=x^2$ non-linear
Time-invariant / time-variant	Invariant if $x(t-t_0)\to y(t-t_0)$	$y=x(t-2)$ invariant; $y=t\,x(t)$ variant
Causal / non-causal	Causal if output depends only on present and past input; for LTI $h(t)=0$ for $t<0$	$y=x(t-1)$ causal; $y=x(t+1)$ non-causal
Static / dynamic	Static (memoryless) if output depends only on present input	Resistor divider static; capacitor dynamic
Stable / unstable	BIBO stable if every bounded input gives bounded output; LTI: $\int\lvert h(t)\rvert dt<\infty$	$h=e^{-t}u(t)$ stable; integrator unstable
Invertible / non-invertible	Input can be recovered uniquely from output	$y=2x$ invertible; $y=x^2$ not

Ideal LPF is non-causal

An ideal low-pass filter with cut-off $B$ passes all frequencies below $B$ with constant gain and linear phase, and blocks all others:

H(f)=\text{rect}\left(\frac{f}{2B}\right)e^{-j2\pi ft_0}=\begin{cases}e^{-j2\pi ft_0}, & \lvert f\rvert<B\\0, & \lvert f\rvert>B\end{cases}

Its impulse response is the inverse Fourier transform:

\begin{aligned} h(t)&=\int_{-B}^{B}e^{-j2\pi ft_0}e^{j2\pi ft}df=\frac{\sin 2\pi B(t-t_0)}{\pi(t-t_0)}\\ &=2B\,\text{sinc}\left[2B(t-t_0)\right] \end{aligned}

              h(t)
               |   peak 2B at t = t0
               |       /\
      .  /\    |  /\  /  \  /\    /\  .
 ----'\/--\/---+-'--\/----\/--\/--\/----> t
               0       t0
   <-- non-zero for t < 0 -->

The sinc function extends from $t=-\infty$ to $t=+\infty$ . For any finite delay $t_0$ , $h(t)\ne0$ for $t<0$ , i.e. the filter responds before the impulse is applied at $t=0$ . This violates the causality condition $h(t)=0$ for $t<0$ , so the ideal LPF is non-causal and physically unrealizable.

The same result follows from the Paley–Wiener criterion: a causal filter needs $\int\frac{\lvert\ln\lvert H(f)\rvert\rvert}{1+f^2}df<\infty$ ; since $\lvert H(f)\rvert=0$ over a whole band, $\ln\lvert H(f)\rvert=-\infty$ and the integral diverges. In practice, a large delay $t_0$ with truncated sinc tails gives a good approximation, and practical filters such as Butterworth and Chebyshev approximate the ideal response with causal designs.

✓ Answer checked by Claude Opus 5.5 (high effort)

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