--- title: "Classification of systems; ideal LPF is non-causal" subject: "Communication Systems" chapter: 1 key: "semester_6/communication_systems:classify-systems-ideal-lpf" source: https://www.bctengineer.com/q/semester_6/communication_systems/classify-systems-ideal-lpf site: BCT Engineering --- # Classification of systems; ideal LPF is non-causal Communication Systems · Chapter 1: Introduction - Asked in: 2064 Shrawan (CS I), 4+4 marks - Video: [Causal and Non-Causal Systems](https://www.youtube.com/watch?v=mqwUtn5cip8) — Neso Academy - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_6/communication_systems/classify-systems-ideal-lpf - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1 ## Question Explain how can you classify systems according to their basic properties. Show that an 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. ## Answer 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: $$ H(f)=\text{rect}\left(\frac{f}{2B}\right)e^{-j2\pi ft_0}=\begin{cases}e^{-j2\pi ft_0}, & \lvert f\rvertB\end{cases} $$ Its impulse response is the inverse Fourier transform: $$ \begin{aligned} h(t)&=\int_{-B}^{B}e^{-j2\pi ft_0}e^{j2\pi ft}df=\frac{\sin 2\pi B(t-t_0)}{\pi(t-t_0)}\\ &=2B\,\text{sinc}\left[2B(t-t_0)\right] \end{aligned} $$ ```text 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. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which system is linear? A. y = x^2 B. y = 3x C. y = x + 1 D. y = |x| Answer: B. Only y = 3x satisfies superposition. (Key point: Linearity test) 2. The impulse response of an ideal LPF with cutoff B and delay t0 is A. A delta function B. 2B sinc[2B(t - t0)] C. e^{-t}u(t) D. rect(t/2B) Answer: B. The inverse transform of a rectangular H(f) with linear phase is a sinc centred at t0. (Key point: Ideal LPF impulse response) 3. The ideal LPF is non-causal because A. Its sinc h(t) extends to t < 0, so h(t) is nonzero before the impulse B. It is unstable C. Its gain is not constant D. It has a delay Answer: A. Causality requires h(t) = 0 for t < 0, but the sinc runs from -∞ to +∞ for any finite t0. (Key point: Why ideal LPF is non-causal) 4. The Paley-Wiener criterion rules out the ideal LPF because A. |H(f)| is too large B. |H(f)| = 0 over a band makes ln|H| infinite so the integral diverges C. Its phase is linear D. It is even Answer: B. A causal filter needs the integral of |ln|H||/(1+f^2) to be finite, which fails when H is zero over a whole band. 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