--- title: "Convolution sum; ideal LPF is non-causal" subject: "Signals and Systems" chapter: 5 key: "semester_4/signals_and_systems:convolution-sum-ideal-lpf-noncausal" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/convolution-sum-ideal-lpf-noncausal site: BCT Engineering --- # Convolution sum; ideal LPF is non-causal Signals and Systems · Chapter 5: Continuous-Time Systems - Asked in: 2078 Poush, 6+6 marks - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/convolution-sum-ideal-lpf-noncausal - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-5 ## Question Derive convolution sum. Prove that an ideal low-pass filter is a non-causal system. ## Short answer Using sifting, x[n]=sum x[k]delta[n-k]; LTI properties give y[n]=sum x[k]h[n-k]. Ideal LPF impulse response is sinc(wc t)/(pi t), nonzero for negative t, so it is non-causal and not real-time realizable. ## Answer #### Derivation of the convolution sum Any discrete-time signal can be written as a sum of shifted, scaled unit impulses (sifting property): $$ x[n] = \sum_{k=-\infty}^{\infty} x[k]\,\delta[n-k] $$ e.g. $x[n] = \{2, 3\}$ (starting at $n=0$) is $2\delta[n] + 3\delta[n-1]$. Let the system be LTI with impulse response $h[n]$, i.e. $\delta[n] \to h[n]$. 1. **Time invariance:** $\delta[n-k] \to h[n-k]$. 2. **Scaling (homogeneity):** $x[k]\,\delta[n-k] \to x[k]\,h[n-k]$ ($x[k]$ is just a number for fixed $k$). 3. **Additivity:** sum over all $k$: $$ y[n] = \sum_{k=-\infty}^{\infty} x[k]\,h[n-k] = x[n] * h[n] $$ This is the **convolution sum**. By substituting $m = n - k$, $y[n] = \sum_m h[m]x[n-m]$ (commutative). **Procedure:** fold $h[k]$ to $h[-k]$, shift by $n$, multiply with $x[k]$, add the products; repeat for each $n$. **Example:** $x[n] = \{1, 2\}$, $h[n] = \{1, 1\}$ (both starting at 0): $y[0] = 1$, $y[1] = 1\cdot1 + 2\cdot1 = 3$, $y[2] = 2$, so $y[n] = \{1, 3, 2\}$. #### Ideal LPF is non-causal The ideal low-pass filter passes all frequencies below the cutoff $\omega_c$ with unit gain and rejects all others: $$ H(j\omega) = \begin{cases} 1, & |\omega| \le \omega_c \\ 0, & |\omega| > \omega_c \end{cases} $$ **Impulse response:** The impulse response is the inverse Fourier transform of $H(j\omega)$: $$ \begin{aligned} h(t) &= \frac{1}{2\pi}\int_{-\infty}^{\infty} H(j\omega)e^{j\omega t}\,d\omega = \frac{1}{2\pi}\int_{-\omega_c}^{\omega_c} e^{j\omega t}\,d\omega \\ &= \frac{1}{2\pi}\cdot\frac{e^{j\omega_c t} - e^{-j\omega_c t}}{jt} = \frac{1}{2\pi}\cdot\frac{2j\sin\omega_c t}{jt} \\ &= \frac{\sin\omega_c t}{\pi t} = \frac{\omega_c}{\pi}\,\text{sinc}\!\left(\frac{\omega_c t}{\pi}\right) \end{aligned} $$ ```text H(jw) h(t) 1 ______ wc/pi | | /\ | | _ _ / \ _ _ ----+------+--> w -' '-' '-/----\-' '-' '-> t -wc 0 wc -pi/wc pi/wc ``` **Causality test:** an LTI system is causal if and only if $h(t) = 0$ for $t < 0$ (the output cannot depend on future input, since $y(t) = \int h(\tau)x(t - \tau)d\tau$ uses $x$ at $t - \tau > t$ whenever $h(\tau) \ne 0$ for $\tau < 0$). For the ideal LPF, $h(t) = \dfrac{\sin\omega_c t}{\pi t}$ is an even function, non-zero on both sides of $t = 0$. For example, $$ h\!\left(-\frac{\pi}{2\omega_c}\right) = \frac{2\omega_c}{\pi^2} \ne 0 $$ So $h(t) \ne 0$ for $t < 0$ and the ideal LPF is **non-causal**. Even with a delay $t_0$, $h(t) = \dfrac{\sin\omega_c(t-t_0)}{\pi(t-t_0)}$ still has a tail reaching $t = -\infty$, so no finite delay makes it causal. (The discrete-time ideal LPF, $h[n] = \dfrac{\sin\Omega_c n}{\pi n}$, is non-causal for the same reason.) Hence it cannot be realized in real time. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is a DT signal written to derive the convolution sum? A. x[n] = sum of x[k] δ[n+k] h[k] B. x[n] = x[0] δ[n] C. x[n] = sum of x[k] δ[n-k] D. x[n] = sum of x[n-k] Answer: C. x[n] is a sum of shifted, scaled impulses. (Key point: Sifting representation) 2. Which properties give y[n] = sum x[k]h[n-k]? A. Causality and stability B. Only linearity C. Commutativity only D. Time invariance, homogeneity and additivity Answer: D. δ[n-k]→h[n-k] (time invariance), scaled by x[k], then summed. (Key point: Derivation steps) 3. For x[n]={1,2} and h[n]={1,1} (starting at 0), what is y[n]? A. {1, 3, 2} B. {1, 2, 1} C. {2, 3, 1} D. {1, 3, 3, 2} Answer: A. y[0]=1, y[1]=1+2=3, y[2]=2. (Key point: Example) 4. An LTI system is causal iff: A. h(t) = 0 for t > 0 B. h(t) = 0 for t < 0 C. h(t) is even D. The integral of |h| is finite Answer: B. Otherwise the output uses future inputs. 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