--- title: "Periodic and non-periodic signals; LTI output is convolution" subject: "Communication Systems" chapter: 1 key: "semester_6/communication_systems:periodic-nonperiodic-convolution" source: https://www.bctengineer.com/q/semester_6/communication_systems/periodic-nonperiodic-convolution site: BCT Engineering --- # Periodic and non-periodic signals; LTI output is convolution Communication Systems · Chapter 1: Introduction - Asked in: 2067 Mangsir (CS I), 4+6 marks - Video: [Impulse Response and Convolution](https://www.youtube.com/watch?v=-oNqg6JH0eI) — 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/periodic-nonperiodic-convolution - Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-1 ## Question Define with examples periodic and non-periodic signals. Prove that for a linear time invariant system, the output is the convolution of the input and the impulse response of the system. ## Short answer Periodic signal repeats after fixed T₀ (e.g., cos, square wave). Non-periodic signal does not repeat (e.g., single pulse, speech). For LTI system with impulse response h(t), input x(t)=∫x(τ)δ(t-τ)dτ; by time-invariance and superposition, output y(t)=∫x(τ)h(t-τ)dτ = x(t)*h(t). ## Answer #### Periodic signal A signal is **periodic** if it repeats exactly after a fixed time $T_0$: $$ g(t)=g(t+T_0)\quad\text{for all }t $$ The smallest such $T_0$ is the fundamental period and $f_0=1/T_0$ is the fundamental frequency. Periodic signals are power signals and have line (discrete) spectra given by the Fourier series. **Examples:** $A\cos(2\pi f_ct)$ with $T_0=1/f_c$; a square wave clock; the 50 Hz mains voltage. #### Non-periodic (aperiodic) signal A signal that does **not** repeat for any $T_0$ is non-periodic. It has a continuous spectrum given by the Fourier transform. **Examples:** a single rectangular pulse $\text{rect}(t/T)$; the decaying exponential $e^{-at}u(t)$; a speech sentence; noise. ```text Periodic (square wave) Non-periodic (single pulse) _ _ _ _ ____ | |_| |_| |_| |_ ... ______| |______ <-T0-> ``` #### Output of an LTI system is convolution of input and impulse response Let the system be LTI with impulse response $h(t)$, i.e. $\delta(t)\to h(t)$. **Step 1 – write the input as a sum of impulses.** By the sifting property: $$ x(t)=\int_{-\infty}^{\infty}x(\tau)\,\delta(t-\tau)\,d\tau $$ This expresses $x(t)$ as a continuous sum of impulses located at $t=\tau$ with strengths $x(\tau)\,d\tau$. **Step 2 – time invariance:** $\delta(t-\tau)\to h(t-\tau)$. **Step 3 – homogeneity (scaling):** $x(\tau)\delta(t-\tau)\to x(\tau)h(t-\tau)$, because $x(\tau)$ is a constant for each $\tau$. **Step 4 – additivity (superposition):** since the input is the integral of these terms, the output is the integral of their responses: $$ y(t)=\mathcal{T}\{x(t)\}=\int_{-\infty}^{\infty}x(\tau)\,\mathcal{T}\{\delta(t-\tau)\}\,d\tau=\int_{-\infty}^{\infty}x(\tau)\,h(t-\tau)\,d\tau $$ $$ y(t)=x(t)*h(t) $$ Hence the output of an LTI system is the convolution of the input with the impulse response. Putting $\lambda=t-\tau$ shows that convolution is commutative: $y(t)=\int h(\lambda)x(t-\lambda)d\lambda=h(t)*x(t)$. **Frequency domain:** by the convolution property, $Y(f)=X(f)H(f)$, where $H(f)$ is the transfer function. **Example:** input $x(t)=u(t)$ to $h(t)=e^{-at}u(t)$: $$ y(t)=\int_0^t e^{-a(t-\tau)}d\tau=\frac{1}{a}\left(1-e^{-at}\right),\quad t\ge0 $$ ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. Which condition defines a periodic signal? A. g(t) = g(t + T0) for all t B. g(t) = -g(t) C. g(t) = 0 for t < 0 D. Finite energy Answer: A. A periodic signal repeats exactly after a fixed period T0; periodic signals are power signals with line spectra. (Key point: Periodic signal definition) 2. The first step in proving y(t) = x(t)*h(t) is to write x(t) as A. A Fourier series B. ∫x(τ)δ(t-τ)dτ, a sum of weighted impulses C. x(t)h(t) D. A single impulse Answer: B. The sifting property expresses the input as a continuous sum of impulses with strengths x(τ)dτ. (Key point: Sifting property step) 3. Time invariance in the proof is used to say A. x(τ)δ(t-τ) gives x(τ)h(t-τ) B. δ(t-τ) gives h(t-τ) C. Sum of responses equals response of sum D. h(t) = 0 for t < 0 Answer: B. A delayed impulse gives a delayed impulse response; scaling and additivity (linearity) then give the weighted integral. (Key point: Role of time invariance and linearity) 4. Convolution y(t) = x(t)*h(t) in the time domain corresponds to what in the frequency domain? A. Y(f) = X(f) * H(f) B. Y(f) = X(f) + H(f) C. Y(f) = X(f)H(f) D. Y(f) = X(f)/H(f) Answer: C. By the convolution property, time convolution becomes multiplication of spectra. 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