--- title: "Spectrum of a signal and its samples" subject: "Signals and Systems" chapter: 4 key: "semester_4/signals_and_systems:sampled-spectrum-relation" source: https://www.bctengineer.com/q/semester_4/signals_and_systems/sampled-spectrum-relation site: BCT Engineering --- # Spectrum of a signal and its samples Signals and Systems · Chapter 4: Sampling - Asked in: 2069 Bhadra, 6 marks - Video: [The Spectrum of Sampled Signals](https://www.youtube.com/watch?v=y794OO606eg) — Barry Van Veen - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/signals_and_systems/sampled-spectrum-relation - Whole chapter: https://www.bctengineer.com/md/semester_4/signals_and_systems/chapter-4 ## Question What is sampling. How are spectrum of continuous time signal and its sampled version related? Illustrate with diagram. ## Short answer Sampling converts a continuous-time signal to discrete samples by multiplying with an impulse train. The sampled spectrum is the original spectrum scaled by 1/T and repeated periodically at integer multiples of the sampling frequency ω_s, becoming periodic with period ω_s. ## Answer #### Sampling **Sampling** converts a continuous-time signal $x(t)$ into samples $x(nT)$ taken every $T$ seconds ($\omega_s = 2\pi/T$). It is modelled as multiplication by an impulse train: $$ x_p(t) = x(t)\sum_{n=-\infty}^{\infty}\delta(t-nT) = \sum_n x(nT)\delta(t-nT) $$ #### Relation between the two spectra The impulse train has transform $P(j\omega) = \dfrac{2\pi}{T}\sum_k \delta(\omega - k\omega_s)$. By the multiplication property, $$ \begin{aligned} X_p(j\omega) &= \frac{1}{2\pi}X(j\omega) * P(j\omega) \\ &= \frac{1}{T}\sum_{k=-\infty}^{\infty} X\big(j(\omega - k\omega_s)\big) \end{aligned} $$ So the spectrum of the sampled signal is: - the original spectrum $X(j\omega)$ scaled by $1/T$, - repeated periodically at every integer multiple of $\omega_s$, - hence **periodic in $\omega$ with period $\omega_s$**. ```text (a) Original X(jw), band-limited to wM /\ A / \ -----/----\-----> w -wM wM (b) Sampled Xp(jw), ws > 2wM (height A/T) /\ /\ /\ / \ / \ / \ -/----\----/----\----/----\--> w -ws 0 ws (c) Sampled Xp(jw), ws < 2wM -> overlap (aliasing) ___ ___ ___ / X X \ --/-------------\--> w ``` #### Interpretation - If $\omega_s > 2\omega_M$, the copies do not overlap and the original spectrum can be cut out with an ideal LPF of gain $T$; $x(t)$ is recovered exactly. - If $\omega_s < 2\omega_M$, the copies overlap and the spectrum near $\omega_s/2$ is corrupted (aliasing). - For the discrete-time sequence $x[n] = x(nT)$, the DTFT is $X(e^{j\Omega}) = X_p(j\Omega/T)$, i.e. the same picture with frequency axis scaled so that $\omega_s$ maps to $2\pi$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is sampling modelled mathematically? A. Multiplying x(t) by an impulse train B. Convolving x(t) with an impulse train C. Integrating x(t) over T D. Differentiating x(t) at nT Answer: A. xp(t) = x(t) Σδ(t - nT) = Σ x(nT) δ(t - nT). (Key point: Sampling model) 2. How is Xp(jω) related to X(jω)? A. Identical to X(jω) B. Scaled by T and shifted once C. Scaled by 1/T and repeated at every multiple of ωs D. Multiplied by ωs Answer: C. Xp is (1/T) times periodic repeats of X, hence periodic with period ωs. (Key point: Spectrum relation) 3. What is the period of Xp(jω) in ω? A. ωM B. ωs C. 2ωM D. T Answer: B. Copies are spaced by ωs, so Xp is periodic with period ωs. (Key point: Periodicity) 4. If ωs < 2ωM, what happens near ωs/2? A. Copies overlap and the spectrum is corrupted (aliasing) B. The spectrum is zero C. The spectrum becomes larger but undistorted D. The copies move apart Answer: A. Overlap makes the spectrum near ωs/2 a sum of two copies. 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