---
title: "Prove sampling theorem; aliasing and aperture effect"
subject: "Communication Systems"
chapter: 4
key: "semester_6/communication_systems:sampling-prove-aliasing-aperture"
source: https://www.bctengineer.com/q/semester_6/communication_systems/sampling-prove-aliasing-aperture
site: BCT Engineering
---

# Prove sampling theorem; aliasing and aperture effect

Communication Systems · Chapter 4: Pulse Modulation

- Asked in: 2073 Shrawan (CS II), 5+3 marks
- Video: [Sampling Theorem with proof | Digital Communication IPU University complete DC B.Tech Unit 1 #04](https://www.youtube.com/watch?v=D_koYmWWl7Q) — IP Academy
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_6/communication_systems/sampling-prove-aliasing-aperture
- Whole chapter: https://www.bctengineer.com/md/semester_6/communication_systems/chapter-4

## Question

State and prove sampling theorem. Define aliasing effect and aperture effect.

## Short answer

A band-limited signal with highest frequency W is uniquely determined by samples taken at intervals Ts ≤ 1/(2W) and recovered by an ideal low-pass filter. Aliasing effect is spectral overlap when fs < 2W, causing high frequencies to fold back as false low frequencies; it is prevented by an anti-alias filter and sampling above 2W. Aperture effect is high-frequency attenuation in flat-top sampling due to the sinc-shaped hold pulse, corrected by an equalizer.

## Answer

#### Sampling theorem
A band-limited signal $g(t)$ with no frequency components above $W$ Hz is uniquely determined by its samples taken at uniform intervals $T_s \le 1/(2W)$, and can be recovered from them exactly by an ideal low-pass filter.

#### Proof
**Step 1: spectrum of the sampled signal.** Ideal sampling:
$$
g_\delta(t) = g(t)\,\delta_{T_s}(t) = \sum_n g(nT_s)\delta(t - nT_s)
$$
The periodic impulse train has Fourier series $\delta_{T_s}(t) = \dfrac{1}{T_s}\sum_n e^{j2\pi nf_st}$. Hence
$$
g_\delta(t) = f_s\sum_n g(t)e^{j2\pi nf_st} \;\Longleftrightarrow\; G_\delta(f) = f_s\sum_{n=-\infty}^{\infty}G(f - nf_s)
$$
The spectrum is $G(f)$ repeated at every multiple of $f_s$.

**Step 2: no overlap.** Each copy occupies $nf_s - W$ to $nf_s + W$. Copies $n = 0$ and $n = 1$ do not overlap if $W \le f_s - W$, i.e. $f_s \ge 2W$.

**Step 3: recovery.** Pass $g_\delta(t)$ through an ideal LPF $H(f) = T_s\,\text{rect}(f/2W)$. Only the $n = 0$ term survives: $G_\delta(f)H(f) = G(f)$. In the time domain, with $f_s = 2W$ and $\text{sinc}(x) = \sin x/x$, $h(t) = \text{sinc}(2\pi Wt)$ and
$$
g(t) = \sum_n g(nT_s)\,h(t - nT_s) = \sum_n g\!\left(\frac{n}{2W}\right)\text{sinc}(2\pi Wt - n\pi)
$$
Thus $g(t)$ is fully recovered from its samples, which proves the theorem.

```text
 Gd(f):  /\      /\      /\
    ____/  \____/  \____/  \____
       -fs      0      fs
            |<-LPF->|   (needs fs >= 2W)
```

#### Aliasing effect
When $f_s < 2W$ the spectral copies overlap; components above $f_s/2$ fold back and appear as false low frequencies, and the signal cannot be recovered. Aliasing is prevented by an anti-aliasing LPF and by sampling above $2W$.

#### Aperture effect
In flat-top sampling with pulse width $\tau$, the spectrum is multiplied by $H(f) = \tau\,\text{sinc}(\pi f\tau)$, which attenuates the high message frequencies. This distortion is the aperture effect. It is reduced by a small $\tau$ and corrected by an equalizer $1/H(f)$.

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. In the proof, the impulse train is expanded as a Fourier series, which yields Gd(f) equal to:
   A. G(f)
   B. fs times the sum of G(f - n fs)
   C. Ts times G(f)
   D. G(f) sinc(f tau)
   Answer: B. The series terms e^(j2 pi n fs t) shift G(f) by n fs, so the spectrum repeats. (Key point: Proof step 1)

2. Copies n = 0 and n = 1 do not overlap if:
   A. W <= fs - W, i.e. fs >= 2W
   B. fs <= W
   C. fs = W/2
   D. W >= 2 fs
   Answer: A. The n = 0 copy ends at W and the n = 1 copy begins at fs - W. (Key point: No-overlap condition)

3. Recovery uses an ideal LPF H(f) of:
   A. Gain 1 and cut-off 2W
   B. T_s rect(f/2W)
   C. A high-pass response
   D. Gain fs and cut-off fs
   Answer: B. A gain of Ts cancels the fs factor and leaves G(f) exactly. (Key point: Recovery filter)

4. Which statement is correct?
   A. Aliasing is caused by fs < 2W; aperture effect by the sinc factor in flat-top sampling
   B. Aliasing is caused by flat-top pulses; aperture effect by low fs
   C. Both are fixed by an equalizer
   D. Both occur only in natural sampling
   Answer: A. Aliasing is spectral overlap fixed by an anti-aliasing LPF; aperture effect is sinc loss fixed by a 1/H(f) equalizer. (Key point: Aliasing and aperture)

## Related in Communication Systems

- [Sampling theorem; aliasing and aperture effect remedies](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asampling-theorem-aliasing-aperture-remedy) — Communication Systems, chapter 4
- [Nyquist-Kotelnikov sampling theorem; aliasing and aperture](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Anyquist-kotelnikov-aliasing-aperture) — Communication Systems, chapter 4
- [Aliasing and aperture effects; types of sampling](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aaliasing-aperture-sampling-types) — Communication Systems, chapter 4
- [Aperture and aliasing; sampling 10cos(20πt)cos(200πt) at 250 samples/s](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aaperture-aliasing-10cos20pit-250) — Communication Systems, chapter 4
- [Sub-sampling theory and aperture effect](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Asubsampling-aperture-effect) — Communication Systems, chapter 4
- [Aperture effect correction; bandpass sampling 80–115 kHz](https://www.bctengineer.com/question.md?key=semester_6%2Fcommunication_systems%3Aaperture-correction-bandpass-80-115) — Communication Systems, chapter 4

## Related in other subjects

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- [Sampling theorem; aliasing cause; Nyquist](https://www.bctengineer.com/question.md?key=semester_4%2Fsignals_and_systems%3Asampling-aliasing-cause-nyquist) — Signals and Systems, chapter 4
- [Sampling theorem, Nyquist criterion and aliasing](https://www.bctengineer.com/question.md?key=semester_4%2Finstrumentation_i%3Asampling-nyquist-aliasing) — Instrumentation I, chapter 6
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