Have you seen a car wheel in a movie that seems to spin backwards? That illusion is aliasing — the same error that corrupts a digital signal when it is sampled too slowly. Aliasing, the Nyquist rate and the anti-aliasing filter are among the most frequently asked topics in digital communication and DSP for BCS (EEE/Telecom), BTCL, Teletalk, power-sector and university exams. This post explains them visually with animations and solved numericals.
- Sampling theorem: a band-limited signal with highest frequency fm can be recovered exactly from its samples if fs ≥ 2fm.
- Nyquist rate = 2fm (minimum sampling rate); Nyquist interval = 1/(2fm).
- Aliasing: if fs < 2fm, spectral copies overlap and high frequencies appear as false low frequencies.
- Alias frequency: fa = |f − n·fs| (n = nearest integer to f/fs). Folding frequency = fs/2.
- Prevention: an anti-aliasing low-pass filter before the sampler, and sampling above the Nyquist rate (guard band).
1. Sampling and the Sampling Theorem
Sampling converts a continuous-time signal x(t) into a sequence of values x(nTs) taken every Ts seconds. The sampling frequency is fs = 1/Ts. Sampling is the first step of every analog-to-digital conversion (PCM, digital audio, digital telephony, DSP).
Nyquist–Shannon sampling theorem: a signal that contains no frequency higher than fm Hz is completely described by its samples taken at a rate
fs ≥ 2 fm ⇒ Nyquist rate = 2fm, Nyquist interval Ts = 1/(2fm)The original signal is recovered by passing the samples through an ideal low-pass reconstruction filter with cut-off between fm and fs − fm.
2. What Is Aliasing? (Animation)
Aliasing is the distortion that occurs when a signal is sampled at a rate lower than the Nyquist rate. The samples of a high-frequency signal then look exactly like the samples of a different, lower-frequency signal — its alias. After reconstruction, the receiver outputs that wrong frequency, and the error cannot be removed afterwards.
In the animation the sampling rate is fixed at 10 kHz. While the signal frequency stays below 5 kHz (fs/2), the red “reconstructed” curve matches the blue signal. Once it crosses 5 kHz, the samples (black dots) trace out a slower wave: an 8 kHz tone is mistaken for 2 kHz.
3. Aliasing in the Frequency Domain
Sampling in time creates copies (images) of the original spectrum centred at 0, ±fs, ±2fs, … If fs ≥ 2fm, the copies are separated and a low-pass filter can pick out the original. If fs < 2fm, neighbouring copies overlap; the overlapped (folded) part is aliasing.
4. How to Find the Alias Frequency
falias = | f − n fs |, n = integer that makes falias ≤ fs/2Think of the frequency axis being folded back and forth at fs/2 (called the folding frequency). Any frequency above fs/2 is reflected into the band 0 – fs/2.
| fs = 10 kHz, input f | Calculation | Appears as |
|---|---|---|
| 3 kHz | below 5 kHz | 3 kHz (correct) |
| 6 kHz | |6 − 10| | 4 kHz |
| 8 kHz | |8 − 10| | 2 kHz |
| 10 kHz | |10 − 10| | 0 Hz (DC!) |
| 13 kHz | |13 − 10| | 3 kHz |
| 17 kHz | |17 − 20| | 3 kHz |
5. How to Prevent Aliasing
- Anti-aliasing (pre-alias) filter: an analog low-pass filter placed before the sampler that removes all components above fs/2. Without it, out-of-band noise and interference will fold into the signal band.
- Sample faster than the Nyquist rate (oversampling): practical filters are not brick-wall, so a guard band is kept between fm and fs − fm.
- Telephone speech: band-limited to 3.4 kHz, sampled at 8 kHz (Nyquist rate 6.8 kHz, guard band 1.2 kHz).
- Audio CD: 20 kHz audio sampled at 44.1 kHz (guard band 4.1 kHz).
- Use proper sample-and-hold and fast ADCs so that the actual sampling rate is what the design assumes.
Related effect: practical flat-top sampling causes the aperture effect (high-frequency attenuation), which is different from aliasing — see Aperture Effect vs Aliasing.
6. Real-Life Examples of Aliasing
- Wagon-wheel effect: a film camera samples at 24 frames/s; spokes that rotate nearly one spoke-gap per frame appear to stand still or turn backwards.
- Helicopter blades / fans looking frozen in phone videos.
- Moiré patterns when photographing fine stripes or a TV screen (spatial aliasing).
- Stroboscope: uses aliasing deliberately to “freeze” rotating machines for speed measurement.
7. Solved Problems
Problem 1. x(t) = 3 cos(2000πt) + 5 cos(6000πt) + cos(12000πt). Find the Nyquist rate. If x(t) is sampled at 5 kHz, which frequencies appear at the output?
- Frequencies: 1 kHz, 3 kHz and 6 kHz ⇒ fm = 6 kHz ⇒ Nyquist rate = 12 kHz.
- With fs = 5 kHz (folding at 2.5 kHz): 1 kHz → 1 kHz; 3 kHz → |3 − 5| = 2 kHz; 6 kHz → |6 − 5| = 1 kHz.
- Output contains only 1 kHz and 2 kHz — the 6 kHz component has merged with the genuine 1 kHz tone and can never be separated.
Problem 2. Find the Nyquist rate of (a) x(t) = sinc(200t) and (b) x(t) = sinc²(200t), where sinc(x) = sin(πx)/(πx).
- (a) sinc(200t) has a rectangular spectrum up to 100 Hz ⇒ fm = 100 Hz ⇒ Nyquist rate = 200 Hz.
- (b) Squaring in time = convolution in frequency ⇒ bandwidth doubles to 200 Hz ⇒ Nyquist rate = 400 Hz.
Problem 3. Two signals band-limited to 1 kHz and 2 kHz are multiplied. Find the Nyquist rate of the product.
Multiplication ⇒ bandwidths add: 1 + 2 = 3 kHz ⇒ Nyquist rate = 6 kHz. (If they were added, the bandwidth would be max(1, 2) = 2 kHz ⇒ 4 kHz.)
8. Practice MCQs (click to see answer)
1. Aliasing occurs when the sampling frequency is: (a) Greater than 2fm (b) Equal to 2fm (c) Less than 2fm (d) Infinite
2. Nyquist rate for a signal band-limited to 4 kHz is: (a) 2 kHz (b) 4 kHz (c) 8 kHz (d) 16 kHz
3. An anti-aliasing filter is a: (a) High-pass filter after sampler (b) Low-pass filter before sampler (c) Band-stop filter after DAC (d) All-pass filter
4. A 7 kHz tone is sampled at 10 kHz. It appears as: (a) 7 kHz (b) 3 kHz (c) 17 kHz (d) 10 kHz
5. Telephone speech is sampled at: (a) 4 kHz (b) 6.8 kHz (c) 8 kHz (d) 44.1 kHz
6. Folding frequency is equal to: (a) fs (b) 2fs (c) fs/2 (d) fm/2
7. Nyquist interval for a 5 kHz band-limited signal is: (a) 0.1 ms (b) 0.2 ms (c) 1 ms (d) 10 µs
8. Nyquist rate of sinc²(100t) is: (a) 100 Hz (b) 200 Hz (c) 50 Hz (d) 400 Hz
9. The wagon-wheel effect in movies is an example of: (a) Quantization noise (b) Aliasing (c) Aperture effect (d) Jitter
10. The gap between fm and (fs − fm) when sampling above the Nyquist rate is called: (a) Guard band (b) Dead band (c) Side band (d) Stop band
9. বাংলায় দ্রুত রিভিশন
- স্যাম্পলিং উপপাদ্য: সর্বোচ্চ কম্পাঙ্ক fm হলে নমুনায়ন হার fs ≥ 2fm হতে হবে। 2fm-কে নাইকুইস্ট রেট বলে।
- অ্যালিয়াসিং: fs < 2fm হলে উচ্চ কম্পাঙ্ক ভুলভাবে নিম্ন কম্পাঙ্ক হিসেবে দেখা দেয়; স্পেকট্রামের কপিগুলো একটির উপর আরেকটি পড়ে।
- অ্যালিয়াস কম্পাঙ্ক = |f − n·fs|; ফোল্ডিং কম্পাঙ্ক = fs/2।
- প্রতিরোধ: স্যাম্পলারের আগে অ্যান্টি-অ্যালিয়াসিং লো-পাস ফিল্টার এবং নাইকুইস্ট রেটের বেশি হারে স্যাম্পলিং (গার্ড ব্যান্ড)।
- উদাহরণ: টেলিফোন ৮ kHz, অডিও CD ৪৪.১ kHz; সিনেমায় চাকা উল্টো ঘোরা অ্যালিয়াসিং-এর উদাহরণ।
Related Posts
- Sampling Theorem for Bandpass Signals
- Aperture Effect vs Aliasing
- PCM vs DM vs ADM
- Why Is Modulation Needed?
References: Simon Haykin, Communication Systems; B. P. Lathi, Modern Digital and Analog Communication Systems; Proakis & Manolakis, Digital Signal Processing. Images and animations: original work by Totalecer.