Sampling Theory —
Nyquist, Aliasing & Reconstruction
CD Audio (1980s): Audio bandwidth ≈ 20 kHz. Nyquist rate = 40 kHz. CD standard: fs = 44.1 kHz (10% above Nyquist for filter rolloff). 16-bit samples → 1.4 Mbps data rate.
Medical ECG: Heart rate signal bandwidth ≈ 150 Hz. Standard ECG sampling: 250–1000 Hz (well above Nyquist). Anti-aliasing LPF at 75 Hz before sampling to prevent muscle artifact aliasing.
Phone calls (GSM/4G): Voice bandwidth = 300–3400 Hz (telephony). Nyquist rate = 6800 Hz. Sampled at 8 kHz → 64 kbps (PCM voice). Your phone's codec applies a 3.4 kHz anti-aliasing filter before sampling every 125 μs.
Stroboscopic effect (aliasing in video): A fan spinning at 30 Hz recorded at 30 fps appears stationary — it completes one full rotation per frame. This is spatial/temporal aliasing. The same effect makes car wheels appear to spin backward in movies.
Complete ADC→DAC chain: (1) Analog signal → (2) Anti-aliasing LPF removes components above fₘ → (3) Sampler creates x[n]=x(nT) at rate fs≥2fₘ → (4) Digital data → (5) Reconstruction LPF → (6) Perfect analog output x̂(t)=x(t).
1. Why Sampling? CT → DT
Digital systems can only process discrete sequences, not continuous-time waveforms. Sampling converts a CT signal x(t) into a DT sequence x[n] = x(nT) by measuring x(t) at uniform intervals T seconds apart. The key question: How fast must we sample to avoid losing information?
2. Nyquist-Shannon Sampling Theorem
x₁(t) has BW = f₁ Hz, x₂(t) has BW = f₂ Hz:
x₁(t) + x₂(t): Nyquist rate = 2·max(f₁, f₂)
x₁(t) · x₂(t): Bandwidth = f₁ + f₂ → Nyquist rate = 2(f₁+f₂) [multiplication = convolution in freq]
x(t²): time compression by 2 → BW doubles → Nyquist rate doubles
3. Spectrum of Sampled Signal
Sampling in time domain is multiplication by an impulse train. In frequency domain, this causes periodic replication of the spectrum.
Adequate sampling keeps spectral copies separated (top case). Under-sampling causes overlap (aliasing) — the reconstruction LPF cannot separate them, so higher-frequency components appear at wrong frequencies.
4. Aliasing — Cause & Effect
Aliasing occurs when the sampling rate is below the Nyquist rate, causing spectral replicas to overlap. A signal at frequency f appears at an alias frequency |f − kfs| after sampling.
Telephone bandwidth is limited to 3.4 kHz for a reason: The public switched telephone network (PSTN) uses 8 kHz sampling. Nyquist says maximum frequency = fs/2 = 4 kHz. A practical anti-aliasing LPF with rolloff limits to 3.4 kHz (not 4 kHz), leaving 600 Hz guard band for the filter transition.
If a voice signal at 5 kHz were allowed through: alias = |5000 − 8000| = 3000 Hz. The 5 kHz sound would masquerade as 3 kHz — wrong pitch! This is why high-quality phone calls (HD Voice, VoLTE) use 16 kHz sampling and 7 kHz bandwidth — much better voice quality.
5. Ideal Reconstruction
Each sample x[nT] is multiplied by a shifted sinc function and the results are summed. The sinc functions have zeros at all other sample points, so they interpolate perfectly between samples. This is the theoretical basis — practical systems use ZOH (staircase) or polynomial interpolation instead of the ideal sinc.
6. Practical Sampling — ZOH, Natural, Flat-Top
| Sampling Method | Description | Spectrum Effect | Application |
|---|---|---|---|
| Ideal (Impulse) Sampling | Multiply by impulse train. Instantaneous samples. | Periodic replicas, no distortion within replica | Theoretical analysis |
| Natural Sampling | Gate pulses follow signal shape during sampling interval τ. | Replicas with sinc (due to pulse width) envelope: sinc(nτ/T) | Analog multiplexer switches |
| Flat-Top (Sample & Hold) | Sample held constant (flat top) during pulse interval τ. Rectangular pulse shape. | Spectrum distorted by sinc(ωτ/2): high-freq roll-off | ADC input stage (S&H circuit) |
| ZOH (Zero-Order Hold) | Each sample held for full period T (staircase output). | H_ZOH(jω) = T·sinc(ωT/2)·e^(−jωT/2): sinc distortion + linear phase | DAC output stage |
ZOH Frequency Response
7. Sampling Rates in Real Systems
| Application | Signal BW (fₘ) | Nyquist Rate | Actual fs | Reason for higher fs |
|---|---|---|---|---|
| Telephone (PSTN) | 3.4 kHz | 6.8 kHz | 8 kHz | Filter transition band margin |
| CD Audio | 20 kHz | 40 kHz | 44.1 kHz | 10% margin + historical reasons |
| DAT / Studio Audio | 20 kHz | 40 kHz | 48 kHz | Professional standard |
| Hi-Res Audio | 40 kHz | 80 kHz | 96/192 kHz | Higher fidelity, easier anti-aliasing |
| Medical ECG | 150 Hz | 300 Hz | 500–1000 Hz | High oversampling for accuracy |
| Seismic monitoring | 100 Hz | 200 Hz | 500 Hz | Capture all relevant frequencies |
GATE Questions — Sampling Theory (8)
x(t) has components at f₁ = 1000 Hz and f₂ = 2000 Hz. fs = 3000 Hz.
Nyquist rate = 2 × 2000 = 4000 Hz. Since fs = 3000 < 4000 → aliasing!
f₁ = 1000 Hz: |1000| < fs/2 = 1500 → No aliasing for 1000 Hz ✓
f₂ = 2000 Hz: 2000 > fs/2 = 1500 → Alias = |2000 − 3000| = 1000 Hz
Answer: (C) — 2000 Hz component aliases to 1000 Hz
sinc(2Wt) ↔ (1/2W)rect(f/2W): bandwidth of sinc(2Wt) = W Hz.
sinc(100πt) = sinc(2·50·πt): bandwidth = 50 Hz
sinc(200πt) = sinc(2·100·πt): bandwidth = 100 Hz
Multiplication in time → convolution in frequency: BW = 50 + 100 = 150 Hz
Nyquist rate = 2 × 150 = 300 Hz
Answer: (C) 300 Hz
Signal at 5 kHz, sampled at fs = 8 kHz. Nyquist limit = 4 kHz.
5 kHz > 4 kHz → aliasing. Alias = |f − fs| = |5000 − 8000| = 3000 Hz = 3 kHz
Answer: (C) 3 kHz
cos(50πt) has frequency 25 Hz. cos(300πt) has frequency 150 Hz.
Product: x(t) = 10cos(50πt)cos(300πt) = 5[cos(250πt) + cos(350πt)]
Using: 2cosA·cosB = cos(A+B) + cos(A-B):
Maximum frequency = 175 Hz → Nyquist rate = 2 × 175 = 350 Hz
Answer: (C) 350 Hz
After ideal sampling (impulse train), spectrum is (1/T)·X(jω) repeated at multiples of ωs.
Reconstruction filter: select the baseband copy → LPF with cutoff at ωs/2 = πfs (or fs/2 in Hz), gain = T.
The gain T compensates for the 1/T factor in the sampled spectrum.
Answer: (B) LPF with cutoff fs/2 and gain T = 1/fs
Flat-top sampling: each sample is held at constant amplitude for pulse duration τ.
This introduces a sinc(ωτ/2) envelope distortion — high frequencies are attenuated (aperture effect).
It is a LINEAR (amplitude/phase) distortion → can be corrected by an equalizer with response 1/sinc(ωτ/2).
Answer: (C) Aperture effect (linear, correctable)
sinc(100t) = sin(100πt)/(100πt). This is sinc(2W·πt) with 2W = 100 → W = 50 Hz? Let's use the standard: sinc(Wt) = sin(Wt)/(Wt) has FT = (π/W)rect(ω/2W). Bandwidth = W/2π Hz when using ω, or W/(2π) Hz.
sinc(100t) — bandwidth using convention: FT = (1/100)rect(f/100)? Depends on convention. For GATE purposes: sinc²(100t) → bandwidth = 100/π Hz... Let's use the simpler approach.
x(t) = sinc²(100t). In freq domain: X(f) = triangle(f/100) [since sinc² ↔ triangle]. Bandwidth = 100 Hz (max non-zero frequency at ±100)? Actually for sinc²(Wt) where sinc(Wt)=sin(Wt)/(Wt): bandwidth is W/π. But GATE typically uses: sinc(100t) has bandwidth 100/(2π)... This is convention-dependent. Most GATE solutions use BW of sinc²(100t) = 100/π ≈ 31.8 Hz → Nyquist rate ≈ 200/π Hz. Most accepted GATE answer is 200 Hz based on treating the bandwidth as 100 Hz for the function sinc(100πt).
Using the standard GATE convention where sinc(2πfₘt) has BW = fₘ, and sin(100πt)/(100πt) = sinc(100πt) → fₘ = 50 Hz → sinc²(100πt) has BW = 100 Hz → Nyquist rate = 200 Hz.
Answer: (B) 200 Hz
At exactly fs = 2fₘ, spectral replicas just touch at ωs/2 = ωₘ — they neither overlap nor have a gap.
Theoretically, an ideal brick-wall LPF with exactly unity gain for |f|≤fₘ and zero for |f|>fₘ can separate them.
Practically: an ideal brick-wall LPF is non-causal (sinc impulse response extending to −∞). Practical filters have finite transition bands and cannot achieve perfect brick-wall cutoff → reconstruction error at the boundary.
This is why practical systems use fs = 1.1 to 2× the theoretical Nyquist rate.
Answer: (C) Theoretically possible but practically requires an ideal brick-wall filter (non-realizable)