GATE Electrical Engineering - Complete Master Library

 

📘 Sampling Theorem – Nyquist Rate & Aliasing

Sampling converts a continuous-time signal into discrete-time signal. To reconstruct original signal without distortion, proper sampling rate is required.

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🔹 1. Sampling Process

Continuous signal:

x(t)

Sampled signal:

x_s(t) = x(t) Σ δ(t − nT)

Where T = Sampling period Sampling frequency:

f_s = 1/T

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🔹 2. Nyquist Sampling Theorem

If a signal is band-limited to:

|f| ≤ f_m

Then sampling frequency must satisfy:

f_s ≥ 2 f_m

Minimum rate: Nyquist Rate = 2 f_m

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🔹 3. Aliasing

If:

f_s < 2 f_m

Then spectral overlap occurs → distortion. This phenomenon is called:

Aliasing

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🔹 4. Frequency Domain View

Sampling in time → Replication in frequency. Spectrum repeats every:

ω_s = 2π f_s

If repetitions overlap → Aliasing.

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🔹 5. Reconstruction

Original signal recovered using: Low Pass Filter (LPF) Cutoff frequency:

f_m

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🔹 6. Example 1

Given: Signal bandwidth = 5 kHz Find minimum sampling frequency. Solution:

f_s ≥ 2 × 5 f_s ≥ 10 kHz

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🔹 7. Example 2

If sampling frequency = 8 kHz Signal bandwidth = 5 kHz Since:

8 < 10

Aliasing occurs.

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🔹 8. Practical Points

• Always use anti-aliasing filter before sampling
• Sampling theorem applies to band-limited signals
• Nyquist rate is minimum safe rate

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🎯 GATE Important Points

• Nyquist rate = 2 f_m
• Aliasing when sampling frequency too low
• Frequency replication concept important
• Direct numerical questions common

Proper Sampling Prevents Information Loss