Nyquist Criterion and DFT Sampling

Nyquist Criterion and DFT Sampling: Understanding the Connection

1. Nyquist Sampling Theorem (Continuous-Time Signals)

The Nyquist theorem states that to perfectly reconstruct a continuous-time signal from its samples, the sampling frequency fs must satisfy:

f_s ≥ 2 f_max

Where fmax is the maximum frequency present in the signal.

• Applies to continuous-time (analog) signals
• Ensures no aliasing (no loss of frequency information)
• The minimum fs = 2 f_max is called the Nyquist rate

2. Discrete Fourier Transform (DFT) Sampling

For a discrete signal x[n] of length N, the DFT is defined as:

X[k] = Σ_n=0^N-1 x[n] e^{-j 2π kn / N},   k = 0,1,...,N-1

• Applies to discrete-time signals
• Requires N frequency samples to fully represent a signal of length N
• Assumes the signal was correctly sampled (Nyquist already satisfied)

3. Why Nyquist and DFT are Not Contradictory

Concept	Domain	Purpose
Nyquist	Continuous-time	Ensures correct sampling (no aliasing)
DFT	Discrete-time	Represents finite-length sequence fully

Connection: Nyquist ensures the discrete signal is correct, DFT ensures its frequency representation is complete.

4. Example: Connecting Nyquist + DFT

Step 1: Continuous-time signal

x(t) = cos(2π · 100 t),   f_max = 100 Hz

Step 2: Apply Nyquist criterion

f_s ≥ 2 f_max = 200 Hz

Sampling interval: T_s = 1/f_s = 1/200

Discrete signal: x[n] = x(n T_s) = cos(π n)

Step 3: Finite-length sequence for DFT

N = 4, x[n] = [1, -1, 1, -1]

Step 4: Compute DFT

• X[0] = 1 - 1 + 1 - 1 = 0
• X[1] = 0
• X[2] = 4 (peak frequency)
• X[3] = 0

Step 5: Interpretation of Peak Frequency

The peak frequency (X[2] = 4) indicates the dominant frequency component of the discrete signal. In this context:

• The original signal had a 100 Hz component
• The peak in the DFT shows that frequency is represented accurately after sampling
• Its location in the DFT corresponds to k = 2, which maps to the correct frequency bin

Step 6: If Nyquist is Violated

Sampling at f_s < 2 f_max (e.g., 150 Hz) causes aliasing → DFT shows incorrect frequency.

5. Summary

• Nyquist = capture the signal correctly
• DFT = represent the captured signal fully in frequency domain