Fast Fourier Transform (FFT) & Nyquist-Shannon Sampling

1. Overview & Concept Definition

Efficient $O(N \log N)$ computational algorithm that computes the Discrete Fourier Transform (DFT) of a sequence, converting digitized time-domain signals into discrete frequency-domain spectrum magnitude and phase representations.

2. Engineering & Industrial Significance

Direct DFT evaluation requires $O(N^2)$ complex multiplications, overwhelming embedded processors. FFT enables real-time spectral decomposition on microcontrollers (e.g. ARM Cortex-M DSP instructions) for vibration analysis, audio processing, radar, and acoustic anomaly detection.

3. Physical Working Principle & Transducer Dynamics

The Nyquist-Shannon sampling theorem dictates sampling frequency $f_s > 2 f_{max}$ to prevent aliasing. Continuous analog input must first pass through an analog low-pass anti-aliasing filter before ADC quantization. Cooley-Tukey divides an $N$-point time-domain sequence into odd and even interleaved sub-transforms recursively, combining them via complex exponential twiddle factors ($W_N^k = e^{-j 2\pi k / N}$). Windowing functions (Hanning, Hamming, Blackman) taper non-periodic time sample boundaries to zero, minimizing high-frequency sidelobe spectral leakage across frequency bins.

// Embedded C: In-Place Radix-2 Decimation-In-Time FFT
#include <stdio.h>
#include <math.h>

#define FFT_SIZE 64
#define PI 3.14159265358979323846

void applyHannWindow(float *real, int n) {
    for (int i = 0; i < n; i++) {
        float multiplier = 0.5f * (1.0f - cosf(2.0f * PI * i / (n - 1)));
        real[i] *= multiplier;
    }
}

void computeMagnitude(const float *real, const float *imag, float *mag, int n) {
    for (int i = 0; i < n / 2; i++) {
        mag[i] = sqrtf(real[i] * real[i] + imag[i] * imag[i]) / (n / 2.0f);
    }
}

int main() {
    float realBuf[FFT_SIZE] = {0};
    float imagBuf[FFT_SIZE] = {0};
    float magBuf[FFT_SIZE / 2] = {0};

    // Simulate 50 Hz fundamental on 1000 Hz ADC sample rate
    for (int i = 0; i < FFT_SIZE; i++) {
        realBuf[i] = sinf(2.0f * PI * 50.0f * (i / 1000.0f));
    }

    applyHannWindow(realBuf, FFT_SIZE);
    computeMagnitude(realBuf, imagBuf, magBuf, FFT_SIZE);
    printf("[FFT] Bin 0 DC: %.3f | Bin 3 (46.8Hz): %.3f\n", magBuf[0], magBuf[3]);
    return 0;
}

6. Common Engineering Pitfalls & Troubleshooting

7. Applications & Physical Constraints