Lab 3.1 — FFT Windows and Spectral Leakage

Lab 3.1 — FFT Windows and Spectral Leakage

Goal

Understand how FFT window selection changes spectral leakage, frequency resolution, amplitude interpretation and measurement confidence.

This lab connects directly to SDR measurement work: if the FFT window and frequency axis are wrong, the measured spectrum can look convincing but lead to incorrect engineering conclusions.

Theory

A finite-length FFT observes only a limited time window of the signal. If the signal frequency does not fall exactly on an FFT bin, the spectrum spreads into nearby bins. This is spectral leakage.

Important concepts:

• sampling rate Fs;
• FFT length N;
• bin spacing df = Fs / N;
• coherent vs non-coherent tone frequency;
• rectangular, Hann, Hamming and Blackman windows;
• main-lobe width;
• side-lobe suppression;
• amplitude correction.

Experiment

Generate two complex tones:

1. a coherent tone exactly on an FFT bin;
2. a non-coherent tone between FFT bins.

For each tone, compute spectra using:

• rectangular window;
• Hann window;
• Hamming window;
• Blackman window.

Compare:

• peak bin;
• leakage level;
• apparent amplitude;
• frequency estimate error;
• ability to see a weak nearby tone.

Python implementation

Minimum expected script structure:

import numpy as np
import matplotlib.pyplot as plt

fs = 2.4e6
n = 4096
t = np.arange(n) / fs

f_bin = 250 * fs / n
f_off = (250.35) * fs / n

x = np.exp(1j * 2 * np.pi * f_off * t)

windows = {
    "rectangular": np.ones(n),
    "hann": np.hanning(n),
    "hamming": np.hamming(n),
    "blackman": np.blackman(n),
}

freq = np.fft.fftshift(np.fft.fftfreq(n, d=1/fs))

for name, w in windows.items():
    xw = x * w
    spec = np.fft.fftshift(np.fft.fft(xw))
    mag_db = 20 * np.log10(np.maximum(np.abs(spec) / np.sum(w), 1e-12))
    plt.plot(freq / 1e3, mag_db, label=name)

plt.grid(True)
plt.xlabel("Frequency, kHz")
plt.ylabel("Magnitude, dBFS")
plt.legend()
plt.show()

MATLAB implementation

Minimum expected script structure:

fs = 2.4e6;
N = 4096;
t = (0:N-1).' / fs;

f_bin = 250 * fs / N;
f_off = 250.35 * fs / N;

x = exp(1j * 2*pi*f_off*t);

windows = {
    'rectangular', ones(N, 1);
    'hann', hann(N);
    'hamming', hamming(N);
    'blackman', blackman(N)
};

freq = fftshift((-floor(N/2):ceil(N/2)-1).' * fs / N);

figure; hold on;
for k = 1:size(windows, 1)
    name = windows{k, 1};
    w = windows{k, 2};
    spec = fftshift(fft(x .* w));
    magDb = 20*log10(max(abs(spec) / sum(w), 1e-12));
    plot(freq/1e3, magDb, 'DisplayName', name);
end

grid on;
xlabel('Frequency, kHz');
ylabel('Magnitude, dBFS');
legend('Location', 'best');

C++ / FPGA bridge

This lab is not only about plotting. The same FFT/window discipline appears in FPGA and embedded diagnostics.

Engineering bridge:

Concept	Software view	FPGA / hardware view
Window coefficients	floating-point vector	ROM or coefficient memory
Multiplication by window	vector multiply	DSP slices or fixed-point multiplier
FFT length	array size	latency, memory and resource cost
Leakage	plot artifact	real measurement limitation
Amplitude correction	divide by window sum	fixed gain compensation

Expected plots

Produce at least:

1. coherent tone spectrum with multiple windows;
2. non-coherent tone spectrum with multiple windows;
3. zoomed view around the tone;
4. optional weak-tone detection case.

Report checklist

• [ ] State Fs, N, df and tone frequency.
• [ ] Explain whether the tone is coherent with FFT bins.
• [ ] Compare main-lobe width for each window.
• [ ] Compare side-lobe suppression.
• [ ] Explain which window is best for amplitude measurement.
• [ ] Explain which window is best for detecting a weak nearby tone.
• [ ] Add a short note on FPGA cost of applying a window.

Engineering conclusion template

For this signal and FFT length, the ______ window gives the best leakage suppression,
but it increases the main-lobe width. For SDR measurements, this means that window
choice must be documented together with Fs, FFT length and amplitude normalization.