Fourier Transform Basics
Sample two sinusoids plus white noise and read their amplitudes from a windowed FFT: bin spacing, Nyquist aliasing, scalloping and the noise floor.
Nyquist Frequency f_s/2
50 Hz
record length T = N/f_s = 2.56 s
Bin Spacing Δf = f_s/N
0.3906 Hz
ENBW 1 bins (Boxcar)
Tone 1 Read Amplitude
0.7448
true 1 at 10 Hz, bin 25.6
Tone 2 Read Amplitude
0.4977
true 0.5 at 25 Hz, bin 64
Sampled signal
- x(t)
One-sided amplitude spectrum
- Amplitude
Model
X_k = Σ wₙxₙ e^(−2πikn/N), amplitude = 2|X_k|/Σwₙ, f_k = k·f_s/N
noise floor (RMS) = 2σ√(Σwₙ²)/Σwₙ = 0.0125
Two sinusoids plus Gaussian white noise of standard deviation σ (fixed seed) are sampled N times at f_s and transformed with an FFT. Dividing by the window sum corrects the coherent gain, so a tone that falls exactly on a bin reads its true amplitude with any window. A tone between bins leaks into its neighbours and reads low (scalloping, up to 3.9 dB for no window, 1.4 dB for Hanning); tapered windows trade a wider peak for lower leakage far from it. Tones above f_s/2 fold back to |f − f_s·round(f/f_s)|.
Model and references
Exact. The equations are exact within the stated assumptions, or are a standard's tabulated values. The results are as good as the inputs. About the tiers
Discrete Fourier transform of the sampled record with coherent-gain correction; the signal is synthetic (two tones, Gaussian noise with a fixed seed).
- Harris F. J. (1978). On the use of windows for harmonic analysis with the discrete Fourier transform. Proc. IEEE 66, 51–83. doi:10.1109/PROC.1978.10837
- Oppenheim A. V., Schafer R. W. (2010). Discrete-Time Signal Processing, 3rd ed., chs. 4 and 10. Pearson.
- Cooley J. W., Tukey J. W. (1965). An algorithm for the machine calculation of complex Fourier series. Math. Comput. 19, 297–301. doi:10.1090/S0025-5718-1965-0178586-1