Signal processing: sampled energy, spectra and quantization

Sample rate is Hz and time is seconds. Signals may be complex for DFT, RMS, energy, power and SNR. Real FFT helpers require real data. energy is sum(abs(x)^2) without sample-period weighting; it is a discrete signal measure, not joules. For an approximation to a continuous integral multiply by 1/fs. power is the mean squared amplitude and rms is its square root.

Spectral conventions

DFT is the NumPy unnormalized forward transform; inverse includes 1/N. windowed_spectrum and Spectrum return raw magnitudes, not calibrated one-sided amplitudes. PSD uses SciPy periodogram density scaling, units amplitude^2/Hz. frequency_response returns frequencies in the supplied fs units (default 2*pi means rad/sample). Digital cutoff parameters use Hz with fs in Hz. The BA IIR representation can be numerically fragile for high order; this package does not yet expose second-order-section filtering.

import numpy as np
from engineeringkit.signal_processing import dft, energy
x = np.exp(2j*np.pi*np.arange(16)/16)
assert np.isclose(energy(x), energy(dft(x))/len(x))

Tests check Parseval, PSD integration, Butterworth cutoff gain 1/sqrt(2), FIR DC gain and sinc reconstruction at the sample nodes. Truncated sinc reconstruction is not an exact infinite bandlimited interpolator away from sample nodes.

ADC and SNR

Quantization is uniform mid-rise with 2**bits bins on [vmin,vmax]. Levels are at bin centers, endpoints saturate, and the maximum in-range error is half a step. Bits are restricted to 1..52 so floating-point level indexing is meaningful. SNR uses mean squared magnitudes for complex values and matching signal/noise shapes. Nonzero signal with zero noise returns +infinity; zero signal with nonzero noise returns -infinity; zero/zero is rejected as undefined.