Skip to content

Taking a record to a lower sample rate

A record sampled at a rate fs holds frequencies up to half that rate, the Nyquist frequency, and no higher. Content above it does not vanish when the record is sampled. It lands at a lower frequency, the alias frequency |n fs - F| for whichever whole number n brings it into the band from zero to fs / 2, and once it is in the samples nothing in the record can tell it from real content. A 4,600 Hz tone on a channel sampled at 5,120 Hz lands at 520 Hz, at its full height, and its samples are those of a 520 Hz tone with its sign reversed.

Keeping every q-th sample of a record is sampling it again at a q-th of the rate, and the same rule applies: everything the record holds between the new Nyquist frequency and the old one folds into the new band. So a record is filtered before it is decimated, and visualdynamics.core.resample does the two in that order.

The span and the dropped lines

A real filter cannot go from flat to fully down in no distance, so the band a record is trusted over stops short of the Nyquist frequency. Analyzers and controllers commonly keep the alias-free span, the sample rate divided by 2.56. A frame of 1,024 samples has 512 lines up to the Nyquist frequency; 1,024 / 2.56 = 400 of them lie inside the span and are kept, and the other 112, in the filter's roll-off, are computed and dropped. That division is where the line counts 400, 800 and 1,600 on an analyzer's menu come from.

Frame Lines to Nyquist Kept Dropped Resolution at 5,120 Hz
1,024 512 400 112 5 Hz
2,048 1,024 800 224 2.5 Hz
4,096 2,048 1,600 448 1.25 Hz

alias_free_span(rate), line_counts(frame) and alias_frequency(frequency, rate) give the arithmetic on its own.

Resampling

resample(history, rate), or history.resample(rate), filters every channel through a zero-phase Butterworth low-pass (anti_alias_filter), then keeps every q-th sample. The new rate must divide the record's into a whole factor of at least 2; a rate that does not is refused with the nearest two that do. The filter runs forward and backward, as the Filter reading's does, so nothing moves in time and the edge transients land at the record's ends. The result keeps the records' descriptions, the source's clock at every q-th sample (a record that starts at 3.25 s still does), and the shock windows, which are in seconds from the record's start. It does not carry the averaging, since an averaging counts its frame in samples, which spans another time at the new rate. The channels are filtered and decimated one at a time, so the record filtered at the full rate is never held whole beside it. A channel holding a sample that is not finite is refused, naming the channel and the count: the filter's recursion runs forward and backward, so one such sample would make every sample of its channel so.

The filter's corner is at 1.2 times the span (CORNER), 2,400 Hz at 5,120 Hz, among the dropped lines and below the new Nyquist frequency. A zero-phase pair is 6.02 dB down at its nominal corner, so a corner on the span would take the span's top line 6 dB down and 0.9 of the span 0.65 dB down; at 1.2 spans the span is flat to 0.1 dB. The lowest frequency that can fold into the span is the new rate less the span (3,120 Hz at 5,120 Hz, 1.56 spans), 1.3 corners up. The default design order is 12 (ORDER), which run forward and backward is 55.8 dB down there taking 51,200 Hz to 5,120 Hz, 55.1 dB taking 81,920 Hz to 5,120 Hz, and never less than 54.7 dB at any factor, past 50 dB of attenuation where content would fold. Eleventh order just gets there at large factors, and tenth falls short at about 46 dB.

resample(history, rate, anti_alias=False) keeps every q-th sample with no filter and issues an AliasWarning every time. Use it only for a record known to be band-limited below the new span, because whether it was is not something the record can show.

The order cannot be reversed. A low-pass applied after the decimation finds the alias already inside its pass band, at a frequency where real content could be, and leaves it where it is. The cure has to come before the samples are thrown away.

An example

A record at 81,920 samples a second holds a 0.1 g tone at 300 Hz on a light floor, and a 1 g sensor resonance at 4,600 Hz. Taken to 5,120 Hz:

import numpy as np

from visualdynamics.core.data import TimeHistory
from visualdynamics.core.resample import read_resample
from visualdynamics.plot import plot_resample

rate, g = 81920.0, 9.80665
t = np.arange(16 * 8192) / rate
floor = 0.005 * np.random.default_rng(14).standard_normal(t.size)
y = 0.1 * np.sin(2 * np.pi * 300.0 * t) + np.sin(2 * np.pi * 4600.0 * t)
history = TimeHistory(t, (y + floor)[None] * g, response_dof=['1Z+'],
                      ordinate_dim=['acceleration'], ordinate_unit=['g'])

slow = history.resample(5120.0)          # filtered, then every 16th sample
reading = read_resample(history, 5120.0, frequencies=[4600.0])
print(reading.describe())
plot_resample(history, 5120.0, frequencies=[4600.0], path='resample.png')

read_resample resamples one channel three ways, filtered first, decimated bare, and decimated then low-passed through the same filter, and turns each and the record as recorded into an autospectrum on one line spacing (Hann frames, half overlapped, 1,024 samples at the new rate by default). Each frequency given above the new Nyquist frequency is folded, and the level each spectrum has where it lands is read; with none given, the record's tallest line above the new Nyquist frequency is the one folded.

Decimated bare, the 4,600 Hz resonance stands at 520 Hz at 0.0667 g²/Hz, the height of a 1 g tone on a 5 Hz Hann line, and the whole floor between 2,560 and 40,960 Hz has folded on top of the real one, sixteen times as high. Low-passed at 2,400 Hz after the decimation, the 520 Hz line is unchanged. Filtered first, the resonance is gone: the line at 520 Hz is the floor, 79.8 dB below the alias. The 300 Hz tone comes through all three at its height.

The record's autospectrum at 81,920 Hz, thin, with the resonance at 4,600 Hz marked by a hollow triangle and joined by a dotted line to a filled triangle at 520 Hz on the dashed bare decimation; the dotted curve low-passed after the decimation lies on the dashed one inside the span, and the heavy curve filtered first runs well below both; the span at 2,000 Hz and the Nyquist frequency at 2,560 Hz are marked, and the 112 dropped lines between them are hatched

Reading What it is
source_rate, rate, factor the record's rate, the new one, and how many samples make one
nyquist, span, corner half the new rate, the alias-free span, and the filter's corner, 1.2 spans
lines, kept, dropped, resolution the spectra's lines to the Nyquist frequency, inside the span and above it, and their spacing
gain_at_span, lowest_fold, gain_at_fold the filter's zero-phase gain at the span's top, the lowest frequency that folds into the span, and the gain there
folds each frequency folded: where it lands, whether inside the span or among the dropped lines, and the level at the alias decimated bare, low-passed after, and filtered first, with the last two in dB against the bare
spectra the four autospectra the levels are read from

In the app

Select a time history and choose Resample on the bar. The plot shows the first selected channel's autospectrum as the example draws it, and the panel beside it sets the decimation factor (the new rate is the record's over it), the filter's order, the frame, and the frequencies to fold, the record's tallest line above the new Nyquist frequency when the field is empty. Under them it reads out the rates, the span, the lines kept and dropped, the resolution, and the filter's gain at the span and at the lowest frequency that folds into it, and under those each fold in words, with a note where there is nothing to read: a channel of zeros, or a record with no line above the new Nyquist frequency. A channel of more than 400,000 samples is read with the progress bar up. Every edit redraws at once, the settings are kept when another record is selected, and the reading is journaled as its plot_resample call. The reading shows what a resampling would do; history.resample(rate) makes the record.

Where judgment lives

  • The new rate. Choose it from the highest frequency that matters, at 2.56 times it, not from the specification's top line. A channel whose peak is wanted, a shock's or an SRS's, needs about ten times its filter frequency, since a peak is a single sample.
  • Content the record never held. Resampling with a filter keeps anything above the new span out of the result, but it cannot remove an alias that was already in the samples when the record was made. A narrow peak that may be one is settled by recording again at another rate and seeing whether it moves.
  • anti_alias=False. It is right only for a record already filtered below the new span, and the warning is there because the record cannot say whether it was.