THD calculation seems off by a lot

[ftab]

I'm having trouble getting a good THD / THD+N calculation from either THD() or THDN().

If you have a moment, could you take a look at this sample code/data and tell me why?

I'm trying to get a distortion figure on a 1khz sine wave on a device where an HP 8920b told me the THD was about 0.056%

However I am fresh out of 8920s to share and need to concoct as good a result as I can using...an 8 bit USB scope and a Raspberry Pi with Python!

Output:

Frequency: 1000.729392 Hz
fundamental amplitude: 300737.188
36118.519 117.290 24.098 128.721 52.875 156.427 66.371 25.883 140.883 73.748 129.345 87.715 131.846 
THD: 12.387467%
0.9079618095454111

import matplotlib.pyplot as plt
import numpy as np
from waveform_analysis import thd

t = np.arange(0.0, 0.005, 1/1000000)
s = [1, 1, 2, 3, 3, 4, 4, 6, 7, 7, 9, 9, 10, 11, 12, 11, 12, 13, 14, 15, 16, 16, 17, 18, 18, 19, 20, 20, 22, 23, 24, 24, 25, 25, 25, 27, 28, 29, 29, 29, 30, 32, 31, 33, 33, 34, 35, 35, 36, 37, 38, 40, 39, 40, 41, 41, 42, 43, 43, 44, 45, 45, 46, 46, 48, 49, 49, 49, 50, 51, 52, 52, 53, 54, 55, 56, 56, 56, 57, 57, 59, 58, 58, 60, 61, 62, 62, 63, 64, 65, 65, 65, 66, 67, 68, 69, 68, 69, 71, 71, 72, 71, 72, 73, 73, 74, 74, 75, 77, 77, 78, 78, 79, 80, 80, 80, 80, 81, 82, 83, 83, 82, 83, 84, 86, 85, 85, 86, 87, 88, 88, 89, 89, 90, 91, 91, 91, 92, 94, 94, 93, 92, 94, 96, 94, 95, 95, 96, 97, 97, 98, 97, 99, 100, 100, 100, 101, 102, 103, 102, 101, 102, 103, 104, 104, 103, 104, 105, 106, 105, 105, 105, 106, 108, 107, 106, 108, 109, 110, 109, 110, 110, 111, 112, 111, 110, 111, 111, 112, 111, 112, 112, 112, 113, 112, 112, 113, 114, 115, 114, 114, 116, 116, 117, 116, 116, 117, 116, 117, 116, 116, 117, 117, 118, 115, 116, 118, 118, 118, 117, 117, 119, 120, 120, 119, 119, 119, 120, 120, 118, 119, 120, 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-30, -30, -30, -28, -28, -27, -26, -26, -24, -24, -24, -23, -22, -20, -20, -20, -19, -17, -17, -17, -15, -15, -14, -14, -13, -13, -12, -11, -10, -10, -10, -8, -8, -6, -5, -5, -4, -2, -2, -2, 0]
thd.THD(s, 1000000)
thdn = thd.THDN(s, 1000000)
print(thdn)
fig, ax = plt.subplots()
ax.plot(t, s)
ax.set(xlabel='time (s)', ylabel='counts', title='Channel 1')
ax.grid()
plt.show()

[endolith]

The problem is likely in these hardcoded lines:

    lowermin = int(true_i * 0.9)
    uppermin = int(true_i * 1.1)

They are not eliminating the entire fundamental when used with the flattop window:

Changing them to

    lowermin = int(true_i * 0.3)
    uppermin = int(true_i * 2.6)

removes the fundamental entirely, and then it produces:

THD function output:
----
Frequency: 1000.729392 Hz
fundamental amplitude: 300737.188
36118.519 117.290 24.098 128.721 52.875 156.427 66.371 25.883 140.883 73.748 129.345 87.715 131.846 
THD: 12.387467%
----
THD+N: 0.0094%

The THD function isn't right at all, as mentioned in the other issue.

import matplotlib.pyplot as plt
import numpy as np
from waveform_analysis import thd

t = np.arange(0.0, 0.005, 1/1000000)
s = [1, 1, 2, 3, 3, 4, 4, 6, 7, 7, 9, 9, 10, 11, 12, 11, 12, 13, 14, 15, 16, 16, 17, 18, 18, 19, 20, 20, 22, 23, 24, 24, 25, 25, 25, 27, 28, 29, 29, 29, 30, 32, 31, 33, 33, 34, 35, 35, 36, 37, 38, 40, 39, 40, 41, 41, 42, 43, 43, 44, 45, 45, 46, 46, 48, 49, 49, 49, 50, 51, 52, 52, 53, 54, 55, 56, 56, 56, 57, 57, 59, 58, 58, 60, 61, 62, 62, 63, 64, 65, 65, 65, 66, 67, 68, 69, 68, 69, 71, 71, 72, 71, 72, 73, 73, 74, 74, 75, 77, 77, 78, 78, 79, 80, 80, 80, 80, 81, 82, 83, 83, 82, 83, 84, 86, 85, 85, 86, 87, 88, 88, 89, 89, 90, 91, 91, 91, 92, 94, 94, 93, 92, 94, 96, 94, 95, 95, 96, 97, 97, 98, 97, 99, 100, 100, 100, 101, 102, 103, 102, 101, 102, 103, 104, 104, 103, 104, 105, 106, 105, 105, 105, 106, 108, 107, 106, 108, 109, 110, 109, 110, 110, 111, 112, 111, 110, 111, 111, 112, 111, 112, 112, 112, 113, 112, 112, 113, 114, 115, 114, 114, 116, 116, 117, 116, 116, 117, 116, 117, 116, 116, 117, 117, 118, 115, 116, 118, 118, 118, 117, 117, 119, 120, 120, 119, 119, 119, 120, 120, 118, 119, 120, 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print('THD function output:\n----')
thd.THD(s, 1000000)
print('----')
thdn = thd.THDN(s, 1000000)
print(f'THD+N: {thdn:.4f}%')
fig, ax = plt.subplots()
ax.plot(t, s)
ax.set(xlabel='time (s)', ylabel='counts', title='Channel 1')
ax.grid()
plt.show()

[endolith]

REW's method, for reference:

If the Use AES17-2015 standard notch option is selected in the Distortion settings the fundamental figure will be the power within a one octave span around the fundamental frequency. If that option is not selected it will be the power in the main lobe of the fundamental.

AES17-1998 says:

4.3.6 All total harmonic distortion plus noise (THD + N) type distortion analyzers used for measurements in this standard shall utilize a notch filter having an electrical Q of at least 1 and not more than 5.

8.5 Total harmonic distortion and noise (THD + N) Harmonic distortion and noise is the ratio of the output noise and distortion level to the output signal level. Both levels shall include all harmonic, inharmonic and noise components.

(Implying that AES spec is THDR.)

[endolith]

This is your actual spectrum, which doesn't have very clearly defined harmonics (green) and has noise at higher frequencies with a peak around -48 dB:

-48 dB should be 0.41% THD, so I'm confused.

[ftab]

We got some power supply switching noise at the higher frequencies that I was trying to clear out of the data with a 20khz LPF before running it through THDN, but this scope only gives 8 bits of resolution and the LPF can't seem to clean it up 😞

I was previously using a scope with 16 bit resolution (an old Picoscope), and I was able to clear the switching noise out from the waveform, and then THDN was giving reasonable-sounding results... but the age of the scope and Picoscope's lack of arm64 support made it painful to try to use and we decided to switch.

We're now using a VDS-1022 which has a library that is much more Python-friendly, but the data is noisier as a result of the resolution.

What's the drawback to changing those two lines?

    lowermin = int(true_i * 0.3)
    uppermin = int(true_i * 2.6)

If I plug those in so that the THDN measurement seems better, would I then mistake some slightly distorted waveforms as good?

edit:

from scipy.signal import butter, lfilter

def butter_lowpass_filter(data, cutoff, fs, order=5):
    b, a = butter(order, cutoff, fs=fs, btype='low', analog=False)
    y = lfilter(b, a, data)
    return y

...

# after s = [1, 1, 2...
s = butter_lowpass_filter(s, 20000, 1000000, 2)

[endolith]

What's the drawback to changing those two lines?

There isn't one; they shouldn't be hardcoded like that in the first place. It should ideally use a width that eliminates the main lobe regardless of signal length, window function, FFT size, etc.

If I plug those in so that the THDN measurement seems better, would I then mistake some slightly distorted waveforms as good?

Actually... the window function main lobe overlaps the 2nd and 3rd harmonics in the graph above, doesn't it? So it isn't able to measure meaningfully in this case, regardless of whether you change those lines.

I think it needs a longer recording with more cycles? A different window function with narrower main lobe would also help, as it retains more of the signal in the middle of the window.

So ideally it would notice when the window bandwidth is too wide like this and produce a useful error message? 🤔

[ftab]

I should probably try to play around with my capture settings a bit, then. I can only get 5000 samples at a time from this scope. At 1mhz sample rate, that's only .005 s...and then the low pass filter screws up the beginning and end, so I have to chop off the first .0005 and last .0005 to end up with 4000 usable samples (4 periods of sine wave) to try to pass to THDN()

It has a range of sample rates from 2.5/s to 100,000,000/s, I'll see if I can get some useful data at lower sample rates.

Here's one I took at 500khz sample rate (now including lpf and fft), from which I can get 9 sine wave periods:

import matplotlib.pyplot as plt
import numpy as np
from waveform_analysis import thd

from scipy.signal import butter, lfilter

def butter_lowpass_filter(data, cutoff, fs, order=5):
    b, a = butter(order, cutoff, fs=fs, btype='low', analog=False)
    y = lfilter(b, a, data)
    return y

# Borrowing FFT plot function from https://stackoverflow.com/a/53925342
def fftPlot(sig, dt=None, plot=True):
    # Here it's assumes analytic signal (real signal...) - so only half of the axis is required

    if dt is None:
        dt = 1
        t = np.arange(0, sig.shape[-1])
        xLabel = 'samples'
    else:
        t = np.arange(0, sig.shape[-1]) * dt
        xLabel = 'freq [Hz]'

    if sig.shape[0] % 2 != 0:
        warnings.warn("signal preferred to be even in size, autoFixing it...")
        t = t[0:-1]
        sig = sig[0:-1]

    sigFFT = np.fft.fft(sig) / t.shape[0]  # Divided by size t for coherent magnitude

    freq = np.fft.fftfreq(t.shape[0], d=dt)

    # Plot analytic signal - right half of frequence axis needed only...
    firstNegInd = np.argmax(freq < 0)
    freqAxisPos = freq[0:firstNegInd]
    sigFFTPos = 2 * sigFFT[0:firstNegInd]  # *2 because of magnitude of analytic signal

    if plot:
        plt.figure()
        plt.plot(freqAxisPos, np.abs(sigFFTPos))
        plt.xlabel(xLabel)
        plt.ylabel('mag')
        plt.title('Analytic FFT plot')
        plt.show()

    return sigFFTPos, freqAxisPos

t = np.arange(0.0, 0.009, 1/500000)
# Actual data capture from scope at 500khz
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# Low pass filter to get rid of power supply switching noise; remove half of first period and half of last period which the lpf glitches out at
s = butter_lowpass_filter(s, 20000, 500000, 2)[250:-250]

#print('THD function output:\n----')
#thd.THD(s, 500000)
print('----')
thdn = thd.THDN(s, 500000) * 100.0
print(f'THD+N: {thdn:.4f}%')
fig, ax = plt.subplots()
ax.plot(t, s)
ax.set(xlabel='time (s)', ylabel='counts', title='Channel 1')
ax.grid()
plt.show()

fftPlot(s, 1/500000)

I don't know what I was doing wrong with the LPF initially that it didn't seem to do anything...this looks like a much cleaner signal

How many periods of the sine wave should I aim for?

Do I need to catch it on the rising edge of a 0-crossing, or will any window work as long as it's an integer number of sine periods?

edit: note: to get percentage from THDN(), I multiplied by 100, stop me if I was wrong about that one

edit edit:

So ideally it would notice when the window bandwidth is too wide like this and produce a useful error message? 🤔

Yes, that would be fantastic

possibly final edit: python snippet with the raw data after LPF

s = [-1.09977277e+02, -1.10507087e+02, -1.10971565e+02, -1.11392150e+02, -1.11812933e+02, -1.12278165e+02, -1.12776409e+02, -1.13284526e+02, -1.13772410e+02, -1.14220043e+02, -1.14642262e+02, -1.15024106e+02, -1.15343888e+02, -1.15603083e+02, -1.15832680e+02, -1.16069341e+02, -1.16311665e+02, -1.16558370e+02, -1.16808321e+02, -1.17073908e+02, -1.17362982e+02, -1.17626351e+02, -1.17817553e+02, -1.17934597e+02, -1.18020128e+02, -1.18092360e+02, -1.18138051e+02, -1.18176059e+02, -1.18220008e+02, -1.18292493e+02, -1.18394463e+02, -1.18471573e+02, -1.18500420e+02, -1.18480538e+02, -1.18414207e+02, -1.18318863e+02, -1.18194920e+02, -1.18017478e+02, -1.17798661e+02, -1.17589256e+02, -1.17397637e+02, -1.17228724e+02, -1.17058034e+02, -1.16841783e+02, -1.16591793e+02, -1.16304798e+02, -1.15967124e+02, -1.15585255e+02, -1.15192815e+02, -1.14813938e+02, -1.14438094e+02, -1.14084459e+02, -1.13751930e+02, -1.13425243e+02, -1.13120109e+02, -1.12806309e+02, -1.12436408e+02, -1.12006873e+02, 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[endolith]

My plots are in dB:

… 
freqs = rfftfreq(len(windowed)*5, 1/fs)
spec = 20*log10(abs(f))
plt.figure()
plt.semilogx(freqs, spec)
plt.axvline(frequency, color='red')
for h in range(2, 10):
    plt.axvline(frequency * h, color='green')
peak = max(spec)
residual = max(spec[20*5:])
print(f'Highest peak: {peak:.2f} dB')
print(f'Highest residual: {residual:.2f} dB')
plt.axhline(residual, color='gray')
print(f'Difference: {peak - residual:.2f} dB')
# Which is 0.4076149 %%

[endolith]

The THD function should be correct now, too, though note that THD uses THDF while THD+N uses THDR. In the future I would like both functions to support both as an option.