Delta sigma modulator harmonics appearing with a sine wave input

Question

I am simulating a first order delta sigma modulator (DSM) used as an ADC. The input to my DSM is a full scale sine wave. The power spectral density I expect from the DSM output is as follows : The sinusoid peak as well as some noise that is shaped to the higher frequencies.

In order to avoid any spectral leakage, I define my sinusoid from the FFT bin it will occupy as well as the number of samples for the FFT (this is also suggested for simulations in the book "Understanding delta sigma data converters" by R. Schreier and G. C. Temes).

bin = 15
Nfft = 8192
t = np.arange(0, Nfft)
signal = np.sin(2*np.pi*bin/Nfft*t)

Then I pass it through the DSM, for which the code is as follows :

def mod1(signal):

    Nfft = signal.size

    y = np.zeros(Nfft)
    x = np.zeros(Nfft)
    v = np.zeros(Nfft)
    for i in range(1, Nfft):
        v[i] = np.sign(y[i-1])
        x[i] = signal[i] - v[i]
        y[i] = x[i-1] + y[i-1]

    return v

Here is my issue : When I choose Nfft as a multiple of the frequency bin, I will see harmonics of sine frequency appearing. These do not appear when Nfft is not multiple of the frequency bin. If these harmonics should really appear and are not some artifact, as they are placed at k*bin, they should also fall exactly into a bin and avoid spectral leakage regardless of the number of fft samples.

Here is a plot illustrating this issue. On the top left, a sine wave and its DSM modulation. On the bottom left, their PSD when the number of samples is not a multiple of the frequency bin. On this side, the noise shaping can clearly be seen.

The same plots can be found on the right when the number of samples is a multiple of the frequency bin. On this side, the harmonics can be seen.

So which PSD is more correct and why do these harmonic appear only in the particular case of the number of FFT samples being a multiple of the fundamental frequency bin ?

Here is the full code to perform the plots :

import numpy as np 
import matplotlib.pyplot as plt
import scipy.signal as sg 


def generate_sine_from_bin(bin, Nfft): 
    t = np.arange(0, Nfft)
    signal = np.sin(2*np.pi*bin/Nfft*t)

    return signal

bin1 = 15
Nfft1 = 8192

bin2 = 15
Nfft2 = 9000

signal1 = generate_sine_from_bin(bin1, Nfft1)
signal2 = generate_sine_from_bin(bin2, Nfft2)

def mod1(signal):

    Nfft = signal.size

    y = np.zeros(Nfft)
    x = np.zeros(Nfft)
    v = np.zeros(Nfft)
    for i in range(1, Nfft):
        v[i] = np.sign(y[i-1])
        x[i] = signal[i] - v[i]
        y[i] = x[i-1] + y[i-1]

    return v

v1 = mod1(signal1)
v2 = mod1(signal2)


ws1 = sg.windows.hann(signal1.size, sym = True)
ws2 = sg.windows.hann(signal2.size, sym = True)

def compute_psd(signal, w):
    w1 = np.linalg.norm(w, 1)
    V =np.abs(np.fft.fft(signal*w)/(w1/2))**2
    return V


Ps1 = compute_psd(signal1, ws1)
Ps2 = compute_psd(signal2, ws2)
P1 = compute_psd(v1, ws1)
P2 = compute_psd(v2, ws2)

fs1 = np.linspace(0, 1, Nfft1)
f1 = np.linspace(0, 1, Nfft1)
fs2 = np.linspace(0, 1, Nfft2)
f2 = np.linspace(0, 1, Nfft2)

fig, ax = plt.subplots(2, 2, figsize = (10, 5))

ax[0, 0].set_title(f"sine wave bin={bin1} Nfft={Nfft1}")
ax[0, 0].plot(signal1)
ax[0, 0].plot(v1)
ax[0, 0].legend(["signal", "dsm"])
ax[0, 0].set_xlabel("Samples")


ax[1, 0].set_title("DSM PSD (Hann window)")
ax[1, 0].semilogx(fs1, 10*np.log10(Ps1))
ax[1, 0].semilogx(f1, 10*np.log10(P1))
ax[1, 0].legend(["signal", "dsm"])
ax[1, 0].set_xlabel("Normalized freq")
ax[1, 0].set_ylim([-100, 50])


ax[0, 1].set_title(f"sine wave bin={bin2} Nfft={Nfft2}")
ax[0, 1].plot(signal2)
ax[0, 1].plot(v2)
ax[0, 1].legend(["signal", "dsm"])
ax[0, 1].set_xlabel("Samples")

ax[1, 1].set_title("DSM PSD (Hann window)")
ax[1, 1].semilogx(fs2, 10*np.log10(Ps2))
ax[1, 1].semilogx(f2, 10*np.log10(P2))
ax[1, 1].legend(["signal", "dsm"])
ax[1, 1].set_xlabel("Normalized freq")
ax[1, 1].set_ylim([-100, 50])

plt.tight_layout()
plt.show()

Thank you for your help !

Answer 1

The harmonics are evidence of actual repetition of patterns in the signal and don't have to do with the length of the FFT, but the frequency of the test tone. Sigma Delta modulators, and in particular a first-order Sigma Delta, are prone to such "pattern noise", which is very dependent on the input waveform. To see how the length does not change the spurs, try changing the length of the FFT for the same signal2 as generated (truncate it to a shorter length); this will increase the width of the tones shown due to spectral leakage but will have little effect on the actual frequency and magnitude for peak of the harmonics (assuming the FFT used is long enough to have sufficient frequency resolution).

I demonstrate this below by first repeating the OP's spectrum for signal2 in the upper plot and then creating in the lower plot the spectrum for signal2 using the same FFT length as was done for signal1.

Any first order sigma delta converter is prone to higher level spurs due to "pattern noise" which is dependent on the frequency of the test tone. In general, any spur in a spectrum plot is indicative of repeating patterns in the waveform. This can be reduced through use of dithering and using a higher order sigma-delta converter which helps to further randomize the generated patterns with longer time duration before repeating.

As an aside, I wouldn't recommend using the hann window for spectral plots due to this sensitivity to spectral leakage as demonstrated here. I recommend instead using the Kaiser window where $\beta$ can be adjusted to balance lobe width with dynamic range. The same spectrum plots were repeated below using a Kaiser window with $\beta=14$. We see they each have a wider lobe width than the OP's plots, but the results are more consistent regardless of test tone being centered on FFT bins or not. Given a general application would typically not involve bin centered tones, this should be more desirable in most cases.