Oversampling — aic2026

OVERSAMPLING

OSR

Bits

Sample ÷

Use the script's adc() instead

The filter is oversample(): a running sum of OSR samples, i.e. a boxcar, i.e. sinc. It is the cheapest low-pass there is. B bits means 2^B levels; the script's adc() counts fractional bits and gives 2^B+1+1, one bit more than it says.

SIGNAL

f_1 = f_c/2^

Amplitude

Noise

Record N = 2^

VIEW

Time window

Hann window before the FFT

Coherent sampling (integer t)

WHAT YOU ARE LOOKING AT

Quantization noise power is set by the LSB alone: Δ²/12, whatever the sample rate. Spread that fixed power over a wider band and there is less of it in any given slice of spectrum. That is the whole idea of oversampling: run the converter faster than you need, then throw the part of the band you do not need away.

Doubling the sample rate halves the noise density, so filtering back down to the original band buys 3 dB, or half a bit. The SNR-vs-OSR panel plots that directly: measured in-band SNR against OSR, with the ideal 3 dB per octave line for comparison. Ten bits of ADC plus an OSR of 4 gets you about the same SNR as eleven bits at Nyquist.

The filter here is the crudest possible: oversample() adds up OSR consecutive samples. That is a boxcar in time, so a sinc in frequency, with zeros at every multiple of f_s/OSR — count the notches in the filtered spectrum. It is a bad low-pass filter, and it still works, which tells you how much slack oversampling gives you.

THINGS WORTH TRYING

Sweep OSR from 1 to 8 and watch the in-band SNR readout climb 3 dB each time it doubles, tracking the purple line. The zeros of the sinc response march down with it.
Drop f_1 to f_c/2¹⁰ first. Now the tone stays inside the band all the way to OSR = 64 and the SNR curve has room to run: about 18 dB, or three bits, for six doublings. The curve stops where the tone would leave the band.
Take bits down to 1 and raise the OSR. A one-bit converter with enough oversampling is still a converter — badly behaved, because the error is correlated with the signal rather than white, but the trend is there. Fixing that correlation is what noise shaping is for; see the sigma-delta example.
Look at where the tone sits relative to the first sinc null. Move f_1 up until it approaches the band edge and watch the boxcar start attenuating the signal itself. A real decimator uses something flatter in the passband than one sinc.

THE PYTHON

#- Oversample
OSR = 4

def oversample(x,OSR):
    N = len(x)
    y = np.zeros(N)
    for n in range(0,N):
        for k in range(0,OSR):
            m = n+k
            if(m < N):
                y[n] += x[m]
    return y

y_on = oversample(y_sn,OSR)

Source: ex/osr.py. The running sum is not decimated afterwards, so the plots stay on the same frequency axis and the filter shape is easy to see.