0.11 Oversampling

Oversampling

The principle of oversampling is simple: run ADC and DAC (Analog $\leftrightarrow$ digital converters) at a sampling rate $f_e$ well beyond Nyquist, say at $f_e=\beta f_0$ where $f_0>2B$ is a sufficient sampling rate.

We call $\beta >1$ the oversampling rate (OSR); it is usually an integer. For audio CD, $\beta =4$ (data is sampled at $f_e=$ 176'100 samples/sec but stored on the disc at $f_0=$ 44'100 samples/sec).

To understand the benefits of oversampling, one has to recall the typical schema [link] of digital signal processing (DSP). First, in the ADC and DAC parts oversampling reduces the losses ofquality due to working too close at Nyquist rate: sampling and reconstruction occur with a cutoff-frequency $f_e$ far from the Nyquist rate $2B$ allowing for simple filter design or even omission of filtering. Second, as an additional benefit, oversampling reducesnoise created by quantification, sampling or other sources of errors.

Note: down- and upsampling before and after processing (DSP: e.g.: de-noising, detection, enhancement, storage, transmission) allowsfor efficient computation and/or transmission at low rate.

Noise reduction under oversampling

In the context of oversampling, the signal has been sampled (and thereby quantized) at $f_e=\beta f_0$ , with $f_0>2B$ and OSR $\beta >>1$ . Thus, we may decimate the quantized signal by a factor $\beta$ and still keep the full signal quality. Decimation consists of digital low-pass filtering atcutoff frequency $1/\left(2\beta \right)$ (which corresponds to $f_0/2$ ), followed by downsampling.

Low-pass filtering and downsampling will not change the signal since $f_0>2B$ , and thus decimation will not change the signal power (cpre. [link] , [link] , [link] magenta). However, low-pass filtering will reduce the power of the noise ; the factor of reduction is $\beta$ if an ideal filter is used. We offer several arguments for this fact.

Spectral picture of noise reduction (ideal filter). During the first step, low-pass filtering, the high frequencies of the power spectrum of the noise are cut away:

After low-pass filtering, the noise is correctly band-limited (at a loss of power); consequently,downsampling will not change the power any further.

Spectral picture of noise reduction (digital filter). When using a digital filter $b=(b_1,...,b_q)$ with energy $E_b=b_1^2+...+b_q^2$ to filter the samples ${\epsilon }_n$ , the spectrum will change through filtering roughly as

Here, we used that the DFT of a well-designed digital filter $b$ is roughly a rectangle of width $1/\beta$ , meaning that a fraction $1/\beta$ of the samples ${\widehat{b}}_k$ at the center take some constant value $\lambda$ , the others are zero. It is easy to verify that $\lambda =\sqrt{E_b\beta }$ in order for the energy of $b$ to be $E_b$ (see Comment 8 ). Consequently, the portion $1/\beta$ of the $K$ samples ${\widehat{\epsilon }}_n$ will be multiplied by $\lambda$ , the others will be set to zero. Using [link] we see that $S(k/K)$ is multiplied by ${\lambda }^2=E_b\beta$ for $-K/\left(2\beta \right)\le k\le K/\left(2\beta \right)$ , and set to zero otherwise.

Comment 8 Let us denote by $q$ the number of samples of the DFT of $b$ . For a well designed digital filter there are roughly $q/\beta$ samples equal to some constant $\lambda$ , the remaining samples are zero. Since the DFT increases energy by factor equal to the sample size $q$ , we get