# Oversampling

In the digital signal processing , the oversampling ( English oversampling [ oʊvɚsæmplɪŋ ]) a special form of sampling . An over-sampling is present when a signal with a higher sampling rate is processed, as for the display of the signal bandwidth is required. The counterpart is subsampling .

Oversampling a signal can have advantages in some signal processing applications. Some of these applications are, for example, digital-to-analog converters (DAC), analog-to-digital converters (ADC) and switched capacitor filters (SC filters).

## Use of oversampling in digital-to-analog conversion

Oversampling with f s to improve the SNR in the yellow frequency range used

Oversampling is used in the digital-to-analog conversion to

• Remove frequencies above half the target sampling frequency,
• to avoid overriding subsequent processing stages due to interference signals ,
• in order to avoid interfering signals through intermodulation of these interfering signals lying outside the transmission band, which represent audible non-harmonic distortions with narrowband signals (e.g. telephones),
• improve the signal-to-noise ratio (SNR) and linearity of digital-to-analog converters. This property is in sigma-delta converters necessary and is also known as noise shaping ( English Noise Shaping ), respectively. The figure on the right shows the noise shaping for various sigma-delta converters with orders 1 to 3. The oversampling, which is represented by the light blue bar up to frequency f s , improves the signal-to-noise ratio in the useful frequency range shown in yellow and shifts the noise into higher frequency components. As a result, these upper frequency ranges with increased noise, since they do not carry any information, can be filtered out by a subsequent low-pass filter and the oversampling by a sampling rate conversion to f 0

The oversampling is done by converting the sampling rate from the given source sampling frequency to the desired target sampling frequency, which is usually freely selectable and can be selected appropriately. Usually, a multiple of the source sampling frequency is extrapolated, the factor usually being a power of two.

Oversampling in the digital sector became known through the oversampling specification in early CD players . The claim was a selling point that later lost its meaning. Technically, the oversampling is inserted into the data stream of digital audio data after the read error correction and before the digital filter, in that the incoming data sets are multiplied by the oversampling factor before they are used by the digital filter for calculation. The digital filter, which is designed to be larger and faster by a factor, generates a number of calculation results that is increased by a factor, which are sent to the adapted digital-analog converter, converted there and output to an inexpensive analog filter. Newer developments combine an upsampling circuit with the oversampling filter, which makes even simpler analog filters possible.

## Use of oversampling in analog-to-digital converters

Oversampling is used in analog-digital conversion to

• Remove frequencies above half the target sampling frequency (partial shifting of the filtering from the analog to the (easier-to-use) digital range)
• to avoid irreparable aliasing errors in the target signal
• To improve the signal-to-noise ratio and the linearity of analog-to-digital converters
• This property is necessary for sigma-delta converters

In practice, a signal that typically has a useful bandwidth of with is sampled with instead of with . The necessary analog anti-aliasing filter then has, instead of a transition area from, the larger transition area from , which is much easier to implement. ${\ displaystyle \ alpha f_ {s}}$${\ displaystyle 0 {,} 3 \ leq \ alpha \ leq 0 {,} 48}$${\ displaystyle n \ cdot f_ {s}}$${\ displaystyle f_ {s}}$${\ displaystyle \ alpha \ cdot f_ {s} \ ldots (1- \ alpha) \ cdot f_ {s}}$${\ displaystyle \ alpha \ cdot f_ {s} \ ldots (n- \ alpha) \ cdot f_ {s}}$

The input audio signal with 1.76 kHz (red line) is digitized to a 1-bit data stream using pulse density modulation (box, blue = 1, white = 0) using the target sampling frequency 176.4 kHz, corresponding to the source sampling frequency 44.1 kHz with 4-fold oversampling.

The oversampling is done by converting the sampling rate from the mostly freely selectable source sampling frequency to the desired target sampling frequency. This is done by low-pass filtering with subsequent decimation of the sampling points.

## practice

In practice, integer frequency ratios, preferably powers of two, are used in oversampling. This reduces the computing effort. With higher-order oversampling , the necessary sampling rate conversion is often carried out in several stages. ${\ displaystyle (k \ geq 8)}$

Higher sampling rates are achieved by removing the sum and difference bands in the frequency range in the case of odd multiples of the sampling frequency. As a result, twice as many samples occur in the time domain, so the sampling rate is doubled. This process is called double oversampling . With quadruple oversampling , the sum and difference bands are removed even with even multiples, except for 4 * n, of the sampling frequency.

According to the condition of the Nyquist-Shannon sampling theorem , the sampling rate must be more than twice the highest signal frequency that occurs in order to allow error-free reconstruction . The theorem assumes ideal antialiasing and reconstruction filters.

In practice, this requires filters that have a high slope and high attenuation (e.g. the filter on a CD player must drop by approx. 100 dB between 20 kHz and 22.05 kHz). With analog technology, filters with such requirements are technically extremely complex and expensive. Oversampling, on the other hand, allows the filtering to be shifted from the analog to the digital range. The filtering is done with a digital filter, at the output only a very simple analog filter is necessary.

Oversampling does not lead to higher data rates and higher memory consumption. This procedure is used when reading out and not when writing data. A side effect is that oversampling improves the signal-to-noise ratio, for example during CD playback . The noise power is evenly distributed over a larger frequency interval by oversampling.

## Note

Antialiasing is often incorrectly referred to as oversampling .

## literature

• Dieter Stotz: Computer-aided audio and video technology . 2nd Edition. Springer Verlag 2011, ISBN 978-3-642-23252-7