Editing Oversampling

{{about|oversampling in signal processing|oversampling in data analysis|Oversampling and undersampling in data analysis}}
{{short description|Sampling higher than the Nyquist rate}}
{{Use American English|date = March 2019}}

In [[signal processing]], '''oversampling''' is the process of [[sampling (signal processing)|sampling]] a signal at a sampling frequency significantly higher than the [[Nyquist rate]]. Theoretically, a bandwidth-limited signal can be perfectly reconstructed if sampled at the Nyquist rate or above it. The Nyquist rate is defined as twice the [[Bandwidth (signal processing)|bandwidth]] of the signal. Oversampling is capable of improving [[Resolution (audio)|resolution]] and [[signal-to-noise ratio]], and can be helpful in avoiding [[aliasing]] and [[phase distortion]] by relaxing [[anti-aliasing filter]] performance requirements.

A signal is said to be oversampled by a factor of ''N'' if it is sampled at ''N'' times the Nyquist rate.

==Motivation==
There are three main reasons for performing oversampling: to improve anti-aliasing performance, to increase resolution and to reduce noise.

===Anti-aliasing===
Oversampling can make it easier to realize analog [[anti-aliasing filter]]s.<ref name=AD-oversample>{{cite web|last1=Kester|first1=Walt|title=Oversampling Interpolating DACs|url=https://www.analog.com/media/en/training-seminars/tutorials/MT-017.pdf|publisher=Analog Devices|access-date=17 January 2015}}</ref> Without oversampling, it is very difficult to implement filters with the sharp cutoff necessary to maximize use of the available bandwidth without exceeding the [[Nyquist limit]]. By increasing the bandwidth of the sampling system, design constraints for the anti-aliasing filter may be relaxed.<ref>{{cite magazine |url=http://www.audioholics.com/education/audio-formats-technology/upsampling-vs-oversampling-for-digital-audio |title=Upsampling vs. Oversampling for Digital Audio |quote=Without increasing the sample rate, we would need to design a very sharp filter that would have to cutoff &#91;sic&#93; at just past 20kHz and be 80-100dB down at 22kHz. Such a filter is not only very difficult and expensive to implement, but may sacrifice some of the audible spectrum in its [[roll-off]]. |author=Nauman Uppal |date=30 August 2004 |access-date=6 October 2012 |magazine=[[Audioholics]]}}</ref> Once sampled, the signal can be [[digital filter|digitally filtered]] and [[downsampled]] to the desired sampling frequency. In modern [[integrated circuit]] technology, the digital filter associated with this downsampling is easier to implement than a comparable [[analog filter]] required by a non-oversampled system.

===Resolution===
In practice, oversampling is implemented in order to reduce cost and improve performance of an [[analog-to-digital converter]] (ADC) or [[digital-to-analog converter]] (DAC).<ref name="AD-oversample" /> When oversampling by a factor of N, the [[dynamic range]] also increases a factor of N because there are N times as many possible values for the sum. However, the signal-to-noise ratio (SNR) increases by <math alt="Square root of N">\sqrt{N}</math>, because summing up uncorrelated noise increases its amplitude by <math alt="Square root of N">\sqrt{N}</math>, while summing up a coherent signal increases its average by N. As a result, the SNR increases by <math alt="Square root of N">\sqrt{N}</math>. 

For instance, to implement a 24-bit converter, it is sufficient to use a 20-bit converter that can run at 256 times the target sampling rate. Combining 256 consecutive 20-bit samples can increase the SNR by a factor of 16, effectively adding 4 bits to the resolution and producing a single sample with 24-bit resolution.<ref name=sillabs>{{cite web|title=Improving ADC Resolution by Oversampling and Averaging |url=https://www.silabs.com/Support%20Documents/TechnicalDocs/an118.pdf |publisher=Silicon Laboratories Inc |access-date=17 January 2015}}</ref>{{efn|While with N{{=}}256 there is an increase in dynamic range by 8 bits, and the level of coherent signal increases by a factor of N, the noise changes by a factor of <math alt{{=}}"Square root of N">\sqrt{N}</math>{{=}}16, so the net SNR improves by a factor of 16, 4 bits or 24&nbsp;dB.}}

The number of samples required to get <math>n</math> bits of additional data precision is

:<math>\mbox{number of samples} = (2^n)^2 = 2^{2n}.</math>

To get the mean sample scaled up to an integer with <math>n</math> additional bits, the sum of <math>2^{2n}</math> samples is divided by <math>2^n</math>:

:<math>\mbox{scaled mean} = \frac{ \sum\limits^{2^{2n}-1}_{i=0} 2^n \text{data}_i}{2^{2n}} = \frac{\sum\limits^{2^{2n}-1}_{i=0} \text{data}_i}{2^n}.</math>

This averaging is only effective if the [[signal]] contains sufficient [[uncorrelated noise]] to be recorded by the ADC.<ref name="sillabs" /> If not, in the case of a stationary input signal, all <math>2^n</math> samples would have the same value and the resulting average would be identical to this value; so in this case, oversampling would have made no improvement.  In similar cases where the ADC records no noise and the input signal is changing over time, oversampling improves the result, but to an inconsistent and unpredictable extent.

Adding some [[dither]]ing noise to the input signal can actually improve the final result because the dither noise allows oversampling to work to improve resolution.  In many practical applications, a small increase in noise is well worth a substantial increase in measurement resolution. In practice, the dithering noise can often be placed outside the frequency range of interest to the measurement, so that this noise can be subsequently filtered out in the digital domain—resulting in a final measurement, in the frequency range of interest, with both higher resolution and lower noise.<ref>{{cite book |last1=Holman |first1=Tomlinson |title=Sound for Film and Television |date=2012 |publisher=CRC Press |isbn=9781136046100 |pages=52–53 |url=https://books.google.com/books?id=gazpAwAAQBAJ&pg=PA52 |access-date=4 February 2019}}</ref>

===Noise===
If multiple samples are taken of the same quantity with uncorrelated noise{{efn|A system's signal-to-noise ratio cannot necessarily be increased by simple oversampling since noise samples are partially correlated (only some portion of the noise due to sampling and analog-to-digital conversion will be uncorrelated).}} added to each sample, then because, as discussed above, uncorrelated signals combine more weakly than correlated ones, averaging ''N'' samples reduces the [[noise power]] by a factor of ''N''. If, for example, we oversample by a factor of 4, the signal-to-noise ratio in terms of power improves by factor of four which corresponds to a factor of two improvement in terms of voltage.

Certain kinds of ADCs known as [[delta-sigma converter]]s produce disproportionately more [[Quantization (signal processing)|quantization]] noise at higher frequencies. By running these converters at some multiple of the target sampling rate, and [[low-pass filter]]ing the oversampled signal down to half the target sampling rate, a final result with ''less'' noise (over the entire band of the converter) can be obtained. Delta-sigma converters use a technique called [[noise shaping]] to move the quantization noise to the higher frequencies.

==Example==
Consider a signal with a bandwidth or highest frequency of ''B''&nbsp;= 100&nbsp;[[Hz]]. The [[sampling theorem]] states that sampling frequency would have to be greater than 200&nbsp;Hz. Sampling at four times that rate requires a sampling frequency of 800&nbsp;Hz. This gives the anti-aliasing filter a [[transition band]] of 300&nbsp;Hz ((''f''<sub>s</sub>/2)&nbsp;− ''B''&nbsp;= (800&nbsp;Hz/2)&nbsp;− 100&nbsp;Hz&nbsp;= 300&nbsp;Hz) instead of 0&nbsp;Hz if the sampling frequency was 200&nbsp;Hz. Achieving an anti-aliasing filter with 0&nbsp;Hz transition band is unrealistic whereas an anti-aliasing filter with a transition band of 300&nbsp;Hz is not difficult.

==Reconstruction==
The term oversampling is also used to denote a process used in the reconstruction phase of digital-to-analog conversion, in which an intermediate high sampling rate is used between the digital input and the analog output. Here, digital interpolation is used to add additional samples between recorded samples, thereby converting the data to a higher sample rate, a form of [[upsampling]]. When the resulting higher-rate samples are converted to analog, a less complex and less expensive analog [[reconstruction filter]] is required. Essentially, this is a way to shift some of the complexity of reconstruction from analog to the digital domain. Oversampling in the ADC can achieve some of the same benefits as using a higher sample rate at the DAC.

== See also ==
*[[Oversampled binary image sensor]]
*[[Supersampling]]
*[[Undersampling]]

==Notes==
{{Notelist}}

==References==
{{Reflist}}

==Further reading==
*{{cite book |author=John Watkinson |title=The Art of Digital Audio |year=1994 |publisher=Focal Press |isbn=0-240-51320-7}}

{{DSP}}

[[Category:Digital signal processing]]
[[Category:Information theory]]