Upsampling
Encyclopedia
Upsampling is the process of increasing the sampling rate
Sample rate conversion
Sample rate conversion is the process of converting a signal from one sampling rate to another, while changing the information carried by the signal as little as possible...

 of a signal. For instance, upsampling raster images such as photographs means increasing the resolution of the image.

The upsampling factor (commonly denoted by L) is usually an integer or a rational fraction greater than unity. This factor multiplies the sampling rate
Sampling rate
The sampling rate, sample rate, or sampling frequency defines the number of samples per unit of time taken from a continuous signal to make a discrete signal. For time-domain signals, the unit for sampling rate is hertz , sometimes noted as Sa/s...

 or, equivalently, divides the sampling period. For example, if compact disc
Compact Disc
The Compact Disc is an optical disc used to store digital data. It was originally developed to store and playback sound recordings exclusively, but later expanded to encompass data storage , write-once audio and data storage , rewritable media , Video Compact Discs , Super Video Compact Discs ,...

 audio is upsampled by a factor of 5/4 then the resulting sampling rate goes from 44,100 Hz
Hertz
The hertz is the SI unit of frequency defined as the number of cycles per second of a periodic phenomenon. One of its most common uses is the description of the sine wave, particularly those used in radio and audio applications....

 to 55,125 Hz.

Filtering

For an aesthetically pleasing upsample, an interpolation
Interpolation
In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....

 filter is required; in both upsampling and downsampling
Downsampling
In signal processing, downsampling is the process of reducing the sampling rate of a signal. This is usually done to reduce the data rate or the size of the data....

, such a low-pass filter
Low-pass filter
A low-pass filter is an electronic filter that passes low-frequency signals but attenuates signals with frequencies higher than the cutoff frequency. The actual amount of attenuation for each frequency varies from filter to filter. It is sometimes called a high-cut filter, or treble cut filter...

 implements anti-aliasing
Anti-aliasing
In digital signal processing, spatial anti-aliasing is the technique of minimizing the distortion artifacts known as aliasing when representing a high-resolution image at a lower resolution...

.

Upsampling process

Consider a discrete signal
Discrete signal
A discrete signal or discrete-time signal is a time series consisting of a sequence of qualities...

  on a radian frequency digital frequency
Digital frequency
Digital frequency is the analogue for discrete signals as frequency is to continuous signals.Since a discrete signal is a sequence it contains no direct information as to determine the frequency of the corresponding continuous signal.Just like in frequency, a digital frequency can have values in...

 range.

Upsampling by integer factor

Let L denote the upsampling factor.
  1. Add L-1 zeros between each sample in . Or, equivalently define
  2. Filter with a low-pass filter which, theoretically, should be the sinc filter
    Sinc filter
    In signal processing, a sinc filter is an idealized filter that removes all frequency components above a given bandwidth, leaves the low frequencies alone, and has linear phase...

     with frequency cut off at


The second step calls for the use of a perfect low-pass filter, which is not implementable.
When choosing a realizable low-pass filter this will have to be considered and it will have aliasing
Aliasing
In signal processing and related disciplines, aliasing refers to an effect that causes different signals to become indistinguishable when sampled...

 effects. These aliases can be removed to a reasonable extent by a finite impulse response low pass filter. The presence of zeros in the sequence which is passed through the filter can be used to reduce the complexity of the filter implementation. The original filter can be split to subfilters and the output of each of these subfilters is sequentially tapped to obtain the filtered output sequence.

Upsampling by rational fraction

Let L/M denote the upsampling factor.
  1. Upsample by a factor of L
  2. Downsample
    Downsampling
    In signal processing, downsampling is the process of reducing the sampling rate of a signal. This is usually done to reduce the data rate or the size of the data....

     by a factor of M


Note that upsampling requires an interpolation filter after increasing the data rate and that downsampling requires a filter before decimation.
These two filters can be combined into a single filter.
Since both interpolation and anti-aliasing filters are low-pass filters, the filter with the smallest bandwidth is more restrictive and, thus, can be used in place of both filters.
Since the rational fraction L/M is greater than unity when , the single low-pass filter should have cutoff at cycles per intermediate sample, the Nyquist frequency
Nyquist frequency
The Nyquist frequency, named after the Swedish-American engineer Harry Nyquist or the Nyquist–Shannon sampling theorem, is half the sampling frequency of a discrete signal processing system...

 of the input sample rate.

See also

  • Downsampling
    Downsampling
    In signal processing, downsampling is the process of reducing the sampling rate of a signal. This is usually done to reduce the data rate or the size of the data....

  • Oversampling
    Oversampling
    In signal processing, oversampling is the process of sampling a signal with a sampling frequency significantly higher than twice the bandwidth or highest frequency of the signal being sampled...

  • Sampling (information theory)
  • Signal (information theory)
  • Data conversion
    Data conversion
    Data conversion is the conversion of computer data from one format to another. Throughout a computer environment, data is encoded in a variety of ways. For example, computer hardware is built on the basis of certain standards, which requires that data contains, for example, parity bit checks....

  • Interpolation
    Interpolation
    In the mathematical field of numerical analysis, interpolation is a method of constructing new data points within the range of a discrete set of known data points....

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