Upsampling
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digital signal processing Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are ...
, upsampling, expansion, and interpolation are terms associated with the process of resampling in a
multi-rate digital signal processing Sample-rate conversion, sampling-frequency conversion or resampling is the process of changing the sampling rate or sampling frequency of a discrete signal to obtain a new discrete representation of the underlying continuous signal. Application ar ...
system. ''Upsampling'' can be synonymous with ''expansion'', or it can describe an entire process of ''expansion'' and filtering (''interpolation''). When upsampling is performed on a sequence of samples of a ''signal'' or other continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a higher rate (or
density Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematical ...
, as in the case of a photograph). For example, if
compact disc The compact disc (CD) is a Digital media, digital optical disc data storage format that was co-developed by Philips and Sony to store and play digital audio recordings. In August 1982, the first compact disc was manufactured. It was then rele ...
audio at 44,100 samples/second is upsampled by a factor of 5/4, the resulting sample-rate is 55,125.


Upsampling by an integer factor

Rate increase by an integer factor ''L'' can be explained as a 2-step process, with an equivalent implementation that is more efficient: #Expansion: Create a sequence, x_L comprising the original samples, x separated by ''L'' − 1 zeros.  A notation for this operation is:  x_L = x . #Interpolation: Smooth out the discontinuities with a
lowpass filter A low-pass filter is a filter that passes signals with a frequency lower than a selected cutoff frequency and attenuates signals with frequencies higher than the cutoff frequency. The exact frequency response of the filter depends on the filter ...
, which replaces the zeros. In this application, the filter is called an interpolation filter, and its design is discussed below. When the interpolation filter is an FIR type, its efficiency can be improved, because the zeros contribute nothing to its
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
calculations. It is an easy matter to omit them from both the data stream and the calculations. The calculation performed by a multirate interpolating FIR filter for each output sample is a dot product: where the ''h'' sequence is the impulse response of the interpolation filter, and ''K'' is the largest value of ''k'' for which ''h'' 'j'' + ''kL''is non-zero. In the case ''L'' = 2, ''h'' can be designed as a
half-band filter In digital signal processing, half-band filters are widely used for their efficiency in multi-rate applications. A half-band filter is a low-pass filter that reduces the maximum bandwidth of sampled data by a factor of 2 (one octave). When multipl ...
, where almost half of the coefficients are zero and need not be included in the dot products. Impulse response coefficients taken at intervals of ''L'' form a subsequence, and there are ''L'' such subsequences (called phases) multiplexed together. Each of ''L'' phases of the impulse response is filtering the same sequential values of the ''x'' data stream and producing one of ''L'' sequential output values. In some multi-processor architectures, these dot products are performed simultaneously, in which case it is called a polyphase filter. For completeness, we now mention that a possible, but unlikely, implementation of each phase is to replace the coefficients of the other phases with zeros in a copy of the ''h'' array, and process the \scriptstyle x_L /math>  sequence at L times faster than the original input rate. Then ''L-1'' of every ''L'' outputs are zero. The desired ''y'' sequence is the sum of the phases, where ''L-1'' terms of the each sum are identically zero.  Computing ''L-1'' zeros between the useful outputs of a phase and adding them to a sum is effectively decimation. It's the same result as not computing them at all. That equivalence is known as the ''second Noble identity''. It is sometimes used in derivations of the polyphase method.


Interpolation filter design

Let X(f) be the
Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...
of any function, x(t), whose samples at some interval, T, equal the x /math> sequence. Then the
discrete-time Fourier transform In mathematics, the discrete-time Fourier transform (DTFT) is a form of Fourier analysis that is applicable to a sequence of values. The DTFT is often used to analyze samples of a continuous function. The term ''discrete-time'' refers to the ...
(DTFT) of the x /math> sequence is the
Fourier series A Fourier series () is a summation of harmonically related sinusoidal functions, also known as components or harmonics. The result of the summation is a periodic function whose functional form is determined by the choices of cycle length (or ''p ...
representation of a
periodic summation In signal processing, any periodic function s_P(t) with period ''P'' can be represented by a summation of an infinite number of instances of an aperiodic function s(t), that are offset by integer multiples of ''P''. This representation is called pe ...
of X(f): When T has units of seconds, f has units of hertz (Hz). Sampling L times faster (at interval T/L) increases the periodicity by a factor of L: which is also the desired result of interpolation. An example of both these distributions is depicted in the first and third graphs of Fig 2. When the additional samples are inserted zeros, they decrease the sample-interval to T/L. Omitting the zero-valued terms of the Fourier series, it can be written as: : \sum_ x(nT/L)\ e^, which is equivalent to regardless of the value of L. What L determines is the DTFT periodicity of a digital filter implemented at the higher data-rate. The second graph depicts a lowpass filter and L=3, resulting in the desired spectral distribution (third graph). The filter's bandwidth is the
Nyquist frequency In signal processing, the Nyquist frequency (or folding frequency), named after Harry Nyquist, is a characteristic of a sampler, which converts a continuous function or signal into a discrete sequence. In units of cycles per second ( Hz), it ...
of the original x /math> sequence.  In units of Hz that value is \tfrac,  but filter design applications usually require normalized units. (see Fig 2, table)


Upsampling by a fractional factor

Let ''L''/''M'' denote the upsampling factor, where ''L'' > ''M''. #Upsample by a factor of ''L'' #
Downsample In digital signal processing, downsampling, compression, and decimation are terms associated with the process of ''resampling'' in a multi-rate digital signal processing system. Both ''downsampling'' and ''decimation'' can be synonymous with ''com ...
by a factor of ''M'' Upsampling requires a lowpass filter after increasing the data rate, and downsampling requires a lowpass filter before decimation. Therefore, both operations can be accomplished by a single filter with the lower of the two cutoff frequencies. For the ''L'' > ''M'' case, the interpolation filter cutoff,  \tfrac ''cycles per intermediate sample'', is the lower frequency.


See also

* Downsampling *
Multi-rate digital signal processing Sample-rate conversion, sampling-frequency conversion or resampling is the process of changing the sampling rate or sampling frequency of a discrete signal to obtain a new discrete representation of the underlying continuous signal. Application ar ...
*
Half-band filter In digital signal processing, half-band filters are widely used for their efficiency in multi-rate applications. A half-band filter is a low-pass filter that reduces the maximum bandwidth of sampled data by a factor of 2 (one octave). When multipl ...
*
Oversampling In signal processing, oversampling is the process of 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 o ...
*
Sampling (information theory) In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples". A sample is a value of the signal at a point in time and/or spac ...
*
Signal (information theory) In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The ''IEEE Transactions on Signal Processing'' ...
*
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 tha ...
*
Interpolation In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points. In engineering and science, one often has a n ...
*
Poisson summation formula In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a ...


Notes


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References


Further reading

* * (discusses a technique for bandlimited interpolation) * {{DSP Digital signal processing Signal processing