Indigital signal processing,downsampling,subsampling,compression, anddecimation are terms associated with the process ofresampling in amulti-rate digital signal processing system. Bothdownsampling anddecimation can be synonymous withcompression, or they can describe an entire process of bandwidth reduction (filtering) and sample-rate reduction.^[1][2] When the process is performed on a sequence of samples of asignal or a continuous function, it produces an approximation of the sequence that would have been obtained by sampling the signal at a lower rate (ordensity, as in the case of a photograph).

Decimation is a term that historically means theremoval of every tenth one.^[a] But in signal processing,decimation by a factor of 10 actually meanskeeping only every tenth sample. This factor multiplies the sampling interval or, equivalently, divides the sampling rate. For example, ifcompact disc audio at 44,100 samples/second isdecimated by a factor of 5/4, the resulting sample rate is 35,280. A system component that performs decimation is called adecimator. Decimation by an integer factor is also calledcompression.^[3][4]

Rate reduction by an integer factorM can be explained as a two-step process, with an equivalent implementation that is more efficient:^[5]

1. Reduce high-frequency signal components with a digitallowpass filter.
2. Decimate the filtered signal byM; that is, keep only everyM^th sample.

Step 2 alone creates undesirablealiasing (i.e. high-frequency signal components will copy into the lower frequency band and be mistaken for lower frequencies). Step 1, when necessary, suppresses aliasing to an acceptable level. In this application, the filter is called ananti-aliasing filter, and its design is discussed below. Also seeundersampling for information about decimatingbandpass functions and signals.

When the anti-aliasing filter is anIIR design, it relies on feedback from output to input, prior to the second step. WithFIR filtering, it is an easy matter to compute only everyM^th output. The calculation performed by a decimating FIR filter for then^th output sample is adot product:^[b]

${\displaystyle y[n]=\sum _{k=0}^{K-1}x[nM-k]\cdot h[k],}$

where theh[•] sequence is the impulse response, andK is its length. x[•] represents the input sequence being downsampled. In a general purpose processor, after computingy[n], the easiest way to computey[n+1] is to advance the starting index in thex[•] array byM, and recompute the dot product. In the caseM=2,h[•] can be designed as ahalf-band filter, where almost half of the coefficients are zero and need not be included in the dot products.

Impulse response coefficients taken at intervals ofM form a subsequence, and there areM such subsequences (phases) multiplexed together. The dot product is the sum of the dot products of each subsequence with the corresponding samples of thex[•] sequence. Furthermore, because of downsampling byM, the stream ofx[•] samples involved in any one of theM dot products is never involved in the other dot products. ThusM low-order FIR filters are each filtering one ofM multiplexedphases of the input stream, and theM outputs are being summed. This viewpoint offers a different implementation that might be advantageous in a multi-processor architecture. In other words, the input stream is demultiplexed and sent through a bank of M filters whose outputs are summed. When implemented that way, it is called apolyphase 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 theh[•] array, process the originalx[•] sequence at the input rate (which means multiplying by zeros), and decimate the output by a factor ofM. The equivalence of this inefficient method and the implementation described above is known as thefirst Noble identity.^[6][c] It is sometimes used in derivations of the polyphase method.

LetX(f) be theFourier transform of any function,x(t), whose samples at some interval,T, equal thex[n] sequence. Then thediscrete-time Fourier transform (DTFT) is aFourier series representation of aperiodic summation ofX(f):^[d]

${\displaystyle \underset{\text{DTFT}}{\underbrace{\sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}\stackrel{x[n]}{\overbrace{x(nT)}}{\mathrm{e}}^{-\mathrm{i}2\pi fnT}}}=\frac{1}{T}\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}X(f-\frac{k}{T}).}$

WhenT has units of seconds,${\displaystyle f}$ has units ofhertz. ReplacingT withMT in the formulas above gives the DTFT of the decimated sequence,x[nM]:

${\displaystyle \sum _{n=-\mathrm{\infty }}^{\mathrm{\infty }}x(n\cdot MT){\mathrm{e}}^{-\mathrm{i}2\pi fn(MT)}=\frac{1}{MT}\sum _{k=-\mathrm{\infty }}^{\mathrm{\infty }}X\left(f-\frac{k}{MT}\right).}$

The periodic summation has been reduced in amplitude and periodicity by a factor ofM.  An example of both these distributions is depicted in the two traces of Fig 1.^[e][f][g]Aliasing occurs when adjacent copies ofX(f) overlap. The purpose of the anti-aliasing filter is to ensure that the reduced periodicity does not create overlap. The condition that ensures the copies ofX(f) do not overlap each other is: so that is the maximumcutoff frequency of anideal anti-aliasing filter.^[A]

LetM/L denote the decimation factor,^[B] where:M, L ∈${\displaystyle \mathbb{Z}}$; M > L.

1. Increase (resample) the sequence by a factor ofL. This is calledUpsampling, orinterpolation.
2. Decimate by a factor ofM

Step 1 requires a lowpass filter after increasing (expanding) the data rate, and step 2 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 theM > L case, the anti-aliasing filter cutoff, ${\displaystyle \frac{0.5}{M}}$cycles per intermediate sample, is the lower frequency.

• Upsampling
• Posterization
• Sample-rate conversion
• Aliasing
• Visvalingam–Whyatt algorithm

• Proakis, John G. (2000).Digital Signal Processing: Principles, Algorithms and Applications (3rd ed.). India: Prentice-Hall.ISBN 8120311299.
• Lyons, Richard (2001).Understanding Digital Signal Processing. Prentice Hall. p. 304.ISBN 0-201-63467-8.Decreasing the sampling rate is known as decimation.
• Antoniou, Andreas (2006).Digital Signal Processing. McGraw-Hill. p. 830.ISBN 0-07-145424-1.Decimators can be used to reduce the sampling frequency, whereas interpolators can be used to increase it.
• Milic, Ljiljana (2009).Multirate Filtering for Digital Signal Processing. New York: Hershey. p. 35.ISBN 978-1-60566-178-0.Sampling rate conversion systems are used to change the sampling rate of a signal. The process of sampling rate decrease is called decimation, and the process of sampling rate increase is called interpolation.
• T. Schilcher. RF applications in digital signal processing//" Digital signal processing". Proceedings, CERN Accelerator School, Sigtuna, Sweden, May 31-June 9, 2007. - Geneva, Switzerland: CERN (2008). - P. 258. - DOI: 10.5170/CERN-2008-003.[1]
• Sliusar I.I., Slyusar V.I., Voloshko S.V., Smolyar V.G. Next Generation Optical Access based on N-OFDM with decimation.// Third International Scientific-Practical Conference "Problems of Infocommunications. Science and Technology (PIC S&T'2016)". – Kharkiv. - October 3 –6, 2016.[2]
• Saska Lindfors, Aarno Pärssinen, Kari A. I. Halonen. A 3-V 230-MHz CMOS Decimation Subsampler.// IEEE transactions on circuits and systems— Vol. 52, No. 2, February 2005. – P. 110.

• Digital signal processing
• Signal processing

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