Editing Discrete-time Fourier transform (section)

== References ==
{{reflist|1|refs=
<ref name=Proakis>
{{cite book |last1=Proakis |first1=John G. |last2=Manolakis |first2=Dimitri G. |title=Digital Signal Processing: Principles, Algorithms and Applications |place=New Jersey |publisher=Prentice-Hall International |year=1996 |edition=3 |language=en |id=sAcfAQAAIAAJ |isbn=9780133942897 |bibcode=1996dspp.book.....P |url-access=registration |url=https://archive.org/details/digitalsignalpro00proa
}}</ref>

<ref name=Rabiner>
{{cite book |last1=Rabiner |first1=Lawrence R. |author1-link=Lawrence Rabiner |last2=Gold |first2=Bernard |date=1975 |title=Theory and application of digital signal processing |page=59 (2.163) |location=Englewood Cliffs, NJ |publisher=Prentice-Hall, Inc. |isbn=978-0139141010 |url-access=registration |url=https://archive.org/details/theoryapplicatio00rabi
}}</ref>

<ref name=Oppenheim>
{{Cite book |last1=Oppenheim |first1=Alan V. |author-link=Alan V. Oppenheim |last2=Schafer |first2=Ronald W. |author2-link=Ronald W. Schafer |last3=Buck |first3=John R. |title=Discrete-time signal processing |year=1999 |publisher=Prentice Hall |location=Upper Saddle River, N.J. |isbn=0-13-754920-2 |edition=2nd |chapter=4.2, 8.4 |url-access=registration |url=https://archive.org/details/discretetimesign00alan |quote=samples of the Fourier transform of an aperiodic sequence x[n] can be thought of as DFS coefficients of a periodic sequence obtained through summing periodic replicas of x[n].
}}&nbsp;
</ref>

<ref name=Crochiere>
{{cite book |last1=Crochiere |first1=R.E. |last2=Rabiner |first2=L.R. |title=Multirate Digital Signal Processing |year=1983 |chapter=7.2 |pages=313–326 |publisher=Prentice-Hall |location=Englewood Cliffs, NJ |isbn=0136051626 |url=https://kupdf.net/download/multirate-digital-signal-processing-crochiere-rabiner_58a7065b6454a7e80bb1e993_pdf
}}</ref>

<ref name=Rao>
{{cite book |title=Signals and Systems |author=Rao, R. |isbn=9788120338593 |url=https://books.google.com/books?id=4z3BrI717sMC |publisher=Prentice-Hall Of India Pvt. Limited |year=2008
}}</ref>

<ref name=Matlab>
{{cite web|url=https://www.mathworks.com/help/signal/ref/periodogram.html |title=Periodogram power spectral density estimate - MATLAB periodogram
}}</ref>

<ref name=Gumas>
{{cite journal|last=Gumas |first=Charles Constantine |date=July 1997 |title=Window-presum FFT achieves high-dynamic range, resolution |journal=Personal Engineering & Instrumentation News |pages=58–64 |url=http://www.chipcenter.com/dsp/DSP000315F1.html |url-status=bot: unknown |archive-url=https://web.archive.org/web/20010210052902/http://www.chipcenter.com/dsp/DSP000315F1.html |archive-date=2001-02-10
}}</ref>

<ref name=Lillington>
{{cite web |last=Lillington |first=John |title=Comparison of Wideband Channelisation Architectures 
|url=http://pdfs.semanticscholar.org/f3af/7acf516685ad04b3ebe19eb79db4dc3feb39.pdf |archive-url=https://web.archive.org/web/20190308132844/http://pdfs.semanticscholar.org/f3af/7acf516685ad04b3ebe19eb79db4dc3feb39.pdf |url-status=dead |archive-date=2019-03-08 |page=4 (fig 7) |access-date=2020-09-06
|publisher=International Signal Processing Conference |date=March 2003 |location=Dallas
|s2cid=31525301 |quote=The "Weight Overlap and Add" or WOLA or its subset the "Polyphase DFT", is becoming more established and is certainly very efficient where large, high quality filter banks are required.
}}</ref>

<ref name=Lillington2>
{{cite web |last1=Lillington |first1=John |title=A Review of Filter Bank Techniques - RF and Digital |website=armms.org |publisher=Libra Design Associates Ltd |access-date=2020-09-06 |location=Isle of Wight, UK |page=11 |url=https://www.armms.org/media/uploads/16_lillington_a-review-of-filter-bank-techniques--.pdf |quote=Fortunately, there is a much more elegant solution, as shown in Figure 20 below, known as the Polyphase or WOLA (Weight, Overlap and Add) FFT.
}}</ref>

<ref name=Chennamangalam>
{{cite web |last=Chennamangalam| first=Jayanth |title=The Polyphase Filter Bank Technique |publisher=CASPER Group |date=2016-10-18 |url=https://casper.berkeley.edu/wiki/The_Polyphase_Filter_Bank_Technique |access-date=2016-10-30
}}</ref>

<ref name=Lyons>
{{cite web |last=Lyons |first=Richard G. |title=DSP Tricks: Building a practical spectrum analyzer |publisher=EE Times |date=June 2008 |url=https://www.embedded.com/dsp-tricks-building-a-practical-spectrum-analyzer |access-date=2024-09-19
}} &nbsp; Note however, that it contains a link labeled ''weighted overlap-add structure'' which incorrectly goes to [[Overlap-add method]].
</ref>

<ref name=Dahl>
{{cite thesis |last=Dahl |first=Jason F. |date=2003-02-06 |title=Time Aliasing Methods of Spectrum Estimation |type=Ph.D. |publisher=Brigham Young University |url=http://scholarsarchive.byu.edu/cgi/viewcontent.cgi?article=1049&context=etd |access-date=2016-10-31
}}</ref>

<ref name=Wang>
{{cite book |first1=Hong |last1=Wang |first2=Youxin |last2=Lu |first3=Xuegang |last3=Wang |title=2006 CIE International Conference on Radar |chapter=Channelized Receiver with WOLA Filterbank |pages=1–3 |date=16 October 2006 |doi=10.1109/ICR.2006.343463 |publisher=IEEE |location=Shanghai, China |isbn=0-7803-9582-4
|s2cid=42688070 }}</ref>

<ref name=Lin>
{{cite journal |last1=Lin |first1=Yuan-Pei |last2=Vaidyanathan |first2=P.P. |title=A Kaiser Window Approach for the Design of Prototype Filters of Cosine Modulated Filterbanks |journal=IEEE Signal Processing Letters |volume=5 |issue=6 |pages=132–134 |date=June 1998 |url=http://authors.library.caltech.edu/6891/1/LINieeespl98.pdf |access-date=2017-03-16|doi=10.1109/97.681427 |bibcode=1998ISPL....5..132L
|s2cid=18159105 }}</ref>

<ref name=Harrisbook>
{{cite book |last=Harris |first=Frederic J. |title=Multirate Signal Processing for Communication Systems |publisher=Prentice Hall PTR |location=Upper Saddle River, NJ |date=2004-05-24 |chapter=9 |pages=226–253 |isbn=0131465112
}}</ref>

<ref name=Harris>
{{cite journal |doi=10.1109/PROC.1978.10837 |last=Harris |first=Fredric J. |title=On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform |journal=Proceedings of the IEEE |volume=66 |issue=1 |pages=51–83 |date=Jan 1978 |url=http://web.mit.edu/xiphmont/Public/windows.pdf |citeseerx=10.1.1.649.9880|bibcode=1978IEEEP..66...51H
|s2cid=426548 }}</ref>

<ref name="Hochgürtel">
{{citation |last1=Hochgürtel |first1=Stefan |title=Efficient implementations of high-resolution wideband FFT-spectrometers and their application to an APEX Galactic Center line survey |url=https://bonndoc.ulb.uni-bonn.de/xmlui/bitstream/handle/20.500.11811/5701/3251.pdf
 |publisher=Rhenish Friedrich Wilhelms University of Bonn |access-date=2024-09-19 |location=Bonn |chapter=2.5 |pages=26–31 |date=2013 |bibcode=2013PhDT.......427H |quote=To perform M-fold WOLA for an N-point DFT, M·N real input samples a<sub>j</sub> first multiplied by a window function w<sub>j</sub> of same size
}}</ref>

<ref name=Prandoni>
{{cite book |last1=Prandoni |first1=Paolo |last2=Vetterli |first2=Martin |title=Signal Processing for Communications |date=2008 |publisher=CRC Press |location=Boca Raton, FL |isbn=978-1-4200-7046-0 |edition=1 |url=https://www.sp4comm.org/docs/sp4comm.pdf |access-date=4 October 2020 |pages=72, 76 |quote=the DFS coefficients for the periodized signal are a discrete set of values for its DTFT
}}</ref>

<!--
<ref name=JOS>
{{cite web |last=Smith |first=Julius O. 
| title=`Introduction to Digital Filters with Audio Applications 
| publisher=Center for Computer Research in Music and Acoustics (CCRMA), Stanford University 
| date = 2007-09-01
| url=https://ccrma.stanford.edu/~jos/filters06/Units.html
| website = ccrma.stanford.edu
| access-date = 2024-10-01
| quote = It can be helpful to think ``seconds per sample'' and ``samples per second,'' where ``samples'' is a dimensionless quantity (pure number) included for clarity.
}}</ref>
-->
}}
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