Goertzel algorithm - Revision history

188.255.182.246: /* External links */

2025-05-12T12:24:38Z

External links

← Previous revision		Revision as of 12:24, 12 May 2025
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	* [http://www.embedded.com/design/configurable-systems/4006427/A-DSP-algorithm-for-frequency-analysis A DSP algorithm for frequency analysis]		* [http://www.embedded.com/design/configurable-systems/4006427/A-DSP-algorithm-for-frequency-analysis A DSP algorithm for frequency analysis]
	* [https://www.embedded.com/the-goertzel-algorithm The Goertzel Algorithm by Kevin Banks]		* [https://www.embedded.com/the-goertzel-algorithm The Goertzel Algorithm by Kevin Banks]
	* [https://www.beis.de/Elektronik/Filter/Goertzel/Goertzel_en.html Analysis of the Goertzel Algorithm by Uwe Beis] in which he compares it to analog 2nd order Chebyshev ~~lowpass~~ filter		* [https://www.beis.de/Elektronik/Filter/Goertzel/Goertzel_en.html Analysis of the Goertzel Algorithm by Uwe Beis] in which he compares it to analog 2nd order [[Chebyshev_filter\| Chebyshev low pass filter]]

	[[Category:FFT algorithms]]		[[Category:FFT algorithms]]

188.255.182.246: /* External links */

2025-05-12T12:22:51Z

External links

← Previous revision		Revision as of 12:22, 12 May 2025
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	* [http://www.embedded.com/design/configurable-systems/4006427/A-DSP-algorithm-for-frequency-analysis A DSP algorithm for frequency analysis]		* [http://www.embedded.com/design/configurable-systems/4006427/A-DSP-algorithm-for-frequency-analysis A DSP algorithm for frequency analysis]
	* [https://www.embedded.com/the-goertzel-algorithm The Goertzel Algorithm by Kevin Banks]		* [https://www.embedded.com/the-goertzel-algorithm The Goertzel Algorithm by Kevin Banks]
	* [https://www.beis.de/Elektronik/Filter/Goertzel/Goertzel_en.html Analysis of the Goertzel Algorithm by Uwe Beis] in which he compares it to analog 2nd order ~~Chebycheff~~ lowpass filter		* [https://www.beis.de/Elektronik/Filter/Goertzel/Goertzel_en.html Analysis of the Goertzel Algorithm by Uwe Beis] in which he compares it to analog 2nd order Chebyshev lowpass filter

	[[Category:FFT algorithms]]		[[Category:FFT algorithms]]

188.255.182.246: /* External links */

2025-05-12T12:21:21Z

External links

← Previous revision		Revision as of 12:21, 12 May 2025
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	* [http://www.embedded.com/design/configurable-systems/4006427/A-DSP-algorithm-for-frequency-analysis A DSP algorithm for frequency analysis]		* [http://www.embedded.com/design/configurable-systems/4006427/A-DSP-algorithm-for-frequency-analysis A DSP algorithm for frequency analysis]
	* [https://www.embedded.com/the-goertzel-algorithm The Goertzel Algorithm by Kevin Banks]		* [https://www.embedded.com/the-goertzel-algorithm The Goertzel Algorithm by Kevin Banks]
	* [https://www.beis.de/Elektronik/Filter/Goertzel/Goertzel_en.html Analysis of the Goertzel Algorithm by Uwe Beis] in which he compares ~~Goertzel's algorithm~~ to analog 2nd order Chebycheff lowpass filter		* [https://www.beis.de/Elektronik/Filter/Goertzel/Goertzel_en.html Analysis of the Goertzel Algorithm by Uwe Beis] in which he compares it to analog 2nd order Chebycheff lowpass filter

	[[Category:FFT algorithms]]		[[Category:FFT algorithms]]

188.255.182.246: /* External links */

2025-05-12T12:20:17Z

External links

← Previous revision		Revision as of 12:20, 12 May 2025
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	* [http://www.embedded.com/design/configurable-systems/4006427/A-DSP-algorithm-for-frequency-analysis A DSP algorithm for frequency analysis]		* [http://www.embedded.com/design/configurable-systems/4006427/A-DSP-algorithm-for-frequency-analysis A DSP algorithm for frequency analysis]
	* [https://www.embedded.com/the-goertzel-algorithm The Goertzel Algorithm by Kevin Banks]		* [https://www.embedded.com/the-goertzel-algorithm The Goertzel Algorithm by Kevin Banks]
	* [https://www.beis.de/Elektronik/Filter/Goertzel/Goertzel_en.html Analysis of the Goertzel ~~algorithm~~ by Uwe Beis] in which he compares Goertzel's algorithm to analog 2nd order Chebycheff lowpass filter		* [https://www.beis.de/Elektronik/Filter/Goertzel/Goertzel_en.html Analysis of the Goertzel Algorithm by Uwe Beis] in which he compares Goertzel's algorithm to analog 2nd order Chebycheff lowpass filter

	[[Category:FFT algorithms]]		[[Category:FFT algorithms]]

188.255.182.246: /* External links */

2025-05-12T12:19:53Z

External links

← Previous revision		Revision as of 12:19, 12 May 2025
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	* [http://www.embedded.com/design/configurable-systems/4006427/A-DSP-algorithm-for-frequency-analysis A DSP algorithm for frequency analysis]		* [http://www.embedded.com/design/configurable-systems/4006427/A-DSP-algorithm-for-frequency-analysis A DSP algorithm for frequency analysis]
	* [https://www.embedded.com/the-goertzel-algorithm The Goertzel Algorithm by Kevin Banks]		* [https://www.embedded.com/the-goertzel-algorithm The Goertzel Algorithm by Kevin Banks]
			* [https://www.beis.de/Elektronik/Filter/Goertzel/Goertzel_en.html Analysis of the Goertzel algorithm by Uwe Beis] in which he compares Goertzel's algorithm to analog 2nd order Chebycheff lowpass filter

	[[Category:FFT algorithms]]		[[Category:FFT algorithms]]

ThinkEmi at 10:32, 5 November 2024

2024-11-05T10:32:45Z

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	The particular filtering structure chosen for the Goertzel algorithm is the key to its efficient DFT calculations. We can observe that only one output value <math>y[N]</math> is used for calculating the DFT, so calculations for all the other output terms are omitted. Since the FIR filter is not calculated, the IIR stage calculations <math>s[0], s[1]</math>, etc. can be discarded immediately after updating the first stage's internal state.		The particular filtering structure chosen for the Goertzel algorithm is the key to its efficient DFT calculations. We can observe that only one output value <math>y[N]</math> is used for calculating the DFT, so calculations for all the other output terms are omitted. Since the FIR filter is not calculated, the IIR stage calculations <math>s[0], s[1]</math>, etc. can be discarded immediately after updating the first stage's internal state.

	This seems to leave a paradox: to complete the algorithm, the FIR filter stage must be evaluated once using the final two outputs from the IIR filter stage, while for computational efficiency the IIR filter iteration discards its output values. This is where the properties of the direct-form filter structure are applied. The two internal state variables of the IIR filter provide the last two values of the IIR filter output, which are the terms required to evaluate the FIR filter stage.		This seems to leave a paradox: to complete the algorithm, the FIR filter stage must be evaluated once using the final two outputs from the IIR filter stage, while for computational efficiency the IIR filter iteration discards its output values. This is where the properties of the direct-form filter structure are applied. The two internal state variables of the IIR filter provide the last two values of the IIR filter output, which are the terms required to evaluate the FIR filter stage.

	== Applications ==		== Applications ==

EldritchMoth: Removing cleanup tag; formatting seems consistent with the example at WP:DISPLAYTAG

2024-10-04T15:02:24Z

Removing cleanup tag; formatting seems consistent with the example at WP:DISPLAYTAG

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	== The algorithm ==		== The algorithm ==
	{{cleanup\|section\|reason=inconsistent styles between equations and labels\|date=February 2014}}
	The main calculation in the Goertzel algorithm has the form of a [[digital filter]], and for this reason the algorithm is often called a ''Goertzel filter''. The filter operates on an input sequence <math>x[n]</math> in a cascade of two stages with a parameter <math>\omega_0</math>, giving the frequency to be analysed, normalised to [[radian]]s per sample.		The main calculation in the Goertzel algorithm has the form of a [[digital filter]], and for this reason the algorithm is often called a ''Goertzel filter''. The filter operates on an input sequence <math>x[n]</math> in a cascade of two stages with a parameter <math>\omega_0</math>, giving the frequency to be analysed, normalised to [[radian]]s per sample.

David Eppstein: Christian Reinsch

2024-03-02T07:42:54Z

Christian Reinsch

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	== Numerical stability ==		== Numerical stability ==
	It can be observed that the [[Pole (complex analysis)\|poles]] of the filter's [[Z transform]] are located at <math>e^{+j \omega_0}</math> and <math>e^{-j \omega_0}</math>, on a circle of unit radius centered on the origin of the complex Z-transform plane. This property indicates that the filter process is [[Marginal stability\|marginally stable]] and vulnerable to [[Numerical stability\|numerical-error accumulation]] when computed using low-precision arithmetic and long input sequences.<ref>{{cite journal\|last1=Gentleman\|first1=W. M.\|title=An error analysis of Goertzel's (Watt's) method for computing Fourier coefficients\|journal=The Computer Journal\|date=1 February 1969\|volume=12\|issue=2\|pages=160–164\|doi=10.1093/comjnl/12.2.160\|doi-access=free}}</ref> A numerically stable version was proposed by Christian Reinsch.<ref>{{Citation \|first1=J. \|last1=Stoer \|first2=R. \|last2=Bulirsch \|date=2002 \|title= Introduction to Numerical Analysis \|publisher= Springer \|isbn=9780387954523}}</ref>		It can be observed that the [[Pole (complex analysis)\|poles]] of the filter's [[Z transform]] are located at <math>e^{+j \omega_0}</math> and <math>e^{-j \omega_0}</math>, on a circle of unit radius centered on the origin of the complex Z-transform plane. This property indicates that the filter process is [[Marginal stability\|marginally stable]] and vulnerable to [[Numerical stability\|numerical-error accumulation]] when computed using low-precision arithmetic and long input sequences.<ref>{{cite journal\|last1=Gentleman\|first1=W. M.\|title=An error analysis of Goertzel's (Watt's) method for computing Fourier coefficients\|journal=The Computer Journal\|date=1 February 1969\|volume=12\|issue=2\|pages=160–164\|doi=10.1093/comjnl/12.2.160\|doi-access=free}}</ref> A numerically stable version was proposed by [[Christian Reinsch]].<ref>{{Citation \|first1=J. \|last1=Stoer \|first2=R. \|last2=Bulirsch \|date=2002 \|title= Introduction to Numerical Analysis \|publisher= Springer \|isbn=9780387954523}}</ref>

	== DFT computations ==		== DFT computations ==

Ulfran: /* Computational complexity */ made clearer that log2 is used in the complexity of an FFT

2023-06-30T08:00:21Z

Computational complexity: made clearer that log2 is used in the complexity of an FFT

← Previous revision		Revision as of 08:00, 30 June 2023
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	: To compute a single [[Discrete Fourier transform\|DFT]] bin <math>X(f)</math> for a complex input sequence of length <math>N</math>, the Goertzel algorithm requires <math>2 N</math> multiplications and <math>4\ N</math> additions/subtractions within the loop, as well as 4 multiplications and 4 final additions/subtractions, for a total of <math>2 N + 4</math> multiplications and <math>4 N + 4</math> additions/subtractions. This is repeated for each of the <math>M</math> frequencies.		: To compute a single [[Discrete Fourier transform\|DFT]] bin <math>X(f)</math> for a complex input sequence of length <math>N</math>, the Goertzel algorithm requires <math>2 N</math> multiplications and <math>4\ N</math> additions/subtractions within the loop, as well as 4 multiplications and 4 final additions/subtractions, for a total of <math>2 N + 4</math> multiplications and <math>4 N + 4</math> additions/subtractions. This is repeated for each of the <math>M</math> frequencies.

	* In contrast, using an [[Fast Fourier transform\|FFT]] on a data set with <math>N</math> values has complexity <math>O(KN\~~log~~ N)</math>.		* In contrast, using an [[Fast Fourier transform\|FFT]] on a data set with <math>N</math> values has complexity <math>O(KN\log_2(N))</math>.

	: This is harder to apply directly because it depends on the FFT algorithm used, but a typical example is a radix-2 FFT, which requires <math>2 \log_2(N)</math> multiplications and <math>3 \log_2(N)</math> additions/subtractions per [[Discrete Fourier transform\|DFT]] bin, for each of the <math>N</math> bins.		: This is harder to apply directly because it depends on the FFT algorithm used, but a typical example is a radix-2 FFT, which requires <math>2 \log_2(N)</math> multiplications and <math>3 \log_2(N)</math> additions/subtractions per [[Discrete Fourier transform\|DFT]] bin, for each of the <math>N</math> bins.

Alexf: Reverted edit by Csvidyalay (talk) to last version by 81.215.12.55

2023-05-31T19:29:53Z

Reverted edit by Csvidyalay (talk) to last version by 81.215.12.55

← Previous revision		Revision as of 19:29, 31 May 2023
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	Like the DFT, the Goertzel algorithm analyses one selectable frequency component from a [[discrete signal]].<ref>{{Citation \|last=Mock \|first=P. \|url=http://focus.ti.com/lit/an/spra168/spra168.pdf \|title=Add DTMF Generation and Decoding to DSP-μP Designs \|journal=EDN \|date=March 21, 1985 \|issn=0012-7515 }}; also found in DSP Applications with the TMS320 Family, Vol. 1, Texas Instruments, 1989.</ref><ref>{{Citation \|first=Chiouguey J. \|last=Chen \|url=http://focus.ti.com/lit/an/spra066/spra066.pdf \|title=Modified Goertzel Algorithm in DTMF Detection Using the TMS320C80 DSP\| publisher=Texas Instruments \|series=Application Report \|id=SPRA066 \|date=June 1996 }}</ref><ref>{{Citation \|last=Schmer \|first=Gunter \|url=http://focus.ti.com/lit/an/spra096a/spra096a.pdf \|title=DTMF Tone Generation and Detection: An Implementation Using the TMS320C54x \|publisher=Texas Instruments \|series=Application Report \|id=SPRA096a \|date= May 2000 }}</ref> Unlike direct DFT calculations, the Goertzel algorithm applies a single [[Real number\|real-valued]] coefficient at each iteration, using real-valued arithmetic for real-valued input sequences. For covering a full spectrum (except when using for continuous stream of data where coefficients are reused for subsequent calculations, which has computational complexity equivalent of [[sliding DFT]]), the Goertzel algorithm has a [[Computational complexity theory\|higher order of complexity]] than [[fast Fourier transform]] (FFT) algorithms, but for computing a small number of selected frequency components, it is more numerically efficient. The simple structure of the Goertzel algorithm makes it well suited to small processors and embedded applications.		Like the DFT, the Goertzel algorithm analyses one selectable frequency component from a [[discrete signal]].<ref>{{Citation \|last=Mock \|first=P. \|url=http://focus.ti.com/lit/an/spra168/spra168.pdf \|title=Add DTMF Generation and Decoding to DSP-μP Designs \|journal=EDN \|date=March 21, 1985 \|issn=0012-7515 }}; also found in DSP Applications with the TMS320 Family, Vol. 1, Texas Instruments, 1989.</ref><ref>{{Citation \|first=Chiouguey J. \|last=Chen \|url=http://focus.ti.com/lit/an/spra066/spra066.pdf \|title=Modified Goertzel Algorithm in DTMF Detection Using the TMS320C80 DSP\| publisher=Texas Instruments \|series=Application Report \|id=SPRA066 \|date=June 1996 }}</ref><ref>{{Citation \|last=Schmer \|first=Gunter \|url=http://focus.ti.com/lit/an/spra096a/spra096a.pdf \|title=DTMF Tone Generation and Detection: An Implementation Using the TMS320C54x \|publisher=Texas Instruments \|series=Application Report \|id=SPRA096a \|date= May 2000 }}</ref> Unlike direct DFT calculations, the Goertzel algorithm applies a single [[Real number\|real-valued]] coefficient at each iteration, using real-valued arithmetic for real-valued input sequences. For covering a full spectrum (except when using for continuous stream of data where coefficients are reused for subsequent calculations, which has computational complexity equivalent of [[sliding DFT]]), the Goertzel algorithm has a [[Computational complexity theory\|higher order of complexity]] than [[fast Fourier transform]] (FFT) algorithms, but for computing a small number of selected frequency components, it is more numerically efficient. The simple structure of the Goertzel algorithm makes it well suited to small processors and embedded applications.

	The Goertzel algorithm is a simple and efficient way to compute the DFT of a signal at a single frequency. It is often used in applications where only a few frequency components are of interest, such as DTMF tone recognition and spectrum analysis.<ref>{{Cite web \|last= \|first= \|date=2023-05-31 \|title=Explain About The Goertzel Algorithm In Digital Signal Processing. \|url=https://csvidyalay.in/explain-about-the-goertzel-algorithm/ \|url-status=live \|access-date=2023-05-31 \|website=Csvidyalay \|language=en-US}}</ref>

	The Goertzel algorithm can also be used "in reverse" as a sinusoid synthesis function, which requires only 1 multiplication and 1 subtraction per generated sample.<ref>{{Citation \|last1=Cheng \|first1=Eric \|last2=Hudak \|first2=Paul \|url=http://haskell.cs.yale.edu/wp-content/uploads/2011/01/AudioProc-TR.pdf \|archive-url=https://web.archive.org/web/20170328022024/http://haskell.cs.yale.edu/wp-content/uploads/2011/01/AudioProc-TR.pdf \|url-status=dead \|archive-date=2017-03-28 \|title=Audio Processing and Sound Synthesis in Haskell \|date=January 2009 }}</ref>		The Goertzel algorithm can also be used "in reverse" as a sinusoid synthesis function, which requires only 1 multiplication and 1 subtraction per generated sample.<ref>{{Citation \|last1=Cheng \|first1=Eric \|last2=Hudak \|first2=Paul \|url=http://haskell.cs.yale.edu/wp-content/uploads/2011/01/AudioProc-TR.pdf \|archive-url=https://web.archive.org/web/20170328022024/http://haskell.cs.yale.edu/wp-content/uploads/2011/01/AudioProc-TR.pdf \|url-status=dead \|archive-date=2017-03-28 \|title=Audio Processing and Sound Synthesis in Haskell \|date=January 2009 }}</ref>

← Previous revision		Revision as of 08:00, 30 June 2023
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	: To compute a single [[Discrete Fourier transform\|DFT]] bin <math>X(f)</math> for a complex input sequence of length <math>N</math>, the Goertzel algorithm requires <math>2 N</math> multiplications and <math>4\ N</math> additions/subtractions within the loop, as well as 4 multiplications and 4 final additions/subtractions, for a total of <math>2 N + 4</math> multiplications and <math>4 N + 4</math> additions/subtractions. This is repeated for each of the <math>M</math> frequencies.		: To compute a single [[Discrete Fourier transform\|DFT]] bin <math>X(f)</math> for a complex input sequence of length <math>N</math>, the Goertzel algorithm requires <math>2 N</math> multiplications and <math>4\ N</math> additions/subtractions within the loop, as well as 4 multiplications and 4 final additions/subtractions, for a total of <math>2 N + 4</math> multiplications and <math>4 N + 4</math> additions/subtractions. This is repeated for each of the <math>M</math> frequencies.

	* In contrast, using an [[Fast Fourier transform\|FFT]] on a data set with <math>N</math> values has complexity <math>O(KN\~~log~~ N)</math>.		* In contrast, using an [[Fast Fourier transform\|FFT]] on a data set with <math>N</math> values has complexity <math>O(KN\log_2(N))</math>.

	: This is harder to apply directly because it depends on the FFT algorithm used, but a typical example is a radix-2 FFT, which requires <math>2 \log_2(N)</math> multiplications and <math>3 \log_2(N)</math> additions/subtractions per [[Discrete Fourier transform\|DFT]] bin, for each of the <math>N</math> bins.		: This is harder to apply directly because it depends on the FFT algorithm used, but a typical example is a radix-2 FFT, which requires <math>2 \log_2(N)</math> multiplications and <math>3 \log_2(N)</math> additions/subtractions per [[Discrete Fourier transform\|DFT]] bin, for each of the <math>N</math> bins.

← Previous revision		Revision as of 19:29, 31 May 2023
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	Like the DFT, the Goertzel algorithm analyses one selectable frequency component from a [[discrete signal]].<ref>{{Citation \|last=Mock \|first=P. \|url=http://focus.ti.com/lit/an/spra168/spra168.pdf \|title=Add DTMF Generation and Decoding to DSP-μP Designs \|journal=EDN \|date=March 21, 1985 \|issn=0012-7515 }}; also found in DSP Applications with the TMS320 Family, Vol. 1, Texas Instruments, 1989.</ref><ref>{{Citation \|first=Chiouguey J. \|last=Chen \|url=http://focus.ti.com/lit/an/spra066/spra066.pdf \|title=Modified Goertzel Algorithm in DTMF Detection Using the TMS320C80 DSP\| publisher=Texas Instruments \|series=Application Report \|id=SPRA066 \|date=June 1996 }}</ref><ref>{{Citation \|last=Schmer \|first=Gunter \|url=http://focus.ti.com/lit/an/spra096a/spra096a.pdf \|title=DTMF Tone Generation and Detection: An Implementation Using the TMS320C54x \|publisher=Texas Instruments \|series=Application Report \|id=SPRA096a \|date= May 2000 }}</ref> Unlike direct DFT calculations, the Goertzel algorithm applies a single [[Real number\|real-valued]] coefficient at each iteration, using real-valued arithmetic for real-valued input sequences. For covering a full spectrum (except when using for continuous stream of data where coefficients are reused for subsequent calculations, which has computational complexity equivalent of [[sliding DFT]]), the Goertzel algorithm has a [[Computational complexity theory\|higher order of complexity]] than [[fast Fourier transform]] (FFT) algorithms, but for computing a small number of selected frequency components, it is more numerically efficient. The simple structure of the Goertzel algorithm makes it well suited to small processors and embedded applications.		Like the DFT, the Goertzel algorithm analyses one selectable frequency component from a [[discrete signal]].<ref>{{Citation \|last=Mock \|first=P. \|url=http://focus.ti.com/lit/an/spra168/spra168.pdf \|title=Add DTMF Generation and Decoding to DSP-μP Designs \|journal=EDN \|date=March 21, 1985 \|issn=0012-7515 }}; also found in DSP Applications with the TMS320 Family, Vol. 1, Texas Instruments, 1989.</ref><ref>{{Citation \|first=Chiouguey J. \|last=Chen \|url=http://focus.ti.com/lit/an/spra066/spra066.pdf \|title=Modified Goertzel Algorithm in DTMF Detection Using the TMS320C80 DSP\| publisher=Texas Instruments \|series=Application Report \|id=SPRA066 \|date=June 1996 }}</ref><ref>{{Citation \|last=Schmer \|first=Gunter \|url=http://focus.ti.com/lit/an/spra096a/spra096a.pdf \|title=DTMF Tone Generation and Detection: An Implementation Using the TMS320C54x \|publisher=Texas Instruments \|series=Application Report \|id=SPRA096a \|date= May 2000 }}</ref> Unlike direct DFT calculations, the Goertzel algorithm applies a single [[Real number\|real-valued]] coefficient at each iteration, using real-valued arithmetic for real-valued input sequences. For covering a full spectrum (except when using for continuous stream of data where coefficients are reused for subsequent calculations, which has computational complexity equivalent of [[sliding DFT]]), the Goertzel algorithm has a [[Computational complexity theory\|higher order of complexity]] than [[fast Fourier transform]] (FFT) algorithms, but for computing a small number of selected frequency components, it is more numerically efficient. The simple structure of the Goertzel algorithm makes it well suited to small processors and embedded applications.

	The Goertzel algorithm is a simple and efficient way to compute the DFT of a signal at a single frequency. It is often used in applications where only a few frequency components are of interest, such as DTMF tone recognition and spectrum analysis.<ref>{{Cite web \|last= \|first= \|date=2023-05-31 \|title=Explain About The Goertzel Algorithm In Digital Signal Processing. \|url=https://csvidyalay.in/explain-about-the-goertzel-algorithm/ \|url-status=live \|access-date=2023-05-31 \|website=Csvidyalay \|language=en-US}}</ref>

	The Goertzel algorithm can also be used "in reverse" as a sinusoid synthesis function, which requires only 1 multiplication and 1 subtraction per generated sample.<ref>{{Citation \|last1=Cheng \|first1=Eric \|last2=Hudak \|first2=Paul \|url=http://haskell.cs.yale.edu/wp-content/uploads/2011/01/AudioProc-TR.pdf \|archive-url=https://web.archive.org/web/20170328022024/http://haskell.cs.yale.edu/wp-content/uploads/2011/01/AudioProc-TR.pdf \|url-status=dead \|archive-date=2017-03-28 \|title=Audio Processing and Sound Synthesis in Haskell \|date=January 2009 }}</ref>		The Goertzel algorithm can also be used "in reverse" as a sinusoid synthesis function, which requires only 1 multiplication and 1 subtraction per generated sample.<ref>{{Citation \|last1=Cheng \|first1=Eric \|last2=Hudak \|first2=Paul \|url=http://haskell.cs.yale.edu/wp-content/uploads/2011/01/AudioProc-TR.pdf \|archive-url=https://web.archive.org/web/20170328022024/http://haskell.cs.yale.edu/wp-content/uploads/2011/01/AudioProc-TR.pdf \|url-status=dead \|archive-date=2017-03-28 \|title=Audio Processing and Sound Synthesis in Haskell \|date=January 2009 }}</ref>