# 3.2 Goertzel's algorithm

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Goertzel's algorithm reduces the cost of computing a single DFT frequency sample by almost a factor of two.It is useful for situations requiring only a few DFT frequencies.

Some applications require only a few DFT frequencies. One example is frequency-shift keying (FSK) demodulation, in which typically two frequencies are used to transmit binary data; another example is DTMF , or touch-tone telephone dialing, in which a detection circuit must constantly monitor the line fortwo simultaneous frequencies indicating that a telephone button is depressed. Goertzel's algorithm reduces the number of real-valued multiplications by almost a factor of two relative to direct computation via the DFT equation . Goertzel's algorithm is thus useful forcomputing a few frequency values; if many or most DFT values are needed, FFT algorithms that compute all DFT samples in $O(N\lg N)$ operations are faster. Goertzel's algorithm can be derived by converting the DFT equation into an equivalent form as a convolution, which can be efficiently implemented as a digital filter. For increased clarity, in the equations below the complex exponential is denoted as $e^{-(i\frac{2\pi k}{N})}={W}_{N}^{k}$ . Note that because ${W}_{N}^{-Nk}$ always equals 1, the DFT equation can be rewritten as a convolution, or filtering operation:

$X(k)=\sum_{n=0}^{N-1} x(n)\times 1{W}_{N}^{nk}=\sum_{n=0}^{N-1} x(n){W}_{N}^{-Nk}{W}_{N}^{nk}=\sum_{n=0}^{N-1} x(n){W}_{N}^{\left(N-n\right)\left(-k\right)}=((({W}_{N}^{-k}x(0)+x(1)){W}_{N}^{-k}+x(2)){W}_{N}^{-k}+\dots +x(N-1)){W}_{N}^{-k}$
Note that this last expression can be written in terms of a recursive difference equation $y(n)={W}_{N}^{-k}y(n-1)+x(n)$ where $y(-1)=0$ . The DFT coefficient equals the output of the difference equation at time $n=N$ : $X(k)=y(N)$ Expressing the difference equation as a z-transform and multiplying both numerator and denominator by $1-{W}_{N}^{k}z^{-1}$ gives the transfer function $\frac{Y(z)}{X(z)}=H(z)=\frac{1}{1-{W}_{N}^{-k}z^{-1}}=\frac{1-{W}_{N}^{k}z^{-1}}{1-({W}_{N}^{k}+{W}_{N}^{-k})z^{-1}-z^{-2}}=\frac{1-{W}_{N}^{k}z^{-1}}{1-2\cos \left(\frac{2\pi k}{N}\right)z^{-1}-z^{-2}}$ This system can be realized by the structure in

We want $y(n)$ not for all $n$ , but only for $n=N$ . We can thus compute only the recursive part, or just the left side of the flow graph in , for $n$

0 1 N
, which involves only a real/complex product rather than a complex/complex product as in a direct DFT , plus one complex multiply to get $y(N)=X(k)$ .
The input $x(N)$ at time $n=N$ must equal 0! A slightly more efficient alternate implementation that computes the full recursion only through $n=N-1$ and combines the nonzero operations of the final recursion with the final complex multiply can be found here , complete with pseudocode (for real-valued data).
If the data are real-valued, only real/real multiplications and real additions are needed until the final multiply.
The computational cost of Goertzel's algorithm is thus $2N+2$ real multiplies and $4N-2$ real adds, a reduction of almost a factor of two in the number of real multiplies relative to direct computation via the DFT equation.If the data are real-valued, this cost is almost halved again.

For certain frequencies, additional simplifications requiring even fewer multiplications are possible. (For example, for the DC( $k=0$ ) frequency, all the multipliers equal 1 and only additions are needed.)A correspondence by C.G. Boncelet, Jr. describes some of these additional simplifications.Once again, Goertzel's and Boncelet's algorithms are efficient for a few DFT frequency samples; if more than $\lg N$ frequencies are needed, $O(N\lg N)$ FFT algorithms that compute all frequencies simultaneously will be more efficient.

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
how did you get the value of 2000N.What calculations are needed to arrive at it
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