# 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.

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
learn
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Damian
yes that's correct
Professor
I think
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The nanotechnology is as new science, to scale nanometric
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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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what king of growth are you checking .?
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what school?
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research.net
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sciencedirect big data base
Ernesto
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