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54-521: Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Goertzel . If an internal link led you here, you may wish to change the link to point directly to the intended article. Goertzel algorithm - an algorithm used in digital signal processing Gerald Goertzel - author of the Goertzel algorithm Ben Goertzel - an American researcher in

108-405: A discrete signal . Unlike direct DFT calculations, the Goertzel algorithm applies a single 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 ),

162-512: A discrete-time filter be given by: governed by the parameter a {\displaystyle a} , a real number with 0 < | a | < 1 {\displaystyle 0<|a|<1} . H ( z ) {\displaystyle H(z)} is stable and causal with a pole at a {\displaystyle a} . The time-domain impulse response can be shown to be given by: where u ( n ) {\displaystyle u(n)}

216-481: A software object that maintains the filter state between updates, with the final power result accessed after the other processing is done. The case of real-valued input data arises frequently, especially in embedded systems where the input streams result from direct measurements of physical processes. When the input data are real-valued, the filter internal state variables sprev and sprev2 can be observed also to be real-valued, consequently, no complex arithmetic

270-421: A DFT. A simple but inelegant expedient is to extend the input sequence x [ n ] {\displaystyle x[n]} with one more artificial value x [ N ] = 0 {\displaystyle x[N]=0} . We can see from equation (9) that the mathematical effect on the final result is the same as removing term x [ N ] {\displaystyle x[N]} from

324-484: A digital IIR filter is to perform z-transform on the analog signal that has been sampled and converted to T(s) by Laplace, which is usually simplified to Pay attention to the fact that there is a multiplier T appearing in the formula. This is because even if the Laplace transform and z-transform for the unit pulse are 1, the pulse itself is not necessarily the same. For analog signals, the pulse has an infinite value but

378-623: A non-trivial denominator, describing those feedback terms. The transfer function of an FIR filter, on the other hand, has only a numerator as expressed in the general form derived below. All of the a i {\displaystyle a_{i}} coefficients with i > 0 {\displaystyle i>0} (feedback terms) are zero and the filter has no finite poles . The transfer functions pertaining to IIR analog electronic filters have been extensively studied and optimized for their amplitude and phase characteristics. These continuous-time filter functions are described in

432-415: A particular frequency response requirement. This is particularly true when the requirement is not one of the usual cases (high-pass, low-pass, notch, etc.) which have been studied and optimized for analog filters. Also FIR filters can be easily made to be linear phase (constant group delay vs frequency)—a property that is not easily met using IIR filters and then only as an approximation (for instance with

486-428: A set of specifications can be accomplished with a lower order ( Q in the above formulae) IIR filter than would be required for an FIR filter meeting the same requirements. If implemented in a signal processor, this implies a correspondingly fewer number of calculations per time step; the computational savings is often of a rather large factor. On the other hand, FIR filters can be easier to design, for instance, to match

540-401: A unit circle in the z {\displaystyle z} -plane. The poles are defined as the values of z {\displaystyle z} which make the denominator of H ( z ) {\displaystyle H(z)} equal to 0: Clearly, if a j ≠ 0 {\displaystyle a_{j}\neq 0} then the poles are not located at

594-418: Is This relationship is used in the Laplace transfer function of any analog filter or the digital infinite impulse response (IIR) filter T(z) of the analog filter. The bilinear transform essentially uses this first order approximation and substitutes into the continuous-time transfer function, H a ( s ) {\displaystyle H_{a}(s)} That is which is used to calculate

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648-518: Is marginally stable and vulnerable to numerical-error accumulation when computed using low-precision arithmetic and long input sequences. A numerically stable version was proposed by Christian Reinsch . For the important case of computing a DFT term, the following special restrictions are applied. Making these substitutions into equation (6) and observing that the term e + j 2 π k = 1 {\displaystyle e^{+j2\pi k}=1} , equation (6) then takes

702-1194: Is a better approximation for any input than the impulse invariant. Step invariant solves the problem of the same sample values when T(z) and T(s) are both step inputs. The input to the digital filter is u(n), and the input to the analog filter is u(t). Apply z-transform and Laplace transform on these two inputs to obtain the converted output signal. Perform z-transform on step input Z [ u ( n ) ] = z z − 1 {\displaystyle Z[u(n)]={\dfrac {z}{z-1}}} Converted output after z-transform Y ( z ) = T ( z ) U ( z ) = T ( z ) z z − 1 {\displaystyle Y(z)=T(z)U(z)=T(z){\dfrac {z}{z-1}}} Perform Laplace transform on step input L [ u ( t ) ] = 1 s {\displaystyle L[u(t)]={\dfrac {1}{s}}} Converted output after Laplace transform Y ( s ) = T ( s ) U ( s ) = T ( s ) s {\displaystyle Y(s)=T(s)U(s)={\dfrac {T(s)}{s}}} The output of

756-735: Is a property applying to many linear time-invariant systems that are distinguished by having an impulse response h ( t ) {\displaystyle h(t)} that does not become exactly zero past a certain point but continues indefinitely. This is in contrast to a finite impulse response (FIR) system, in which the impulse response does become exactly zero at times t > T {\displaystyle t>T} for some finite T {\displaystyle T} , thus being of finite duration. Common examples of linear time-invariant systems are most electronic and digital filters . Systems with this property are known as IIR systems or IIR filters . In practice,

810-410: Is a special case of a conformal mapping, often used to convert a transfer function H a ( s ) {\displaystyle H_{a}(s)} of a linear, time-invariant (LTI) filter in the continuous-time domain (often called an analog filter) to a transfer function H d ( z ) {\displaystyle H_{d}(z)} of a linear, shift-invariant filter in

864-454: Is equally valid to apply equation (11) and calculate the signal power from term y [ N ] {\displaystyle y[N]} or to apply equation (2) and calculate the signal power from term y [ N − 1 ] {\displaystyle y[N-1]} . Both cases lead to the following expression for the signal power represented by DFT term X [ k ] {\displaystyle X[k]} : In

918-446: Is often larger for an FFT, and the practical advantage favours the Goertzel algorithm even for M {\displaystyle M} several times larger than log 2 ⁡ ( N ) {\displaystyle \log _{2}(N)} . As a rule-of-thumb for determining whether a radix-2 FFT or a Goertzel algorithm is more efficient, adjust the number of terms N {\displaystyle N} in

972-516: Is often restricted to the range 0 to π (see Nyquist–Shannon sampling theorem ); using a value outside this range is not meaningless, but is equivalent to using an aliased frequency inside this range, since the exponential function is periodic with a period of 2π in ω 0 {\displaystyle \omega _{0}} . The second-stage filter can be observed to be a FIR filter , since its calculations do not use any of its past outputs. Z-transform methods can be applied to study

1026-595: Is required in the first IIR stage. Optimizing for real-valued arithmetic typically is as simple as applying appropriate real-valued data types for the variables. After the calculations using input term x [ N − 1 ] {\displaystyle x[N-1]} , and filter iterations are terminated, equation (11) must be applied to evaluate the DFT term. The final calculation uses complex-valued arithmetic, but this can be converted into real-valued arithmetic by separating real and imaginary terms: Comparing to

1080-453: Is the unit step function . It can be seen that h ( n ) {\displaystyle h(n)} is non-zero for all n ≥ 0 {\displaystyle n\geq 0} , thus an impulse response which continues infinitely. The main advantage digital IIR filters have over FIR filters is their efficiency in implementation, in order to meet a specification in terms of passband, stopband, ripple, and/or roll-off. Such

1134-434: Is the simplest IIR filter design method. It is the most accurate at low frequencies, so it is usually used in low-pass filters. For Laplace transform or z-transform, the output after the transformation is just the input multiplied by the corresponding transformation function, T(s) or T(z). Y(s) and Y(z) are the converted output of input X(s) and input X(z), respectively. When applying the Laplace transform or z-transform on

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1188-497: 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 s [ 0 ] , s [ 1 ] {\displaystyle s[0],s[1]} , etc. can be discarded immediately after updating the first stage's internal state. This seems to leave a paradox: to complete the algorithm,

1242-472: The Bessel filter ). Another issue regarding digital IIR filters is the potential for limit cycle behavior when idle, due to the feedback system in conjunction with quantization. Impulse invariance is a technique for designing discrete-time infinite-impulse-response (IIR) filters from continuous-time filters in which the impulse response of the continuous-time system is sampled to produce the impulse response of

1296-570: The Laplace domain . Desired solutions can be transferred to the case of discrete-time filters whose transfer functions are expressed in the z domain, through the use of certain mathematical techniques such as the bilinear transform , impulse invariance , or pole–zero matching method . Thus digital IIR filters can be based on well-known solutions for analog filters such as the Chebyshev filter , Butterworth filter , and elliptic filter , inheriting

1350-413: The poles of the filter's Z transform are located at e + j ω 0 {\displaystyle e^{+j\omega _{0}}} and e − j ω 0 {\displaystyle e^{-j\omega _{0}}} , on a circle of unit radius centered on the origin of the complex Z-transform plane. This property indicates that the filter process

1404-415: The pseudocode below, the real-valued input data is stored in the array x and the variables sprev and sprev2 temporarily store output history from the IIR filter. Nterms is the number of samples in the array, and Kterm corresponds to the frequency of interest, multiplied by the sampling period. It is possible to organise the computations so that incoming samples are delivered singly to

1458-410: 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

1512-616: The FIR filter stage. Examining equation (6), a final IIR filter pass to calculate term y [ N ] {\displaystyle y[N]} using a supplemental input value x [ N ] = 0 {\displaystyle x[N]=0} applies a complex multiplier of magnitude 1 to the previous term y [ N − 1 ] {\displaystyle y[N-1]} . Consequently, y [ N ] {\displaystyle y[N]} and y [ N − 1 ] {\displaystyle y[N-1]} represent equivalent signal power. It

1566-514: The Goertzel algorithm has a 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 can also be used "in reverse" as a sinusoid synthesis function, which requires only 1 multiplication and 1 subtraction per generated sample. The main calculation in

1620-576: 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 x [ n ] {\displaystyle x[n]} in a cascade of two stages with a parameter ω 0 {\displaystyle \omega _{0}} , giving the frequency to be analysed, normalised to radians per sample. The first stage calculates an intermediate sequence, s [ n ] {\displaystyle s[n]} : The second stage applies

1674-420: The analog filter is y(t), which is the inverse Laplace transform of Y(s). If sampled every T seconds, it is y(n), which is the inverse conversion of Y(z).These signals are used to solve for the digital filter and the analog filter and have the same output at the sampling time. The following equation points out the solution of T(z), which is the approximate formula for the analog filter. The bilinear transform

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1728-461: The area is 1 at t=0, but it is 1 at the discrete-time pulse t=0, so the existence of a multiplier T is required. Step invariance is a better design method than impulse invariant. The digital filter has several segments of input with different constants when sampling, which is composed of discrete steps. The step invariant IIR filter is less accurate than the same input step signal to the ADC. However, it

1782-486: The characteristics of those solutions. Digital filters are often described and implemented in terms of the difference equation that defines how the output signal is related to the input signal: where: A more condensed form of the difference equation is: To find the transfer function of the filter, we first take the Z-transform of each side of the above equation to obtain: After rearranging: We then define

1836-708: The data set upward to the nearest exact power of 2, calling this N 2 {\displaystyle N_{2}} , and the Goertzel algorithm is likely to be faster if FFT implementations and processing platforms have a significant impact on the relative performance. Some FFT implementations perform internal complex-number calculations to generate coefficients on-the-fly, significantly increasing their "cost K per unit of work." FFT and DFT algorithms can use tables of pre-computed coefficient values for better numerical efficiency, but this requires more accesses to coefficient values buffered in external memory, which can lead to increased cache contention that counters some of

1890-417: The discrete-time domain. The bilinear transform is a first-order approximation of the natural logarithm function that is an exact mapping of the z -plane to the s -plane. When the Laplace transform is performed on a discrete-time signal (with each element of the discrete-time sequence attached to a correspondingly delayed unit impulse), the result is precisely the Z transform of the discrete-time sequence with

1944-534: The discrete-time system. Impulse invariance is one of the commonly used methods to meet the two basic requirements of the mapping from the s-plane to the z-plane. This is obtained by solving the T(z) that has the same output value at the same sampling time as the analog filter, and it is only applicable when the inputs are in a pulse. Note that all inputs of the digital filter generated by this method are approximate values, except for pulse inputs that are very accurate. This

1998-481: The end of the tapped delay line, the system has no further memory of that impulse and has returned to its initial state; its impulse response beyond that point is exactly zero. Although almost all analog electronic filters are IIR, digital filters may be either IIR or FIR. The presence of feedback in the topology of a discrete-time filter (such as the block diagram shown below) generally creates an IIR response. The z domain transfer function of an IIR filter contains

2052-438: The field of artificial intelligence Retrieved from " https://en.wikipedia.org/w/index.php?title=Goertzel&oldid=375261482 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages Goertzel algorithm Like the DFT, the Goertzel algorithm analyses one selectable frequency component from

2106-451: The filter updates at term N − 1 {\displaystyle N-1} and immediately applying equation (2) rather than equation (11) misses the final filter state updates, yielding a result with incorrect phase. 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 y [ N ] {\displaystyle y[N]}

2160-596: The following filter to s [ n ] {\displaystyle s[n]} , producing output sequence y [ n ] {\displaystyle y[n]} : The first filter stage can be observed to be a second-order IIR filter with a direct-form structure. This particular structure has the property that its internal state variables equal the past output values from that stage. Input values x [ n ] {\displaystyle x[n]} for n < 0 {\displaystyle n<0} are presumed all equal to 0. To establish

2214-511: The following form: We can observe that the right side of equation (9) is extremely similar to the defining formula for DFT term X [ k ] {\displaystyle X[k]} , the DFT term for index number k {\displaystyle k} , but not exactly the same. The summation shown in equation (9) requires N + 1 {\displaystyle N+1} input terms, but only N {\displaystyle N} input terms are available when evaluating

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2268-400: The impulse response, even of IIR systems, usually approaches zero and can be neglected past a certain point. However the physical systems which give rise to IIR or FIR responses are dissimilar, and therein lies the importance of the distinction. For instance, analog electronic filters composed of resistors, capacitors, and/or inductors (and perhaps linear amplifiers) are generally IIR filters. On

2322-417: The initial filter state so that evaluation can begin at sample x [ 0 ] {\displaystyle x[0]} , the filter states are assigned initial values s [ − 2 ] = s [ − 1 ] = 0 {\displaystyle s[-2]=s[-1]=0} . To avoid aliasing hazards, frequency ω 0 {\displaystyle \omega _{0}}

2376-455: The inverse tangent function. Since complex signals decompose linearly into real and imaginary parts, the Goertzel algorithm can be computed in real arithmetic separately over the sequence of real parts, yielding y r [ n ] {\displaystyle y_{\text{r}}[n]} , and over the sequence of imaginary parts, yielding y i [ n ] {\displaystyle y_{\text{i}}[n]} . After that,

2430-459: The numerical advantage. Both algorithms gain approximately a factor of 2 efficiency when using real-valued rather than complex-valued input data. However, these gains are natural for the Goertzel algorithm but will not be achieved for the FFT without using certain algorithm variants specialised for transforming real-valued data . Infinite impulse response Infinite impulse response ( IIR )

2484-499: The origin of the z {\displaystyle z} -plane. This is in contrast to the FIR filter where all poles are located at the origin, and is therefore always stable. IIR filters are sometimes preferred over FIR filters because an IIR filter can achieve a much sharper transition region roll-off than an FIR filter of the same order. Let the transfer function H ( z ) {\displaystyle H(z)} of

2538-432: The other hand, discrete-time filters (usually digital filters) based on a tapped delay line employing no feedback are necessarily FIR filters. The capacitors (or inductors) in the analog filter have a "memory" and their internal state never completely relaxes following an impulse (assuming the classical model of capacitors and inductors where quantum effects are ignored). But in the latter case, after an impulse has reached

2592-435: The power-spectrum application, the only difference are the calculation used to finish: This application requires the same evaluation of DFT term X [ k ] {\displaystyle X[k]} , as discussed in the previous section, using a real-valued or complex-valued input stream. Then the signal phase can be evaluated as taking appropriate precautions for singularities, quadrant, and so forth when computing

2646-474: The properties of the filter cascade. The Z transform of the first filter stage given in equation (1) is The Z transform of the second filter stage given in equation (2) is The combined transfer function of the cascade of the two filter stages is then This can be transformed back to an equivalent time-domain sequence, and the terms unrolled back to the first input term at index n = 0 {\displaystyle n=0} : It can be observed that

2700-567: The substitution of where T {\displaystyle T} is the numerical integration step size of the trapezoidal rule used in the bilinear transform derivation; or, in other words, the sampling period. The above bilinear approximation can be solved for s {\displaystyle s} or a similar approximation for s = ( 1 / T ) ln ⁡ ( z ) {\displaystyle s=(1/T)\ln(z)} can be performed. The inverse of this mapping (and its first-order bilinear approximation)

2754-461: The summation, thus delivering the intended DFT value. However, there is a more elegant approach that avoids the extra filter pass. From equation (1), we can note that when the extended input term x [ N ] = 0 {\displaystyle x[N]=0} is used in the final step, Thus, the algorithm can be completed as follows: The last two mathematical operations are simplified by combining them algebraically: Note that stopping

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2808-523: The transfer function to be: The transfer function allows one to judge whether or not a system is bounded-input, bounded-output (BIBO) stable . To be specific, the BIBO stability criterion requires that the ROC of the system includes the unit circle. For example, for a causal system, all poles of the transfer function have to have an absolute value smaller than one. In other words, all poles must be located within

2862-437: The two complex-valued partial results can be recombined: In the complexity order expressions, when the number of calculated terms M {\displaystyle M} is smaller than log ⁡ N {\displaystyle \log N} , the advantage of the Goertzel algorithm is clear. But because FFT code is comparatively complex, the "cost per unit of work" factor K {\displaystyle K}

2916-442: The unit impulse, the result is 1. Hence, the output results after the conversion are Now the output of the analog filter is just the inverse Laplace transform in the time domain. If we use nT instead of t, we can get the output y(nT) derived from the pulse at the sampling time. It can also be expressed as y(n) This discrete time signal can be applied z-transform to get T(z) The last equation mathematically describes that

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