Non-Parametric Estimation of the Periodic Signal Parameters in the Frequency Domain

Parameters estimations of periodic signals ( ) g t , where frequency of the investigated component is the key parameter, play a fundamental role in a variety of applications: impedance measurement, power quality estimation, radar, A/D testing, etc. The problem of evaluating the spectral performance of a given periodic signal reduces to the parameter estimation of each spectral component (frequency, amplitude, and phase) in the presence of noise. Estimation methods can be classified as parametric (D’Antona & Ferrero, 2006) and nonparametric (Agrež, 2002). Parametric methods are model-based and have very good selectivity and statistical efficiency, but require computationally intensive algorithms and very good ‘model agreement’ with a real multi-component signal. For this reason, such methods are unsuitable for many estimation problems. A better approach is to use nonparametric methods, which estimate the spectral parameters of interest by evaluating first the discrete Fourier transform (DFT) of the signal and then the parameters of the particular component. As we are dealing with periodic signals, the integral frequency transformation with the kernel j ft e− 2π is, in principle, the best approximation to periodicity of the signal. Analysis of the frequency spectrum provides the opportunity to see systematic periodicities in the presence of the reduced random noise by integration. Many of these estimations are based on coherent sampling; that is, on the accurate synchronization of the signal and the sampling rate, and on the collection of a number of samples belonging to an integer number of the signal periods. However, the normal situation for signal parameter estimation is non-coherent, or quasi-coherent sampling, and in such a sampled signal there can also be spurious components. When failing to observe an integer number of periods of even a single tone, the tone energy is spread over the whole frequency axis, and the leakage from neighboring components can significantly bias estimations of the component parameters.

In words, T Δ and W Δ cannot simultaneously be arbitrary small. In relation to measurements this is interpreted to imply that the uncertainty in the determination of a frequency, is of the order of magnitude of the reciprocal of the time taken to measure it.
The second principle, that of the limited changes of signals, limits the design of the signal shape. The more smoothly and slowly a function changes, the more rapidly its transformation changes and vice versa (Seibert, 1986). In practice this means that the spectrum of the signal should essentially vanish for frequencies greater than some frequency max f , and that the tails of the pulse in the time domain must die sufficiently rapidly that the tail of a large pulse will not seriously distort another smaller pulse at an adjacent time instant. The quantitative expression of this principle can be derived from Parseval's theorem and the differentiation properties of the Fourier transformation Both fundamental principles limit the accuracy of parameter estimations and depend upon the measurement time. Here this is taken to mean the relative time to measure a periodic signal, or the number of repetitions of the periodicity in it: where the measurement time determines the basic frequency resolution in the frequency domain fT = meas 1 Δ .
A finite time of measurement is a source of dynamic errors, which are shown as leakage parts of the measurement window spectrum, convolved on the spectrum of the measuredsampled signal (Fig. 1). The sampled analogue multi-frequency signal ( ) g t , by ( ) f t = sampling 1 Δ , can be written as follows: The normalized moment of inertia about the center of gravity of the signal distribution.
The normalized first momentcenter of gravity of the signal distribution.
where m f , m A , and m ϕ are frequency, amplitude, and phase of one component with index m among M + 1, respectively. Index k is the current time index of the successive samples t Δ apart. Tones of the sampled signal do not generally coincide with the basic set of the periodic components of the DFT, which is the most well-known, non-parametric method for frequency decomposition of signals (Harris, 1978). Using N samples of the signal (4), the DFT at the spectral line i is given by: where m i is an integer value and the displacement term m δ is caused by the non-coherent sampling. The DFT coefficients surrounding one signal component are due to both the short-range leakage and the long-range leakage contributions from the second term of the investigated component, and from both terms of other components (7) (Fig. 1). Fig. 1. The short-range leakage influences (a) and long-range leakage influences (b) on the amplitude DFT coefficients (rectangular window; m = 6.3 θ ) The long-range leakage contributions can be reduced in several ways: by increasing the measurement time, by using windows with a faster reduction of the side lobes than with a rectangular window (like the Hann window, Rife-Vincent windows, Dolph-Chebychev windows, etc., Fig. 2), or by using the multi-point interpolated DFT algorithms and a window with known behavior of the spectrum (Agrež, 2002). For the sake of analytical simplicity, cosine-class windows are frequently used (Belega & Dallet, 2009;Novotný & Sedláček, 2010). The three basic classes of cosine windows were defined RV1, RV2, and RV3. For analyses, the first two classes are interesting.
Windows of the class RV1 (Fig. 2: curves a, b, and c) are designed for maximization of the window spectrum side-lobes fall-off b − θ , based on the number of the time domain window derivatives zeroes at the window ends (Novotný & Sedláček, 2010): When the order b is 1 (RV1-1), the coefficient a 0,1 is 1 and the equation (8)  13 2) e x p a n d t h e window transform main-lobe and reduce the spectral leakage.
Windows of the class RV2 ( Fig. 2: curves d and e) are designed for minimization of the window spectrum main-lobe width, for a given maximum level of the side-lobes relative where d is the exponent of the damping. They are the Taylor approximation to the Dolph-Chebychev windows and give good results when spectral components are very close (Andria et al., 1989).

Parameters estimations
It is important how the used window works in the estimation. Especially in the first step if we have an iterative procedure of estimation. Estimation of the first step has the nonparametric nature since there is no information about the signal at the beginning.

Frequency estimation
Parameters of the measurement component can be non-parametric estimates by means of the interpolation. From the comparative study (Schoukens et al., 1992) it can be concluded that the key for estimating the three basic parameters is in determining the position of the measurement component mm m i = − δ θ , between DFT coefficients m Gi () and m Gi + (1 ) , surrounding the component m ( Fig. 1.). Estimation can be done by multi-point estimations (Agrež, 2002) and using windows with known spectra, like the Hann window: for 2-points estimation: is the sign of displacement and can be estimated by the difference of the phase DFT coefficients: The three largest, local DFT coefficients can be used in the three-point interpolation and in this case, long-range leakage contributions can be considered. Portions ( ) i Δ (7) of the longrange leakage tails have the following properties: they decrease with increasing frequency and they change sign at successive coefficients ( )

ΔΔ Δ
, so that the ratio of coefficients can be expressed as: The numerators and the denominators in (9) and (11) have the form where the amplitude DFT coefficients are added with suitable weights. The form of the denominator in (11) construction of the Hann window spectrum with the Dirac delta function ( ) D *, a n d t h e spectrum of the rectangular window (Harris, 1978): rect.

11
1 11 24 4 θ θ but instead of the rectangular window, the Hann window can be used: From the point of view of leakage, the denominator is a sum of the weighted leakages. We can get the weights with a triple subtraction of the long-range leakage tails: The numerator is a subtraction of the sum of the first two, from the sum of the last two DFT coefficients In this case the long-range leakage tails are also reduced.
It is appropriate to form multi-point interpolations on an odd number of coefficients, in order to have symmetry around the largest local coefficient Gi . In a five-point interpolation with the Hann window, similar averages are used as in the three-point interpolation. The quotient is used to eliminate the amplitude influence of the investigated component.
An estimation of the periodic parameter by the interpolation of the DFT gives the same effect as the reduction of spectrum tails. The meaning of the interpolation is the weighted summation of the amplitude coefficients, or better, symmetrical subtraction of the successive adjacent leakage parts of the window spectrum (14). The idea for long-range leakage reduction by summation of the adjacent weighted DFT coefficients, is at the core of the construction of the cosine class windows. Weights for forming the Hann window and the Rife-Vincent Class I windows from the rectangular window, are obtained by repeated convolution of the two-point weight pairs ( 1, 1 ), that is by repeated subtraction of the neighboring pairs of the spectrum leakage tails. Binomial weights ( ) r j 2 ( r = 1, 2, ... is a number of coefficients of one half; jr = 0, 1, , 2 … ) can be obtained from a Pascal triangle.
The displacement estimations with the multi-point ( r =+ = 21 3 , 5 , 7 , … η ) interpolations of the DFT using the Hann window can be written as:

Non-Parametric Estimation of the Periodic Signal Parameters in the Frequency
) is the current index. The first term of the numerator in (17)

Reduction of systematic error
The results of the simulations, where the relative frequency was changed, show that the systematic contribution of the error -the estimation bias -decreases ( Fig The "absolute" form of the standard deviation (18) is usually related to the values of the DFT coefficients of interest. In coherent sampling the largest local amplitude DFT coefficient is equal to represents the normalized peak signal gain of the window w(k) (Solomon 1992). The relative form of the standard deviation can be written as: The root of the equivalent noise bandwidth ENBW (Harris, 1978) is a factor determining the size of the standard deviation when using different windows: ENBW ENBW >= rect. 1 .
Distortions of the DFT coefficients and their number in an interpolation have a significant influence on the uncertainty of the displacement estimation.  As the standard deviations of the amplitude coefficients are almost equal , it is possible to formulate the expression for the standard deviation of displacement. For the three-point interpolation using the Hann window (11), it can be expressed as: The same mathematical procedure can be used for other higher multi-point interpolations (17).  (22). If one wants to compare it with the unbiased Cramér-Rao lower bound (CRB) for the estimation of the frequency (Petri, 2002) the relationship has to be reexpressed: This form is larger than f CRB, σ for the frequency estimation taking into account all measurement information (Fig. 4).
Errors in relative frequency estimations with different numbers of interpolation points have normal distributions. The standard deviation of the three-point frequency estimation, which has the lowest standard deviation in the vicinity of the integer values of the relative frequency, is about 2.2 times higher than ).

A trade-off between bias and uncertainty
If we reduce the leakage tails, or systematic errors by the interpolation, we apparently widen the estimation main-lobe W est Δ ( ENBW >1 ), and the noise in the estimation increases in comparison to the CRB. For example, the noise of the cosine windows increases 30dB (Harris, 1978). At the same time, the systematic errors ( E max θ ) decrease with increasing numbers of points. Increasing the number of the used DFT coefficients is reasonable until the systematic error drops under the noise error. After this point, by increasing the relative frequency m θ , or with spacing between the two frequency components ( m ∝ ⋅ 2 θ ), it is logical to decrease the number of interpolation points.
The criterion for selecting one of the algorithms could be the minimum common uncertainty of the estimation considering both contributions: The effective value of the systematic contribution is obtained by dividing the maximal error by the square root of two, since systematic errors are phase dependent with a sine like shape.
The borders of relative frequency where one interpolation can pass over another depend upon the number b of bits of the A/D converter. With a 10-bit A/D converter ( SNR ≈ 62dB ), which is frequently used in industrial environments, it is convenient to use the three-point DFT interpolation with the Hann window, in the interval << Multi-point interpolations present worse results: at lower θ owing to systematic error ("window width"), and at higher values of θ owing to the larger noise sensibility. The solid line at the top of Fig. 5 shows where different multi-point interpolations can be used to achieve the best results of the one-component frequency estimation.

Amplitude estimation
From the behavior of the systematic error of the frequency estimation (Fig. 3), it can be concluded that it is better to use the Hann, or some higher order cosine window for the estimation, if the window spectrum is analytically known. When the displacement m δ for the specific component is determined, it is easy to get the amplitude using the Hann window (9a) and neglecting the long-range contribution ( ) m i Δ in (7): As in the case of the frequency estimation, with the summation of the DFT coefficients, we subtract the long-range leakage tails and reduce their influences. We get the weights for the three-point summation with the triple subtraction of the long-range leakage tails (14).
In this manner, we can get the amplitude of the signal by summing the largest three local DFT coefficients around the signal component following the result of (26): www.intechopen.com the amplitude estimation with the three-point interpolation ( m A 3H ) can be expressed as follows:  (14), or summations of the DFT coefficients, then three subtractions of the obtained and reduced tails as in (26), and so on. After rearrangement the amplitude can be expressed with the weighted five largest coefficients: The same procedure can be used for the seven-point interpolation: The relative error () ( ) eA AA * = − 1 ( A * = 1 is the true value of the amplitude) drops with increasing relative frequency and with the number of the interpolation points ( Fig. 6: 0 ethe amplitude is estimated only with the largest coefficient, 1 e -estimation with (25), e 3estimation with (30), etc; The same testing conditions as for Fig. 3). Comparing figures 6 and 7 shows the importance of the frequency estimation accuracy. If we know the value of the frequency on the three-point interpolation accuracy level, then the amplitude estimation is reasonable with the three-point interpolation. The accuracy of the amplitude estimation can be improved, if the frequency is better estimated (e.g., by the multipoint interpolations).

Influence of noise on the amplitude estimation
Uncertainty of the component amplitude estimation mainly depends on the uncertainties of the amplitude DFT coefficients. Equation (19) is valid for all amplitude coefficients of the DFT that are large enough and sufficiently (half of the main lobe width) moved away from the margins of the spectral field ( The price for the effective leakage reduction is in the increase of the estimation uncertainties, related to the unbiased CRB fixed by the signal-to-noise-ratio for a particular component. In Fig. 8, there are standard uncertainties of the amplitude estimation related to the CRB (33) (Petri, 2002) for the three-point estimation. The distortions of the DFT coefficients and the number of points in the interpolation, have significant influence on the uncertainty of displacement m δ , and amplitude m A , succesively. As with the frequency estimation, the systematic errors decrease when increasing the number of points. Increasing the number of DFT coefficients used in the interpolation is reasonable, until the systematic error drops under the noise error (Fig. 9). After this point, by increasing the relative frequency m θ (the number of periods of the measured signal in the measurement interval), or increasing spacing between two frequency components (

Phase estimation
The second parameter of the signal component, besides the amplitude of the frequency main lobe, is the phase, i.e. the time position of the signal structure. As with previous estimations, the function ( ) W θ has to be analytically known. For the rectangular window with a large number of points N>> 1 , the following equation is valid, where the Dirichlet kernel is used (Harris, 1978): www.intechopen.com The component phase m ϕ is referred to as the start point of the window (not the middle point as is the case with the frequency and amplitude).
As N is usually large N >> 1 , and considering (6)  Gi − 1 (Fig. 10). The largest side coefficient may commonly be expressed as:  As we know, the spectrum with the Hann window can be obtained by shifting and weighing summations of the rectangular window spectrum (12). Using (34) in (12), it can be written: The largest DFT coefficient can be deduced from (5) and (40)

Reduction of the systematic error
In the first approximation using the rectangular window, the second term in (36) and (38) Another possibility is to estimate the component phase only by the phase DFT coefficient itself, where m ϕ is referred to the middle point of the measurement window. However, this method has the same weak point as (44) and (45), since it doesn't consider the long-range contributions of the window (Fig. 14d).
We can improve the estimation by considering the long-range contributions. Because the disturbing angle components ( ) * Δϕ in (37) and (39) are small, they can be exchanged by sine functions and approximated by quotients: Here, the maximal amplitude values * from (36) and (38) The multiplication of (37) The systematic errors of the phase estimations m E = − 0 ϕ ϕ ( 0 ϕ -is the true value of the phase) are phase dependent (Fig. 11: The error curves are very close to the sine like functions). In simulations, the absolute maximum values of the errors at a given relative frequency have been searched when phase has been changed in intervals −≤ ≤ 22 π ϕπ (Fig. 12). The estimation errors drop with the increasing relative frequency.  (49), c -by (51); * * * a, b, c-θ is estimated by (9) Using the Hann window, the expressions for phase have the same forms as for the rectangular window when the second term in (41) and (43) (42) and (43) and when the sign is negative s = −1 , the estimation is done with The phase estimation can be improved further with averaging of the estimations by (57) and (58). With this averaging we get the three-point estimation (Fig. 14c The best results are obtained from the three-point estimation using the Hann window, when the frequency is known (Fig. 14c: ( ) E − ≤⋅ ≈°⇐> 5 max 1.7 10 rad 1m 5.5 ϕ θ ). If the frequency has to be estimated, the overall error increases, but it is still under the error level of the one-point estimation (Fig. 14: <⇐> * ba 5 . 5 θ ). In Fig. 14, for error curve * b , the frequency is estimated by the three-point interpolation (11).