The work in this chapter is an attempt to propose the techniques on Non parametric spectrum estimation problems and is reported in the following sections.
4.1Statement of the problem:
Though the non parametric spectral estimation has good dynamic performance, it has few drawbacks such as spectral leakage effects due to windowing, requires long data sequences to obtain the necessary frequency resolution, assumption of auto correlation estimate for the lags greater than length of the sequences to be zero which limits the quality of the power spectrum and the assumption of available data are periodic with period N which may not be realistic. Hence alternative must be explored to reduce the spectral leakage effects, to decease the uncertainty in the low frequency regions, to improve the frequency resolution, to reduce variance with the increased percentage of overlapping data samples a consistent spectral estimate with minimum amount of bias and variance.
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The study of spectral leakage effects methods have been discussed by many authors. In this work, a non parametric power spectrum estimation method on nonuniform and uneven data sequences using Lomb Transforms and resampling, linear interpolation and cubic interpolation methods. The simulation results show the reduction in spectral leakage, improved spectral estimation accuracy and shifting of frequency peaks towards the low frequency region. The simulation results show the good argument with the published work.
To reduce the spectral leakage effects and to resolve the spectral peaks at higher frequencies of non uniform data sequences, a nonparametric power spectrum estimation method using prewhitening and post coloring technique is proposed. The combination of nonparametric with parametric method as preprocessor is proposed in large active range situations. The simulation results show the good argument with the published work.
To reduce the variance of a spectral estimate, a non parametric spectral estimation method based on circular overlapping of samples is proposed. The existing Welch nonparametric power spectrum estimation method has increased variance with the increased percentage of overlapping of samples. Welch estimate uses the linear overlapping of the samples. Hence the Welch estimate is not a consistent spectral estimate. To overcome this, circular overlapping of samples is proposed. The variance of the proposed estimate decreases with increased percentage of circular overlapping of samples, the spectral variance is found to be nonmonotonically decreasing function. The simulation results show the robustness of proposed estimate with the existing Welch estimate in the published work.
4.2 Power Spectrum Estimation of nonuniform data sequences using nonlinear overlapping of samples.
We consider nonparametric techniques of spectrum estimation. These methods are based on the idea of estimating the autocorrelation sequence of a random process from a set of measured data, and then taking the Fourier transform to obtain an estimate of the power spectrum. We begin with the periodogram, a nonparametric method first introduced by Schuster in 1898 in his study of periodicities in sunspot numbers [47, 49]. As we will see, although the periodogram is easy to compute, it is limited in its ability to produce an accurate estimate of the power spectrum, particularly for short data records. We will then examine a number of modifications to the periodogram that have been proposed to improve its statistical properties. These include the modified periodogram, Bartlett's method, Welch's method, and the Blackman-Tukey method.
The Power spectrum estimation of randomly spaced samples using nonparametric methods is well known and is reconsidered in this algorithm. Commonly used nonparametric methods are Periodogram, Modified Periodogram and Welch methods. The Periodogram and Modified Periodograms are asymptotically inconsistent spectral estimators for non uniform samples, Welch method is a consistent estimate for the random samples and the method is revisited for getting low normalised variances using different percentage of circular overlapping of the samples than the existing Bartlett's method of estimation. Although Circular overlap causes a discontinuity on the random process; it is shown that for a normally distributed ergodic weakly stationary random process the power spectrum estimate is asymptotically unbiased. The variance of the proposed estimate decreases due to circular overlapping of the samples. Further an expression is derived for the lowest reachable variance with respect to different fractions of overlap of samples.
Periodogram: The power spectrum of a wide- sense stationary random process is the Fourier transform of the autocorrelation sequence,
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Therefore, spectrum estimation is, in some sense, an autocorrelation estimation problem. An autocorrelation ergodic process and an unlimited amount of data, the autocorrelation sequence may, in theory, be determined using the time-average
However, if is only measured over a finite interval, say n = 0,1, …, N - 1, then the autocorrelation sequence must be estimated using, for example, Eq. (8.2) with a finite sum,
In order to ensure that the values of that fall outside the interval [0, N - 1] are excluded from the sum, Eq. (8.3) will be rewritten as follows:
With the values of for defined using conjugate symmetry, and set equal to zero for Taking the discrete-time Fourier transform of leads to an estimate of the power spectrum known as the periodogram,
Modified Periodogram: we saw that the periodogram is proportional to the squared magnitude of the Fourier transform of the windowed signal
Instead of applying a rectangular window Eq. (6) suggests the possibility of using other data windows. A rectangular window has a narrow main lobe compared to other window and, therefore, produces the least amount of spectral smoothing; it has relatively large side lobes that may lead to masking of weak narrowband components. The periodogram of a process that is windowed with a general window is called a modified periodogram and is given by
Where is the length of the window and
is a constant that, as we will see, is defined so that will be asymptotically unbiased.
Bartlett's method: we look at Bartlett's method of periodogram averaging, which, unlike either the periodogram or the modified periodogram, produces a consistent estimate of the power spectrum . The motivation for this method comes from the observation that the expected value of the periodogram converges to as the data record length goes to infinity,
Therefore, if we can find a consistent estimate of the mean, , then this estimate will be a consistent estimate of .
In our discussion of the sample mean, we saw how averaging a set of uncorrelated measurements of a random variable yields a consistent estimate of the mean, This suggests that we consider estimating the power spectrum of a random process by periodogram averaging. Thus, let for be uncorrelated realizations of a random process over the interval with the periodogram of
the average of these periodograms is
Power spectrum estimation with Nonlinear Overlapping of samples:
Another way to enforce the variance to decrease is by averaging. Bartlett's Method divides the signal of length N into K segments, with each segment having length L=N/K. The Periodogram Method is then applied to each of the K segments. The average of the resulting estimated power spectra is taken as the estimated power spectrum. One can show that the variance is reduced by a factor K, but a price in spectral resolution is paid . The Welch Method, , eliminates the trade off between spectral resolution and variance in the Bartlett Method by allowing the segments to over lap.
Furthermore, the truncation window can also vary. Essentially, the Modified Periodogram Method is applied to each of the overlapping segments and averaged out. However, the Welch estimator only uses regular overlap as illustrated in Figure1. at the signal x(n) with n=0,1,2 ,…, N-1 is an ergodic weakly stationary, , Gaussian process. We divide the signal x(n) in K independent segments such that every segment has length L = N/K. Further, we extract different overlapping sub-records by following the scheme of Figure 2. The i-th overlapping sub-record of the signal x (n) satisfies,
With the fraction of overlap, n=0,…,L-1 and where denotes ) modulo N imposing circular overlap. As we can see from Figure 2, circular overlap has the property that every time sample is an equal number of times member of a sub record, further the different sub records need to overlap an integer number of time samples. One can show that the two properties are respectively satisfied if
We define the discrete Fourier transform of the i-th sub-record as
where we corrected the phase to refer all sub records to the same time origin. In the beginning of the section, we assumed the signal x(n) to be ergodic and weakly stationary. Next, we apply Wold's decomposition theorem. Wold's theorem, , states that a weakly stationary Gaussian signal can be interpreted as filtered white Gaussian noise where the filter has a square summable impulse response, but not necessarily stable in the BIBO (Bounded Input Bounded Output) sense, see Figure 3. In the sequel of this paper,
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Figure 3.Filter response and spectral estimate
we shall assume the signal x(n)=h(n) * e(n) where e(n) is zero mean white Gaussian noise with unit variance. The impulse response h(n) is square summable such that Further, for a time window w(n) with n=0 , … ,L-1, we de fine the modified ,as
By averaging over the different overlapping sub records in equation (12), the Power Spectrum Estimator with circular overlap is defined as,
In the next section, we study the statistical properties of equation (13).
POWER SPECTRUM ESTIMATOR WITH CIRCULAR OVERLAP:
Besides the analysis of some classical statistical properties like the expected value and the variance of the PSE (4), we show that circular overlap works. From Figure 2, it is clear that the overlapping sub- records formed by the end of the last segment and the beginning of the first segment introduce a discontinuity in the signal. We need to examine the influence of this discontinuity on the statistical properties of the PSE with circular overlap.
The following result can be shown, ,
where denotes the normalized DFT of the filter h(n) as explained and the convergence in equation (14) is in Mean Square sense,. Convergence in Mean Square implies that the first and second moment converges as well. Therefore,
Where E[.] denotes the Expected value . From the statistical computations it is shown that, e(n) is zero mean and unit variance white noise. Therefore,
proving that the PSE with circular overlap is asymptotically unbiased. This is the same result as for the PSE with Welch's method, using the setup from Figure 1. We can show that the bias vanishes asymptotically as Welch estimate. In the case of circular overlap, the filter characteristic suffers from Leakage effects where the Welch estimate does not. For instance, it can be shown that for 80% overlap, a reduction of the bias with 10% requires 5 times longer records for the PSE with Circular Overlap than for the Welch PSE. This is the price to pay; however, we show in the next subsection that a significant reduction in standard deviation on the Welch PSE can be achieved by applying Circular Overlap. In the last subsection, we show with extensive simulations that by using a Hanning window, we can expect a reduction in Mean square error of approximately 20% for free.
The effect of nonlinear overlap on proposed Estimate:
We discuss the effect of overlap on the proposed estimate as follows.
(i) The variance is non-increasing as a function of the fraction of overlap.
(ii) The variance converges to a non-zero lower bound if the fraction of overlap tends to 1.
We examine the lowest reachable level, with respect to all fractions of overlap, for the variance of the PSE with Circular Overlap. The variance is non increasing as a function of the fraction of overlap, thus the lowest variance is achieved as the fraction of overlap tends to 1. Therefore, we examine the behavior of the variance, for since this is the largest possible fraction of overlap for a record of length L. We showed, , that for and L sufficiently large that,
Where denotes the normalized DFT of w(n) at .In the classical case, the variance for Bartlett's PSE equals expression (17) for a fraction of overlap .Therefore, it is clear that by using Circular Overlap one gains a factor , in variance with respect to the Bartlett estimator. Unfortunately, a (Frequency domain) closed form expression like (17) does not exist for the variance of the Welch PSE. In the next sub-section we compare the PSE with circular overlap with the classical Bartlett PSE (without overlap).
The flow chart for the PSE with circular overlaps is as shown in the figure 4.
Design a Chebyshev Type-I/Butterworth filter h (n)
Process White noise e(n) over the filter to get random data x(n)
Segment the random data into number of segments (k) using using Hanning window
Circular overlap the segments for the desired % of overlapping(r)
Evaluate the PSE of the individual segments and add all segments.
Is Variance < True variance
Get the consistent estimate with circular overlaps
Run for number of iterations i =10000
Figure 4.1: Flow chart of PSE with circular overlapping method.
Figure 4.1 Filter response of Gaussian noise of zero mean and
Figure 4.2 Average realizations 0f 100 simulations of PSE with circular overlap with non overlapping and 3 segments (chebyshev type-I filter)
Figure 4.3 Average realizations 0f 100 simulations Welch estimate
with non overlapping and 3 segments (chebyshev type-I filter) .
Figure 4.4 The true spectrum estimation of W.S.S Gaussian Ergodic
Figure 4.5 Comparison of spectral estimates for 0% overlapping of
Figure 4.6 Average realizations 0f 100 simulations of PSE with circular overlap with non overlapping and 3 segments (chebyshev type-I filter).
4.7 Average realizations 0f 100 simulations Welch estimate
with 50% overlapping and 3 segments (chebyshev type-I filter) .
4.8 The true spectrum estimation of W.S.S Gaussian Ergodic
Figure 4.9 Comparison of spectral estimates for 50% overlapping
4.10 Average realizations 0f 100 simulations of PSE with circular
overlap with non overlapping and 3 segments (chebyshev type-I filter).
4.11 Average realizations 0f 100 simulations Welch estimate
with 80% overlapping and 3 segments (chebyshev type-I filter) .
4.12 The true spectrum estimation of W.S.S Gaussian Ergodic
Figure 4.13 Comparison of spectral estimates for 80% overlapping
Figure 4.14 Average realizations 0f 100 simulations of PSE with circular
overlap with 100% overlaps and 3 segments (Chebyshev type-I filter).
4.15 Average realizations 0f 100 simulations Welch estimate
with 100% overlapping and 3 segments (chebyshev type-I filter) .
4.16 The true spectrum estimation of W.S.S Gaussian Ergodic
Figure 4.17 Comparison of spectral estimates for 100% overlapping
Figure 4.18 Average realizations of 100 simulations of PSE with circular overlap, Welch and true estimates with 60% overlap and 4 segments (chebyshev type-I filter cutoff frequency of 0.25).
Figure 4.19 Average realizations of 100 simulations of PSE with circular overlap, Welch and true estimates with70.5% overlap and 4 segments (chebyshev type-I filter with cutoff frequency of 0.25).
Figure 4.19 Average realizations of 100 simulations of PSE with circular overlap, Welch and true estimates with76.6% overlap and 4 segments (chebyshev type-I filter with cutoff frequency of 0.25).
Figure 4.20 Average realizations 0f 100 simulations of PSE with circular
overlap with non overlapping and 3 segments (Butterworth filter).
Figure 4.21 Average realizations 0f 100 simulations Welch estimate
with 0% overlapping and 3 segments (Butterworth filter) .
Figure 4.22 the true spectrum estimation of W.S.S Gaussian Ergodic
Figure 4.23 Comparison of spectral estimates for 0% overlapping
Figure 4.23 Average realizations 0f 100 simulations of PSE with circular
overlap with non overlapping and 3 segments (Butterworth filter).
Figure 4.24 Average realizations 0f 100 simulations Welch estimate
with 50% overlapping and 3 segments (Butterworth filter) .
Figure 4.25 The true spectrum estimation of W.S.S Gaussian Ergodic
Figure 4.26 Comparison of spectral estimates with 50% overlapping
Figure 4.27 Average realizations 0f 100 simulations of PSE with circular
overlap with 80% overlapping and 3 segments (Butterworth filter).
Figure 4.28 Average realizations 0f 100 simulations Welch estimate
with 80% overlapping and 3 segments (Butterworth filter) .
Figure 4.29 The true spectrum estimation of W.S.S Gaussian Ergodic
Figure 4.30 Comparison of spectral estimates with 80% overlapping
Figure 4.31 Average realizations 0f 100 simulations of PSE with circular
overlap with 100% overlapping and 3 segments (Butterworth filter).
Figure 4.32 Average realizations 0f 100 simulations Welch estimate
with 100% overlapping and 3 segments (Butterworth filter) .
Figure 4.33 The true spectrum estimation of W.S.S Gaussian Ergodic
Figure 4.34 Comparison of spectral estimates with 100% overlapping
As the percentage overlapping of the samples is increases the variance is reduced as the case of chebyshev filter from Figure 4.2 to Figure 4.19 to a minimum quantity and also for the case of Butterworth filter as shown from Figure 4.20 to Figure 4.34, the bias is also approaches to true value hence the power spectrum estimation with nonlinear overlapping of samples is said to be a consistent estimate.
4.3 Power Spectrum estimation in wide dynamic range:
Power spectral density (PSD) estimation techniques are widely used in many applications, such as sonar, radar, geophysics and biomedicine. Similar to single channel power spectral density (PSD) estimation, there are basically two broad categories of MPSD estimators. One is the nonparametric approach, among which the Fourier-based estimators are the most popular [1-3]. The other is the parametric method, which assumes a model for the data. Spectral estimation then becomes a problem of estimating the parameters in the assumed model. The most commonly used model is the autoregressive (AR) model because accurate estimates of the AR parameters can be found by solving a set of linear equations [1,3,5]. Similar to the single channel case for short data records the Fourier-based methods can suffer from significant bias problems while AR model-based methods can suffer from inaccuracies in the model as well as from imprecise model order selection. Furthermore, some effective AR model-based approaches cannot be easily extended to the multichannel case [1,5]. In addition, as pointed out by Jenkins and Watts , spurious cross-correlation or spurious cross-spectral content may arise unless a prewhitening filter is applied before PSD estimation. One such prewhitening filter was suggested by Thomson  for single channel PSD estimation.
The filter system function is given by and the filter parameters a , a , a …a[p]can be estimated from the data using any AR-model based method. Denoting the output of this FIR filter by u[n], a Fourier-based estimator is then used to generate the PSD estimate. F=inally, the PSD estimate of the original data is found as 
where are the estimated AR filter parameters. We term this the AR prewhitened (ARPW) spectral estimator.
Because of the inconsistency of the definitions in the literature concerning MPSD estimation, the following definitions will be made. A complex multichannel sequence x[n] is defined as the complex vector x where represents the data observed at the output of the ith channel and L is the number of channels. For a wide sense stationary (WSS) multichannel random process, the autocorrelation function (ACF) at lag k is defined as the matrix function
where E[·] is the mathematical expectation, the superscript H denotes conjugate transpose and is the cross-correlation function (CCF) between and at lag
For multichannel data of N samples, the sample vector, which is , is defined as
The multichannel autocorrelation matrix of order is defined as (5)
From definition (2) it is seen that so is hermitian. Because the multichannel process is assumed to be wide sense stationary, is also block Toeplitz. The power spectral density matrix or cross-spectral matrix is defined as
The diagonal elements Pii(f) are the PSDs of the individual channels or auto-PSDs, while the off-diagonal elements Pil(f) for are the cross-PSDs between and , which are defined as
The magnitude squared coherence (MSC) between channels and is a quantity that indicates whether the spectral amplitude of the process at a particular frequency in channel is associated with large or small spectral amplitude at the same frequency in channel . It is defined (7)
A classic Fourier-based spectral estimator is the periodogram, which is given as the matrix
where the Fourier transform is the L - 1 vector
The th order AR model is defined as
AR coefficient matrices and is the excitation white noise or and is the excitation noise covariance matrix with being the discrete delta function.
The ARPW estimator given in (1) is readily extended to the multichannel case. With the notations defined above, the multichannel version of (1) is
are the estimated multichannel AR filter parameters. In addition to reducing spurious cross-spectral content, this prewhitened spectral estimator also gives an auto-PSD spectral estimate with much less bias than a Fourier-based spectral estimator. This is because the prewhitener reduces the dynamic range of the PSD. However, this estimator is still inferior to the method proposed in this paper. Instead of the FIR prewhitening filter, the proposed estimator uses a prewhitening matrix, which is essentially a time-varying filter that is less susceptible to end effects. The new estimator for a single channel PSD has been proposed in , while in this paper it is extended to MPSD estimation. Assume the signal
where is an vector, Ac is an complex amplitude to be estimated, f0 is a known frequency, and is a complex Gaussian noise vector with zero mean and known covariance matrix . The estimate of Ac is 
where is the data sample vector given by (4) and with being an identity matrix. The covariance matrix of this estimator is 
(15) Therefore, for a general WSS multichannel random process the MVSE is defined as
where is the estimated ACM of and
x1(n)The proposed system for the power spectrum estimator is as shown below
Prewhiten matrix Filter
Estimated model order (EEF) criteria
Estimated model order (EEF) criteria
Figure 4.35 Proposed system for MARMPW spectral estimator
The proposed algorithm for the power spectrum estimator is as explained below
1. Choose a maximum AR model order p max.
2. using the EEF model order estimator of (28) obtain pˆ.
3. Estimate the AR model parameters for the different order AR models
In the simulations to follow we have used the autocorrelation or Yule-Walker method and so the AR model parameters for all the lower order models are available.
4. Segment the data into K equal length blocks, with
5. Construct the matrix using the estimated parameters.
6. Compute the PSD of each segment by then average them to get the final estimate.
To evaluate the effectiveness of above algorithm, consider an AR (4) process which has a wide dynamic range and a nonparametric Welch estimate that suffers from leakage problem. The simulation results are as shown in the following figures. Using the system function of the model and white Gaussian noise of zero mean and unit variance , input sequence is generated for different lengths of 64,128,256and 512.The signal samples are Prewhitened using the system model to increase the spectral flatness to avoid leakage problem.
Figure 4.36 True estimate of AR(4) process for N=64.
Figure 4.37 Welch estimate, the two peaks are not resolve for N=64.
Figure 4.38 Prewhitening estimate, two peaks are resolved for N=64.
Figure 4.39 Comparison of three estimates for N=64.
Figure 4.40 True estimate of AR(4) process for N=128.
Figure 4.41 Welch estimate, the two peaks are not resolve for N=128.
Figure 4.42 Prewhitening estimate, two peaks are resolved for N=128.
Figure 4.43 Comparison of three estimates for N=128.
Figure 4.44 True estimate of AR(4) process for N=256.
Figure 4.45 Welch estimate, the two peaks are not resolve for N=256.
Figure 4.46 Prewhitening estimate, two peaks are resolved for N=256.
Figure 4.47 Comparison of three estimates for N=256.
Figure 4.48 True estimate of AR(4) process for N=512.
Figure 4.49 Welch estimate, the two peaks are not resolve for N=512.
Figure 4.50 Prewhitening estimate, two peaks are resolved for N=64.
Figure 4.51 Comparison of three estimates for N=512.
As we know, the nonparametric methods do not resolve the two peaks in true spectrum and suffers from leakage at high frequencies. Hence the combination of nonparametric with parametric resolves two peaks with ease also follows the true spectrum. Therefore the use of parametric method as a preprocessor is highly recommended in the wide dynamic range of spectral estimations.
4.4 Power spectrum estimation using interpolation techniques:
The power spectrum estimation of uneven and non uniformly sampled random sequences can be carried out by least squares periodogram (LSP), and coherent sampling methods. The proposed algorithm for the estimation of nonuniform random sequences uses the interpolation methods as resampling methods thus eliminates the low pass filtering effects and also Lomb transforms as well as weighted least squares methods are suggested.
The computational methods used in digital computer for the evaluation of the of a library functions, such as sin(x), cos (x), requires polynomial approximations using Taylor series. The data required to construct a Taylor polynomial is the value of the function f(x) and its derivative. The main disadvantage of this method is to know the higher order derivatives which are hard to compute.
In statistical signal analysis and scientific analysis arise the situations where the function y=f(x) is available only for N+1 tabulated data values, and a technique is needed to approximate the function f(x) at nontabulated abscissas. If there is a significant amount of errors in the tabulated values the curve fitting techniques are used.On the other hand if the points are known to have a high degree of accuracy, then a polynomial curve p(x) that passes through them is considered. When the polynomial approximation is considered within an interval, the approximation p(x) is called an interpolated value. If the approximation is considered outside the interval, then p(x) is called an extrapolated value. Polynomials are used to design algorithms to approximate functions, for numerical differentiation, for numerical integration and for making computer drawn curves that must pass through specified points.
Polynomial interpolation for a set N+1 points is generally not quite approximated. A polynomial of degree N can have N-1 number of maxima and minima, and the graph can wiggle in order to pass through the points. Another method is to piece together the graphs of lower degree polynomials and interpolate between the successive nodes. The set of the function forms a piecewise polynomial curve. Interpolation is a technique of making a perfect approximation of given function with in the interval of values.
The different interpolation techniques involve linear interpolation and cubic spline interpolation. The linear interpolation involves the linear relationship of the interval values whereas the cubic interpolation involves the nonlinear relationship of the interval values. The different cubic spline technique involves clamped spline, natural spline, extrapolated spline, parabolically terminated spline and endpoint curvature adjusted spline. A practical feature of splines is the minimum of oscillatory behavior that they posses.
Unevenly and Nonuniformly sampled data sequences are not sampled with the Nyquist rates. Hence these data sequences require resampling methods. Linear interpolation and cubicspline interpolation techniques are employed as resampling methods to estimate the power of the nonuniform data sequences.
The least squares spectral analysis is a technique of estimating the power spectrum of nonuniformly sampled sequences based on the least squares fit of sinusoids to data sequences.In this method data sequences are approximated using the weighted sum of sinusoidal frequency components using a linear regression method or a least squares fit method. The number of sinusoids that are used for approximation should be less than or equal to the number of data samples.
A data vector Y can be represented as a weighted sum of sinusoids as. The elements of the matrix A can be calculated by evaluating each function at the sampling time instants, with the weighted vector. The weighted vector x is chosen such that to minimize the sum of squared errors in approximating the data vector Y and the solution for X is given as a closed form where the matrix is a diagonal matrix. Then the power spectrum estimate using the least squares spectral analysis can be given by
The algorithm for power spectrum estimation using interpolation techniques is as follows.
Nonuniform data sequences are generated for different lengths of 'N'.
Interpolate the data sequences with in the interval of values.
By applying the Linear as well as Cubic Spline interpolation techniques with in the interval, the data sequence is approximated as x (n).
The power spectrum of the interpolated data sequence x(n) is calculated employing the nonparametric methods.
The Lomb periodogram is evaluated for the nonuniform data sequence where is defined by
The weighted least squares periodogram can be given by
, where 'a' and 'b' are data
The simulations results for the nonuniformly data sequences of white Gaussian noise, real sinusoids in Gaussian noise and narrow band sinusoidal components in broad band noise are illustrated as follows.