Frequency and power estimation of sinusoids using eigen-approach

Frequency and power estimation of sinusoidal signals, has captured the attention of researchers due to its important applications. This paper involves a frequency and power estimation method of unidentified number of source sinusoidal signals using Eigen technique. The eigenvector corresponding to the correlation matrix’s minimum eigenvalue yields the minimum power of the FIR filter’s output power. The corresponding roots are determined using the filter polynomial and root elimination technique is used to determine the pseudospectrum. Consequently, by deploying the pseudospectrum and filter polynomial, the real power spectrum is derived.


Introduction
Frequency and power evaluation of several sinusoids embedded in wide-band signals has been an important issue in engineering [1] [2]. There has been extensive research regarding different estimation approaches ; as estimation methods have many uses in various domains such as biomedical engineering, image processing, radar applications, and communications [1] [3]. Nevertheless, performance effectiveness of such approaches depends on the estimator's precision and complexity.
Eigen-approach methods in the literature grabbed the researchers' attention due to estimating with a high resolution; nevertheless, the majority of such techniques require the exact number of signals in order to distinct signal and noise subspace, which is a main drawback. ESPRIT, MUSIC, and Pisarenko harmonic decomposition (Phd) are examples of these techniques. [4] [5]. These methods are considered poor estimation techniques for certain applications of which the number of source signals is unknown.
Additionally, the usage of random search optimization processes towards estimating frequencies and power of sinusoidal signals has been offered; the author in [6] uses a population based stochastic method, Particle Swarm Optimization (PSO) to estimate diverse sinusoids' frequencies along with their real power, without previous knowledge of the signals' number, amplitude, nor phase [7]. Furthermore, random search is employed to evaluate the direction-of-arrival and power estimation of source signals [8].
The current paper presents a non-parametric method that estimates the frequency and power of several sinusoids without knowing the number of signals in advance using an eigen-approach. The transfer function of the filter with weight vectors of the eigenvector conforming to the lowest eigenvalue of the correlation matrix produces the filter's minimum output power. Moreover, the transfer function's roots are placed on the unit circle [9]. Consequently, the angle of the roots reflects probable signal occurrences on the unit circle. On the basis of this criteria, the locations of the roots and the real signal power are calculated by excluding one root at a time and recalculating the output signal power.

Problem formulation
The received signal that arrives at the filter constitutes of sinusoidal signals alongside basic white gaussian noise and can be expressed as: where s(k) is the summation of M sinusoidal signals and g(k) denotes the noise signal. The input signal x(k) at sample (k) is : A m ,f m and represents the sinusoidal signal's amplitude, frequency, and the phase. Sampling frequency is presented by fs and g(k) is additive white noise with mean of zero and power of . The correlation matrix can be formulated as: where is the correlation matrix of the input Identity matrix , I, consists of (N+1) * (N+1) elements. Symbol N is the FIR filter order and V is the (N + 1) * M delay matrix. Matrix V is expressed as: is a column vector resembling the delay of the normalized angular frequency, , i.e., The FIR filter's output is made of the weighted linear arrangement of current input and previous values of the input. Using a for the coefficients of the filter a= 1 of order (N+1), the filter's output ,y k , is Using K snapshots (assuming ergodic random processes signals) and x(k) containing the sampled data, the output's power can be computed as Considering the order of the filter, N, is more or equal to the signals' number M, then the output power of the noise signal, Pg, can be stated as P g , 0 0 (2019) https://doi.org/10.1051/matecconf /201927201008 MATEC Web of Conferences 272 ICFMCE 2018 10 8 The filter coefficient vector, a0 , resembles the roots on the unit circle that overlap with M signals' frequencies. The noise power is evaluated as follows: Using the method discussed in [9] [10] the problem can be expressed as follows: H XX Minimize a aR a (11) subject to 1 The autocorrelation Matrix is a Toeplitz matrix [11] and the optimization problem is resolved using a Lagrange multiplier.
To minimalize the fitness function given in (11), the eigenvector a0 that corresponds to the minimum eigenvalue λ should be found; therefore the resultant power will be minimized to the noise power level as shown below:

Frequency and power estimation
Eigen technique is utilized to minimize the output power function. The filter's transfer function is expressed as follows: where is the n th root angular frequency. When N roots coincide with M sources, the fitness function is minimized.
By excluding one of the roots and recalculating the corresponding coefficient vector, the output power at the frequency of the excluded root is estimated. As a result, the estimated power indicates the presence of actual signal or noise.
The method for both frequency and power evaluation is clarified as follows: The output power at the frequency of the excluded root can be written as follows: Once the angular frequencies are found, the estimated real power spectrum is achieved , 0 0 (2019) https://doi.org/10.1051/matecconf /201927201008 MATEC Web of Conferences 272 ICFMCE 2018 10 8 by plotting P n versus . Therefore, the exact number of the incoming source signals can be determined from the estimated power spectrum.

Results
To determine the validity of the proposed approach, this involved analyzing an eighth order FIR filter. Consequently, we use eight roots to estimate the frequencies of utmost eight input signals. To find the output's minimum received power, Eigen technique is used. Equations (14) and (17) are used to find the roots and the source signals' power, respectively.
Two scenarios of input source signals are simulated with zero dBm noise level ( ) and 100 samples to show the effectiveness of the proposed method. The first scenario constitutes three input signals of 10 dB SNR with two closely spaced signals. As for the second scenario, eight input signals with 10 dB and 6 dB SNR levels are used. Figure 1 shows the estimated frequencies at 0.1,0.125, and 0.6 cycles/sample with 10 dBm power levels. Eight power values at the estimated locations are calculated. Relating the output power values and the estimated noise power level; the real signal frequency is estimated. The estimated normalized frequencies are 0.1001, 0.1234 and 0.5998 cycles/sample. The proposed method estimates the power of the three input signals as shown in Figure 1 with accurate power levels of 9.8629 , 10.1505 , and 9.7772 dBm. Figure 2 shows the pseudospectrum for the same three signals at the same frequencies of Figure 1, using the MUSIC algorithm with 100 snapshots. Comparing the results shown by Figures 1 and 2, the estimated frequencies using the proposed method are as accurate as the MUSIC algorithm. Nevertheless, MUSIC algorithm is a parametric approach; needs the number of input signals in advance.  For the second scenario, eight source signals having frequencies of 0.1, 0.125, 0.25, 0.35, 0.45, 0.65, 0.80, and 0.90 cycles/sample respectively are simulated. Figure 3 shows that the estimated frequencies and the real power of the eight signals are estimated accurately.
For comparison reasons, Figure 4 shows the pseudospectrum using the MUSIC algorithm when the same eight signals are assumed. All signal frequencies are resolved, yet the closely spaced ones are not as accurate.   Table 1 shows the estimated frequencies for the proposed approach and for the MUSIC Algorithm. Both are accurate in estimating the frequency; however, the proposed approach estimates the exact power of these signals as well. To determine the influence of different signal-to-noise ratios on both frequency and power estimation, an input sinusoidal signal of 0.15 cycles/sample frequency with different SNR levels is considered. Table 2 shows the estimated frequency, , and the estimated power, , for different SNR and 100 samples. The results show a good level of accuracy.

Conclusions
Due to the importance of frequency and power estimation of sinusoidal signals, a new technique has been proposed in this paper using the eigen-approach. The frequencies of unknown input sinusoidal signals along with their real power values are estimated. The Transfer function of the FIR filter with the eigenvector corresponding to the minimum eigenvalue of the correlation matrix produces the minimum output power of the FIR filer. From the roots' locations of the transfer function, the estimated power is calculated by removing the roots one at a time and recalculating the output power. The new technique estimates the signals' frequencies and power accurately without any need for prior estimate of the number of input source signals.