Application of RELAX in Line Spectrum Estimation

Abstract

In the field of signal processing, line spectrum estimation is a classic research topic and is widely applied in the fields of communications, radar, sonar, seismology, etc. [1–14]. In these areas, the signal being processed can often be represented using a sinusoidal signal model. As a result, scholars around the world have attempted to solve the parameter estimation problem of sinusoidal signals, focusing primarily on issues of accuracy and computational complexity. The earliest paper regarding this subject can be traced back to an article published by Prony in 1795 [15].

Appendices

Appendix 3.1: CRB for Sinusoidal Signal Parameter Estimation

When the additive noise is a zero-mean Gaussian white noise, [40, 41] give the CRB of sinusoidal signal parameter estimation. The CRB of the corresponding sinusoidal parameter estimation is given below when the additive noise follows a zero-mean Gaussian color noise with unknown covariance matrix $\varvec{Q}$.

The observation data vector can be expressed as

$$ \varvec{y} = \varvec{\varOmega \alpha } + \varvec{e} $$

(3.79)

where, for a one-dimensional sinusoidal signal, $\varvec{y}$ is defined by (2.80); $\varvec{\varOmega}$ equals $\varvec{A}$ in (2.84). For a two-dimensional sinusoidal signal, $\varvec{y}$ is defined by (3.17); $\varvec{\varOmega}= \left[ {\varvec{a}(f_{1} ) \otimes \overline{\varvec{a}} (\bar{f}_{1} ),\varvec{a}(f_{2} ) \otimes \overline{\varvec{a}} (\bar{f}_{2} ), \ldots ,\varvec{a}(f_{P} ) \otimes \overline{\varvec{a}} (\bar{f}_{P} )} \right]$ and $\varvec{e}$ are defined as a noise vector; Let $\varvec{Q}$ denote the covariance matrix for noise vector $\varvec{e}$.

The variables in the likelihood function of $\varvec{y}$ have unknown elements in $\varvec{Q}$, sinusoidal signal frequencies, and the real and imaginary parts of the sinusoidal signal amplitude. The $(i,j)$ th element of the Fisher information matrix for Gaussian distribution parameter estimation can be given by the following extended Slepian-Bangs formula [57] (see Appendix 1.2)

$$ \left\{ \varvec{J} \right\}_{ij} = {\text{trace}}\left( {\varvec{Q}^{ - 1} \varvec{Q}_{i}^{\prime } \varvec{Q}^{ - 1} \varvec{Q}_{j}^{\prime } } \right) + 2\text{Re} \left[ {\left( {\varvec{\alpha}^{\text{H}}\varvec{\varOmega}^{\text{H}} } \right)^{\prime }_{i} \varvec{Q}^{ - 1} \left( {\varvec{\varOmega \alpha }} \right)^{\prime }_{j} } \right] $$

(3.80)

where $\varvec{X}_{i}^{\prime }$ denotes the gradient of $\varvec{X}$ with respect to the ith unknown parameter; ${\text{trace(}}\varvec{X} )$ denotes the trace of $\varvec{X}$; ${\text{Re(}}\varvec{X} )$ denotes the real part of $\varvec{X}$, and ${\text{Im(}}\varvec{X} )$ denotes the imaginary part of $\varvec{X}$. Note that since $\varvec{Q}$ is not dependent on the parameters in $(\varvec{\varOmega \alpha })$, and $(\varvec{\varOmega \alpha })$ is not dependent on the elements in $\varvec{Q}$, $\varvec{J}$ is a block diagonal matrix. This simple finding means that as long as we replace $\varvec{\varOmega}$ with $\varvec{Q}^{ - 1/2}\varvec{\varOmega}$ and $\varvec{\varOmega}_{i}^{\prime }$ with $\varvec{Q}^{ - 1/2}\varvec{\varOmega}_{i}^{\prime }$ in the CRB formula in the case of Gaussian white noise, we can obtain the CRB for sinusoidal signal parameter estimation in the case of colored noise.

Let

$$ \varvec{G} = 2\varvec{\varOmega}^{\text{H}} \varvec{Q}^{ - 1}\varvec{\varOmega} $$

(3.81)

$$ \varvec{\varDelta } = 2\varvec{\varOmega}^{\text{H}} \varvec{Q}^{ - 1} \varvec{DP} $$

(3.82)

where for one-dimensional sinusoidal signal $\varvec{P} = {\text{diag}}(\varvec{\alpha})$, $\varvec{D}$’s pth column ($p = 1,\;2,\; \ldots ,\;P$) is $\partial \varvec{a}(f_{p} )/\partial f_{p}$. For two-dimensional sinusoidal signal, $\varvec{P} = {\text{diag}}(\varvec{\alpha}) \otimes \varvec{I}_{2}$, where $\varvec{I}_{2}$ is a two-dimensional identity matrix; $\varvec{D}$’s $(2p - 1)$ th column and $(2p)$ th column are $\partial \left[ {\varvec{a}(f_{p} ) \otimes \bar{\varvec{a}}(\bar{f}_{p} )} \right]/\partial {f}_{p}$, and $\partial \left[ {\varvec{a}(f_{p} ) \otimes \bar{\varvec{a}}(\bar{f}_{p} )} \right]/\partial \bar{f}_{p}$ respectively.

Next, we let

$$ \varvec{\varGamma}= 2\text{Re} \left[ {\varvec{P}^{\text{H}} \varvec{D}^{\text{H}} \varvec{Q}^{ - 1} \varvec{DP}} \right] $$

(3.83)

$$ \varvec{\eta}= \left[ {\begin{array}{*{20}c} {\text{Re}^{\text{T}} (\varvec{\alpha}),} & {\text{Im}^{\text{T}} (\varvec{\alpha}),} & {\varvec{f}^{\text{T}} } \\ \end{array} } \right]^{\text{T}} $$

(3.84)

For a one-dimensional sinusoidal signal, $\varvec{f} = [f_{1} ,\;f_{2} ,\; \ldots ,\,f_{P} ]^{\text{T}}$; for a two-dimensional sinusoidal signal, $\varvec{f} = [f_{1} ,\; \, \bar{f}_{1} ,\; \, f_{2} ,\;\bar{f}_{2} ,\; \ldots ,\;f_{P} ,\;\bar{f}_{P} ]^{\text{T}}$. Similar to the results obtained by [18, 40], the CRB of $\varvec{\eta}$ can be obtained as

$$ {\text{CRB}}\left(\varvec{\eta}\right){ = }\left[ {\begin{array}{*{20}c} {\text{Re} \left( \varvec{G} \right)} & { - \text{Im} \left( \varvec{G} \right)} & {\text{Re} \left( \varvec{\varDelta } \right)} \\ {\text{Im} \left( \varvec{G} \right)} & {\text{Re} \left( \varvec{G} \right)} & {\text{Im} \left( \varvec{\varDelta } \right)} \\ {\text{Re}^{\text{T}} \left( \varvec{\varDelta } \right)} & {\text{Im}^{\text{T}} \left( \varvec{\varDelta } \right)} &\varvec{\varGamma}\\ \end{array} } \right]^{ - 1} $$

(3.85)

Notice when $\varvec{Q} = \sigma^{2} \varvec{I}$, the above results are consistent with the results in [40, 41].

Note that when the additive noise follows an AR or ARMA random process, the CRB of the sinusoidal parameter estimate can also be calculated by (3.85), where the noise covariance matrix $\varvec{Q}$ can be calculated using AR or ARMA noise model parameters [18].

Appendix 3.2: CRB for Exponentially Decaying Sinusoidal Signal Parameter Estimation

When the additive noise is a zero-mean colored noise random process and the covariance matrix $\varvec{Q}$ is unknown, the CRB of the exponentially decaying sinusoidal signal parameters is given below. Assuming that the observation data vector can be expressed as

$$ \varvec{y} = \varvec{\varPhi \alpha } + \varvec{e} $$

(3.86)

The definition of $\varvec{y}$ is shown in (3.31), and $\varvec{e}$ is a noise vector. $\varvec{\varPhi}$ has the following form

$$ \varvec{\varPhi}= \left[ {\varvec{\varphi }_{1} ,\varvec{\varphi }_{2} , \ldots ,\varvec{\varphi }_{P} } \right] $$

(3.87)

The definition of $\varvec{\varphi }_{{{\kern 1pt} p}}$ is shown in (3.32), and we have

$$ \varvec{\alpha}= \left[ {\alpha_{1} ,\alpha_{2} , \ldots ,\alpha_{P} } \right]^{\text{T}} $$

(3.88)

The unknown parameters for the likelihood function of $\varvec{y}$ are the following: elements in $\varvec{Q}$, the real and imaginary parts of the amplitude, the attenuation factor and the frequency. The $(i,\;j)$ th element of the Fisher information matrix can be given by the following extended Slepian-Bangs formula [57] (see Appendix 1.2)

$$ \left\{ \varvec{J} \right\}_{ij} = {\text{trace}}\left( {\varvec{Q}^{ - 1} \varvec{Q}_{i}^{\prime } \varvec{Q}^{ - 1} \varvec{Q}_{j}^{\prime } } \right) + 2\text{Re}\left[ {\left( {\varvec{\alpha}^{\text{H}}\varvec{\varPhi}^{\text{H}} } \right)^{\prime }_{i} \varvec{Q}^{ - 1} \left( {\varvec{\varPhi \alpha }} \right)^{\prime }_{j} } \right] $$

(3.89)

Note that since $\varvec{Q}$ is not dependent on the parameters in $(\varvec{\varPhi \alpha })$, and $(\varvec{\varPhi \alpha })$ is not dependent on the elements in $\varvec{Q}$, $\varvec{J}$ is a block diagonal matrix. Then, we know that the CRB of the attenuated sinusoidal signal parameter estimation can be determined by the second term in the right of Eq. (3.89). Let

$$ \varvec{G}_{11} = 2\varvec{\varPhi}^{\text{H}} \varvec{Q}^{ - 1}\varvec{\varPhi} $$

(3.90)

$$ \varvec{G}_{12} = 2\varvec{\varPhi}^{\text{H}} \varvec{Q}^{ - 1} \varvec{C\varPhi \varTheta } $$

(3.91)

$$ \varvec{G}_{21} = \varvec{G}_{12}^{\text{H}} $$

(3.92)

$$ \varvec{G}_{22} = 2\varvec{\varTheta}^{\text{H}}\varvec{\varPhi}^{\text{H}} \varvec{CQ}^{ - 1} \varvec{C\varPhi \varTheta } $$

(3.93)

where $\varvec{C} = {\text{diag}}\left\{ {0,\;1,\; \ldots ,\;N} \right\}$, $\varvec{\varTheta}= {\text{diag}}\left\{\varvec{\alpha}\right\}$. Let

$$ \varvec{\eta}= \left[ {\begin{array}{*{20}c} {\text{Re}^{\text{T}} \left(\varvec{\alpha}\right),} & {\text{Im}^{\text{T}} \left(\varvec{\alpha}\right),} & {\varvec{d}^{\text{T}} ,} & {\varvec{\omega}^{\text{T}} } \\ \end{array} } \right]^{\text{T}} $$

(3.94)

where

$$ \varvec{d} = \left[ {d_{1} ,d_{2} , \ldots ,d_{P} } \right]^{\text{T}} $$

(3.95)

$$ \varvec{\omega}= \left[ {\omega_{1} ,\omega_{2} , \ldots ,\omega_{P} } \right]^{\text{T}} $$

(3.96)

We can obtain the formula to calculate CRB [19]

$$ \varvec{CRB}(\varvec{\eta}) = \left[ {\begin{array}{*{20}c} {\text{Re} (\varvec{G}_{11} )} & { - \text{Im} (\varvec{G}_{11} )} & {\text{Re} (\varvec{G}_{12} )} & { - \text{Im} (\varvec{G}_{12} )} \\ {\text{Im} (\varvec{G}_{11} )} & {\text{Re} (\varvec{G}_{11} )} & {\text{Im} (\varvec{G}_{12} )} & {\text{Re} (\varvec{G}_{12} )} \\ {\text{Re} (\varvec{G}_{21} )} & { - \text{Im} (\varvec{G}_{21} )} & {\text{Re} (\varvec{G}_{22} )} & { - \text{Im} (\varvec{G}_{22} )} \\ {\text{Im} (\varvec{G}_{21} )} & {\text{Re} (\varvec{G}_{21} )} & {\text{Im} (\varvec{G}_{22} )} & {\text{Re} (\varvec{G}_{22} )} \\ \end{array} } \right]^{ - 1} $$

(3.97)

Appendix 3.3: CRB for Arbitrary Envelope Sinusoidal Signal Parameter Estimation

Next, we analyze the performance of NLS on the frequency estimation of arbitrary envelope sinusoidal waves. The following procedure gives the estimated asymptotic variance in (3.68).

Theorem: Assuming that in (3.47) $x(n)$ follows a Gaussian stationary random process and $e(n)$ follows a Gaussian white noise with zero mean and $\sigma_{e}^{2}$ variance; and the asymptotic variance of $N^{{{3 \mathord{\left/ {\vphantom {3 2}} \right. \kern-0pt} 2}}} \times (\hat{\omega } - \omega )$ is given by the following equation [20]

$$ \begin{aligned} \mathop {\lim }\limits_{N \to \infty } N^{3} \xi \left( {\hat{\omega } - \omega } \right)^{2} & = \frac{{6\sigma_{e}^{2} }}{{r_{a} \left( 0 \right)}}\left[ {1 + \frac{1}{2}\frac{{\sigma_{e}^{2} }}{{r_{a} \left( 0 \right)}}} \right] \\ & = \frac{6}{\text{SNR}}\left[ {1 + \frac{1}{2}{\text{SNR}}_{{}}^{ - 1} } \right] \\ \end{aligned} $$

(3.98)

where $a(t) = |\alpha |x(n)$ and ${\text{SNR}} = r_{a} (0)/\sigma_{e}^{2}$. It should be noted that the variance of the NLS frequency estimation method described above does not depend on the parameter model of the envelope [65,66,67]; instead it relies on $r_{a} (0)$ and $r_{a} (0) \triangleq {\text{E[}}a^{2} (n) ]$.

The estimation variance given on $\hat{\omega }$ described above is compared with the CRB. If the assumed covariance matrix for $a(n)$ follows a Gaussian stationary process that can be parameterized by a finite dimensional vector $\varvec{\theta}$, and when the SNR is higher, the CRB can be expressed as [1]

$$ {\text{CRB}} = \frac{1}{{N^{3} }}\frac{{6\sigma_{e}^{2} }}{{r_{a} (0)}} $$

(3.99)

When the SNR is high, we can conclude that the variance of the NLS frequency estimation is very close to the CRB. It is observed that, as compared to (3.99), the NLS frequency estimation variance in (3.98) does not depend on a particular model of envelope, nor does it need to have $a(n)$ follow a finite dimensional parametric model [20, 68].