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New Procedure for Estimation of Amplitude and Phase of Analog Multiharmonic Signal Based on the Differential Irregular Samples

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Abstract

This paper is concerned with the estimation of amplitude and phase of an analog multi-harmonic signal based on a series of differential values of the signal. To this end, assuming that the signal fundamental frequency is known beforehand (i.e., estimated in an independent stage), a complexity-reduced scheme is proposed here. The reduction in complexity is achieved owing to completely new analytical and summarised expressions that enable quick estimation at a low numerical error. The proposed algorithm for the calculation of the unknown parameters requires O((2M)2) flops, while the straightforward solution of the obtained equations takes O((2M)3) flops, where M is number of harmonic coefficients. It is proved that the estimation performance of the proposed algorithm can attain Cramer-Rao lower bound (CRLB) for sufficiently high signal-to-noise ratios. It is applied in signal reconstruction, spectral estimation, system identification, as well as in other important signal processing problems. The paper investigates the errors related to the signal parameter estimation, and computer simulation demonstrates the accuracy of these algorithms.

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Acknowledgments

The author wishes to thank to the Ministry of Education and Science of the Republic of Serbia for its support of this work provided within the projects 42009 and OI-172057.

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Authors and Affiliations

  1. Faculty of Technical Sciences Čačak, University of Kragujevac, Svetog Save 65, 32000, Čačak, Serbia

    Predrag B. Petrović

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  1. Predrag B. Petrović
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Correspondence to Predrag B. Petrović.

Appendix

Appendix

For 1 ≤ p ≤ 2M ∧ 1 ≤ q ≤ M:

$$ \begin{array}{l}{\mathbf{F}}_p^q={\left|\begin{array}{ccccccccc}\hfill \cos {\varphi}_1\hfill & \hfill \dots \hfill & \hfill \cos \left(q-1\right){\varphi}_1\hfill & \hfill \cos \left(q+1\right){\varphi}_1\hfill & \hfill \dots \hfill & \hfill \cos M{\varphi}_1\hfill & \hfill \sin {\varphi}_1\hfill & \hfill \dots \hfill & \hfill \sin M{\varphi}_1\hfill \end{array}\right|}_p=\\ {}=\frac{e^{-M\frac{\pi }{2}i}}{2^{2M-1}}{\left|\begin{array}{ccccccccc}\hfill {e}^{\varphi_1i}+{e}^{-{\varphi}_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{\left(q-1\right){\varphi}_1i}+{e}^{-\left(q-1\right){\varphi}_1i}\hfill & \hfill {e}^{\left(q+1\right){\varphi}_1i}+{e}^{-\left(q+1\right){\varphi}_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{M{\varphi}_1i}+{e}^{-M{\varphi}_1i}\hfill & \hfill {e}^{\varphi_1i}-{e}^{-{\varphi}_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{M{\varphi}_1i}-{e}^{-M{\varphi}_1i}\hfill \end{array}\right|}_p=\\ {}=\frac{e^{-M\frac{\pi }{2}i}}{2^M}{\left|\begin{array}{ccccccccccccc}\hfill {e}^{-{\varphi}_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{-\left(q-1\right){\varphi}_1i}\hfill & \hfill {e}^{-\left(q+1\right){\varphi}_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{-M{\varphi}_1i}\hfill & \hfill {e}^{\varphi_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{\left(q-1\right){\varphi}_1i}\hfill & \hfill {e}^{q{\varphi}_1i}-{e}^{-q{\varphi}_1i}\hfill & \hfill {e}^{\left(q+1\right){\varphi}_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{M{\varphi}_1i}\hfill \end{array}\right|}_p=\\ {}=\frac{e^{-M\frac{\pi }{2}i}}{2^M}\left\{\begin{array}{l}{\left|\begin{array}{ccccccccc}\hfill {e}^{-{\varphi}_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{-\left(q-1\right){\varphi}_1i}\hfill & \hfill {e}^{-\left(q+1\right){\varphi}_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{-M{\varphi}_1i}\hfill & \hfill {e}^{\varphi_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{M{\varphi}_1i}\hfill \end{array}\right|}_p+\\ {}+{\left(-1\right)}^M{\left|\begin{array}{ccccccccc}\hfill {e}^{-{\varphi}_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{-M{\varphi}_1i}\hfill & \hfill {e}^{\varphi_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{\left(q-1\right){\varphi}_1i}\hfill & \hfill {e}^{\left(q+1\right){\varphi}_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{M{\varphi}_1i}\hfill \end{array}\right|}_p\end{array}\right\}=\\ {}=\frac{{\left(-1\right)}^{\frac{M\left(M+1\right)}{2}}}{2^M}{e}^{-M\frac{\pi }{2}i}\left\{\begin{array}{l}{\left|\begin{array}{ccccccccc}\hfill {e}^{-M{\varphi}_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{-{\varphi}_1i}\hfill & \hfill {e}^{\varphi_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{\left(q-1\right){\varphi}_1i}\hfill & \hfill {e}^{\left(q+1\right){\varphi}_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{M{\varphi}_1i}\hfill \end{array}\right|}_p-\\ {}-{\left|\begin{array}{ccccccccc}\hfill {e}^{-M{\varphi}_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{-\left(q+1\right){\varphi}_1i}\hfill & \hfill {e}^{-\left(q-1\right){\varphi}_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{-{\varphi}_1i}\hfill & \hfill {e}^{\varphi_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{M{\varphi}_1i}\hfill \end{array}\right|}_p\end{array}\right\}=\\ {}=\frac{{\left(-1\right)}^{\frac{M\left(M+1\right)}{2}}}{2^M}{e}^{-M\frac{\pi }{2}i}{e}^{-M\left({\varphi}_1+\dots +{\varphi}_{p-1}+{\varphi}_{p+1}+\dots +{\varphi}_{2M}\right)i}\left\{\begin{array}{l}{\left|\begin{array}{ccccccccc}\hfill 1\hfill & \hfill \dots \hfill & \hfill {e}^{\left(M-1\right){\varphi}_1i}\hfill & \hfill {e}^{\left(M+1\right){\varphi}_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{\left(M+q-1\right){\varphi}_1i}\hfill & \hfill {e}^{\left(M+q+1\right){\varphi}_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{2M{\varphi}_1i}\hfill \end{array}\right|}_p-\\ {}-{\left|\begin{array}{ccccccccc}\hfill 1\hfill & \hfill \dots \hfill & \hfill {e}^{\left(M-q-1\right){\varphi}_1i}\hfill & \hfill {e}^{\left(M-q+1\right){\varphi}_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{\left(M-1\right){\varphi}_1i}\hfill & \hfill {e}^{\left(M+1\right){\varphi}_1i}\hfill & \hfill \dots \hfill & \hfill {e}^{2M{\varphi}_1i}\hfill \end{array}\right|}_p\end{array}\right\}=\\ {}=\frac{{\left(-1\right)}^{\frac{M\left(M+1\right)}{2}}}{2^M}{e}^{-M\frac{\pi }{2}i}{e}^{-M\left({\varphi}_1+\dots +{\varphi}_{p-1}+{\varphi}_{p+1}+\dots +{\varphi}_{2M}\right)i}{\left({\varDelta}_{2M+1,M+1}^{\left(p,M+q\right)}-{\varDelta}_{2M+1,M+1}^{\left(p,M-q+1\right)}\right)}_{x_j={e}^{\varphi_ji}}\end{array} $$
(24)

It follows that:

$$ {\mathbf{F}}_p^q=\frac{{\left(-1\right)}^{\frac{M\left(M+1\right)}{2}}}{2^M}{e}^{-M\frac{\pi }{2}i}{e}^{-M\left({\varphi}_1+\dots +{\varphi}_{p-1}+{\varphi}_{p+1}+\dots +{\varphi}_{2M}\right)i}{\left({\varDelta}_{2M+1,M+1}^{\left(p,M+q\right)}-{\varDelta}_{2M+1,M+1}^{\left(p,M-q+1\right)}\right)}_{x_j={e}^{\varphi_ji}}\kern0.5em for\kern0.5em 1\le p\le 2M\wedge 1\le q\le M $$
(25)

where Δ (r,s)2M + 1,M + 1 is the determinant obtained from Δ 2M + 1,M + 1 after the r row and s column have been eliminated.

$$ \begin{array}{l}{\varDelta}_{2M+1,M+1}=\left|\begin{array}{ccccccc}\hfill 1\hfill & \hfill {x}_1\hfill & \hfill \dots \hfill & \hfill {x}_1^{M-1}\hfill & \hfill {x}_1^{M+1}\hfill & \hfill \dots \hfill & \hfill {x}_1^{2M}\hfill \\ {}\hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill & \hfill \dots \hfill \\ {}\hfill 1\hfill & \hfill {x}_{2M}\hfill & \hfill \dots \hfill & \hfill {x}_{2M}^{M-1}\hfill & \hfill {x}_{2M}^{M+1}\hfill & \hfill \dots \hfill & \hfill {x}_{2M}^{2M}\hfill \end{array}\right|\Rightarrow \\ {}{\varDelta}_{2M+1,M+1}=\left({\displaystyle \prod_{j=k+1}^{2M}{\displaystyle \prod_{k=1}^{2M-1}\left({x}_j-{x}_k\right)}}\right)\left({x}_1{x}_2\dots {x}_{2M}\right){\displaystyle \sum \frac{1}{{\left({x}_1{x}_2\dots {x}_{2M}\right)}_M}}\end{array} $$
(26)

When we determinate Δ (r,s)2M + 1,M + 1 , we must eliminate r row and s column from Δ 2M + 1,M + 1, and if Δ 2M + 1,M + 1 is developed by r row, what we obtain is that D r,s  = (−1)r + s Δ (r,s)2M + 1,M + 1 is the coefficient in x s − 1 r (1 ≤ s ≤ M), i.e. the coefficient beside x s r if M + 1 ≤ s ≤ 2M.

$$ {\varDelta}_{2M+1,M+1}={\left(-1\right)}^r\left({x}_r-{x}_1\right)\dots \left({x}_r-{x}_{r-1}\right)\left({x}_r-{x}_{r+1}\right)\dots \left({x}_r-{x}_{2M}\right)\left({\displaystyle \prod_{\begin{array}{l}1\le j\le 2M\\ {}k,j\ne r\end{array}}{\displaystyle \prod_{\begin{array}{l}1\le k\le 2M\hfill \\ {}k\prec j\hfill \end{array}}\left({x}_j-{x}_k\right)}}\right)\left({x}_1{x}_2\dots {x}_{2M}\right){\displaystyle \sum \frac{1}{{\left({x}_1{x}_2\dots {x}_{2M}\right)}_M}} $$
(27)

Here is:

$$ \begin{array}{l}\left({x}_1{x}_2\dots {x}_{2M}\right){\displaystyle \sum \frac{1}{{\left({x}_1{x}_2\dots {x}_{2M}\right)}_M}}=\left({x}_1\dots {x}_{r-1}{x}_{r+1}\dots {x}_{2M}\right)\cdot {x}_r\cdot \left({\displaystyle \sum \frac{1}{{\left({x}_1\dots {x}_{r-1}{x}_{r+1}\dots {x}_{2M}\right)}_M}+\frac{1}{x_r}{\displaystyle \sum \frac{1}{{\left({x}_1\dots {x}_{r-1}{x}_{r+1}\dots {x}_{2M}\right)}_{M-1}}}}\right)\Rightarrow \\ {}{\varDelta}_{2M+1,M+1}^{\left(r,s\right)}=\left({\displaystyle \prod_{\begin{array}{l}1\le j\le 2M\\ {}k,j\ne r\end{array}}{\displaystyle \prod_{\begin{array}{l}1\le k\le 2M\hfill \\ {}k\prec j\hfill \end{array}}\left({x}_j-{x}_k\right)}}\right)\left({x}_1\dots {x}_{r-1}{x}_{r+1}\dots {x}_{2M}\right)\left\{\begin{array}{l}{\displaystyle \sum {\left({x}_1\dots {x}_{r-1}{x}_{r+1}\dots {x}_{2M}\right)}_{2M-s}\cdot {\displaystyle \sum \frac{1}{{\left({x}_1\dots {x}_{r-1}{x}_{r+1}\dots {x}_{2M}\right)}_{M-1}}}}-\\ {}-{\displaystyle \sum {\left({x}_1\dots {x}_{r-1}{x}_{r+1}\dots {x}_{2M}\right)}_{2M-s+1}\cdot {\displaystyle \sum \frac{1}{{\left({x}_1\dots {x}_{r-1}{x}_{r+1}\dots {x}_{2M}\right)}_M}}}\end{array}\right\}\kern0.5em for\kern0.5em 1\le s\le M\end{array} $$
(28)

If we introduce the following symbols:

$$ \begin{array}{l}{V}_t={\displaystyle \sum {\left({x}_1\dots {x}_{r-1}{x}_{r+1}\dots {x}_{2M}\right)}_t}\left|{}_{x_j={e}^{\varphi_ji}}\right.\\ {}{\overline{V}}_t={\displaystyle \sum \frac{1}{{\left({x}_1\dots {x}_{r-1}{x}_{r+1}\dots {x}_{2M}\right)}_t}}\left|{}_{x_j={e}^{\varphi_ji}}\right.\end{array} $$
(29)

It follows that:

$$ \begin{array}{l}{\varDelta}_{2M+1,M+1}^{\left(p,M-q+1\right)}\left({x}_t={e}^{\varphi_ti}\right)=\left({\displaystyle \prod_{\begin{array}{l}1\le j\le 2M\\ {}k,j\ne p\end{array}}{\displaystyle \prod_{\begin{array}{l}1\le k\le 2M\hfill \\ {}k\prec j\hfill \end{array}}\left({x}_j-{x}_k\right)}}\right)\left({x}_1\dots {x}_{p-1}{x}_{p+1}\dots {x}_{2M}\right)\left({V}_{M+q-1}{\overline{V}}_{M-1}-{V}_{M+q}{\overline{V}}_M\right)\left|{}_{x_t={e}^{\varphi_ti}}\right.\kern0.5em for\kern0.5em 1\le q\le M-1,\kern0.5em while\kern0.5em for\\ {}q=M\Rightarrow {V}_{2M}=0;{V}_{2M-1}=\left({x}_1\dots {x}_{p-1}{x}_{p+1}\dots {x}_{2M}\right)\end{array} $$
(30)

We can write that:

$$ \begin{array}{l}{\varDelta}_{2M+1,M+1}^{\left(p,M+q\right)}\left({x}_t={e}^{\varphi_ti}\right)=-\left({\displaystyle \prod_{\begin{array}{l}1\le j\le 2M\\ {}k,j\ne p\end{array}}{\displaystyle \prod_{\begin{array}{l}1\le k\le 2M\hfill \\ {}k\prec j\hfill \end{array}}\left({x}_j-{x}_k\right)}}\right)\left({x}_1\dots {x}_{p-1}{x}_{p+1}\dots {x}_{2M}\right)\left({V}_{M-q-1}{\overline{V}}_{M-1}-{V}_{M-q}{\overline{V}}_M\right)\left|{}_{x_t={e}^{\varphi_ti}}\right.\kern0.5em for\kern0.5em 1\le q\le M-2,\kern0.5em while\kern0.5em for\\ {}q=M-1\Rightarrow {V}_0=1\kern0.5em and\kern0.5em for\kern0.5em q=M\Rightarrow {V}_{-1}=0;{V}_0=1\end{array} $$
(31)

From this, it follows that:

$$ {\mathbf{F}}_p^q=\frac{{\left(-1\right)}^{\frac{M\left(M+1\right)}{2}}}{2^M}{e}^{-M\frac{\pi }{2}i}{e}^{-M\left({\varphi}_1+\dots +{\varphi}_{p-1}+{\varphi}_{p+1}+\dots +{\varphi}_{2M}\right)i}\left({\displaystyle \prod_{\begin{array}{l}1\le j\le 2M\\ {}k,j\ne p\end{array}}{\displaystyle \prod_{\begin{array}{l}1\le k\le 2M\hfill \\ {}k\prec j\hfill \end{array}}\left({x}_j-{x}_k\right)}}\right)\left({x}_1\dots {x}_{p-1}{x}_{p+1}\dots {x}_{2M}\right)\left\{\begin{array}{l}\left({V}_{M+q}{\overline{V}}_M-{V}_{M+q-1}{\overline{V}}_{M-1}\right)+\\ {}+\left({V}_{M-q}{\overline{V}}_M-{V}_{M-q-1}{\overline{V}}_{M-1}\right)\end{array}\right\}\left|{}_{x_t={e}^{\varphi_ti}}\right. $$
(32)
$$ \left({\displaystyle \prod_{\begin{array}{l}1\le j\le 2M\\ {}k,j\ne p\end{array}}{\displaystyle \prod_{\begin{array}{l}1\le k\le 2M\hfill \\ {}k\prec j\hfill \end{array}}\left({x}_j-{x}_k\right)}}\right)\left|{}_{x_t={e}^{\varphi_ti}}\right.={\left(-1\right)}^p\frac{{\displaystyle \prod_{j=k+1}^{2M}{\displaystyle \prod_{k=1}^{2M-1}\left({e}^{\varphi_ji}-{e}^{\varphi_ki}\right)}}}{{\displaystyle \prod_{\begin{array}{l}1\le k\le 2M\\ {}k\ne p\end{array}}\left({e}^{\varphi_pi}-{e}^{\varphi_ki}\right)}} $$
(33)

We can write that:

$$ \begin{array}{l}{\displaystyle \prod_{j=k+1}^{2M}{\displaystyle \prod_{k=1}^{2M-1}\left({e}^{\varphi_ji}-{e}^{\varphi_ki}\right)}}={\left(-1\right)}^M{2}^{M\left(2M-1\right)}{e}^{-M\frac{\pi }{2}i}{e}^{\frac{2M-1}{2}\left({\varphi}_1+\dots +{\varphi}_{2M}\right)i}{\displaystyle \prod_{j=k+1}^{2M}{\displaystyle \prod_{k=1}^{2M-1} \sin \frac{\varphi_j-{\varphi}_k}{2}}}\\ {}{\displaystyle \prod_{\begin{array}{l}1\le k\le 2M\\ {}k\ne p\end{array}}\left({e}^{\varphi_pi}-{e}^{\varphi_ki}\right)}={2}^{2M-1}{e}^{\frac{2M-1}{2}\pi i}{e}^{\frac{2M-1}{2}{\varphi}_pi}{e}^{\frac{1}{2}\left({\varphi}_1+\dots {\varphi}_{p-1}+{\varphi}_{p+1}+\dots +{\varphi}_{2M}\right)i}{\displaystyle \prod_{\begin{array}{l}1\le k\le 2M\\ {}k\ne p\end{array}} \sin \frac{\varphi_p-{\varphi}_k}{2}}\Rightarrow \\ {}\left({\displaystyle \prod_{\begin{array}{l}1\le j\le 2M\\ {}k,j\ne p\end{array}}{\displaystyle \prod_{\begin{array}{l}1\le k\le 2M\hfill \\ {}k\prec j\hfill \end{array}}\left({x}_j-{x}_k\right)}}\right)\left|{}_{x_t={e}^{\varphi_ti}}\right.={\left(-1\right)}^p{e}^{-\left(M-1\right)\frac{\pi }{2}i}{2}^{2{M}^2-3M+1}{e}^{\left(M-1\right)\left({\varphi}_1+\dots {\varphi}_{p-1}+{\varphi}_{p+1}+\dots +{\varphi}_{2M}\right)i}\frac{{\displaystyle \prod_{j=k+1}^{2M}{\displaystyle \prod_{k=1}^{2M-1} \sin \frac{\varphi_j-{\varphi}_k}{2}}}}{{\displaystyle \prod_{\begin{array}{l}1\le k\le 2M\\ {}k\ne p\end{array}} \sin \frac{\varphi_p-{\varphi}_k}{2}}}\Rightarrow \\ {}{\mathbf{F}}_p^q={\left(-1\right)}^p{\left(-1\right)}^{\frac{M\left(M-1\right)}{2}}i\cdot {2}^{2{M}^2-4M+1}\frac{{\displaystyle \prod_{j=k+1}^{2M}{\displaystyle \prod_{k=1}^{2M-1} \sin \frac{\varphi_j-{\varphi}_k}{2}}}}{{\displaystyle \prod_{\begin{array}{l}1\le k\le 2M\\ {}k\ne p\end{array}} \sin \frac{\varphi_p-{\varphi}_k}{2}}}\cdot \left\{\left({V}_{M+q}{\overline{V}}_M-{V}_{M+q-1}{\overline{V}}_{M-1}\right)+\left({V}_{M-q}{\overline{V}}_M-{V}_{M-q-1}{\overline{V}}_{M-1}\right)\right\}\end{array} $$
(34)

It follows that:

$$ \frac{{\mathbf{X}}_p^q}{{\mathbf{X}}_{2M}}=\frac{{\left(-1\right)}^q}{2^{2M-1}}\cdot \frac{i}{{\displaystyle \prod_{\begin{array}{l}1\le k\le 2M\\ {}k\ne p\end{array}} \sin \frac{\varphi_p-{\varphi}_k}{2}}}\left\{\left({V}_{M+q}{\overline{V}}_M-{V}_{M+q-1}{\overline{V}}_{M-1}\right)+\left({V}_{M-q}{\overline{V}}_M-{V}_{M-q-1}{\overline{V}}_{M-1}\right)\right\}\cdot \frac{1}{{\displaystyle {\sum}_M \cos \left\{\frac{1}{2}\left({\varphi}_1+\dots +{\varphi}_{2M}\right)-{\left({\varphi}_1+\dots +{\varphi}_{2M}\right)}_M\right\}}} $$
(35)

If we introduce the following symbols:

$$ \begin{array}{l}{C}_t={e}^{\frac{1}{2}\left({\varphi}_1+\dots +{\varphi}_{p-1}+{\varphi}_{p+1}+\dots +{\varphi}_{2M}\right)i}{\overline{V}}_t={A}_t+i{B}_t\\ {}{A}_t^{(p)}={A}_t={\displaystyle {\sum}_t \cos \left\{\frac{1}{2}\left({\varphi}_1+\dots +{\varphi}_{p-1}+{\varphi}_{p+1}+\dots +{\varphi}_{2M}\right)-{\left({\varphi}_1+\dots +{\varphi}_{p-1}+{\varphi}_{p+1}+\dots +{\varphi}_{2M}\right)}_t\right\}}\\ {}{B}_t^{(p)}={B}_t={\displaystyle {\sum}_t \sin \left\{\frac{1}{2}\left({\varphi}_1+\dots +{\varphi}_{p-1}+{\varphi}_{p+1}+\dots +{\varphi}_{2M}\right)-{\left({\varphi}_1+\dots +{\varphi}_{p-1}+{\varphi}_{p+1}+\dots +{\varphi}_{2M}\right)}_t\right\}}\Rightarrow \\ {}\left({V}_{M+q}{\overline{V}}_M-{V}_{M+q-1}{\overline{V}}_{M-1}\right)+\left({V}_{M-q}{\overline{V}}_M-{V}_{M-q-1}{\overline{V}}_{M-1}\right)={\overline{C}}_{M+q}{C}_M-{\overline{C}}_{M+q-1}{C}_{M-1}+{\overline{C}}_{M-q}{C}_M-{\overline{C}}_{M-q-1}{C}_{M-1}\\ {}{\overline{C}}_{M+q}={A}_{M+q}-i{B}_{M+q}\\ {}{A}_{M+q}={A}_{M-q-1};{B}_{M+q}=-{B}_{M-q-1}\Rightarrow \\ {}{\overline{C}}_{M+q}={C}_{M-q-1};{\overline{C}}_{M+q-1}={C}_{M-q};{C}_{M-1}={A}_{M-1}+i{B}_{M-1};{A}_{M-1}={A}_M;{B}_{M-1}=-{B}_M\Rightarrow \\ {}\left({V}_{M+q}{\overline{V}}_M-{V}_{M+q-1}{\overline{V}}_{M-1}\right)+\left({V}_{M-q}{\overline{V}}_M-{V}_{M-q-1}{\overline{V}}_{M-1}\right)={A}_M\left({C}_{M-q-1}-{C}_{M-q}+{\overline{C}}_{M-q}-{\overline{C}}_{M-q-1}\right)+i{B}_M\left({C}_{M-q-1}+{C}_{M-q}+{\overline{C}}_{M-q}+{\overline{C}}_{M-q-1}\right)=\\ {}=2i{A}_M\left({B}_{M-q-1}-{B}_{M-q}\right)+2i{B}_M\left({A}_{M-q-1}+{A}_{M-q}\right)\end{array} $$
(36)

It follows that:

$$ \frac{{\mathbf{X}}_p^q}{{\mathbf{X}}_{2M}}=\frac{{\left(-1\right)}^{q+1}}{2^{2M-2}}\cdot \frac{A_M^{(p)}\left({B}_{M-q-1}^{(p)}-{B}_{M-q}^{(p)}\right)+{B}_M^{(p)}\left({A}_{M-q-1}^{(p)}+{A}_{M-q}^{(p)}\right)}{{\displaystyle \prod_{\begin{array}{l}1\le k\le 2M\\ {}k\ne p\end{array}} \sin \frac{\varphi_p-{\varphi}_k}{2}}{\displaystyle {\sum}_M \cos \left\{\frac{1}{2}\left({\varphi}_1+\dots +{\varphi}_{2M}\right)-{\left({\varphi}_1+\dots +{\varphi}_{2M}\right)}_M\right\}}};\kern0.5em for\kern0.5em 1\le q\le M-2 $$
(37)

In addition, for:

$$ \begin{array}{l}q=M-1\Rightarrow {V}_0=1;{C}_0={e}^{\frac{1}{2}\left({\varphi}_1+\dots +{\varphi}_{p-1}+{\varphi}_{p+1}+\dots +{\varphi}_{2M}\right)i}\\ {}{A}_0= \cos \frac{1}{2}\left({\varphi}_1+\dots +{\varphi}_{p-1}+{\varphi}_{p+1}+\dots +{\varphi}_{2M}\right);{B}_0= \sin \frac{1}{2}\left({\varphi}_1+\dots +{\varphi}_{p-1}+{\varphi}_{p+1}+\dots +{\varphi}_{2M}\right)\\ {}q=M\Rightarrow {V}_{-1}=0\Rightarrow {C}_{-1}=0;{A}_{-1}={B}_{-1}=0\kern0.5em \left({A}_0\kern0.5em and\kern0.5em {B}_0\kern0.5em are\kern0.5em the\kern0.5em same\kern0.5em as\kern0.5em before\right)\end{array} $$
(38)

Now, we can determine co-factors F q p for 1 ≤ p ≤ 2M; M + 1 ≤ q ≤ 2M, and F M + q p for 1 ≤ q ≤ M. As above, we obtain that:

$$ {\mathbf{F}}_p^q=\frac{{\left(-1\right)}^{\frac{M\left(M+1\right)}{2}}}{2^M}{e}^{-\left(M-1\right)\frac{\pi }{2}i}{e}^{-M\left({\varphi}_1+\dots +{\varphi}_{p-1}+{\varphi}_{p+1}+\dots +{\varphi}_{2M}\right)i}{\left({\varDelta}_{2M+1,M+1}^{\left(p,M+q\right)}-{\varDelta}_{2M+1,M+1}^{\left(p,M-q+1\right)}\right)}_{x_j={e}^{\varphi_ji}} $$
(39)

From this, it follows that:

$$ \frac{{\mathbf{X}}_p^{M+q}}{{\mathbf{X}}_{2M}}=\frac{{\left(-1\right)}^{q+1}}{2^{2M-2}}\cdot \frac{A_M^{(p)}\left({A}_{M-q}^{(p)}-{A}_{M-q-1}^{(p)}\right)+{B}_M^{(p)}\left({B}_{M-q}^{(p)}+{B}_{M-q-1}^{(p)}\right)}{{\displaystyle \prod_{\begin{array}{l}1\le k\le 2M\\ {}k\ne p\end{array}} \sin \frac{\varphi_p-{\varphi}_k}{2}\cdot }{\displaystyle {\sum}_M \cos \left\{\frac{1}{2}\left({\varphi}_1+\dots +{\varphi}_{2M}\right)-{\left({\varphi}_1+\dots +{\varphi}_{2M}\right)}_M\right\}}} $$
(40)

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Petrović, P.B. New Procedure for Estimation of Amplitude and Phase of Analog Multiharmonic Signal Based on the Differential Irregular Samples. J Sign Process Syst 81, 11–27 (2015). https://doi.org/10.1007/s11265-014-0892-1

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