New procedure for harmonics estimation based on Hilbert transformation

Abstract

The paper suggests a new procedure for the estimation of the parameters (amplitude and phase) for a complex periodic signal, based on the concept of analytic signal (AS). Starting from the supposition that the signal fundamental frequency has been estimated in advance at an independent stage, new analytical relations are derived for fast calculation of the unknown parameters, entailing a low numerical error. In this manner a method has been defined with a potential for successful implementation in various signal processing applications such as signal reconstruction, spectral estimation, measuring and monitoring in power systems. To review the real performance of the proposed algorithm, simulation procedures have been used to compare its measurement precision with the results obtained using fast Fourier transform (FFT) and continuous wavelet transformation (CWT) implementations. It has been found that the proposed algorithm yields a high level of precision in calculating unknown parameters of the processed signal, delivering all the necessary calculations in real time.

Appendix

The algebraic cofactor corresponding to element $u_\mathrm{a} (t_l )$ of determinant $\Delta _k$ is

$$\begin{aligned} \Delta _{kl}= & {} \left| \begin{array}{cccccc} {a_0 }&{} \cdots &{} {a_{k-1} e^{j2(k-1)\pi ft_1 }}&{} {a_{k+1} e^{j2(k+1)\pi ft_1 }}&{} \cdots &{} {a_M e^{j2M\pi ft_1 }} \\ \vdots &{} &{} \vdots &{} \vdots &{} &{} \vdots \\ {a_0 }&{} \cdots &{} {a_{k-1} e^{j2(k-1)\pi ft_{l-1} }}&{} {a_{k+1} e^{j2(k+1)\pi ft_{l-1} }}&{} \cdots &{} {a_M e^{j2M\pi ft_{l-1} }} \\ {a_0 }&{} \cdots &{} {a_{k-1} e^{j2(k-1)\pi ft_{l+1} }}&{} {a_{k+1} e^{j2(k+1)\pi ft_{l+1} }}&{} \cdots &{} {a_M e^{j2M\pi ft_{l+1} }} \\ \vdots &{} &{} \vdots &{} \vdots &{} &{} \vdots \\ {a_0 }&{} \cdots &{} {a_{k-1} e^{j2(k-1)\pi ft_{M+1} }}&{} {a_{k+1} e^{j2(k+1)\pi ft_{M+1} }}&{} \cdots &{} {a_M e^{j2M\pi ft_{M+1} }} \\ \end{array} \right| \end{aligned}$$

(23)

$$\begin{aligned} \Delta _{kl}= & {} a_0 \cdots a_{k-1} a_{k+1} \cdots a_M \nonumber \\&\left| \begin{array}{cccccc} 1&{} \cdots &{} {e^{j2(k-1)\pi ft_1 }}&{} {e^{j2(k+1)\pi ft_1 }}&{} \cdots &{} {e^{j2M\pi ft_1 }} \\ \vdots &{} &{} \vdots &{} \vdots &{} &{} \vdots \\ 1&{} \cdots &{} {e^{j2(k-1)\pi ft_{l-1} }}&{} {e^{j2(k+1)\pi ft_{l-1} }}&{} \cdots &{} {e^{j2M\pi ft_{l-1} }} \\ 1&{} \cdots &{} {e^{j2(k-1)\pi ft_{l+1} }}&{} {e^{j2(k+1)\pi ft_{l+1} }}&{} \cdots &{} {e^{j2M\pi ft_{l+1} }} \\ \vdots &{} &{} \vdots &{} \vdots &{} &{} \vdots \\ 1&{} \cdots &{} {e^{j2(k-1)\pi ft_{M+1} }}&{} {e^{j2(k+1)\pi ft_{M+1} }}&{} \cdots &{} {e^{j2M\pi ft_{M+1} }} \\ \end{array} \right| \end{aligned}$$

(24)

Let

$$\begin{aligned}&e^{j2\pi ft_l }+\cdots +e^{j2k\pi ft_l } +\cdots +e^{j2M\pi ft_l }=-p_1 \nonumber \\&e^{j2\pi ft_l }e^{j4\pi ft_l }+e^{j2\pi ft_l } e^{j6\pi ft_l }\nonumber \\&\quad + \cdots +e^{j2(M-1)\pi ft_l }e^{j2M\pi ft_l }=p_2 \nonumber \\&\cdots \nonumber \\&e^{j2\pi ft_l }\cdots e^{j2(M-1)\pi ft_l }+\cdots +e^{j4\pi ft_l }\nonumber \\&\quad + \cdots +e^{j2M\pi ft_l }=(-1)^{M-1}p_{M-1} \nonumber \\&e^{j2\pi ft_l }\cdots e^{j2M\pi ft_l }=(-1)^{M}p_M \end{aligned}$$

(25)

If we introduce following notation (for every complex number z)

$$\begin{aligned} P(z;l)= & {} \prod _{m=1, m\ne l}^{M+1} {\left( {z-e^{j2k\pi ft_m }} \right) } \nonumber \\= & {} z^{M}+p_1 z^{M-1}+\cdots +p_{M-1} z+p_M \end{aligned}$$

(26)

It’s follows that $P(e^{j2k\pi ft_l };l)=0$. Then, for every $m=1,2,\ldots ,l-1,l+1,\ldots ,M+1$ holds

$$\begin{aligned} e^{j2M\pi ft_m }= & {} -p_1 e^{j2(M-1)\pi ft_m }\nonumber \\&-\cdots -p_{M-1} e^{j2\pi ft_m }-p_M \end{aligned}$$

(27)

and therefore

$$\begin{aligned} \Delta _{kl}= & {} a_0 \cdots a_{k-1} a_{k+1} \cdots a_M \cdot \nonumber \\&\cdot \left| \begin{array}{cccccc} 1&{} \cdots &{} {e^{j2(k-1)\pi ft_1 }}&{} {e^{j2(k+1)\pi ft_1 }}&{} \cdots &{} {-p_1 e^{j2(M-1)\pi ft_1 }-\cdots -p_{M-1} e^{j2\pi ft_1 }-p_M } \\ \vdots &{} &{} \vdots &{} \vdots &{} &{} \vdots \\ 1&{} \cdots &{} {e^{j2(k-1)\pi ft_{l-1} }}&{} {e^{j2(k+1)\pi ft_{l-1} }}&{} \cdots &{} {-p_1 e^{j2(M-1)\pi ft_{l-1} }-\cdots -p_{M-1} e^{j2\pi ft_{l-1} }-p_M } \\ 1&{} \cdots &{} {e^{j2(k-1)\pi ft_{l+1} }}&{} {e^{j2(k+1)\pi ft_{l+1} }}&{} \cdots &{} {-p_1 e^{j2(M-1)\pi ft_{l+1} }-\cdots -p_{M-1} e^{j2\pi ft_{l+1} }-p_M } \\ \vdots &{} &{} \vdots &{} \vdots &{} &{} \vdots \\ 1&{} \cdots &{} {e^{j2(k-1)\pi ft_{M+1} }}&{} {e^{j2(k+1)\pi ft_{M+1} }}&{} \cdots &{} {-p_1 e^{j2(M-1)\pi ft_{M+1} }-\cdots -p_{M-1} e^{j2\pi ft_{M+1} }-p_M } \\ \end{array} \right| \end{aligned}$$

(28)

$$\begin{aligned} \Delta _{kl}= & {} a_0 \cdots a_{k-1} a_{k+1} \cdots a_M \cdot \nonumber \\&\cdot \left| \begin{array}{cccccc} 1&{} \cdots &{} {e^{j2(k-1)\pi ft_1 }}&{} {e^{j2(k+1)\pi ft_1 }}&{} \cdots &{} {-p_{M-k} e^{j2k\pi ft_1 }} \\ \vdots &{} &{} \vdots &{} \vdots &{} &{} \vdots \\ 1&{} \cdots &{} {e^{j2(k-1)\pi ft_{l-1} }}&{} {e^{j2(k+1)\pi ft_{l-1} }}&{} \cdots &{} {-p_{M-k} e^{j2k\pi ft_{l-1} }} \\ 1&{} \cdots &{} {e^{j2(k-1)\pi ft_{l+1} }}&{} {e^{j2(k+1)\pi ft_{l+1} }}&{} \cdots &{} {-p_{M-k} e^{j2k\pi ft_{l+1} }} \\ \vdots &{} &{} \vdots &{} \vdots &{} &{} \vdots \\ 1&{} \cdots &{} {e^{j2(k-1)\pi ft_{M+1} }}&{} {e^{j2(k+1)\pi ft_{M+1} }}&{} \cdots &{} {-p_{M-k} e^{j2k\pi ft_{M+1} }} \\ \end{array}\right| \end{aligned}$$

(29)

$$\begin{aligned} \Delta _{kl}= & {} a_0 \cdots a_{k-1} a_{k+1} \cdots a_M \cdot \left( {-p_{M-k} } \right) \cdot \left( {-1} \right) ^{M-k-1}\cdot \nonumber \\&\cdot \left| \begin{array}{ccccccc} 1&{} \cdots &{} {e^{j2(k-1)\pi ft_1 }}&{} {e^{j2k\pi ft_1 }}&{} {e^{j2(k+1)\pi ft_1 }}&{} \cdots &{} {e^{j2M\pi ft_1 }} \\ \vdots &{} &{} \vdots &{} \vdots &{} \vdots &{} &{} \vdots \\ 1&{} \cdots &{} {e^{j2(k-1)\pi ft_{l-1} }}&{} {e^{j2k\pi ft_{l-1} }}&{} {e^{j2(k+1)\pi ft_{l-1} }}&{} \cdots &{} {e^{j2M\pi ft_{l-1} }} \\ 1&{} \cdots &{} {e^{j2(k-1)\pi ft_{l+1} }}&{} {e^{j2k\pi ft_{l+1} }}&{} {e^{j2(k+1)\pi ft_{l+1} }}&{} \cdots &{} {e^{j2M\pi ft_{l+1} }} \\ \vdots &{} &{} \vdots &{} \vdots &{} \vdots &{} &{} \vdots \\ 1&{} \cdots &{} {e^{j2(k-1)\pi ft_{M+1} }}&{} {e^{j2k\pi ft_{M+1} }}&{} {e^{j2(k+1)\pi ft_{M+1} }}&{} \cdots &{} {e^{j2M\pi ft_{M+1} }} \\ \end{array} \right| \end{aligned}$$

(30)

$$\begin{aligned} \Delta _{kl}= & {} \left( {-1} \right) ^{M-k}p_{M-k} a_0 \cdots a_{k-1} a_{k+1} \cdots a_M \cdot V_M \nonumber \\&\left( {e^{j2\pi ft_1 },\ldots ,e^{j2k\pi ft_{l-1} },e^{j2k\pi ft_{l+1} },\ldots ,e^{j2k\pi ft_{M+1} }} \right) \end{aligned}$$

(31)

$$\begin{aligned} \Delta _{kl}= & {} \left( {-1} \right) ^{M-k}p_{M-k} a_0 \cdots a_{k-1} a_{k+1} \nonumber \\&\cdots a_M \prod _{ \mathop {p,q\ne l}\limits ^{1\le p<q\le M+1 }} {\left( {e^{j2\pi ft_q }-e^{j2\pi ft_p }} \right) } \end{aligned}$$

(32)