Frequency estimation using distributed adaptive algorithm with noisy regressor

Abstract

A smart discrete Fourier transform based modified diffusion technique is proposed in this paper that estimates the frequency in presence of regressor noise. This technique explores iterative relationship between the consecutive DFT in the form of smart DFT, and employs least mean square algorithm to measure and track the frequency. Multiple sensor nodes are employed to explore the time and space diversity following diffusion technique to improve frequency estimation. The classical diffusion algorithm is modified to address the presence of noise in regressor data. The efficacy of the modified algorithm is presented here following mathematical derivation to match both theoretical and simulation results.

Appendices

Appendix A Proof of the mean for input regressor

The DFT equation is as follows

$$\begin{aligned} {X_{DFT}} = \sum \limits _{i = 0}^{L - 1} {x\left( i \right) .{e^{ - j\frac{{2\pi .i}}{L}}}} \end{aligned}$$

(A1)

The mean is derived for the first L-DFT samples of the signal, where each of DFT sample is evaluated based on its L succeeding samples

$$\begin{aligned}&E\left[ {{X_{DFT}}} \right] = {\mu _{{X_{DFT}}}} \nonumber \\&\quad = E\left[ \begin{array}{l} \sum \limits _{i = 0}^{L - 1} {x\left( i \right) .{e^{ - j\frac{{2\pi .i}}{L}}} }\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, +\sum \limits _{i = 0}^{L - 1} {x\left( {i + 1} \right) .{e^{ - j\frac{{2\pi .i}}{L}}} } \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, +\cdots + \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \sum \limits _{i = 0}^{L - 1} {x\left( {i + L - 1} \right) .{e^{ - j\frac{{2\pi .i}}{L}}}} \end{array} \right] \end{aligned}$$

(A2)

Expanding each of the $x\left( i \right)$ sample with its L consecutive points, the above equation is rewritten as

$$\begin{aligned} \begin{array}{l} E\left[ \begin{array}{l} x\left( 0 \right) .{w^0} + x\left( 1 \right) .{w^1} +\cdots + x\left( {L - 1} \right) {w^{L - 1}}\\ \,\,\,\,\,\,\,\,\,\, + x\left( 1 \right) .{w^0} + x\left( 2 \right) .{w^1} +\cdots + x\left( L \right) {w^{L - 1}}\\ \,\,\,\,\,\,\,\,\,\, +\cdots + \\ \,\,\,\,\,\,\,\,\,\, + x\left( {L - 1} \right) .{w^0} + x\left( L \right) .{w^1} +\cdots + x\left( {2L - 1} \right) {w^{L - 1}} \end{array} \right] \end{array} \end{aligned}$$

(A3)

Rearranging the above equation in terms of common Twiddle factor, and assuming the signal to be periodic with L-points, the above equation is simplified to 0.

$$\begin{aligned} \begin{array}{l} E\left[ \begin{array}{l} \left( {x\left( 0 \right) + x\left( 1 \right) +\cdots + x\left( {L - 1} \right) } \right) {w^0}\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {x\left( 1 \right) + x\left( 1 \right) +\cdots + x\left( L \right) } \right) {w^1}\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, +\cdots +\\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, + \left( {x\left( {L - 1} \right) + x\left( L \right) +\cdots + x\left( {2L - 1} \right) } \right) {w^{L - 1}} \end{array} \right] = 0 \end{array} \end{aligned}$$

(A4)

Appendix B Proof of the variance for input regressor

$$\begin{aligned} {\mathop {\mathrm{var}}} \left( {{X_{DFT}}} \right) = E\left[ {{{\left( {{X_{DFT}}} \right) }^2}} \right] - {\left( {E\left[ {{X_{DFT}}} \right] } \right) ^2} = \sigma _{{X_{DFT}}}^2 \end{aligned}$$

(B1)

Evaluating the first part of the above equation

$$\begin{aligned} E \left[ {{{\left( {{X_{DFT}}} \right) }^2}} \right] = E\left[ {\left( {\sum \limits _{i = 0}^{L - 1} {x\left( i \right) .{e^{ - j\frac{{2\pi .i}}{L}}}} } \right) ^*}\left( {\sum \limits _{i = 0}^{L - 1} {x\left( i \right) .{e^{ - j\frac{{2\pi .i}}{L}}}} } \right) \right] \end{aligned}$$

(B2)

Applying the complex conjugate into first part, the equation rewritten as

$$\begin{aligned} E\left[ \left( {\sum \limits _{i = 0}^{L - 1} {{x^*}\left( i \right) .{e^{j\frac{{2\pi .i}}{L}}}} } \right) \left( {\sum \limits _{i = 0}^{L - 1} {x\left( i \right) .{e^{ - j\frac{{2\pi .i}}{L}}}} } \right) \right] \end{aligned}$$

(B3)

The input signal is a periodic signal, so assuming $x\left( i \right)$ as $\cos \left( {\frac{{2\pi k}}{L}} \right)$ in the above equation, it gives

$$\begin{aligned} E\left[ \left( {\sum \limits _{i = 0}^{L - 1} {\cos \left( {\frac{{2\pi i}}{L}} \right) .{e^{j\frac{{2\pi .i}}{L}}}} } \right) \left( {\sum \limits _{k = 0}^{L - 1} {\cos \left( {\frac{{2\pi k}}{L}} \right) .{e^{ - j\frac{{2\pi .k}}{L}}}} } \right) \right] \end{aligned}$$

(B4)

The above equation for L points, simplifies to

$$\begin{aligned} \frac{1}{L}\left( {\sum \limits _{i = k = 0}^{L - 1} {{{\cos }^2}\left( {\frac{{2\pi i}}{L}} \right) .} } \right) = 0.5 = \sigma _{{X_{DFT}}}^2 \end{aligned}$$

(B5)

The weight parameter w is a single tap scalar only. So, regressor matrix dimension will be of $1 \times 1$, and the global input regressor matrix ${{\textbf {R}}}$ will be of dimension of $N \times N$.

Appendix C Computational complexity

The computational complexity of algorithm Eq. (22) is shown in Table 1.

Table 1 Computational complexity