An Enhanced Fractional Least Mean Square Filter Encountering the Specific Unknown System Vector

Xie, Xuetao; Pu, YoungGun; Li, Lei; Wang, Jian

doi:10.1109/tcsii.2021.3113465

Cited by 21 publications

(9 citation statements)

References 19 publications

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“…The idea of enhanced fractional derivative based LMS is recently introduced 45 but not yet explored/exploited for any specific application. We call it as EFDLMS and exploring its application for effective parameter estimation of power signals comprising of multi frequency sine signal.…”

Section: Proposed Methodologymentioning

confidence: 99%

“…We call it as EFDLMS and exploring its application for effective parameter estimation of power signals comprising of multi frequency sine signal. Defining the cost function as

\begin{array}{l} g ((), j) goodbreak= ε^{2} ((), j) \\ g ((), j) goodbreak= {[x (j) - \hat{x} (j)]}^{2} goodbreak= {[x (j) - {boldξ}^{T} (j) \hat{boldυ}]}^{2} . \end{array}

Calculating the fractional derivative of the cost function with respect to

\hat{boldυ}

using Faa di Bruno formula and after doing some simplification as reported in Xie et al, 45 we obtain:

{{}_{α}Δ}_{{\overset{υ}{true}}_{k} (j)}^{η} g (j) = \frac{{[{\overset{υ}{true}}_{k} (j) - α]}^{- η}}{Γ (1 - η)} g (j) + \frac{Γ (1 + η)}{Γ (η) Γ (2)} 1! \times \frac{{[{\overset{υ}{true}}_{k} (j) - α]}^{1 - η}}{Γ (2 - η)} {{}_{α}Δ}_{\hat{x} (j)}^{1} g (j) \frac{1}{1!} ([], {{}_{α}Δ}_{{\overset{υ}{true}}_{k} (j)}^{<...})

…”

Section: Proposed Methodologymentioning

confidence: 99%

“…Introducing the sign function in (10) to avoid fractional extreme points and supporting the square error estimate for convergence improvement. 45 α Δ η…”

Section: Identification Modelmentioning

confidence: 99%

“…Calculating the fractional derivative of the cost function with respect to υ using Faa di Bruno formula and after doing some simplification as reported in Xie et al, 45 we obtain:…”

Section: Identification Modelmentioning

confidence: 99%

“…44 Recently, Pu et al introduced enhanced fractional derivative based fractional LMS, (EFDLMS) that computes the fractional derivative using Faa di Bruno formula. 45 The reported promising features of the EFDLMS motivated us to exploit it for effective parameter estimation of composite sinusoidal signals.…”

mentioning

confidence: 99%

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Enhanced fractional adaptive processing paradigm for power signal estimation

Chaudhary

Khan

Raja

et al. 2022

Math Methods in App Sciences

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Fractional calculus tools have been exploited to effectively model variety of engineering, physics and applied sciences problems. The concept of fractional derivative has been incorporated in the optimization process of least mean square (LMS) iterative adaptive method. This study exploits the recently introduced enhanced fractional derivative based LMS (EFDLMS) for parameter estimation of power signal formed by the combination of different sinusoids. The EFDLMS addresses the issue of fractional extreme points and provides faster convergence speed. The performance of EFDLMS is evaluated in detail by taking different levels of noise in the composite sinusoidal signal as well as considering various fractional orders in the EFDLMS. Simulation results reveal that the EDFLMS is faster in convergence speed than the conventional LMS (i.e., EFDLMS for unity fractional order).

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Section: Proposed Methodologymentioning

confidence: 99%

“…We call it as EFDLMS and exploring its application for effective parameter estimation of power signals comprising of multi frequency sine signal. Defining the cost function as

\begin{array}{l} g ((), j) goodbreak= ε^{2} ((), j) \\ g ((), j) goodbreak= {[x (j) - \hat{x} (j)]}^{2} goodbreak= {[x (j) - {boldξ}^{T} (j) \hat{boldυ}]}^{2} . \end{array}

Calculating the fractional derivative of the cost function with respect to

\hat{boldυ}

using Faa di Bruno formula and after doing some simplification as reported in Xie et al, 45 we obtain:

{{}_{α}Δ}_{{\overset{υ}{true}}_{k} (j)}^{η} g (j) = \frac{{[{\overset{υ}{true}}_{k} (j) - α]}^{- η}}{Γ (1 - η)} g (j) + \frac{Γ (1 + η)}{Γ (η) Γ (2)} 1! \times \frac{{[{\overset{υ}{true}}_{k} (j) - α]}^{1 - η}}{Γ (2 - η)} {{}_{α}Δ}_{\hat{x} (j)}^{1} g (j) \frac{1}{1!} ([], {{}_{α}Δ}_{{\overset{υ}{true}}_{k} (j)}^{<...})

…”

Section: Proposed Methodologymentioning

confidence: 99%

“…Introducing the sign function in (10) to avoid fractional extreme points and supporting the square error estimate for convergence improvement. 45 α Δ η…”

Section: Identification Modelmentioning

confidence: 99%

“…Calculating the fractional derivative of the cost function with respect to υ using Faa di Bruno formula and after doing some simplification as reported in Xie et al, 45 we obtain:…”

Section: Identification Modelmentioning

confidence: 99%

mentioning

confidence: 99%

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