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Cascade–Cascade Least Mean Square (LMS) Adaptive Noise Cancellation


The paper presents a new model of noise cancellation using cascading of cascaded LMS adaptive filters. The model has a combination of ‘\(2N+1\)’ LMS filters for N-stage of adaptive noise cancellation. First LMS filter works as a basic noise canceller, next two work as 1st stage of noise canceller using a cascaded form of LMS filters known as LMS Block-1, and all others have the same arrangement as LMS Block-1 known as LMS Block-2, LMS Block-3, \(\ldots \), LMS Block-N. The LMS Block has a two-stage noise reduction of the additive noise or interference. LMS Block-1 is cascaded noise canceller, determines and reduces noise again after reduction in noise from 1st LMS filter. The analysis and simulation model gives the responses of noise cancellation like error signal, output signal and signal-to-noise ratio with respect to step sizes, filter lengths and initial weight of filters. This paper also shows the simulation of cascade–cascade LMS adaptive noise cancellation for two stages (\(N = 2\)) and gives the higher signal-to-noise ratio at Nth stage with respect to previous stages. It is the next novel point of this paper that no other elements are present in the cascade–cascade LMS adaptive noise cancellation rather than LMS filters as noise canceller.

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Correspondence to Awadhesh Kumar Maurya.



See Tables 4, 5, 6 and Figs. 13, 14, 15.

Table 4 Combination of step sizes (\(\mu _{1}, \mu _{2}, \mu _{3})\)
Table 5 Combination of step sizes: (\(\mu _{1}, \mu _{2})\) at constant \(\mu _{3}\), \((\mu _{2},\, \mu _{3})\) at constant \(\mu _{1}\) and (\(\mu _{1},\, \mu _{3})\) at constant \(\mu _{2}\)
Table 6 Convergence rate of error signal for N-LMS Blocks at \(\mu \ll 1\)
Fig. 13
figure 13

SNRs for single and new model-based cascaded LMS-ANC at \(L=L{'}=L^{\prime \prime }\). a 2, b 4, c 8, d 16, e 32, f 64 and combination of step sizes of LMS filters (\(\mu _{2}, \mu _{3})\) at constant \(\mu _{1}\)

Fig. 14
figure 14

SNRs for single and new model-based cascaded LMS-ANC at \(L=L{'}=L^{\prime \prime }\). a 2, b 4, c 8, d 16, e 32, f 64 and combination of step sizes of LMS filters \((\mu _{1}, \mu _{3})\) at constant \(\mu _{2}\)

Fig. 15
figure 15

SNRs for single and new model-based cascaded LMS-ANC at \(L=L{'}=L^{\prime \prime }\). a 2, b 4, c 8, d 16, e 32, f 64 and combination of step sizes of LMS filters \((\mu _{1}, \mu _{2})\) at constant \(\mu _{3}\)

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Maurya, A.K. Cascade–Cascade Least Mean Square (LMS) Adaptive Noise Cancellation. Circuits Syst Signal Process 37, 3785–3826 (2018).

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