4.2.1 Hammerstein Models and Cascade Models

A very simple class of LIP nonlinear filters can be described with the cascade structure depicted in Fig. 4.2, where a nonlinearity $f\{\cdot \}$ is followed by a linear system with IR vector h. The output depends linearly on the parameters h of the linear system and can be expressed in the notation of Eq. (4.2) as

$y\left(n\right)=f\left\{x\left(n\right)\right\}⁎h(n)=x_{\mathrm{NL}}\left(n\right)⁎h(n).$

If $f\{\cdot \}$ in Fig. 4.2 is a memoryless nonlinearity, the structure is widely known as HM [19], whereas the more general case, where $f\{\cdot \}$ is a nonlinearity with memory, will simply be referred to as CM in the following (the attribute nonlinear will be implied). HMs can, for instance, be employed for loudspeaker linearization (e.g., [20,21]) and AEC (e.g., [22–32]). HMs are simple and powerful models for acoustic echo paths if the major source of nonlinearity is the playback equipment (amplifiers and loudspeakers operating at the limits of their dynamic range) and can be mapped to $f\{\cdot \}$, while the acoustic IR (describing sound propagation through the acoustic environment) can be described by h [22].

4.2.2 Hammerstein Group Models and Cascade Group Models

The term Cascade Group Model (CGM) will refer to models with the structure depicted in Fig. 4.3. Such CGMs are composed of B parallel branches, each of which is a nonlinear CM (see Section 4.2.1) with a branch nonlinearity $f_b\{\cdot \}$ and IR ${\mathbf{h}}_{(b)}$. For memoryless branch nonlinearities $f_b\{\cdot \}$, each branch becomes an HM and the entire CGM will be referred to as HGM in the following. HGMs have successfully been employed to describe both acoustic systems, e.g., in [33–37], and nonacoustic systems, e.g., [38,39]. In their treatment in the literature, HGMs have been addressed by various names, such as the Uryson model [3,39], or are simply treated as an approximation of a particular CGM. The most prominent examples of such HGM/CGM pairs are the so-called power filters [37,40,41] and Volterra filters [19]. Power filters are HGMs with monomes as branch nonlinearities $f_b\{\cdot \}$, e.g.,

$f_1\{\cdot \}=(\cdot ),f_2\{\cdot \}={(\cdot )}^2,f_3\{\cdot \}={(\cdot )}^3,\dots ,.f_B\{\cdot \}={(\cdot )}^B.$

Volterra filters result from power filters by augmenting the latter with additional branches where the branch nonlinearities $f_b\{\cdot \}$ perform time-lagged products of powers of the input (time-lagged products of the power filter branch nonlinearities), e.g., the set

$\begin{array}{c}\hfill f_1\left\{x\left(n\right)\right\}=x\left(n\right),f_2\left\{x\left(n\right)\right\}=x^2(n),f_3\left\{x\left(n\right)\right\}=x\left(n\right)\cdot x(n-1),\hfill \\ \hfill \dots ,f_{2+R}\left\{x\left(n\right)\right\}=x\left(n\right)\cdot x(n-R),f_{3+R}\left\{x\left(n\right)\right\}=x^3\left(n\right),\dots \hfill \end{array}$

leads to a Volterra filter in Diagonal Coordinate Representation (DCR) [6]. The set of IRs ${\mathbf{h}}_{(b)}$ of branches where $f_b\{\cdot \}$ performs (potentially time-lagged) products of o input signal values is called oth-order Volterra kernel and the branches within a kernel are also termed diagonals of the respective kernel. In the aforementioned example, the second-order kernel is modeled by $R+1$ diagonals ($b\in \{2,\dots ,2+R\}$). Volterra filters are universal approximators for a large class of nonlinear systems and have therefore been applied in biology (e.g. [1]), control engineering (e.g., [3]) and communications (e.g., [4–6]), as well as in acoustic signal processing (e.g., [7–16,42,43]). Recently, HGMs and CGMs have also been realized with other sets of branch nonlinearities, such as Fourier basis functions [44], Legendre polynomials [45,46] or Chebyshev polynomials [47]. Also, Functional-Link Adaptive Filters (FLAFs) [34,35,48–50] can be viewed as HGM or CGM structures. All these models can be described by the input/output relationship

$y\left(n\right)=\sum _{b=1}^B\sum _{k=0}^{L-1}\underset{x_b(n-k)}{\underbrace{f_b\left\{x(n-k)\right\}}}{h}_{(b)}\left(k\right)=\sum _{b=1}^B{x}_b\left(n\right)⁎h_{(b)}\left(n\right),$

where $h_{(b)}\left(k\right)$ denotes the kth tap of the linear subsystem in branch b and $x_b\left(n\right)$ is the bth branch signal. Thus, algorithms developed for one particular realization (i.e., set of basis functions $f_b\{\cdot \}$) of CGMs can often immediately be applied to other realizations. This holds for the adaptation algorithms for coefficient estimation described in Sections 4.3 and 4.4, as well as for the structure estimation methods outlined in Section 4.6.

4.2.3 Bilinear Cascade Models

For applications where CGMs with a large number of branches are computationally too demanding, e.g., for AEC in mobile devices with very limited processing or battery power, CM-structured adaptive filters are particularly attractive. Yet, the nonlinearity itself is usually unknown. Therefore, the nonlinearity $f_{}\{\cdot \}$ of a CM is typically modeled by a parametric preprocessor according to

$f_{}\left\{x\left(n\right)\right\}=f_{\text{pp}}\{x\left(n\right)\}=\sum _{b=1}^B{w}_b{f}_b\left\{x\left(n\right)\right\}=x_{\mathrm{pp}}\left(n\right),$

where B nonlinearities $f_b\{\cdot \}$ are combined with weights $w_b$, which are parameters to be identified in addition to the linear subsystem. The subindex “pp” of the nonlinear preprocessor and its output signal are used to emphasize the parametrical structure of the preprocessor. The general structure of a CM with a preprocessor according to Eq. (4.8) is depicted in Fig. 4.4A. Most prominently, $f_b\{\cdot \}$ are chosen as monomes [20,21,24,51–54], e.g., $f_b\{\cdot \}={(\cdot )}^b$, such that the nonlinearity $f_{\text{pp}}\{\cdot \}$ can be seen as a truncated Taylor series expansion. Fourier series- [55] and Legendre series-based approximations [28,30,31,56] have also recently been employed in AEC. Expanding the preprocessor of Fig. 4.4A in the block diagram results in Fig. 4.4B. Obviously, such a system has the input/output relationship

$y\left(n\right)=x_{\mathrm{pp}}\left(n\right)⁎h(n)$

$=h(n)⁎\sum _{b=1}^B\stackrel{y_b\left(n\right)}{\overbrace{w_b\underset{x_b\left(n\right)}{\underbrace{f_b\left\{x\left(n\right)\right\}}}}},$

and the paths from $x_b\left(n\right)$ to $y\left(n\right)$ consist not only of a single linear system, but of cascades of $w_b$ (a single-tap linear system) with the linear system h. Therefore, the output of $y\left(n\right)$ linearly depends on both the parameter vector $\mathbf{w}={[w_1,w_2,\dots ,w_B]}^{\mathrm{T}}$ and the parameter vector h, which is why such systems will be referred to as bilinear¹ CMs in the following. Bilinearity holds regardless of $f_b\{\cdot \}$ being memoryless or with memory and also for general linear filters with IRs ${\mathbf{h}}_{(b)}$ replacing the single-tap filters $w_b$, resulting in CGM preprocessors.² Thus, in the following, all considerations on simple preprocessor systems according to Fig. 4.4A translate to bilinear CMs with CGM preprocessors and vice versa, unless noted otherwise.

4.3.1 NLMS for Cascade Group Models

In this section, the adaptation of CGMs according to Section 4.2.2 by a time-domain and a partitioned-block frequency-domain NLMS will be described in Sections 4.3.1.1 and 4.3.1.2, respectively. Note that the branch nonlinearities will be considered as time-invariant and known, such that only the linear subsystems of ${\mathbf{h}}_{(b)}$ of the CGMs from Section 4.2.2 are adaptive.

4.3.2 Filtered-X Adaptation for Bilinear Cascade Models

In this section, the FX algorithm will first be introduced and discussed on a conceptual level for bilinear CMs, before specifying it in detail and proposing an efficient realization for CGM preprocessors with single-tap branch filters (i.e., parametric preprocessors according to Eq. (4.8)).

The Generic Filtered-X Structure  FX algorithms are frequently employed in Active Noise Control (ANC) [64] for prefilter adaptation. Thereby, the FX algorithm exploits the fact that the ordering of linear time-invariant systems in a cascade can be reversed without altering the cascade's output: for the preprocessor-based CM in Fig. 4.4B, joint linearity and time invariance allow incorporating h into each of the branches, as depicted in Fig. 4.5. Therein, the prefiltered inputs to the former prefilters $w_b$ are often termed FX signals. The representation of the CM in Fig. 4.5 allows one to directly apply an NLMS-type adaptation algorithm for linear MISO systems from Section 4.3.1 to adapt $w_b$, as their inputs are known and their combined output is directly matched to the target signal (in system identification: an unknown system's output). Mathematically, an FX LMS algorithm can also be derived directly like an LMS algorithm as stochastic gradient descent algorithm w.r.t. the partial derivative w.r.t. the prefilters. The exchange of the filtering orders results simply from an advantageous summation order in the gradient calculation. Note that adaptive filters for $w_b$ and h actually violate the time invariance required for exchanging the filter order. However, the time variance appears to be uncritical in practice, where FX algorithms have been employed for ANC for more than three decades [64,65].

Review of Known Algorithms for Filtered-X Adaptation of Bilinear Cascade Models  Many FX-like algorithms have been derived for the adaptation of bilinear LIP nonlinear systems but do not establish the link to the filtered-X algorithms and the exchange of filtering orders. The following description of these methods will highlight the fact that these algorithms can be viewed and implemented as FX algorithms.

In [24], the FX preprocessor adaptation was derived and applied to the adaptation of a polynomial preprocessor with $f_b\{\cdot \}={(\cdot )}^b$—a nonadaptive offline version of this mechanism was proposed in [23], which resembles the iterative version of [51] for AutoRegressive Moving Average (ARMA) models as linear submodels. More recent work on such cascade models considered a generalization of [24] by allowing longer IRs than a single tap [37,66] or considered piece-wise linear functions for $f_b\{\cdot \}$ in conjunction with a modified joint normalization of the linear and nonlinear filter coefficients [27]. In particular, [37] employs a power filter as preprocessor to the linear subsystem. The resulting algorithm corresponds to a time-domain FX algorithm. In [66], a particular CGM preprocessor is discussed, which corresponds to a Volterra filter in triangular representation (see, e.g., [19] for the triangular representation) with memory lengths of $N_1$ and $N_2$ for the kernels of orders 1 and 2, respectively, and no memory at all for higher-order Volterra kernels (becoming a simple memoryless preprocessor³ for these orders). Structures similar to this preprocessor also result from the so-called EVOLutionary Volterra Estimation (EVOLVE), which will be outlined later on in Section 4.6. While evaluating their algorithm for a memoryless preprocessor ($N_1=N_2=0$), [66] also evaluates an extension of [24] by adapting the linear subsystem with a frequency-domain NLMS with unconstrained update. The preprocessor is adapted, as in [24], by a Recursive Least Squares (RLS)-like algorithm based on the FX signals. Still, the two RLS descriptions differ as [66] employs the direct inversion of the involved correlation matrix, whereas [24] makes use of the alternative variant with a recursively computed inverse (see [57] for RLS-adaptive filters). Another extension of [24] is treated in [25] (in German), where the signal flow and benefit of a subband-AEC variant of [24] is discussed on a theoretical level.

Note that none of these approaches describes the preprocessor adaptation explicitly as an FX algorithm.⁴ As opposed to this, the remainder of this section introduces an FX PB-FNLMS algorithm for CMs with CGM preprocessors and tailors this algorithm to the special case of memoryless preprocessors according to Eq. (4.8).

Filtered-X Adaptation of Preprocessors of Partitioned-Block CMs  In the following, consider the structure of Fig. 4.4B as a single-tap CGM, for which DFT-domain branch signal vectors can be computed according to Eq. (4.23). Based on that, preprocessed DFT-domain input signal vectors can be determined by

${\underset{\_}{\mathbf{x}}}_{\mathrm{pp},\mathit{\kappa }}=\sum _{b=1}^B{\hat{w}}_b(\kappa -1){\underset{\_}{\mathbf{x}}}_{(b),\kappa }$

and the DFT-domain FX signal vectors ${\underset{\_}{\mathbf{x}}}_{\mathrm{FX},(b),\mathit{\kappa }}$ (see Fig. 4.5 illustrating the FX signals) can be computed by partitioned convolution in a two-step procedure as

${\underset{\_}{\mathbf{x}}}_{\mathrm{FX},(b),\mathit{\kappa }}^{\circ }=\sum _{p=0}^{P-1}{\underset{\_}{\mathbf{x}}}_{(b),\kappa -p}\odot {\underset{\_}{\hat{\mathbf{h}}}}_{\kappa -1}^{(p)}$

${\underset{\_}{\mathbf{x}}}_{\mathrm{FX},(b),\mathit{\kappa }}=\mathbf{F}\underset{{\mathbf{x}}_{\mathrm{FX},(b),\mathit{\kappa }}}{\underbrace{\left[\begin{array}{c}\hfill [{\mathbf{0}}_M,{\mathbf{I}}_M]{\mathbf{x}}_{\mathrm{FX},(b),\mathit{\kappa }-\mathrm{1}}\hfill \\ \hfill [{\mathbf{0}}_M,{\mathbf{I}}_M]{\mathbf{F}}^{\mathrm{H}}{\underset{\_}{\mathbf{x}}}_{\mathrm{FX},(b),\mathit{\kappa }}^{\circ }\hfill \end{array}\right]}}.$

With the introduced signals, the adaptive filter's output can alternatively be determined by one of the two equations

${\hat{\mathbf{y}}}_{\kappa }={\mathbf{W}}_{01}{\mathbf{F}}^{\mathrm{H}}\sum _{p=0}^{P-1}{\underset{\_}{\mathbf{x}}}_{\mathrm{pp},\kappa -p}{\underset{\_}{\hat{\mathbf{h}}}}_p^{(\kappa )}$

$\approx {\mathbf{W}}_{01}{\mathbf{F}}^{\mathrm{H}}\underset{{\underset{\_}{\mathbf{y}}}_{\kappa }^{\circ }}{\underbrace{\sum _{b=1}^B{\hat{w}}_b(\kappa -1){\underset{\_}{\mathbf{x}}}_{\mathrm{FX},(b),\mathit{\kappa }}}},$

where Eq. (4.33) corresponds to block processing with the signal flow in Fig. 4.4B and Eq. (4.34) corresponds to the signal flow in Fig. 4.5. For time-invariant systems, Eqs. (4.33) and (4.34) are identical. This allows one to compute the error signal vector ${\mathbf{e}}_{\kappa }={\mathbf{y}}_{\kappa }-{\hat{\mathbf{y}}}_{\kappa }$ and its DFT

${\underset{\_}{\mathbf{e}}}_{\kappa }=\mathbf{F}{\mathbf{e}}_{\kappa },$

both of which will be used for NLMS-type adaptation of the parameters. Analogously to Section 4.3.1.2, a PB-FNLMS adaptation of ${\underset{\_}{\hat{\mathbf{h}}}}_{\kappa }^{(p)}$ can be expressed as

${\widehat{\underset{\_}{\mathbf{h}}}}_{\kappa }^{\circ (p)}={\underset{\_}{\hat{\mathbf{h}}}}_{\kappa -1}^{(p)}+{\underset{\_}{\mathbf{x}}}_{\mathrm{norm},\mathrm{pp},\kappa -p}\odot {\underset{\_}{\mathbf{e}}}_{\kappa },$

${\underset{\_}{\hat{\mathbf{h}}}}_{\kappa }^{(p)}=\mathbf{F}\underset{{\hat{\mathbf{h}}}_{\kappa }^{(p)}}{\underbrace{{\mathbf{W}}_{10}{\mathbf{F}}^{\mathrm{H}}\left({\widehat{\underset{\_}{\mathbf{h}}}}_{\kappa }^{\circ (p)}\right)}},$

where ${\underset{\_}{\mathbf{x}}}_{\mathrm{norm},\mathrm{pp},\kappa }$ is the normalized preprocessed input signal vector computed analogously to Eqs. (4.26) and (4.27) (containing the adaptation step size) and where Eq. (4.36) and Eq. (4.37) correspond to the unconstrained and the constrained update, respectively.

Similar to [24], the weights ${\hat{w}}_b\left(\kappa \right)$ are estimated by an NLMS algorithm employing the FX signals as

${\hat{w}}_b\left(\kappa \right)={\hat{w}}_b(\kappa -1)+\frac{{\mu }_{\text{FX}}}{E_b+\delta }\langle {\mathbf{x}}_{\mathrm{FX},(b),\kappa },{\mathbf{e}}_{\kappa }\rangle$

with the adaptation step size ${\mu }_{\text{FX}}$ and

$E_b=\langle {\mathbf{x}}_{\mathrm{FX},(b),\mathit{\kappa }},{\mathbf{x}}_{\mathrm{FX},(b),\mathit{\kappa }}\rangle .$

For the more general case where a partitioned-block CGM with $P_{\text{pp}}$ partitions ${\underset{\_}{\hat{\mathbf{h}}}}_{(b),\kappa }^{(p)}$ replaces the preprocessor with weights ${\hat{w}}_b\left(\kappa \right)$, all linear combinations with the weights, like in Eq. (4.30), have to be replaced by actual partitioned convolutions and PB-FNLMS updates can be computed for the preprocessor according to

${\widehat{\underset{\_}{\mathbf{h}}}}_{(b),\kappa }^{\circ (p)}={\underset{\_}{\hat{\mathbf{h}}}}_{(b),\kappa -1}^{(p)}+{\underset{\_}{\mathbf{x}}}_{\mathrm{FX},\mathrm{norm},(b),\kappa -p}\odot {\underset{\_}{\mathbf{e}}}_{\kappa }$

and

${\underset{\_}{\hat{\mathbf{h}}}}_{(b),\kappa }^{(p)}=\mathbf{F}\underset{{\hat{\mathbf{h}}}_{(b),\kappa }^{(p)}}{\underbrace{{\mathbf{W}}_{10}{\mathbf{F}}^{\mathrm{H}}\left({\widehat{\underset{\_}{\mathbf{h}}}}_{(b),\kappa }^{\circ (p)}\right)}}$

with the normalized DFT-domain FX branch signal vectors ${\underset{\_}{\mathbf{x}}}_{\mathrm{FX},\mathrm{norm},(b),\kappa -p}$ determined from the FX branch signals ${\mathbf{x}}_{\mathrm{FX},(b),\kappa }$ analogously to Eqs. (4.26) and (4.27). This corresponds to a generalization of [37] from time-domain power filter preprocessor adaptation to PB-FNLMS adaptation of general CGM preprocessors.

Thereby, the adaptation of bilinear CMs has been decomposed into two ordinary PB-FNLMS system identification tasks, plus the additional prefiltering operations generating the FX branch signals. All these components can be implemented efficiently using partitioned-block convolution techniques.

4.3.3 Summary of Algorithms

The main algorithms presented in Section 4.3 are summarized in tabular form to support an implementation in practice. The most basic algorithm is the time-domain NLMS algorithm summarized in Table 4.1. The operations listed therein have to be evaluated at each sample index n. For processing an entire block of M subsequent samples, as in block- and frequency-domain processing, all equations in Table 4.1 have to be evaluated sample by sample for each of the block's samples—M times in total. For PB-FNLMS adaptation, the operations for processing a block of M samples are summarized in Table 4.2. The operations listed therein have to be executed for each signal frame κ (once every M samples). In the same way, the operations required for FX adaptation of a bilinear CM with a single-tap CGM as nonlinear preprocessor are listed in Table 4.3.

Table 4.1

Summary of operations of a time-domain NLMS algorithm for CGMs as executed for each sampling instant n

computed quantity	computation formula		equation no.
branch signals	$x_b\left(n\right)=f_b\left\{x\left(n\right)\right\}$	∀b	Eq. (4.11)
time-reversed branch signal vectors	${\check{\mathbf{x}}}_{(b),n}={[x_b\left(n\right),\dots ,x_b(n-L+1)]}^{\mathrm{T}}$	∀b	–
output signal	$\hat{y}\left(n\right)={\sum }_{b=1}^B\langle {\hat{\mathbf{h}}}_{(b),n-1},{\check{\mathbf{x}}}_{(b),n}\rangle$		Eq. (4.12)
error signal	$e\left(n\right)=y\left(n\right)-\hat{y}\left(n\right)$		Eq. (4.13)
frame energy	$E_{x_b}\left(n\right)=\langle {\check{\mathbf{x}}}_{(b),n},{\check{\mathbf{x}}}_{(b),n}\rangle$	∀b	Eq. (4.15)
filter coefficients	${\hat{\mathbf{h}}}_{(b),n}={\hat{\mathbf{h}}}_{(b),n-1}+{\mu }_b\cdot \frac{e\left(n\right)}{E_{x_b}\left(n\right)+\delta }{\check{\mathbf{x}}}_{(b),n}$	∀b	Eq. (4.14)

Table 4.2

Summary of operations of a PB-FNLMS algorithm for CGMs as executed for each frame κ

computed quantity	computation formula		equation no.
branch signals	${\mathbf{x}}_{(b),\kappa }=\left[\begin{array}{c}\hfill [{\mathbf{0}}_M,{\mathbf{I}}_M]{\mathbf{x}}_{(b),\kappa -1}\hfill \\ \hfill {[f_b\left\{x\left(\kappa M\right)\right\},\dots ,f_b\left\{x(\kappa M+M-1)\right\}]}^{\mathrm{T}}\hfill \end{array}\right]$	∀b	Eq. (4.22)
∼ in DFT domain	${\underset{\_}{\mathbf{x}}}_{(b),\kappa }=\mathbf{F}{\mathbf{x}}_{(b),\kappa }$	∀b	Eq. (4.23)
DFT-domain output estimate with cyclic convolution artifacts	${\widehat{\underset{\_}{\mathbf{y}}}}_{\kappa }^{\circ }={\sum }_{b=1}^B{\sum }_{p=0}^{P-1}{\underset{\_}{\mathbf{x}}}_{(b),\kappa -p}\odot {\underset{\_}{\hat{\mathbf{h}}}}_{(b),\kappa -1}^{(p)}$		Eq. (4.24)
DFT-domain error signal	${\underset{\_}{\mathbf{e}}}_{\kappa }=\mathbf{F}({\mathbf{y}}_{\kappa }-{\mathbf{W}}_{01}{\mathbf{F}}^{\mathrm{H}}{\widehat{\underset{\_}{\mathbf{y}}}}_{\kappa }^{\circ })$		Eq. (4.25)
branch PSDs		∀b	Eq. (4.26)
normalized branch signals	${\underset{\_}{\mathbf{x}}}_{\mathrm{norm},(b),\kappa }={\mu }_b{\left({\underset{\_}{\mathbf{x}}}_{(b),\kappa }\right)}^{⁎}\oslash ({\mathbf{s}}_{x_b,\kappa }+\delta )$	∀b	Eq. (4.27)
updated filters	${\widehat{\underset{\_}{\mathbf{h}}}}_{(b),\kappa }^{\circ (p)}={\underset{\_}{\hat{\mathbf{h}}}}_{(b),\kappa -1}^{(p)}+{\underset{\_}{\mathbf{x}}}_{\mathrm{norm},(b),\kappa -p}\odot {\underset{\_}{\mathbf{e}}}_{\kappa }$	∀b,p	Eq. (4.28)
constrained update	${\underset{\_}{\hat{\mathbf{h}}}}_{(b),\kappa }^{(p)}=\mathbf{F}{\mathbf{W}}_{10}{\mathbf{F}}^{\mathrm{H}}\left({\widehat{\underset{\_}{\mathbf{h}}}}_{(b),\kappa }^{\circ (p)}\right)$	∀b,p	Eq. (4.29)

Table 4.3

Summary of operations of an FX algorithm tailored to the adaptation of bilinear CMs with single-tap CGM preprocessors as executed for each frame κ

computed quantity	computation formula		equation no.
branch signals	${\mathbf{x}}_{(b),\kappa }=\left[\begin{array}{c}\hfill [{\mathbf{0}}_M,{\mathbf{I}}_M]{\mathbf{x}}_{(b),\kappa -1}\hfill \\ \hfill {[f_b\left\{x\left(\kappa M\right)\right\},\dots ,f_b\left\{x(\kappa M+M-1)\right\}]}^{\mathrm{T}}\hfill \end{array}\right]$	∀b	Eq. (4.22)
∼ to DFT domain	${\underset{\_}{\mathbf{x}}}_{(b),\kappa }=\mathbf{F}{\mathbf{x}}_{(b),\kappa }$	∀b	Eq. (4.23)
DFT-domain preprocessed signal	${\underset{\_}{\mathbf{x}}}_{\mathrm{pp},\mathit{\kappa }}={\sum }_{b=1}^B{\hat{w}}_b(\kappa -1){\underset{\_}{\mathbf{x}}}_{(b),\kappa }$		Eq. (4.30)
DFT-domain FX branch signals with cyclic convolution artifacts	${\underset{\_}{\mathbf{x}}}_{\mathrm{FX},(b),\mathit{\kappa }}^{\circ }={\sum }_{p=0}^{P-1}{\underset{\_}{\mathbf{x}}}_{(b),\kappa -p}\odot {\underset{\_}{\hat{\mathbf{h}}}}_{\kappa -1}^{(p)}$	∀b,p	Eq. (4.31)
output signal	${\hat{\mathbf{y}}}_{\kappa }={\mathbf{W}}_{01}{\mathbf{F}}^{\mathrm{H}}{\sum }_{b=1}^B{\hat{w}}_b(\kappa -1)\cdot {\underset{\_}{\mathbf{x}}}_{\mathrm{FX},(b),\mathit{\kappa }}^{\circ }$		Eq. (4.33)
error signal	${\mathbf{e}}_{\kappa }={\mathbf{y}}_{\kappa }-{\hat{\mathbf{y}}}_{\kappa }$		–
∼ to DFT domain	${\underset{\_}{\mathbf{e}}}_{\kappa }=\mathbf{F}{\mathbf{e}}_{\kappa }$		Eq. (4.35)
preprocessed signal PSD			–
normalized input	${\underset{\_}{\mathbf{x}}}_{\mathrm{norm},\mathrm{pp},\kappa }={\left({\underset{\_}{\mathbf{x}}}_{\mathrm{pp},\mathit{\kappa }}\right)}^{⁎}\odot ({\mu }_b\oslash ({\mathbf{s}}_{x_{\text{pp}},\kappa }+\delta ))$		–
updated filters	${\widehat{\underset{\_}{\mathbf{h}}}}_{\kappa }^{\circ (p)}={\underset{\_}{\hat{\mathbf{h}}}}_{\kappa -1}^{(p)}+{\underset{\_}{\mathbf{x}}}_{\mathrm{norm},\mathrm{pp},\kappa -p}\odot {\underset{\_}{\mathbf{e}}}_{\kappa }$,	∀p	Eq. (4.40)
constrained update	${\underset{\_}{\hat{\mathbf{h}}}}_{\kappa }^{(p)}=\mathbf{F}\underset{{\hat{\mathbf{h}}}_{\kappa }^{(p)}}{\underbrace{{\mathbf{W}}_{10}{\mathbf{F}}^{\mathrm{H}}\left({\widehat{\underset{\_}{\mathbf{h}}}}_{\kappa }^{\circ (p)}\right)}}$,	∀p	Eq. (4.41)
FX energies	$E_b\approx E_b^{\circ }=\langle {\underset{\_}{\mathbf{x}}}_{\mathrm{FX},(b),\mathit{\kappa }}^{\circ },{\underset{\_}{\mathbf{x}}}_{\mathrm{FX},(\mathrm{b}),\mathit{\kappa }}^{\circ }\rangle$	∀b > 1	Eq. (4.45)
weight update	${\hat{w}}_b\left(\kappa \right)={\hat{w}}_b(\kappa -1)+\frac{{\mu }_{\text{FX}}}{E_b^{\circ }+\delta }\langle {\underset{\_}{\mathbf{x}}}_{\mathrm{FX},(b),\mathit{\kappa }}^{\circ },{\underset{\_}{\mathbf{e}}}_{\kappa }\rangle$	∀b > 1	Eq. (4.46)

These algorithms will be employed as building blocks of the computationally very efficient SA filters in the following section, Section 4.4. Therein, the aforementioned basic algorithms will only be referenced without repeating all equations.

4.4.1 Significance-Aware Decompositions of LIP Nonlinear Systems

In [32], two different SA decompositions have been employed, which will be denoted Serial SA (SSA) decomposition (employed for ESA filtering in [32]) and Parallel SA (PSA) decomposition (employed for so-called “classical” SA filtering in [32]). Both decompositions allow for the estimation of the nonlinearity with a low-complexity nonlinear adaptive filter, as will be explained in the following.

Serial Significance-Aware Decomposition

The SSA decomposition is depicted schematically in Fig. 4.6, where the unknown system is cascaded with an equalizer filter ${\hat{\mathbf{h}}}_{\mathrm{eq}}$. The overall response of this serial connection will ideally be a (delayed) version of the nonlinear preprocessor $f\{\cdot \}$. Thus, an estimate for the physically inaccessible intermediate signal $x_{\mathrm{NL}}\left(n\right)$ is obtained, which allows one to estimate the nonlinearity directly from this decomposition efficiently by a nonlinear model with a short temporal support. An adaptive system using this decomposition will be specified in Section 4.4.2.

Parallel Significance-Aware Decomposition

The PSA decomposition is illustrated in Fig. 4.7. The LIP property of a nonlinear CM is employed to decompose its IR vector into a dominant region ${\mathbf{h}}_{\mathrm{d}}$ (light gray), describing, e.g., direct-path propagation, and a complementary region ${\mathbf{h}}_{\mathrm{c}}$ (dark gray). Augmenting the unknown system with a parallel system modeling the complementary-region output signal component $y_{\mathrm{c}}\left(n\right)$ allows one to compute the dominant-region output component $y_{\mathrm{d}}\left(n\right)=y\left(n\right)-y_{\mathrm{c}}\left(n\right)$. As $f_{}\{\cdot \}$, $x_{\mathrm{NL}}\left(n\right)$, ${\mathbf{h}}_{\mathrm{c}}$, and thereby $y_{\mathrm{c}}\left(n\right)$ are not known during system identification, they have to be replaced by estimates $f_{\text{pp}}\{\cdot \}$, $x_{\mathrm{pp}}\left(n\right)$, ${\hat{\mathbf{h}}}_{\mathrm{c}}$ and ${\hat{y}}_{\mathrm{c}}\left(n\right)$ in practice, as depicted in the upper part of Fig. 4.7. Still, the output of the parallel arrangement yields an estimate $e_{\overline{\mathrm{c}}}\left(n\right)=y\left(n\right)-{\hat{y}}_{\mathrm{c}}\left(n\right)\approx y_{\mathrm{d}}\left(n\right)$ of the dominant-region component of the unknown system (bottom of Fig. 4.7). As the dominant-region component of the system describes the transmission of a significant amount of energy, a nonlinear dominant-region model can be estimated at a high Signal to Noise Ratio (SNR) and will suffer much less from gradient noise than a reverberation tail model. Adaptive systems using this decomposition will be specified in Sections 4.4.3 and 4.4.4.