The NLMS algorithm with time-variant optimum
stepsize derived from a Bayesian network perspective

Machine learning techniques have been widely applied to signal processing tasks since decades [1, 2]. For example, directed graphical models, termed Bayesian networks, have shown to provide a powerful framework for modeling causal probabilistic relationships between random variables [3, 4, 5, 6, 7]. In previous work, the update equations of the Kalman filter and the normalized least mean square (NLMS) algorithm have already been derived from a Bayesian network perspective based on a linear relation between the latent room impulse response (RIR) vector and the observation [8, 9].
The NLMS algorithm is one of the most-widely used adaptive algorithms in speech signal processing and a variety of stepsize adaptation schemes has been proposed to improve its system identification performance [10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21]. In this article, we derive a novel NLMS-like filter adaptation (termed EM-NLMS algorithm) by applying the expectation-maximization (EM) algorithm to a probabilistic model for linear system identification. Based on the conditional independence rules for Bayesian networks, it is shown that the normalized stepsize of the EM-NLMS algorithm is equivalent to the one proposed in [10], which is now commonly accepted as optimum NLMS stepsize rule, see e.g. [22]. As the main difference relative to [10] , the normalized stepsize is here estimated as part of the EM algorithm instead of being approximated by artificially extending the acoustic echo path. For a valid comparison, we review the algorithm of [10] for the linear acoustic echo cancellation (AEC) scenario shown in Fig. 1.

The acoustic path between loudspeaker and microphone at time $n$ is modeled by the linear  finite impulse response (FIR) filter

with time-variant coefficients $h_{\kappa,n}$, where ${\kappa=0,...,M-1}$. The observation equation models the microphone sample $d_{n}$:

with the additive variable $v_{n}$ modeling near-end interferences and the observed input signal vector $\mathbf{x}_{n}=[x_{n},x_{n-1},...,x_{n-M+1}]^{T}$ capturing the time-domain samples $x_{n}$. The iterative estimation of the RIR vector by the adaptive FIR filter $\mathbf{\hat{h}}_{n}$ is realized by the update rule

with the stepsize $\lambda_{n}$ and the error signal

relating the observation $d_{n}$ and its estimate $\hat{d}_{n}=\mathbf{x}_{n}^{T}\mathbf{\hat{h}}_{n-1}$. In [10], the optimal choice of $\lambda_{n}$ has been approximated as:

where $||\cdot||_{2}$ denotes the Euclidean norm and $\mathcal{E}\{\cdot\}$ the expectation operator. As the true echo path $\mathbf{h}_{n}$ is unobservable, so that the numerator in (5) cannot be computed, $\lambda_{n}$ is further approximated by introducing a delay of $N_{T}$ coefficients to the echo path $\mathbf{h}_{n}$. Moreover, a recursive approximation of the denominator in (5) is applied using the forgetting factor $\eta$ [22, 23]. The resulting stepsize approximation

leads to oscillations which have to be addressed by limiting the absolute value of $\lambda_{n}$ [24].

In this article, we derive the EM-NLMS algorithm which applies the filter update of (3) using the stepsize in (5), where $\lambda_{n}$ is estimated in the M Step of the EM algorithm instead of being approximated by using (6).

Throughout this article, the Gaussian probability density function (PDF) of a real-valued length-$M$ vector $\mathbf{z}_{n}$ with mean vector $\boldsymbol{\mu}_{\mathbf{z},n}$ and covariance matrix $\mathbf{C}_{\mathbf{z},n}$ is denoted as

where $|\cdot|$ represents the determinant of a matrix. Furthermore, $\mathbf{C}_{\mathbf{z},n}=C_{\mathbf{z},n}\mathbf{I}$ (with identity matrix $\mathbf{I}$) implies the elements of $\mathbf{z}_{n}$ to be mutually statistically independent and of equal variance $C_{\mathbf{z},n}$.

In this part, we compare the proposed EM-NLMS algorithm to the NLMS algorithm reviewed in Section I and show the equality between the adaptive stepsizes in (5) and (16). We reformulate the stepsize update in (16) by applying the conditional independence rules for Bayesian networks [9]: First, we exploit the equalities

which lead to the following relations:

Second, it can be seen in Fig. 2 that the state vector $\mathbf{h}_{n-1}$ and the uncertainty $\mathbf{w}_{n}$ are statistically independent as they share a head-to-head relationship with respect to the latent vector $\mathbf{h}_{n}$. As a consequence, the numerator in (16) can be rewritten as

Finally, we consider the mean of the squared error signal

which is not conditioned on the microphone signal $d_{n}$. By applying the conditional independence rules to the Bayesian network in Fig. 2, the head-to-head relationship with respect to $d_{n}$ implies $v_{n}$ to be statistically independent from $\mathbf{h}_{n-1}$ and $\mathbf{w}_{n}$, respectively. Consequently, we can rewrite (32) as:

The insertion of (31) and (33) into the stepsize defined in (16) yields the identical expression for $\lambda_{n}$ as in (5). The main difference of the proposed EM-NLMS algorithm is that the model parameters $C_{\mathbf{h},n}$ and $C_{\mathbf{w},n}$ (and consequently the normalized stepsize $\lambda_{n}$) are estimated in the M step of the EM algorithm instead of being approximated using (6).

This section focuses on the experimental verification of the EM-NLMS algorithm (“EM-NLMS”) in comparison to the adaptive stepsize-NLMS algorithm described in Section I (“Adapt. NLMS”) and the conventional NLMS algorithm (“Conv. NLMS”) with a fixed stepsize. An overview of the algorithms including the individually tuned model parameters is shown in Table II. Note the regularization of all three stepsize updates by the additive constant $\epsilon=0.01$ to avoid a division by zero. For the evaluation, we synthesize the microphone signal by convolution of the loudspeaker signal with an RIR vector measured in a room with $T_{60}=100$ ms (filter length $M=512$ at a sampling rate of $16$ kHz). This is realized for both white noise and a male speech signal as loudspeaker signals. Furthermore, background noise is simulated by adding Gaussian white noise at a global signal-to-noise ratio of $20$ dB. The comparison is realized in terms of the stepsize $\alpha_{n}$ and the system distance $\Delta h_{n}$ as a measure for the system identification performance:

The results for white noise as input signal are illustrated in Fig 3. Note that in Fig. 3a) the EM-NLMS shows the best system identification compared to the Adapt. NLMS and the Conv. NLMS. As depicted in Fig. 3b), the stepsize $\alpha_{n}$ of the EM-NLMS and the Adapt. NLMS decreases from a value of $0.5$ with the stepsize of the EM-NLMS decaying more slowly.
For male speech as input signal, we improve the convergence of the Conv. NLMS by setting a fixed threshold to stop adaptation ($\alpha_{n}=0$) in speech pauses. Furthermore, the absolute value of $\lambda_{n}$ for the Adapt. NLMS is limited to 0.5 (for a heuristic justification see [24]). As illustrated in Fig. 4a), the EM-NLMS shows again the best system identification compared to the Adapt. NLMS and the Conv. NLMS. By focusing on a small time frame, we can see in Fig. 4b) that the stepsize $\alpha_{n}$ of the EM-NLMS algorithm is not restricted to the values of $0$ and $0.5$ (as Conv. NLMS) and not affected by oscillations (as Adapt. NLMS).
Note that the only relevant increase in computational complexity of the EM-NLMS relative to the Conv. NLMS is caused by the scalar product $\mathbf{\hat{h}}_{n}^{T}\mathbf{\hat{h}}_{n}$ for the calculation of $C_{\mathbf{w},n}$ (cf. Table II), which seems relatively small compared to other sophisticated stepsize adaptation algorithms.

In this article, we derive the EM-NLMS algorithm from a Bayesian network perspective and show the equality with respect to the NLMS algorithm initially proposed in [10]. As main difference, the stepsize is estimated in the M Step of the EM algorithm instead of being approximated by artificially extending the acoustic echo path. For the derivation of the EM-NLMS algorithm, which is experimentally shown to be promising for the task of linear AEC, we define a probabilistic model for linear system identification and exploit the product and conditional independence rules of Bayesian networks. All together this article exemplifies the benefit of applying machine learning techniques to classical signal processing tasks.