Introduction to Adaptive Filtering

The Kalman filter is just one of many adaptive filtering (or estimation) algorithms. Despite its elegant derivation and often excellent performance, the Kalman filter has two drawbacks:

1. The derivation and hence performance of the Kalman filter depends on the accuracy of the a priori assumptions. The performance can be less than impressive if the assumptions are erroneous.
2. The Kalman filter is fairly computationally demanding, requiring O(P²) operations per sample. This can limit the utility of Kalman filters in high rate real time applications.

As a popular alternative to the Kalman filter, we will investigate the so-called least-mean-square (LMS) adaptive filtering algorithm.

The principle advantages of LMS are

1. No prior assumptions are made regarding the signal to be estimated.
2. Computationally, LMS is very efficient, requiring O(P) per sample.

The price we pay with LMS instead of a Kalman filter is that the rate of convergence and adaptation to sudden changes is slower for LMS than for the Kalman filter (with correct prior assumptions).

Adaptive Filtering Applications

Channel/System Identification

Figure 1

Channel/System Identification

Noise Cancellation

Suppression of maternal ECG component in fetal ECG (Figure 2).

Figure 2: Cancelling maternal heartbeat in fetal electrocardiography (ECG): position of leads.

Figure 3

y is an estimate of the maternal ECG signal present in the abdominal signal (Figure 4).

Figure 4: Results of fetal ECG experiment (bandwidth, 3-35Hz; sampling rate, 256Hz): (a)reference input (chest lead); (b)primary input (abdominal lead); (c)noise-canceller output.

Channel Equalization

Figure 5

Channel Equalization

Adaptive Controller

Figure 6: Here, the reference signal is the desired output. The adaptive controller adjusts the controller gains (filter weights) to keep them appropriate to the system as it changes over time.

Adaptive Controller

Iterative Minimization

Most adaptive filtering alogrithms (LMS included) are modifications of standard iterative procedures for solving minimization problems in a real-time or on-line fashion. Therefore, before deriving the LMS algorithm we will look at iterative methods of minimizing error criteria such as MSE.

Conider the following set-up: x_k:observation y_k:signal to be estimated

Linear estimator

y_k=w₁x_k+w₂x_{<apply><minus></minus>k1</apply>}+…+w_px_{<apply><plus></plus><apply><minus></minus>kp</apply>1</apply>}(1)

Figure 7

Impulse response of the filter: …,0,0,w₁,w₂,…w_p,0,0,…

Vector notation

y_k=x_k^Tw(2)

Where x_k=(

xk

x<apply><minus></minus>k1</apply>

⋮

x<apply><plus></plus><apply><minus></minus>kp</apply>1</apply>

) and w=(

w1

w2

⋮

wp

)

Error signal

ek	=	yk−yk
	=	yk−xkTw

(3)

Assumptions

(x_k,y_k) are jointly stationary with zero-mean.

MSE

E[ek2]	=	E[(yk−xkTw)2]
	=	E[yk2]−2wTE[xkyk]+wTE[xkxkT]w
	=	Ryy−2wTRxy+wTRxxw

(4)

Where R_yy is the variance of y_k², R_xx is the covariance matrix of x_k, and R_xy=E[x_ky_k] is the cross-covariance between x_k and y_k

NOTE: 

The MSE is quadratic in W which implies the MSE surface is "bowl" shaped with a unique minimum point (Figure 8).

Figure 8

Optimum Filter

Minimize MSE:

∂

∂w

 (E[e_k²])=-2R_xy+2R_xxw=0⇒w_opt=R_xx^-1R_xy(5)

Notice that we can re-write Equation 5 as

E[x_kx_k^Tw]=E[x_ky_k](6)

or

E[xk(yk−xkTw)]	=	E[xkek]
	=	0

(7)

Which shows that the error signal is orthogonal to the input x_k (by the orthogonality principle of minimum MSE estimator).

Steepest Descent

Although we can easily determine w_opt by solving the system of equations

R_xxw=R_xy(8)

Let's look at an iterative procedure for solving this problem. This will set the stage for our adaptive filtering algorithm.

We want to minimize the MSE. The idea is simple. Starting at some initial weight vector w₀, iteratively adjust the values to decrease the MSE (Figure 9).

Figure 9

In One-Dimension

We want to move w₀ towards the optimal vector w_opt. In order to move in the correct direction, we must move downhill or in the direction opposite to the gradient of the MSE surface at the point w₀. Thus, a natural and simple adjustment takes the form

w₁=w₀−

1

2

 μ

∂

∂w

 (E[e_k²])|_w=w0(9)

Where μ is the step size and tells us how far to move in the negative gradient direction (Figure 10).

Figure 10

Generalizing this idea to an iterative strategy, we get

w_k=w_k−1−

1

2

 μ

∂

∂w

 (E[e_k²])|_w=wk−1(10)

and we can repeatedly update w: w₀,w₁,…,w_k. Hopefully each subsequent w_k is closer to w_opt. Does the procedure converge? Can we adapt it to an on-line, real-time, dynamic situation in which the signals may not be stationary?

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