HYBRID DESIGN OF FILTERED-X ADAPTIVE ALGORITHM
VIA SAMPLED-DATA CONTROL THEORY
Masaaki Nagahara

Yutaka Yamamoto

Kyoto University
Graduate School of Informatics
nagahara@ieee.org

Kyoto University
Graduate School of Informatics
yy@i.kyoto-u.ac.jp

ABSTRACT
Analysis and design of ¿ltered-x adaptive algorithms are conventionally done by assuming that the transfer function in thesecondary path is a discrete-time system. However, in realsystems such as active noise control, the secondary path is acontinuous-time system. Therefore, such a system should beanalysed and designed as a hybrid system including discreteand continuous- time systems and AD/DA devices. In this article, we propose a hybrid design taking account of continuoustime behaviour of the secondary path via lifting (continuoustime polyphase decomposition) technique in sampled-data control theory.
Index Terms— Active Noise Control, Adaptive ¿lters,
Least mean square methods, Sampled data systems, Functional analysis
1. INTRODUCTION
With the advance of digital technology, it is common to use
digital systems for signal processing. In particular, active
noise control, which we study here, can adopt an advanced
adaptive algorithm, by using the power of fast DSPs (digitalsignal processors) [1].
Fig. 1 shows a standard active noise control system. In
this system, xc (t) represents a noise (or a reference signal)
which enters the duct. The objective here is to eliminate thenoise at the point C. To achieve this, we adopt a digital ¿lter
K(z) with AD (analog-to-digital) and DA (digital-to-analog)
devices. By the discretized signal x of x c (t), the ¿lter K(z)produces another digital signal y, which is converted to ananalog signal by a DA converter, and then a control sound isadded in the duct by a speaker B to cancel the noise.
In active noise control, it is important to compensate thedistortion by the transfer characteristic of the secondary path(from B to C). To compensate this, a standard adaptive algorithm uses a ¿ltered signal of the noise x, and is called ſlteredx algorithm [2]. This ¿lter is a model of the secondary path,and conventionally is a discrete-time one (see e.g., [2, 1]). It
follows that the adaptive ¿lter K(z) optimizes the norm (orthe variance in the stochastic setup) of e(nh), n = 0, 1, 2, . . .

1-4244-1484-9/08/$25.00 ©2008 IEEE353

xc (t)

ACBAD

xK(z)

ye(t)

DAFig. 1. Active noise control system
where h is the sampling time of AD and DA device. Thisis proper if the secondary path is also a discrete-time system. However, in reality, the path is a continuous-time system, and the optimization has to be executed taking accountof the behavior of the continuous-time error signal e(t). Suchan optimization may seem to be dif¿cult because the system
is a hybrid one, which contains continuous- and discrete-timesystems, AD and DA devices.
The same situation has been considered in control systems theory. The modern sampled-data control theory has
been developed in 90’s, which gives an exact design/analysis
method for hybrid systems containing continuous-time plants
and discrete-time controllers [3]. The key idea is lifting. Lifting is a transformation of continuous-time signals to a discretetime signals. The operation can be interpreted as a continuoustime polyphase decomposition. In multirate signal processing,the (discrete-time) polyphase decomposition enables the designer to perform all computations at the lowest rate [4]. In
the same way, by lifting, continuous-time signals or systems
can be represented in the discrete-time domain with no errors(see section 3). The lifting approach is recently applied todigital signal processing [5, 6].
In line with these comtributions, this article will focus ona new design of ¿ltered-x adaptive algorithm which takes account of the continuous-time behavior. As a ¿lter before x,
our adaptive scheme uses an analog (i.e., continuous-time)model of the secondary path. An analog model can be obtained via, for example, acoustic impedance measurement,see [7]. The proposed algorithm involves an integral computation on a ¿nite interval. To execute this computation, weadopt an approximation based on lifting representation. TheICASSP 2008xc (t)ShxK(z)yInstead of the conventional adaptive ¿lter design [8], this problem deals with the continuous-time behavior of the error signal e(t). To solve such a hybrid problem, the lifting approachbased on the sampled-data control theory is very effective.

d(t)

P (s)

e(t)
HhF (s)−w(t)3. LIFTING OPERATOR
Fig. 2. Block diagram of active noise control systemapproximated algorithm can be easily executed by a (linear,time-invariant, and ¿nite dimensional) digital ¿lter.
Throughout this paper, we use the following notations.
We denote by L 2 and L2 [0, h) the Lebesgue space consistingof all square integrable real functions on [0, ∞) and [0, h), respectively. By 2 we denote the set of all real-valued squaresummable sequences on Z + := {0, 1, 2, . . .}. In a Hilbert
space X, the inner product is denoted by ·, · X , and the normby  · X . For a matrix M , we denote its transpose by M  .
By (v)i or (M )ij we denote the i-th or (i, j)-th element of avector v or a matrix M .
2. PROBLEM FORMULATIONIn this section, we formulate the design problem of active
noise control shown in Fig. 1. Consider the block diagram
shown in Fig. 2. In this diagram, P (s) is the transfer function of the primary path from A to C in Fig. 1. The transfer
function of the secondary path from B to C is represented byF (s). Both P (s) and F (s) are continuous-time systems. We
model the AD device by the ideal sampler S h with a sampling
period h, that is,
The DA device is modeled by the zero-order hold H h withthe same period h,(Hh y)(t) :=∞

β0 (t − kh)y[k],t ∈ [0, ∞),

k=0



f = f [0], f[1], f[2], . . . .

(2)This is a discrete-time signal whose values are functions.
The lifting operator L can be interpreted as the polyphasedecomposition [4] of continuous-time signals. In multirate
signal processing, several sampling rates are uni¿ed in thelowest sampling rate by the polyphase decomposition. In thesame way, by lifting, a continuous-time signal is represented
as a discrete-time one.
The space to which the lifted signal (2) belongs is wellde¿ned due to the following lemma.
Lemma 1 Let f be in L2 . Then,∞(Lf )[k]2L2 [0,h) < ∞.

k=0

By this lemma, the following  2 -like space can be de¿ned:
∞22
 := f :f [k]L2 [0,h) < ∞ .
The space 2 is a Hilbert space, and the inner product ·, · 2is de¿ned by

Under the setup, we consider the following optimization
problem.
Problem 1 Find the optimal FIR (ſnite impulse response) ſlter
N−1
ak z −kK(z) =k=0

 ∞J=

where h is a sampling period. For each k, f [k] is a functionin L2 [0, h), and we denote the value of f [k] at θ ∈ [0, h) by
f [k](θ). The lifted signal f is a series of functions in L 2 [0, h),k=0where β0 (t) is the zero-order hold function de¿ned by
1, t ∈ [0, h),β0 (t) :=0, otherwise.

which minimizes(Lf )[k] := f [k] := {f (kh + θ), 0 ≤ θ < h}, k ∈ Z+ ,

Lf 2 :=k ∈ Z+ .

(Sh xc )[k] := xc (kh),In this section, we de¿ne the lifting operator. Consider acontinuous-time signal f (t) ∈ L 2loc . Then, the lifting operator L is de¿ned by

02|e(t)| dt.

(1)

354

u, v2 :=∞u[k], v[k]L2 [0,h) ,u, v ∈ 2 .

k=0

The lifting operator has a good property shown in the following lemma [9].
Lemma 2 For all u, v ∈ L2 , u, vL2 = Lu, Lv2 . Moreover, the lifting operator L is unitary.
By using this property, we derive in the next section an adaptive algorithm taking account of the continuous-time behaviors of analog signals.

4. DESIGN VIA LIFTING METHOD

d(t)
x(t)

xSh

4.1. Hybrid design of adaptive ¿lters
Since the lifting operator L preserves the inner product, theobjective function (1) can be described in the lifted (i.e., discretetime) domain:yK(z)Hhw(t)
F (s)
−e(t)

u
 AdaptiveF(s) Algorithm

HhJ = e, eL2 = e, e2 = d, d2 − 2w, d2 + w, w2 ,where e := Le, d := Ld, and w
 := Lw. Then, by using
a property that the lifted system F H h := LF Hh becomes a
discrete-time, linear, and time-invariant system (see [9]), wehave
w, d2 = F Hh y, d2 =N−1ak z −k u, d2 ,k=0

w, w2 =N−1 N−1

ak al z −k u, z −l u2 ,k=0 l=0where u := F Hh x and z is the shift operator in 2 . Next, thegradient of J with respect to a m (m = 0, 1, . . . , N − 1) isgiven by
∂J= −2z −m u, d2 + 2∂amN−1al z −m u, z −l u2 .

4.2. Approximation of adaptive algorithm
To execute the algorithm (3), we have to calculate the L 2 [0, h)
inner product u[n− m], e[n] L2 [0,h) , m = 0, . . . , N − 1. The
exact value of this is calculated by an integral on [0, h), andis dif¿cult to obtain in practice. Therefore, we introduce anapproximation method for this computation.
First, we split the interval [0, h) into L short intervals[0, h/L), [h/L, 2h/L), . . ., [h − h/L, h). Assume that theerror e is constant on each short interval 1 , we haveu[n − m], e[n]L2 [0,h) = ed [n] ud [n − m],where(ed [n])i := e[n]((i − 1)h/L) = e(nh + (i − 1)h/L) , ih/Lu
[n − m](θ)dθ, i = 1, . . . , L.
(ud [n])i :=(i−1)h/L

l=0

Setting the gradient to zero and solving for a m , the parameter−1a := [a0 , a1 , . . . , aN −1 ] is optimized by a opt = Tuu
Tud ,where(Tuu )ij := z −i+1 u, z −j+1 u2 ,(Tud )i := z −i+1 u, d2 ,Fig. 3. ¿ltered-x adaptive systemi, j = 1, . . . , N

The optimal parameter a opt is given by a product of ma−1
trices Tuuand Tud . However, each element of the matrices
are calculated by an in¿nite sum on k = 0, 1, 2, . . . (or an integral on the whole interval [0, ∞)). Such a calculation is notsuitable for practical processing. Moreover, only an estimateof the secondary path F (s) is available, which are denoted byF(s). Therefore, we use instantaneous estimates for T uu andTud , and adopt an LMS type of algorithm as follows:a[n + 1] = a[n] + μTue [n],n ∈ Z+ ,(3)Then, ud [n] is effectively computed by the following digital¿lter,
ξ[n + 1] = Ad ξ[n] + Bd x[n],Fdud [n] = Cd ξ[n] + Dd x[n],where Ad , Bd , Cd , and Dd are matrices de¿ned by heAτ Bdτ,Ad := eAh , Bd := ih/L(Cd )i :=(i−1)h/L

 ih/L

where u
 := FHh x andu[n − i + 1], e[n]L2 [0,h) ,(Tue [n])i := Then the integral in u
 d [n] can be computed via a state-space
representation of FHh [9]. Assume that a state-space representation of the model F for the secondary path is given byẋ(t) = Ax(t) + Bv(t),F :u(t) = Cx(t), t ∈ [0, ∞).

(Dd )i :=
i = 1, . . . , N.

Fig. 3 shows the block diagram of this ¿ltered-x adaptive system. We should notice that the input x is ¿ltered by a copy
FHh of the secondary path including the hold device H h .

355(i−1)h/L0CeAθ dθ, θ0CeA(θ−τ ) Bdτ dθ,i = 1, . . . , L.

Note that the integrals in B d , Cd , and Dd can be effectively
computed by using matrix exponential, see [10, 9]. We show
the proposed adaptive scheme in Fig. 4.
1 This means that the desired signal d(t) does not contain high frequency
components such that d(t) does not oscillate in each short interval.

0.5

a[n]ShxFd (z)ud Adaptive ed
Se(t)

h/L

Algorithm

0.4

Error |e(t)|2xc (t)Fig. 4. ¿ltered-x adaptive scheme

200.2
0.10Gain (dB)

0.300100200300Time (s)400500í20Fig. 6. Square error |e(t)| 2 ; proposed design (solid) anddiscrete-time design (dotted)

í40
í60
í8007. REFERENCES
2π

46Frequency (rad/s)

810Fig. 5. Freqnecy response of F (s) (solid) and P (s) (dash).
The dotted line shows the Nyquist frequency.

5. SIMULATIONIn this section, we simulate our adaptive ¿ltering. Set thesampling period h = 1. The fast-sampling ratio L is 10. The
transfer functions F (s) and P (s) have peaks at ω = 1, 2, 3, 4and ω = 1.2, 2.4, 3.6, 4.8, respectively. These frequency responses are shown in Fig. 5. Note that these systems havepeaks beyond the Nyquist frequency ω = π. Let the input
xc (t) be the white Gaussian signal with zero-mean and variance 1, and Fig. 6 shows the square error |e(t)| 2 . We compare
our result with a conventional design (i.e., a discrete-time design [1]). Our result shows better response than the conventional one, since our method takes account of the continuoustime characteristic, in particular, high frequency components
beyond the Nyquist frequency.

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6. CONCLUSION[8] S. Haykin, Adaptive Filter Theory, Prentice Hall, 1996.
In this article, we have proposed a hybrid design of ¿ltered-x
adaptive algorithm via lifting method in sampled-data control theory. The proposed algorithm can take account of thecontinuous-time behavior of the error signal. We have also
proposed an approximation of the algorithm, which can beeasily implemented in DSP. A simulation result shows the effectiveness of our method.

356[9] T. Chen and B. Francis, Optimal Sampled-Data ControlSystems, Springer, 1995.
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