OPTIMAL DESIGN OF FRACTIONAL DELAY FIR FILTERSWITHOUT BAND-LIMITING ASSUMPTION
Masaaki Nagahara and Yutaka Yamamoto
Department of Applied Analysis and Complex Dynamical Systems,
Graduate School of Informatics,
Kyoto University,
Kyoto 606-8501, Japan
ABSTRACT
Fractional delay filters are those that are designed to delay the input signals by a fractional amount of the samplingtime. Since the delay is fractional, the intersample behavior of original analog signals become crucial. While theconventional design bases itself on the assumption that theincoming analog signals are fully band-limited up to theNyquist frequency, the present paper applies the modern
sampled-data H ∞ optimization which aims at restoring theintersample behavior without the band-limiting assumption.
It is shown that the optimal FIR filter design is reducible to aconvex optimization described by a linear matrix inequality(LMI). A design example is shown to illustrate the advantage of the proposed method.
1. INTRODUCTION
Fractional delay filters are to delay the input signal by afraction of the sampling period. Such a filter has wide applications in signal processing, including digital communications, speech processing and digital modeling of musicalinstruments [1].
Conventionally, fractional delay filters are designed in
the discrete-time domain by assuming that the incoming
analog signals are fully band-limited up to the Nyquist frequency. However, this assumption is not realistic because
no real analog signals are fully band-limited. Moreover, by
their very nature, such filters should reconstruct intersample signal values. For such a problem, sampled-data control theory [2] provides an optimal platform. This theory isalready confirmed to be effective in some digital filter design problems (e.g., [3]), and it is ideal in dealing with thecontinuous-time behavior.
We thus formulate the design problem of fractional delay FIR filters as a sampled-data H ∞ optimization problem.
That is, we design an FIR filter which minimizes the H ∞ norm of the error system including both continuous- anddiscrete-time signals. We show that this design problem isreducible to a convex optimization with a linear matrix inequality (LMI). A design example is shown to illustrate theadvantage of the proposed method.
Throughout this paper, we denote by L2 the Lebesgue
space consisting of all square integrable real functions on[0, ∞), and by ℓ2 the set of all real-valued square summablesequences on Z+ := {0, 1, 2, . . .}. For linear spaces X andY , we denote by B(X, Y ) the collection of all bounded linear operators of X into Y . AT denotes the transpose of amatrix A.
2. DESIGN PROBLEM
2.1. Fractional delay filters
Consider a continuous-time signal v(t) as shown in Fig. 1(a). The signal v(t) is delayed by the continuous-time delayv(t)(a)v(nT )v(t − D)(b)v(nT − D)(c)(d)
Fig. 1. (a) continuous-time signal v(t), (b) delayed signalv(t − D), (c) sampled signal v(nT ), (d) delayed and sampled signal v(nT − D)operator e−Ds (D > 0) as shown in Fig. 1 (b). Then the

delayed signal v(t − D) is sampled with period T and becomes a discrete-time signal v(nT − D), n ∈ Z+ as shownin Fig. 1 (d).
On the other hand, consider the sampled signal v(nT ),n ∈ Z+ as shown in Fig. 1 (c). Then we define the ideal
fractional delay filter as follows:
Definition 1 The ideal fractional delay filter K id with delay time D is defined by
K id : v(nT ) 7→ v(nT − D).

Note that if D = kT , k ∈ Z+ , the ideal filter K id isthe discrete-time delay z −k . Moreover, if the input analog
signal v(t) is fully band-limited up to the Nyquist frequency
ωN := π/T , the impulse response of the ideal fractional
delay filter is obtained as follows [1]:sin π(nT − D)/T= sinc π(nT − D)/T,π(nT − D)/T(1)
n = 0, ±1, ±2, . . . .

k id [n] =governs the frequency-domain characteristic1 of the input
signal w ∈ L2 . The upper path of the diagram is the ideal
process of the fractional delay filter (the process (a) → (b)→ (d) in Fig. 1), that is, the continuous-time signal v isdelayed by the continuous-time delay e−Ds , sampled withperiod T , and becomes a discrete-time signal zd ∈ ℓ2 . On
the other hand, the lower path is the real process ((a) →(c) → (d) in Fig. 1), that is, the continuous-time signal v issampled with period T , filtered by K(z) to be designed, andbecomes a discrete-time signal ud ∈ ℓ2 .
Put ed := zd − ud (the difference between the ideal output zd and the real output ud ), and let E denote the systemfrom w ∈ L2 to ed ∈ ℓ2 . Then our problem is to find thefilter KD (z) which minimizes the H ∞ norm of the error
system E.
Problem 1 Given a stable, strictly proper F (s), delay timeD > 0, sampling period T > 0, find the digital filter K(z)which minimizes∥ed ∥ℓ2.
w∈L2 ∥w∥L2∥E∥∞ := sup(3)The frequency response of this ideal filter is derived by theFourier transform:
K id (ejωT ) = e−jωT D ,ω ≤ ωN .

(2)

The ideal filter (1) or (2) cannot be realized since thefilter is generally non-causal and infinite dimensional, andhence a conventional design aims at approximating (1) or(2) via a window method, maximally-flat FIR approximations, weighted least-squares approximation, and so forth[1, 4].
These methods are based upon the band-limiting assumption. In practice, however, analog signals always containsome frequency components beyond the Nyquist frequency
as mentioned above. In what follows, we thus formulate
the design problem of fractional delay filters without thisassumption.
2.2. Design problem of fractional delay filters
Consider the block diagram Fig. 2. In this diagram, F (s)
w

F (s)

v

zde−DsTydTK(z)ud + ed−Fig. 2. Error system for designing fractional delay filter
K(z)3. DESIGN OF FRACTIONAL DELAY FIR FILTERS
3.1. Reduction to a finite-dimensional problem
A difficulty in Problem 1 is that the error system E contains
both continuous- and discrete-time dynamics, that is, E is
an infinite-dimensional system. To solve this problem, weemploy the modern sampled-data control theory [2]. Via
this theory, we have the following theorem.
Theorem 1 For the error system E ∈ B(L2 , ℓ2 ), there exists a finite-dimensional discrete-time system Ed such that∥E∥∞ = ∥Ed ∥∞ .
Outline of Proof Introduce the dual operator [5] E ∗ of Esuch that (Ew, v)ℓ2 = (w, E ∗ v)L2 , w ∈ L2 , v ∈ ℓ2 . Since
E is bounded, we have ∥E∥2∞ = ∥EE ∗ ∥. Then, the operator EE ∗ is in B(ℓ2 , ℓ2 ), and there exists a finite-dimensional
discrete-time system Ed ∈ B(ℓ2 , ℓ2 ) such that Ed Ed∗ =EE ∗ , and finally we have ∥E∥2∞ = ∥EE ∗ ∥ = ∥Ed ∥2∞ . For
the precise discussion, see [6].
¤This theorem leads to a finite-dimensional optimization
problem. Our objective is then to find an FIR filter minimizing ∥Ed ∥∞ .
1 Conventionally, F (s) is assumed to be an ideal filter such thatF (jω) = 0, |ω| ≥ ωN .

3.2. Design of fractional delay FIR filter
FIR filters are often preferred to IIR filters because of theiradvantages in respect of implementation. In this section, wewill show the design of the optimal FIR filters according toTheorem 1.
For fixed N ≥ 0, put
K(z) =N∑

4. DESIGN EXAMPLE

an z −n ,n=0

and introduce a state-space realization
K(z) = CK (α)(zI − AK )−1 BK + DK (α),][where α = a0 a1 . . . aN ,(4) 0 100.. ..
.. 
.
.
. , BK = AK = ,.
. . 10100][CK (α) = aN aN −1 . . . a1 , DK (α) = a0 .

We present a design example of fractional delay filters. Thedesign parameters are as follows: the sampling period T =1 [sec], the delay D = 10.8 [sec], and()Lωc
F (s) =, ωc = 0.5, L = 1, 2, 4, 8.
s + ωcNote that F (s) has the cutoff frequency at 0.5 [rad/sec]. Theorder of our FIR filter N = 31. If L = 1, we have theanalytic solution [6]K(z) = z −m (a0 + a1 z −1 ),where D = d + mT , 0 < d < T , m ∈ Z+ andThen the discrete-time system Ed (z) is given as follows [6]:Ed (z) = (C1 − K(z)C2 )(zI − Ad )−1 Bd ,Theorem 2 gives an LMI characterization for the existence of an FIR filter K(z) such that ∥Ed ∥∞ < γ. Whether(6) is satisfied can easily be checked, for example by standard MATLAB (particularly, LMI toolbox) routines [8]. To
obtain the optimal α, minimize γ subject to the LMI (6),which can also easily be accomplished by the software.

(5)

where C1 , C2 , Ad , Bd are matrices. Substitution of (4) in(5) yields Ed (z) = C(α)(zI − A)−1 B, where][ ][BdAd0,, B=A=0
BK C2 AK[]C(α) = C1 − DK (α)C2 −CK (α) .
Note that the transfer function Ed (z) is linear in α. It follows that the design problem of choosing α to minimize∥Ed ∥∞ can be expected to become an LMI. In fact, thebounded real lemma [7] readily yields the following.
Theorem 2 Assume γ > 0. Then ∥Ed ∥∞ < γ if and only
if there exists a matrix P > 0 such that T
A PA − PAT P BC(α)T BT P A−γI + B T P B0  < 0.
(6)
C(α)
0−γI
Proof By the bounded real lemma [7], ∥Ed ∥∞ < γ isequivalent to the condition that there exists a matrix P > 0such that[]T [][] []ABP 0ABP0<.
C(α) 00 I C(α) 00 γ2IThen, this inequality is equivalently converted to (6) by using the Schur complement [7].
¤sinh ωc (T − d)a0 = √,ωc sinh ωc T

a1 = e−ωc T (eωc d − a0 ),which minimizes ∥Ed ∥∞ . Thus, we use this formula when
L = 1 and design by Theorem 2 when L ≥ 2. We compare
these filters with a conventional filter of 31-st order FIR filter by the Kaiser window method [1].
Fig. 3 shows the Bode plots of the filters obtained by
the proposed method and the conventional one. The phase
responses are almost similar, whereas there is a differencein the magnitude. We can see that as L becomes larger,
the frequency response approaches the ideal response (2),and the conventional filter appears best in the context of theconventional design methodology.
To see the difference, we show the frequency responsesof the error system E (see [9]) of the proposed filter (L =1) and the conventional one in Fig. 4. Our filter exhibits
much smaller errors in the high-frequency domain. This isbecause the conventional design does not take into accountthe frequency response of the source analog signals while
the present method does.
Then, we show the time response against rectangularwaves in Fig. 5 ( (a) is the proposed design and (b) is theconventional one). The present method is clearly superior
to the conventional one which shows much ringing at theedges of the waves.
5. CONCLUSION
We have presented a new method of designing fractional delay FIR filters via sampled-data H ∞ optimization. An advantage here is that an analog optimal performance can beSampled-data design

Bode Diagram

Phase (deg)Magnitude (dB)

21.5

L=4L=801−2

L=20.5−6L=1−80Amplitude−40-0.5−360−720-1

−1080
−1440

-1.5

−1800
−2160
−110010203040504050

Time (sec)010(a) H ∞ designFrequency (rad/sec)

Fig. 3. Fractional delay FIR filters: H ∞ design (solid) andconventional design (dots)

Conventional design
1.5

1Frequency response-11Amplitude

0.5

-11.5

Gain (dB)-120-0.5-12.5-1-13-1.5-13.50102030Time (sec)(b) Conventional design-14 -1
1010
Frequency (rad/sec)

0Fig. 5. Time responses
Fig. 4. Frequency response of E: H
conventional design (dash)∞design (solid) andobtained. The design problem can be reduced to a convex
optimization with an LMI, which leads to an easy computation of design. The designed filter exhibits a much moresatisfactory performance than conventional ones.
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