Proceedings of the 42nd IEEE
Conference on Decision and Control
Maui, Hawaii USA, December 2003

FrE13-5
Optimal Design of Fractional Delay Filters

Masaaki Nagahara
Department of Applied Analysis
and Complex Dynamical Systems,
Graduate School of Informatics,
Kyoto University,
Kyoto 606-8501, JAPANnagahara@i.kyoto-u.ac.jp

Yutaka Yamamoto
Department of Applied Analysis
and Complex Dynamical Systems,
Graduate School of Informatics,
Kyoto University,
Kyoto 606-8501, JAPANyy@i.kyoto-u.ac.jp

Abstract— Fractional delay filters are those that are designed
to delay the input signals by a fractional amount of thesampling time. They are widely used in digital communications,speech processing, digital modeling of musical instruments, etc.
Since the delay is fractional, the intersample behavior of input
signals become crucial. While the conventional design bases
itself on the full band-limiting assumption, the present paper
applies the modern sampled-data H ∞ control which aims at
restoring the intersample behavior. It is shown that the optimal
design problem is reducible to a norm-equivalent discretetime problem. A numerical example is shown to illustrate theadvantage of the proposed method.

appear somewhat restrictive, it covers many typical cases andvariations from it can be taken care of by some robustness
properties. A numerical example is shown to illustrate theadvantage of the proposed method.
Throughout this paper, we use the following notations.
L2 [0, ∞) and L2 [0, h) denote the Lebesgue spaces consistingof all square integrable real functions on [0, ∞) and [0, h),respectively. L2 [0, ∞) may be abbreviated to L2 . l2 denotes
the set of all real-valued square summable sequences, and2lL
2 [0,h) the set of all square summable sequences whose
values are in L2 [0, h). <N denotes the N -dimensional vector
space over <. The transfer function
whose
realization is


A B
T
. A denotes the{A, B, C, D} is denoted as
C Dtranspose of a matrix A.

I. INTRODUCTION
Fractional delay filters are to delay the input signal by afraction of the sampling time. Such a filter has wide applications in signal processing, including digital communications,speech processing and digital modeling of musical instruments [2], [6]. For example, in digital communications, thereceived continuous-time pulse sequences should be sampled
at the middle of each pulse to decide the received bit orsymbol value. For this purpose, a fractional delay filter isused to synchronize the sampler with the incoming signal[2], [6].
Conventionally, fractional delay filters are designed in
the discrete-time domain by assuming that the incoming
continuous-time 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 reconstructintersample signal values, and sampled-data control theory
provides an optimal platform for such a problem. This theoryis already confirmed to be effective in some digital filter
design problems [9], [4], [5], and it is ideal in dealing withthe continuous-time behavior.
We thus formulate the design problem of fractional delay
filters as a sampled-data H ∞ control problem. That is, we
design a filter which minimizes the H ∞ -norm of the error
system consisting of a continuous-time delay and a filter with
ideal sampler. We show that this design problem is reducible
to a discrete-time H ∞ one by lifting [8]. We also provide ananalytical solution to the case in which such fractional delays
are time-varying, under the assumption that the underlying
frequency characteristic of the continuous-time input signal
is governed by a low-pass filter of first order. While this may

0-7803-7924-1/03/$17.00 ©2003 IEEE

II. DESIGN PROBLEM OF FRACTIONAL DELAY
FILTERS
A. Fractional delay filters
Consider the continuous-time signal x(t) shown in Fig.
1 (a). The continuous-time signal x(t) is delayed by thecontinuous-time delay operator e−Ds (D > 0) as shown in
Fig. 1 (b). Then the delayed signal x(t − D) is sampledwith sampling period h and becomes a discrete-time signal
xD [n] := x(nh − D), n = 0, 1, 2, . . . (shown in Fig. 1 (d)).
On the other hand, consider the sampled signal x[n] :=x(nh), n = 0, 1, 2, . . . , with sampling period h as shown in
Fig. 1 (c). Then we define the ideal fractional delay filter asfollows:
id
Definition 1: The ideal fractional delay filter KD
withdelay time D is defined byidKD: x[n] = x(nh) 7→ xD [n] = x(nh − D).

Note that if D = kh, k = 0, 1, 2, . . . , the ideal fractional
id
delay filter KD
is the discrete-time delay z −k . Moreover, if
the input analog signal x(t) is fully band-limited up to the
Nyquist frequency ωN := π/h, the impulse response of theideal fractional delay filter is obtained as follows [2], [6]:6539

sin π(n − D)= sinc(n − D),π(n − D)n = 0, ±1, ±2, . . . .

id
kD
[n] =

(1)

x(t)

x(nh)

w
e

LF L−1ve

Sh e−Ds L−1

Sh L−1(a)x(t − D)

(b)

(d)

F (s)

ud + ed−Lifted error system

x(nh − D)

Fig. 1. (a) continuous-time signal x(t), (b) delayed signal x(t − D), (c)sampled signal x(nh), (d) delayed and sampled signal x(nh − D)

v

vD

e−Ds
Sh

Fig. 2.

KD(c)Fig. 3.

w

yd

zd

yd

zd

Sh

KD (z)

signal v is sampled by the sampler Sh , filtered by KD (tobe designed), and becomes a discrete-time signal ud ∈ `2 .
Put ed := zd − ud (the difference between the ideal output
zd and the real output ud ), and let Tew denote the systemfrom w ∈ L2 to ed ∈ `2 . Then our problem is to find thefilter KD (z) which minimizes the H ∞ norm of the error
system Tew .
Problem 1: Given a stable, strictly proper F (s), delay
time D > 0, sampling time h > 0 and an attenuation levelγ > 0, find a digital filter KD (z) such thatud + ed−kTew k∞ := supw∈L2

ked k`2
.
kwkL2

(3)

Error system for designing fractional delay filter KD

III. DESIGN OF FRACTIONAL DELAY FILTERS

The frequency response of this ideal filter is derived by the
Fourier transform:
id jωh(e ) = e−jωDh ,KDω ≤ 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 [2], [6].
These methods are based upon the assumption that the
input analog signals are fully band-limited. In practice,
however, analog signals always contain some frequency
components beyond the Nyquist frequency. In what follows,
we thus formulate the design problem of fractional delay
filters by using the sampled-data H ∞ optimization.
B. Design problem of fractional delay filters
Consider the block diagram Fig. 2. In this diagram, F (s)governs the frequency-domain characteristic of the input
signal w ∈ L2 . The upper path of the diagram is the ideal
process of the fractional filter (the process (a) → (b) → (d) in
Fig. 1), that is, the continuous-time signal v is delayed by thecontinuous-time delay e−Ds , sampled by the sampler Sh andbecomes a discrete-time signal zd ∈ `2 . On the other hand,
the lower path is the real process, that is, the continuous-time

A. Lifting model of sampled-data error system
Let {A, B, C, 0} be a minimal realization of F (s) andD = mh + d where m ≥ 0 is an integer and 0 ≤ d < h is areal number. First, we introduce the lifting operator L [8]:2
L : L2 [0, ∞) 3 f 7→ {fe[k](θ)}∞
k=0 ∈ lL2 [0,h) , θ ∈ [0, h),fe[k](·) := f (kh + ·) ∈ L2 [0, h).

We apply lifting to the continuous-time signals w and v andput w
e := Lw, ve := Lv. By lifting the continuous-time
signals, we obtain a discrete-time system shown in Fig. 3
from the sampled-data system Fig. 2.
Then a state-space realization of the lifted error system
shown in Fig. 3 is obtained as follows:

6540




B
x[n + 1] = Ad x[n] +
w[n],
e
0
ed [n] = Ce x[n] − ud [n],yd [n] = Cy x[n],u d = K D yd ,

(4)




eAh
0
0
Ad :=  CeA(h−d)
0
0 ,
0
Bm Am


Z h
A(h−τ )
e
B
w(τ
e
)dτ


0
,
Bw
e := 
 Z h−dA(h−d−τ )
Ce
B w(τ
e )dτ

w
e



Fig. 4.

B
0

 wd

Td

ed

Sampled-data system Tew

0

Ce := [0, 0, Cm ],Cy := [C, 0, 0],

For any u ∈ <,where {Am , Bm , Cm , 0} is a realization of discrete-timedelay z −m , that is,


0
 ..
 .
Am := 
 .
 ..
0

1
..
.

..

.

..

...

.
...

Bm := [0, . . . , 0, 1]T ,| {z }Z heA(h−θ) B (B∗2 u) dθ
B1 B∗2 u =
0
Z h


T
=
eA(h−θ)B χ[0,h−d] (θ)B T eA (h−d−θ)C T u dθ
0
Z h−d
T
eA(h−d−θ)BB T eA (h−d−θ) dθ C T u
= eAd


0 




 m,

1 



0

0

= eAd M (h − d)C T u.

m−1

Cm := [1, 0, . . . , 0].
| {z }
m−1

The proof for B1 B∗1 and B2 B∗2 is similar.

B. Norm-equivalent finite dimensional model
The discrete-time state-space realization (4) has an infinite
dimensional operator B : L2 [0, h) → <N +1 (N is thedimension of A). By introducing the dual operator B∗ :<N +1 → L2 [0, h) of B, we can obtain a finite dimensional
operator (i.e., matrix), which leads to a norm-equivalent finite
dimensional system of the infinite dimensional system (4).
The dual operator B∗ of B is given as follows [7]:
B∗ =



B∗1

B∗2



Remark 1: It is well known that the matrix M (t) can becomputed by using matrix exponential [3]:

By this formula, we can numerically compute the matrix (5)easily.

,T
B∗1 := B T eA (h−θ) ,Using Lemma 1, we obtain a (finite dimensional) discretetime system Tew which is equivalent to the sampled-datasystem Tew with respect to their H ∞ -norm.

TB∗2 := χ[0,h−d] (θ)B T eA (h−d−θ) C T ,θ ∈ [0, h],(1,θ ∈ [0, h − d],χ[0,h−d] (θ) :=0,otherwise.

Theorem 1: For the sampled-data system Tew , there exists
a discrete-time system Tew such that

We have the following lemma:
Lemma 1: The operator BB∗ is a matrix given by
∗

BB =



M (t) :=M (h)T
CM (h − d)eA d

Z t

F11 (t)0T
M (t) = F22(t)F12 (t), 
F12 (t)−A BB T:= expt .
F22 (t)0
AT

eAd M (h − d)C T
CM (h − d)C T

kTew k∞ = kTew k∞ .
,T

eAθ BB T eA θ dθ.

0

Proof: We first prove B1 B∗2 = eAd M (h − d)C T .

(5)

Proof:For the state space equations (4), put wd :=
we and let Td denote the system from wd to edB(see Fig. 4). Then we have Tew = Td
where Td0
is a discrete-time system. From Lemma 1, BB∗ is finite
dimensional. Decomposing BB∗ as BB∗ = Bd BdT with a


6541

B
0

w
ed



Bd
0

 wd

ed

Td

Fig. 5.

Norm-equivalent discrete-time system Tew

ed




yd



 Ad

 Ce
Cy

Bd
0
0
0



For this purpose, we assume that the filter F (s) is thefirst-order low-pass filter:ωc.
F (s) =s + ωcWhile this is certainly restrictive, it still has the advantageof having the capability of reconstructing high-frequency
components beyond the Nyquist frequency, and the variancefrom this first-order frequency distribution is estimated by arobustness argument. See Section V below.
We have the following theorem:
Theorem 2: The optimal filter KD (z) is obtained as follows:ωc e−ωc (h−d) B0 + (z − e−ωc h )B1 (d)KD (z) =,ωc B0 z m+1s
1 − e−2ωc h
(6)
,
B0 :=
2ωc
r
ωc e−ωc d (1 − e−2(h−d)ωc )√B1 (d) :=·.
2
1 − e−2ωc hMoreover, the optimal value of kTew k∞ issωc sinh(ωc d) sinh(ωc (h − d))kTew k∞ = B2 (d) :=.
sinh(ωc h)(7)
Proof:The state space realization of F (s) is{−ωc , 1, ωc, 0}. Therefore by using Lemma 1, we have



ed
w
0 

−I 
 ud
0

- KD (z)Fig. 6.

Norm-equivalent discrete-time system

matrix Bd , we havekTew k2∞ = Td
=
=
=
=



B
0

 2

 ∞ 
∗
B
Td
Td
0
∞
∗


B
B
∗
Td
Td
0
0
∞
T


Bd
Bd
Td∗
Td
0
0
∞

 2
Bd
Td
.
0
∞


B
0

Bd BdT =

1 − e−2ωc h2ωc e−ωc d (1 − e−2ωc (h−d) )2Bdcompletes the proof.
0
A state space realization of Tew is obtained by replacing B
in (4) with Bd .
Thus the sampled-data H ∞ optimization (Problem 1) isequivalently reduced to a discrete-time H ∞ optimization.
The block diagram for the optimization is shown in Fig. 6,
and we can easily find the optimal filter KD (z) by standard
softwares such as MATLAB.
Putting Tew := TdIV. DESIGN OF VARIABLE FRACTIONAL DELAY
FILTERSe−ωc d (1 − e−2ωc (h−d) )2.
2(1 − e−2ωc (h−d) )ωc

By the Cholesky factorization of Bd BdT , we obtain


B0
0
Bd =
,
B1 (d) B2 (d)where B0 , B1 (d) and B2 (d) are the values in (6) and (7).
Then the norm-equivalent discrete-time system (shown in
Fig. 6) is given as follows:



Bd
−1
Tew (z) = (Ce − KD (z)Cy ) (zI − Ad )0
= G1 (z) G2 (z) ,The H ∞ design method in the previous section requires
the delay D = mh + d to be fixed. In some applications, afilter with a variable delay (i.e., the delay can be changed
without redesigning the filter) is desired in accordance withvarying delays. We here give an optimal design for the filter
KD (z) having the delay D (or m and d) as an adaptive
parameter.

6542

G1 (z) := −KD (z)C(zI − eAh )−1 B0no+ z −m−1 CeA(h−d) (zI − eAh )−1 B0 + B1 (d)ω c B0z − e−ωc hωc e−ωc (h−d) B0
+
B
(d)
,
+ z −m−11
z − e−ωc h
G2 (z) := z −m−1 B2 (d).
= −KD (z)Since G2 (z) is independent of KD (z), the optimal value isattained by minimizing G1 (z). It is clearly possible to makeG1 (z) = 0 by taking

F (jω)−ωc h

B0 + (z − e)B1 (d).
ωc B0 z m+1
The minimum value of kTew k∞ is attained as
KD (z) =ωc e−ωc (h−d)ωkTew k∞ = kG2 (z)k∞ = B2 (d).

F∆ (jω)Fig. 7.

Note that the obtained optimal filter consists of a first-orderFIR (Finite Impulse Response) filter and discrete-time delayz −m ; this is because F (s) is of first order. We conjecture that
if F (s) is n-th order, the optimal filter may be the productof an n-th order filter and z −m , since the order of the plant(shown in Fig. 6) is n + m.

Bode Plot
2
Magnitude (dB)0

V. ROBUSTNESS OF DESIGNED FILTERS
Our design requires that the frequency-domain characteristic F (s) be known. In practice, however, F (s) cannot be
identified exactly. Therefore, we partially circumvent thisdefect by discussing the robustness of the filter against
the uncertainty of F (s). Let us assume the unstructured
uncertainty of the following type:
∆
Tew
:=

−Ds

-4
-6

0
-360



-720
-1080
-1440
-1800

Sh e
− KD Sh F∆ ,∆ ∈ ∆ := {∆ : k1 + ∆k∞ ≤ γ} .

-2160
10

0

Frequency (rad/sec)

We have the following proposition:
Proposition 1: For any ∆ ∈ ∆,∆k∞ ≤ γkTew k∞
kTew

-2

-8

Phase (deg)F∆ (s) := F (s)(1 + ∆(s)),

Nominal filter F (s) (solid) and perturbed F∆ (s) (dash)(8)

Proof: By the submultiplicativity of H ∞ -norm, wereadily have∆kTew
k∞ = kTew (1 + ∆)k∞ ≤ kTew k∞ k1 + ∆k∞≤ γkTew k∞ .

By this proposition, the nominal performance kTew k∞ isguaranteed against the perturbation ∆ ∈ ∆ if γ ≤ 1. In somecases, it is possible that γ = 1, in which case the performanceis bounded as illustrated in Fig. 7. This means that if we takeF (s) that covers all possible gain characteristics of the input
analog signals, it gives a bound for the error norm. This atleast partially justifies the choice of the first-order weighting
F (s) in the previous section.
VI. DESIGN EXAMPLE
We present a design example of fractional delay filters.
The design parameters are as follows: the sampling period
h = 1, the delay is D = 10.8, that is, m = 10 and d = 0.8,
and
0.5
F (s) =.
s + 0.5

Fig. 8. Bode plot of filters: sampled-data design (solid) and conventional
design (dash)

Note that F (s) has the cutoff frequency ωc = 0.5.
We compare the optimal filter obtained by Theorem 2 witha conventional filter of 31-st order FIR filter with the Kaiser
window [2].
Fig. 8 shows the Bode plots of the filters obtained by the
sampled-data design and the conventional one. As illustrated
in Fig. 8, the conventional filter is closer to the ideal filter(2) as expected, so that it appears better in the context of theconventional design methodology.
However, the present filter exhibits much smaller errors
in the high-frequency domain as shown in Fig. 9. This isbecause the conventional design does not take into accountthe frequency response of the source analog signals while
the present method does.
To see the difference between the present filter and theconventional one, we show the time response against rectangular waves in Fig. 10 (sampled-data design) and in Fig. 11(conventional design). The present method is clearly superior
to the conventional one which shows much ringing at theedges of the waves.

6543

Conventional design
1.5

-11.5

1

-12

0.5
Amplitude

Gain (dB)Frequency response
-11

-12.5

-13

0

-0.5

-13.5
-1

-14 -1
10

10

0

-1.5

Frequency (rad/sec)

Fig. 9.
Frequency response of Tew : sampled-data design (solid) andconventional design (dash)

Sampled-data design

1

Amplitude

0.5

0

-0.5

-1

0

10

20

30

40

50

Time (sec)Fig. 10.

10Fig. 11.

20
30
Time (sec)40

50Time response (conventional design)

VIII. REFERENCES

1.5

-1.5

0

Time response (sampled-data design)

VII. CONCLUSION
We have presented a new method of designing fractional
delay filters via sampled-data H ∞ optimization. An advantage here is that an analog optimal performance can beobtained. We have also given the H ∞ optimal filter having
delay time variable D as an adaptive variable, when thefrequency distribution of the input analog signal is modeled
as a first-order low-pass filter. While the hypothesis here issomewhat restrictive, we have also shown that a certain error
bound can be obtained for a more general class of weights.
The designed filter exhibits a much more satisfactory performance than conventional ones.

6544[1] T. Chen and B. A. Francis, Optimal Sampled-data
Control Systems, Springer (1995)[2] T. I. Laakso, V. Välimäki, M. Karjalainen and U. K.
Laine, Splitting the unit delay, IEEE Signal Process.
Mag., vol. 13, no. 1, pp. 30–60 (1996)[3] C. F. Van Loan, Computing integrals involving thematrix exponential, IEEE Trans. Autom. Control, AC23 pp. 395–404 (1994)[4] M. Nagahara and Y. Yamamoto, A new design forsample-rate converters, Proc. of 39th Conf. on Decision
and Control, pp. 4296–4301 (2000)[5] M. Nagahara and Y. Yamamoto: Design for digitalcommunication systems via sampled data H ∞ control,
IFAC Workshop on Periodic Control Systems, pp. 211–
216 (2001)[6] V. Välimäki and T. I. Laakso, Principles of fractional
delay filters, IEEE ICASSP’00, pp. 3870–3873 (2000)[7] Y. Yamamoto, On the state space and frequency domaincharacterization of H ∞ -norm of sampled-data systems,
Systems and Control Letters, Vol. 21, No. 2, pp. 163–172 (1993)[8] Y. Yamamoto, A function space approach to sampleddata control systems and tracking problems, IEEE
Trans. Autom. Control, AC-39 pp. 703–712 (1994)[9] Y. Yamamoto and P. P. Khargonekar, Frequency response of sampled-data systems, IEEE Trans. on Automatic Control, AC-41, pp. 166–176 (1996)