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

However

Fractional delay ﬁlters are those that are designed to delay the input

Real analog signals are never fully band-limited. They contain somesignals by a fractional amount of the sampling time. Since the delay isfrequency components beyond the Nyquist frequency.

fractional, the intersample behavior of original analog signals become crucial. While the conventional design bases itself on the assumption that the incoming analog signals are fully band-limited up
to the Nyquist frequency, the present paper applies the modern sampled-data H∞ optimization which aims at restoring the intersample
behavior without the band-limiting assumption. It is shown thatthe optimal FIR ﬁlter design is reducible to a convex optimization described by a linear matrix inequality (LMI). A design example isshown to illustrate the advantage of the proposed method.

How Do Fractional Delay Filters Work?

Nyquist freq.

We Propose• Design with the frequency characteristic of the input analog signals
which are not fully band-limited up to the Nyquist Frequency.
• Optimization with the H∞ norm of the error system which both
analog and digital signals.

ST• Formulated as a sampled-data H∞ optimization Problem.

T
Fractional
Delaye - Ds

Design Problem
Given F( s), D > 0, and T > 0, ﬁnd the digital ﬁlter K(z) minimizing

DDST
D is not an integer multiple of sampling time T.

band-limited up to the Nyquist freST e= KfdST

The solution (the ideal ﬁlter) is given byKfd(e jωT ) = e-jωDT (frequency response)

e-Ds

kfd[ n] = sinc( nT - D ),F(s)e-Ds

STSTSTSTKfd(z)K(z)

edThe analog ﬁlter F( s) governs the frequency-domain characteristicof the input signal w. F( s) is conventionally assumed to be an ideal
lowpass ﬁlter whose cut-off freq. is the Nyquist freq. Here, we do notorsin( πx)sinc( x) := πx2w ∋ L2w

quency, we have the equation:-Ds2-Ds≔(Se- K(z)ST ) F(s)ℰT

Conventional Design
If the input analog signals are fullyw || ℓℰ|||| ℰ ||∞≔ sup || w || L

(impulse response)

The ideal ﬁlter is non-causal and inﬁnite-dimensional.
Conventional design aims at approximating the ideal ﬁlter via(1) Window method (with impulse response)(2) Maximally-ﬂat FIR approximation (with frequency response)(3) Weighted least-squares approximation (with frequency response)

assume such full band-limiting.
The error system ℰ has both continuous- and discrete-time signals.
We discretize the continuous-time signals preserving H∞ norm of theerror system, and ﬁnd the optimizing ﬁlter in the discrete-time domain.
This is a sampled-data H∞ optimization problem.

Reduction to a Finite Dimensional
Linear Time Invariant System

Outline of Proof(1) By bounded real lemma (Kalman-Yakubovich-Popov lemma),|| Ed ||∞ < γ is equivalent to that ∃ P > 0 such that

The error system ℰ is an inﬁnite-dimensional system since(1) ℰ is an operator from L2 to ℓ 2,ABTC(α) 0(2) ℰ contains a time delay.
By using sampled-data control theory, we can obtain a finite-dimensional (FD) linear time-invariant (LTI) system whose H∞ norm isequal to that of the system ℰ.

P0A0IC(α) 0B<P00 γ2I(2) By Schur complement, this is equivalent to ().

�Design Procedure(1) Give a delay time D, sampling time T, analog ﬁlter F( s).
(2) Set FIR order N and compute matrices A, B and C(α).

Theorem 1(3) Find α minimizing γ s.t. P > 0, γ > 0 and the LMI ().
The optimal α in the procedure (3) can be easily obtained by stan-For the error system ℰ ∋ B( L , ℓ ), there exists an FD LTI system Ed2

2

dard linear optimization softwares (e.g., LMI toolbox in MATLAB).

such that

Moreover, if F(s) is a ﬁrst order lowpass ﬁlter, we can obtain the|| ℰ ||∞ = || Ed ||∞optimal filter analytically.

Outline of Proof

Design Example(1) Dual operator ℰ * of ℰ such that ( ℰ w, v ) ℓ = ( w, ℰ *v ) L

22(1) Design parameters: D = 10.8 [sec], T = 1 [sec] and F(s) is(ωc LF(s) = s + ω , ωc = 0.5 [rad/sec], L = 1, 2, 4, 8.
c(3) ℰℰ *∋ B( ℓ 2 , ℓ 2 ), and there exists a finite-dimensional discrete-timesystem Ed ∋ B( ℓ 2 , ℓ 2 ) such that Ed Ed* = ℰℰ *.

(2) Filter length is 32 taps (i.e., N = 31).

2

�2

Magnitude (dB)

Ed is given as a transfer function by()Phase (deg)Discrete-time H Optimization∞Ed(z) = ( C1 - K(z)C2 )( zI - Ad )-1BdBode Diagram
0-11.5−2

L=2−4-12−6

L=1−80−360-13−720-13.5−1440−1800−2160−110-14 -1
10010Frequency (rad/sec)

Fig. 1. Fractional delay FIR ﬁlters: sample-data
H∞ design (solid) and 32-tap FIR ﬁlter by the Kaiser window method (dots)

K(z) = a0 + a1 z-1 + ... + aN z-N

-12.5

−1080

where C1, C2, Ad and Bd are matrices.
Assume K( z) is an FIR ﬁlter,

L=4

L=8

and we can rewrite () as
where C(α), A, B are matrices. Note that C(α) is linearly dependent

0.5

0.5

Reduced Problem

Amplitude

1Amplitude

10-0.5-1-1-1.5-1.5

10203040Time (sec)Fig. 3. Time response against a rectangularwave: sample-data H∞ design

Find the FIR parameter α minimizing

0-0.50jθ-1≔supEC(eIA)B|)(
α|| d ||∞|

50

0Conventional design

1.5

Ed(z) = C(α)( zI - A )-1B

Our problem is then reduced as follows:

10Frequency (rad/sec)

Fig. 2. Frequency response of ℰ : sample-data
H∞ design (solid) and 32-tap FIR ﬁlter by the Kaiser window method (dots)

Sampled-data design

1.5on α = [a0, a1 , ..., aN ], and A and B are independent of α.

Frequency response-11Gain (dB)(4) || ℰ ||∞= || ℰℰ * ||= || Ed Ed* ||= || Ed ||∞2((2) || ℰ ||∞= || ℰℰ * ||20102030Time (sec)4050

Fig. 4. Time response against a rectangularwave: 32-tap FIR ﬁlter by the Kaiser window
method (dots)θ ∋ [0, π)Theorem 2
Assume γ > 0. Then || Ed ||∞ < γ if and only if there exists a matrix
P > 0 such thatTA PA - PTA PBTB PA-γI + B PB

0C(α)

0-γI

We have presented a new method of designing fractional delay FIR ﬁlters via sampled-data H∞ optimization. An advantage here is that anC(α)TTConclusion< 0()analog optimal performance can be obtained. The design problem
can be reduced to a convex optimization with an LMI, which leads toan easy computation of design. The designed ﬁlter exhibits a much
more satisfactory performance than conventional ones.