H ∞ -Optimal Fractional Delay Filters with
Application to Pitch Shifting
Masaaki Nagahara ∗ Yutaka Yamamoto ∗∗∗Graduate School of Informatics, Kyoto University, Kyoto,606-8501, JAPAN (e-mail: nagahara@ieee.org).
∗∗
Graduate School of Informatics, Kyoto University, Kyoto,606-8501, JAPAN (e-mail: yy@i.kyoto-u.ac.jp).
Abstract: Fractional delay ﬁlters are digital ﬁlters to delay discrete-time signals by a fraction of
the sampling period. Since the delay is fractional, the intersample behavior of the original analog
signal becomes crucial. In contrast to the conventional designs based on the Shannon sampling
theorem with the band-limiting hypothesis, the present paper proposes a new approach based on
the modern sampled-data H ∞ optimization which aims at restoring the intersample behavior
beyond the Nyquist frequency. By using the lifting transform or continuous-time polyphasedecomposition, the design problem is equivalently reduced to a discrete-time H ∞ optimization,
which can be eﬀectively solved by numerical computation softwares. Moreover, a closed-form
solution is obtained under an assumption on the original analog signals. Using this closed-form
solution, we introduce a sampling rate conversion with arbitrary conversion rate, and propose
a new pitch shifting method for digital sound synthesis. Design examples are given to illustrate
the advantage of the proposed method.
Keywords: Fractional delay ﬁlter, sampled-data control, H ∞ optimization, digital signal
processing, sampling rate conversion, pitch shifting.
1. INTRODUCTION

Pei (2008)], and minmax (Chebyshev) optimization [Putnam and Smith (1997)].

Fractional delay ﬁlters are digital ﬁlters that are designed
to delay discrete-time signals by a fractional amount of
the sampling period. Such ﬁlters have wide applications
in signal processing, including sampling rate conversion[Ramstad (1984); Smith and Gossett (1984)], nonuniform
sampling [Johansson and Löwenborg (2002); Prendergast
et al. (2004)], wavelet transform [Yu (2007)], digital modeling of musical instruments [Lehtonen and Laakso (2007);
Välimäki et al. (2006)], to name a few. For more applications, see survey papers [Laakso et al. (1996); Välimäki
and Laakso (2000)].

Although these studies are based on the Shannon paradigm,
no real analog signals are fully band-limited, and hence
the assumption is not realistic. It is, therefore, necessary
to design a ﬁlter that takes account of high-frequency components beyond the Nyquist frequency and the intersample
behavior.

Conventionally, fractional delay ﬁlters are designed basedon the Shannon sampling theorem [Shannon (1949); Unser(2000)] for strictly-bandlimited analog signals. By this theory, the optimal ﬁlter coeﬃcients are obtained by samplinga delayed sinc function. This ideal ﬁlter is however not
realizable because of its non-causality and instability, andhence many studies have focused their attention on approximating the ideal ﬁlter by, for example, windowed sincfunction [Cain et al. (1995); Selva (2008)], maximally-ﬂat
FIR approximation [Hermanowicz (1992); Pei and Wang
(2001); Samadi et al. (2004); Hachabiboglu et al. (2007);
Shyu and Pei (2008)], all-pass approximation [Jing (1987)],weighted least-squares [Tarczynski et al. (1997); Shyu and This research is supported in part by the JSPS Grant-in-Aid
for Scientific Research (B) No. 2136020318360203, Grant-in-Aid for
Exploratory Research No. 22656095, and the MEXT Grant-in-Aid
for Young Scientists (B) No. 22760317.

In our recent study [Yamamoto et al. (2012)], we have
proved that sampled-data H ∞ control theory provides anoptimal platform to overcome the frequency limitation
enforced by the Shannon sampling theorem. Based on thisstudy, we formulate the design of fractional delay ﬁlters
as a sampled-data H ∞ optimization problem. That is,
we design a ﬁlter that minimizes the H ∞ norm of theerror system between the ideal fractional delay and an approximated one. In particular, the closed-form formula for
the H ∞ optimal fractional delay ﬁlter is given under theassumption that the underlying frequency characteristic of
the continuous-time input signal is governed by a low-passﬁlter of ﬁrst order.
By using the closed-form formula, we propose a new pitch
shifting method for digital sound synthesis [Roads (1996)].
Pitch shifting is a technique for raising or lowering the original pitch of audio signals. This is often used in synthesizing musical tone from a recorded signal of a musical instrument with a ﬁxed fundamental frequency [Roads (1996)].
We show by simulation that the proposed method outperforms the conventional phase-vocoder method [Portoﬀ(1976); Ellis (2002)].

v(t − D)v(t)w(A) Delay
t

F (s)

t

0

(B) Samplingv(nT )

v(nT − D)(D) FDF

(B)

e−Ds

ST

(C)

0 D
(C) Sampling

v

(A)

ST

ud

(D)
vd

K(z)

ūd + ed−Fig. 2. Error system E for designing fractional delay ﬁlter
K(z). (A)–(D) correspond to those in Fig. 1.
sin π(n − D/T )= sinc(n − D/T ),π(n − D/T )(2)sin(πt)n = 0, ±1, ±2, . . . , sinc(t) :=.
πt
The frequency response of this ideal ﬁlter is given in thefrequency domain as
K id (ejωT ) = e−jωD , ω ≤ ΩNyquist .
(3)
k id [n] =

t

t
0 D
Fig. 1. Fractional delay process: (A) a continuous-time
signal v(t) (top left) is delayed by D > 0. (B) the
delayed signal v(t − D) is sampled at t = nT , n =0, 1, . . .. (C) the signal v(t) is sampled at t = nT ,n = 0, 1, . . .. (D) digital ﬁltering (fractional delayﬁlter, FDF) to produce (or estimate) the sequence{v(nT − D)} from the sampled-data {v(nT )}.
0

2. DESIGN PROBLEM OF FRACTIONAL DELAY
FILTERSIn this section, we review fractional delay ﬁlters with conventional design methods based on the Shannon sampling
theorem. Then, we formulate the design problem as asampled-data H ∞ optimization problem.
2.1 Fractional delay ﬁlters
Consider a continuous-time signal v(t) shown in Fig. 1(top-left ﬁgure). Assume v(t) = 0 for t < 0 (i.e., it is acausal signal). Delaying this signal by D > 0 gives thedelayed continuous-time signal v(t − D) shown in Fig. 1(top-right in Fig. 1). Then by sampling v(t − D) withsampling period T , we obtain the discrete-time signal{v(nT −D)}n∈Z as shown in Fig. 1 (bottom-right in Fig. 1).
Next, let us consider the directly sampled signal {v(nT )}n∈Zof the original analog signal v as shown in Fig. 1 (bottomleft in Fig. 1). The objective of fractional delay ﬁlters
is to reconstruct or estimate the delayed sampled signal {v(nT − D)}n∈Z directly from the sampled data{v(nT )}n∈Z when D is not an integer multiple of T . We
now deﬁne the ideal fractional delay ﬁlter.
Deﬁnition 1. The ideal fractional delay ﬁlter K id withdelay D > 0 is the mapping which produces {v(nT −D)}n∈Z from {v(nT )}n∈Z , that is,
K id : {v(nT )}n∈Z → {v(nT − D)}n∈Z .
Assume for the moment that the original analog signal v isfully band-limited up to the Nyquist frequency, that is 1 ,π(1)
V (jω) = 0, |ω| ≥ ΩNyquist := ,T
where V is the Fourier transform of v. Then the impulse
response of the ideal fractional delay ﬁlter is obtained by[Laakso et al. (1996)]:
1 The symbol := means that the left-hand side is defined by theright-hand side.

Since the impulse response (2) does not vanish at n =−1, −2, . . . and is not absolutely summable, the ideal
ﬁlter is noncausal and unstable, and hence the ideal ﬁlter
is not physically realizable. Conventional designs thus
aim at approximating the impulse response (2) or thefrequency response (3) by a causal and stable ﬁlter via
a window method, maximally-ﬂat FIR approximations,weighted least-squares approximation, and so forth, asmentioned in Section 1.
These methods rely upon the band-limiting assumption(1). In practice, however, real analog signals always containfrequency components beyond the Nyquist frequency, andhence (1) never holds. To overcome this, we formulate thedesign problem of fractional delay ﬁlters without such anassumption by introducing the notion of sampled-data H ∞optimization [Yamamoto et al. (2012)].
2.2 Design problem of fractional delay ﬁlters
Let us consider the error system shown in Fig. 2.
F (s) is a stable and strictly proper transfer function which
deﬁnes the frequency-domain characteristic of the original
analog signal v. More precisely, we assume that the analog
original signal v is in the following subspace of L2 :F L2 := v ∈ L2 : v = F w, w ∈ L2 .
The upper path of the diagram in Fig. 2 is the ideal processof the fractional delay ﬁlter (the process (A) → (B) in
Fig. 1), that is, the continuous-time signal v is delayed by
the continuous-time delay e−Ds , and then sampled by theideal sampler ST with period T > 0 to become an 2 signal
ud := ST e−Ds v, or
ud [n] := ST e−Ds v [n] = v(nT − D), n ∈ Z+ .
On the other hand, the lower path represents the realprocess ((C) → (D) in Fig. 1), that is, the continuoustime signal v is directly sampled with the same period T
to produce a discrete-time signal vd ∈ 2 deﬁned by
vd [n] := (ST v) [n] = v(nT ), n ∈ Z+ .
This signal is then ﬁltered by K(z) to be designed, and we
obtain an estimation signal ūd = K(z)ST v ∈ 2 .

w̃LF L−1ṽ

ud

ST e−Ds L−1

ST L−1

vd

K(z)

w

ūd + ed−Fig. 3. Lifted error system Ẽ
Put ed := ud − ūd (the diﬀerence between the ideal output
ud and the estimation ūd ), and let E denote the error
system from w ∈ L2 to ed ∈ 2 (see Fig. 2). Symbolically,
E is represented by
E = ST e−Ds − K(z)ST F (s).
Then our problem is to ﬁnd a ﬁlter K(z) that minimizes
the H ∞ norm of the error system E:
Problem 1. Given a stable, strictly proper F (s), a delay
time D > 0, and a sampling period T > 0, ﬁnd the digitalﬁlter K(z) that minimizesEw2
E∞ =  ST e−Ds − K(z)ST F (s)∞ = sup.
2 wL2
w∈L
w=0

(4)
3. DESIGN OF FRACTIONAL DELAY FILTERS
The error system E in Fig. 2 contains both continuoustime and discrete-time signals, and hence the system is nottime-invariant; in fact, it is T -periodic [Chen and Francis(1995)]. In this section, we introduce the continuous-time
lifting technique [Yamamoto (1994); Chen and Francis(1995)] to derive a norm-preserving system transformationfrom E to a time-invariant ﬁnite-dimensional discrete-timesystem. In this section, we ﬁx the delay D > 0. For variabledelay case, see Section 3.3.
3.1 Lifted model of sampled-data error system
Let {A, B, C} be a minimal realization [Rugh (1996)] of
F (s):
dxF (t)= AxF (t) + Bw(t), v(t) = CxF (t), t ∈ R+ . (5)dt
We assume A ∈ Rν×ν , B ∈ Rν×1 , C ∈ R1×ν , and xF (0) =0 ∈ Rν (ν is a positive integer). Let D = mT + d wherem ∈ Z+ and d is a real number such that 0 ≤ d < T . First,
we introduce the lifting operator L [Yamamoto (1994);
Chen and Francis (1995)]:
L : L2 [0, ∞) f → {f˜[k](θ)}∞ ∈ 2 := 2 (Z+ , L2 [0, T )),k=0θ ∈ [0, T ), f˜[k](·) := f (kh + ·) ∈ L2 [0, T ).
This operator transforms a continuous-time signal inL2 [0, ∞) to an 2 sequence of functions in L2 [0, T ). We
apply lifting to the continuous-time signals w and v andput w̃ := Lw, ṽ := Lv. By this, the error system in Fig. 2,
is transformed into a time-invariant discrete-time systemẼ shown in Fig. 3. Since the operator L is an isometry, wehave
Ẽ w̃2
E∞ = Ẽ∞ := sup.
(6)
2 w̃2
w̃∈
w̃=0



B
0



wd

E0

ed

Fig. 4. Lifted sampled-data system Ẽ
Then a state-space realization of the lifted error system Ẽ
is given by 
Bx[n + 1] = Ad x[n] +w̃[n],
0(7)
ed [n] = Ce x[n] − ūd [n], vd [n] = Cy x[n],ūd = Kvd ,where the pertinent operators Ad , B, Ce and Cy are
given as follows: First, B is a linear (inﬁnite-dimensional)
operator deﬁned by
B : L2 [0, T ] → Rν+1 ,⎡⎤T
A(T −τ )e
B w̃(τ )dτ⎢⎥0⎥w̃ → B w̃ = ⎢T−d⎣⎦A(T −d−τ )CeB w̃(τ )dτ0

The other operators in (7) are all matrices deﬁned by
⎤
⎡
0 0
eAT
Ad := ⎣CeA(T −d) 0 0 ⎦ ∈ R(ν+m+1)×(ν+m+1) ,0
Bm Am
Ce := [0, 0, Cm ] ∈ R1×(ν+m+1) ,Cy := [C, 0, 0] ∈ R1×(ν+m+1) ,where {Am , Bm , Cm } is a realization of the discrete-timedelay z −m .
3.2 Norm-equivalent ﬁnite dimensional system
The discrete-time state-space realization (7) involves an
inﬁnite dimensional operator B : L2 [0, T ) → Rν+1 whereν is the dimension of the state space of F (s) deﬁnedby (5). Introducing the dual operator [Yamamoto (1993)]
B ∗ : Rν+1 → L2 [0, T ) of B, and composing this with B, we
can obtain a norm-equivalent ﬁnite dimensional system of
the inﬁnite dimensional system (7).
For the state space equations (7) of the lifted system Ẽ,
put
 
B
wd :=
w̃
0
and let E0 denote the system from wd to ed (see Fig. 4).
Note that the system E0 is a ﬁnite-dimensional discretetime system. Since BB ∗ is a positive semi-deﬁnite matrix[Nagahara and Yamamoto (2003)], there exists a matrix
Bd such that BB ∗ = Bd Bd . Deﬁne a ﬁnite-dimensional
discrete-time system by
Bd
Ed := E0.
0
See Fig. 5 for the block diagram of Ed . Then the discretetime system Ed is equivalent to the sampled-data system
E with respect to their H ∞ norm as described in thefollowing theorem [Nagahara and Yamamoto (2003)]:
Theorem 1. Assume that the sampled-data system E is in
B(L2 , 2 ), the set of all bounded linear operators of L2 into
2 . Then, we have Ed ∈ B(2 , 2 ) and E∞ = Ed ∞ .

wd



Bd
0

 wd

E0

ed

v(t)
v(mT )

Fig. 5. Discrete-time system Ed
A state space realization of Ed is obtained by replacing B
in (7) with Bd .
Thus the sampled-data H ∞ optimization (Problem 1) isequivalently reduced to discrete-time H ∞ optimization toﬁnd the optimal K that minimizes
Ed ∞ = G1 − KG2 ∞ ,
where


Bd
G1 (z) := Ce (zI − Ad )−1
,
0


Bd
.
G2 (z) := Cy (zI − Ad )−10
We can easily ﬁnd the optimal ﬁlter K(z) by standard softwares such as MATLAB Robust Control Toolbox [Balaset al. (2005)]. Moreover, the design problem is reducible
to an LMI (linear matrix inequality) by assuming thatthe ﬁlter K(z) is a ﬁxed-order FIR ﬁlter [Nagahara andYamamoto (2005)].
3.3 Closed-form solution under ﬁrst-order assumption
The H ∞ design method in the previous section requires
the delay D = mT + d to be ﬁxed. In some applications,a ﬁlter with a variable delay (i.e., the delay can bechanged without redesigning the ﬁlter). We here givedesign methods for the ﬁlter K(z) having delay D (or mand d) as an adaptive parameter.
To design variable fractional delay ﬁlters, we ﬁrst assume
that the ﬁlter F (s) is a ﬁrst-order low-pass ﬁlter with cutoﬀ
frequency ω = ωc > 0:ωc.
(8)
F (s) =s + ωcUnder this assumption, we have the following theorem.
Theorem 2. Assume that F (s) is given by (8). Then the
optimal ﬁlter K(z) is given by
K(z) = a0 (d)z −m + a1 (d)z −m−1 ,
(9)
where
sinh (ωc (T − d)), a1 (d) := e−ωc T (eωc d − a0 ).
a0 (d) :=sinh(ωc T )(10)
The proof is found in [Nagahara and Yamamoto (2003)].
For higher order F (s), one can obtain the H ∞ optimalﬁlters for ﬁxed delay parameters, say, d1 < d2 < · · · < dM ,via numerical optimization with a linear matrix inequality(LMI) [Nagahara and Yamamoto (2005)]. Then one canobtain an approximate ﬁlter for arbitrarily d ∈ [dk , dk+1 ]by linear combination of the kth and k + 1th ﬁlters. See[Nagahara and Yamamoto (2012)] for more detail.

v(krT )v((m + 1)T )

dk
t
mT
krT
(m + 1)T
Fig. 6. The value v(krT ) is given by shifting v(t) by
dk = (m + 1)T − krT and sampling at t = (m + 1)T .
previous section, and propose a new pitch shifting methodby the sampling rate converter.
4.1 Sampling rate conversion by fractional delay ﬁlter
Let us consider a continuous-time signal {v(t)}t∈R+ . Assume that we are given sampled data v[m] := v(mT ),m ∈ Z+ where T > 0 is a sampling period. Then we
execute sampling rate conversion on this discrete-time
signal. By r, we denote the conversion rate. We assume
r is a positive real number. Then sampling rate conversion
aims at estimating the values of {v(krT )}k∈Z+ .
For this purpose, we adopt a sampling rate conversion
by using fractional delay ﬁlters [Ramstad (1984)]. In
this conversion, we use the sampled-data H ∞ optimal
fractional delay ﬁlter given in Theorem 2. Let us consider
estimation of the value v(krT ) where k is a positive
integer. Assume that the time krT satisﬁes mT < krT ≤(m+1)T where m is a non-negative integer. Let dk := (m+1)T − krT . Then we havev(krT ) = v ((m + 1)T − (m + 1)T + krT )= v ((m + 1)T − dk )= v(t − dk )|t=(m+1)T .
That is, the value v(krT ) is obtained by delaying v(t) by dk
and sampling at time t = (m + 1)T (see Fig. 6). Therefore,
the estimation vr [k] for v(krT ) can be obtained by thefractional delay ﬁlter given in (9) as
vr [k] = a0 (dk )v ((m + 1)T ) + a1 (dk )v(mT ),where a0 (·) and a1 (·) are given in (10). Note that thisﬁlter is a two-tap FIR ﬁlter and the estimation needs
much fewer computation than the conventional upsampler/ﬁlter/downsampler scheme [Vaidyanathan (1993)].
Also we emphasize that the computation load is the samefor arbitrary real rate r while that of the conventional
scheme depends on r. This is an advantage over conventional methods in the case of real-time processing. In
addition, while conventional design of the digital ﬁlter insampling rate conversion depends on the band-limiting
assumption mentioned above, our design can take accountof analog characteristic of input signals.
The algorithm of the proposed sampling rate conversion
is shown in Algorithm 1. In this algorithm, we deﬁnev[m] := v(mT ), m = 0, 1, 2, . . ..
4.2 Pitch shifting

4. APPLICATION TO PITCH SHIFTING
In this section, we introduce sampling rate conversion
by using the H ∞ -optimal fractional ﬁlter given in the

Pitch shifting is a technique for raising or lowering theoriginal pitch of audio signals. To accomplish this, we
can use a sampling rate converter; convert the sampling16000
14000
12000
10000
Amplitude

Algorithm 1 Sampling rate conversion
vr [0] := v[0]k := 1for m = 0, 1, 2, . . . dowhile kr ≤ m + 1 do
d := (m + 1)T − krT
vr [k] := a0 (d)v[m + 1] + a1 (d)v[m]k := k + 1
end while
end for

8000
6000
4000
2000
0
0

(a)

100

200

300
Frequency [Hz]

400

500

600

Fig. 8. Frequency response: original signal with fundamental frequency 110 Hz (dots) and pitch-shifted signal
with fundamental frequency 110 × 25/12 ≈ 146.83 Hz(solid).

(b)(a) Proposed method(c)Fig. 7. Pitch shifting by sampling rate converter: (a)original digital signal, (b) sampling rate conversion,and (c) pitch sifting by processing the signal in (b)with the same sampling rate as (a).
period T to rT with r > 0, and play the converted signal
with the original sampling period T . By this process, wehave a raised (if r > 1) or lowered (if r < 1) pitch. For
example 2 , the pitch 110 Hz, that is A2 note is converted to
D3 note (110 × 25/12 ≈ 146.83 Hz) with r = 2−5/12 . This isanalogous to listening to music by fast- or slow-forwarding
a cassette tape. This process is eﬀectively done by theclosed-form solution in Theorem 2 if the sound is given
by digital data. We illustrate this process in Fig. 7. For
details on the algorithm for pitch shifting, see [Nagahara
et al. (2011)].
We then show an example of pitch shifting. By using
the proposed fractional delay ﬁlter, we shift the pitchof a guitar sound with A2 (110 Hz) to D3 note (110 ×
25/12 ≈ 146.83 Hz). For this, we take r = 2−5/12 .
Fig. 8 shows the frequency responses of the original signal
with fundamental frequency 110 Hz (dots) and the pitchshifted signal. The frequency response of the shifted signal
shows that the processed signal has a valid fundamental
frequency 110 × 25/12 ≈ 146.83 Hz and the harmonics with
110 × 25/12 × n, n = 2, 3, 4.
2

In this article, we consider equal temperament [French (2009)].

(b) Phase vocoder

Fig. 9. Reconstruction error
We show another example of pitch shifting. We shift thenote A2 (110Hz) used above to A3 (220 Hz). Then theshifted signal is again shifted down to the original note A2 .
We measure the error between the original sound and theprocessed one. Figure 9(a) shows the reconstruction error
by the proposed method. For comparison, we processed by
the phase vocoder [Portoﬀ (1976); Ellis (2002)], which is
widely used in pitch shifting. Figure 9(b) shows the errorof the phase vocoder. We can see that the phase vocoder
produces much larger errors than the proposed method.
This shows eﬀectiveness of our method.
One can listen to the processed sounds at the following
web page:
http://www-ics.acs.i.kyoto-u.ac.jp/~nagahara/ps/

5. CONCLUSION
We have presented a new method of designing fractional
delay ﬁlters via sampled-data H ∞ optimization. An advantage here is that an optimal analog performance can beattained. The optimal design problem can be equivalently
transformed to discrete-time H ∞ optimization, which iseasily executed by standard numerical optimization toolboxes. We have also given the H ∞ optimal ﬁlter having
delay time variable D as a variable parameter. In particular, a closed-form solution is given when the frequency
distribution of the input analog signal is modeled as a ﬁrstorder low-pass ﬁlter. Using this closed-form solution, we
have introduced a sampling rate conversion with arbitrary
conversion rate, and proposed a new pitch shifting methodfor digital sound synthesis. Examples have been given toillustrate the advantage of the proposed method.
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