Flexible Wavelets for Music Signal Processing*

Journal of New Music Research 0929-8215/01/3001-013$16.00
2001, Vol. 30, No. 1, pp. 13–22 © Swets & Zeitlinger
Flexible Wavelets for Music Signal Processing*
Gianpaolo Evangelista
Swiss Federal Institute of Technology, Audio-Visual Communication Laboratory, Lausanne, Switzerland
Abstract
Musical signals require sophisticated time-frequency techniques
for their representation. In the ideal case, each
element of the representation is able to capture a distinct
feature of the signal and can be attached either a perceptual
or an objective meaning. Wavelet transforms constitute a
remarkable advance in this field and have several advantages
over Gabor expansions or short-time Fourier methods. However,
application of conventional wavelet bases on musical
signals produces disappointing results for at least two
reasons: (1) the frequency resolution of dyadic wavelets is
one-octave, too poor for any meaningful acoustic decomposition
and (2) pseudoperiodicity or pitch information of
voiced sounds is not exploited. Fortunately, the definition of
wavelet transform can be extended in several directions,
allowing for the design of bases with arbitrary frequency resolution
and for adaptation to time-varying pitch characteristic
in signals with harmonic or even inharmonic structure of
the frequency spectrum. In this paper we discuss refined
wavelet methods that are applicable to musical signal analysis
and synthesis. Flexible wavelet transforms are obtained
by means of frequency warping techniques.
Introduction
* “Sound examples are available in the JNMR Electronic Appendix
(EA), please see ”
The analysis and processing of most features of sound requires
mixed time and frequency characterization. Gabor expansions
and the Short-Time Fourier Transform (STFT) were among
the first techniques to be applied to audio signals in order to
display their time-varying frequency spectrum characteristics.
The derived spectrogram is still in use in many applications.
While the human hearing sense is particularly gifted at classifying
acoustic patterns according to both their time of occurrence
and brightness of their transitory frequency spectrum,
our mathematical models, having to cope with the uncertainty
principle, cannot achieve indefinitely accurate resolution in
both time and frequency domains. Our perception of sound
is likely based on a non-linear frequency scale, as reflected
by both cochlear functional models, psychoacoustic theories
and conventional tone classification, e.g., tempered scale, in
western music. This observed behavior is contrasting with the
frequency resolution of the STFT, which is uniform on any
portion of the frequency axis. In audio signal processing the
choice of the representation is crucial for deriving elegant and
efficient algorithms for sound manipulation, special effects,
coding, feature detection and synthesis. An important goal is
to achieve a close match between our perception and the mathematical
realization and efficiency of the representation.
The introduction of wavelets and multiresolution approximation
geared the analysis tools towards variable timefrequency
resolution with the construction of integral
transforms and expansion bases adjusted on non-uniform
time-frequency grids (Daubechies, 1992) (Mallat, 1998).
Higher frequencies are represented at higher time resolution
and lower frequencies at lower time resolution, while keeping
the uncertainty product constant. On one hand, the integral
wavelet transform is redundant but it does not introduce limitations
on either resolution other than those dictated by the
uncertainty principle. On the other hand the dyadic wavelet
bases are easy to generate and fast algorithms for computing
the expansion coefficients are available but they constrain
the frequency resolution to one octave (Evangelista, 1991;
1997). Wavelet bases based on finer rational frequency
resolution factors lead to difficult design procedures in order
to satisfy regularity constraints (Blu, 1998). Furthermore,
the resolution factor is in practice limited to ratios of small
integers. In contrast, frequency warping techniques allow
for arbitrary choice of analysis bands. This is an important
Accepted: 30 May, 2001
Correspondence: G. Evangelista, Swiss Federal Institute of Technology, Audio-Visual Communication Laboratory, IN-LCAV Ecublens,
CH-1015 Lausanne, Switzerland. E-mail: gianpaolo.evangelista@epfl.ch

14 G. Evangelista
point if we want to adapt the signal representation either
to perceptual scales, e.g., Bark scale, or to objective bands
associated to a single feature. However, computation of
the transform based on general warping maps is not efficient
unless one is able to derive an associated discrete-time
signal processing structure (Evangelista & Cavaliere,
1998d).
In this paper, we present an approach to flexible wavelet
transforms where the frequency resolution of the basis elements
can be designed at will. This is achieved by discretetime
iterated frequency warping. We consider the important
and unique case where each elementary frequency warping
stage is computationally realizable with arbitrary precision.
The main ingredient is the Laguerre transform whose computation
is achieved by means of a chain of digital allpass
filters or dispersive delay line. Iterated Laguerre warping is
applied to wavelets leading to wavelets whose bandwidths
can be assigned by selecting a set of parameters (Evangelista
& Cavaliere, 1998c). Interestingly enough, these wavelets are
still based on a dyadic scheme, which is a natural setting for
iterated band splitting.
Another issue in musical signal processing is that the
frequency spectrum of voiced signals peaks at uniformly or
non-uniformly spaced frequencies. While exact periodicity
is rarely encountered in real-life signals, exploitation of
the pitch information or of pseudoperiodicity is an asset for
their representation. The Fourier transform is particularly
gifted at revealing periodicity, however the STFT represents
pseudoperiodic signals as a superposition of partials whose
dynamics is captured with uniform time resolution. It follows
that transients are blurred if the selected time-resolution is
too poor or frequency resolution is at risk when the analysis
window is too short.
In previous papers (Evangelista, 1993; 1994) the author
suggested methods for exploiting the pitch information
in wavelet representations by introducing the Pitch-
Synchronous Wavelet Transform (PSWT). The signal is first
converted into a sequence of variable-length vectors each
containing the samples of one period of the signal, then the
sequences of components are analyzed by means of an array
of wavelet transforms. This representation is able to capture
period-to-period fluctuations of the signal by means of basis
elements that are comb-like in the frequency domain. Periodic
behavior is trapped in a narrow comb adjusted on the
harmonic grid and transients are represented at several scales
by multiresolution basis elements whose Fourier transforms
are organized in ensembles of sidebands of the harmonics.
Faster transitions are represented, at fine time resolution, by
larger bands lying far from the harmonics. Vice-versa, slow
transitions and quasi-stationary dynamics are represented, at
coarser time resolution, by narrower sidebands lying closer
to the harmonics. In particular, this technique allows for separation
of perceptually and physically distinct phenomena
such as the bow noise and the harmonic resonant component
in a violin tone. Importantly enough, this separation is
achieved by means of a complete and orthogonal set and the
sum of the components exactly reconstructs the original
signal.
The PSWT technique is amenable to further generalization,
essentially by introducing alternate methods for
forming the vector signal, e.g., based on intermediate Gaborlike
or STFT representations, which can be formalized in the
multiwavelets framework.
While the PSWT works efficiently when the frequencies
of the partials are arranged on a harmonic grid, the frequency
spectrum of many sounds, such as drums and piano tones in
the low register, has an inherently inharmonic structure.
From a physical model point of view, this phenomenon may
be explained by the fact that, when the stiffness of the material
is not negligible, dispersive wave propagation occurs
within the medium. Furthermore, coupled 2D modes in
drums have a peculiar distribution of eigenfrequencies. A
partial solution of this problem is provided by frequency
warping techniques that allow for redistribution of eigenfrequencies
into harmonics. Inharmonic signals are first regularized
by means of a frequency map and then analyzed by
means of PSWT, leading to the Frequency Warped PSWT
(Evangelista & Cavaliere, 1998c).
The paper is organized as follows. In section 2 the properties
of one-step frequency warping maps of continuoustime
signals and wavelets are examined. Methods for
frequency warping discrete-time signals and the Laguerre
transform are illustrated in section 3. These methods are
applied to the definition of both discrete-time and continuous-time
warped wavelets in section 4.
Frequency warping and wavelets
Frequency warping
Frequency warping is obtained by mapping the frequency
axis w by means of a suitable real function W(w). Given a
continuous-time signal x(t) with Fourier Transform (FT)
we form the warped signal
i.e., we let
Ú
+•
jw
t
X( w)= x() t e - dt
-•
1 +•
jwt
˜x()= t Ú X( W( w)
) e dw
2p -•
Thus, at any frequency w – the FT of the warped signal has the
same magnitude as the FT of the signal at frequency W(w – ),
i.e., the frequency spectrum of the warped signal is a
deformed version of that of the original signal. Clearly, for
the warping operation to be well defined, the map W(w) must
satisfy a number of requirements. In this paper we are inter-
j w t
˜X( w)= X( ( w)
)= x() t e - W
W
( )
dt
Ú
+•
-•
(1)

Flexible wavelets for music signal processing 15
ested in reversible warping operations defined by a strictly
increasing, invertible, map W of the real axis onto itself. The
map is also assumed to have odd parity: W(-w) = -W(w). In
this way frequency ordering is preserved and warping a real
signal yields a real signal. Notice that, as defined in Equation
1, frequency warping is not energy preserving since
narrow frequency bands may be mapped into broader bands
with equal peak amplitude. In order to preserve energy in any
frequency interval one needs to multiply the right hand side
of Equation 1 by a suitable amplitude scaling function or,
more generally, define a suitable measure of the frequency
axis. In the sequel we will assume that the map is almost
everywhere differentiable, e.g., that W(w) is absolutely continuous,
and define a scaled warped operation on the signal
x(t) as the one producing the FT
where the derivative W¢(w) is positive almost everywhere
since the map is assumed to be strictly increasing. In that case
one obtains energy preservation in any band. In fact, a frequency
band with support [w 0 ,w 1 ] is mapped into a frequency
band with support [W -1 (w 0 ), W -1 (w 1 )] and
1
2p
Ú
From a perceptual point of view this property is important
since the scaled warped signal is properly equalized while in
the unscaled version some parts of the frequency spectrum
are boosted. From a mathematical point of view scaled
warping is associated with an orthogonal operator :
with real kernel
-1
W ( w1)
-1
W ( w0
)
ˆX( w)= W¢( w) X( W( w)
)
2 1 w1
2
ˆX( w) dw
= X w dw
2p
Ú ( )
w0
xt ˆ ()= [ x]()= t Wt ( , t) x( t)
dt
1 +•
j( wt- W( w)
t)
Wt ( ,t)= ¢( w) e dw
2p
Ú W
-•
and inverse operator -1 defined by the kernel
Ú
W - 1
( t, t )= W ( t,
t )
Finally, we remark that, by exploiting duality of the FT one
can define time warping operations in a similar fashion.
Simple warped wavelets
In (Baraniuk & Jones, 1993; 1995) unitary warping operators
were exploited to define warped wavelets. The construction
may be summarized as follows: the signal is
unwarped by means of the inverse warping operator -1 ,
then the expansion coefficients on a dyadic wavelet basis
are computed. Reconstruction is achieved by applying the
warping operator to the wavelet expansion of the
unwarped signal. In other words, given an orthogonal and
complete dyadic wavelet set
+•
-•
(2)
where
y
one computes the expansion coefficients
-
xˆ = 1 x,
y
and reconstructs the signal as follows
xt ()= Â xˆ nm , [ y
nm , ]() t
Since the frequency warping operator is orthogonal we have
-1
xˆ = x, y = x,
y
nm , nm , nm ,
and Equation 4 may be considered as the expansion on the
warped wavelet basis
Indeed this basis is orthogonal since
-1
y , y = y , y = d d
n¢ , m¢ n, m n¢ , m¢
n, m n¢ , n m¢
, m
and complete in view of unitary equivalence.
The FT of the warped wavelets
ˆ
nm , t y
nm ,
y
{ ()} Œ
-n
2 -n
()= t 2 y 2 t-m
nm , 00 ,
nm , nm ,
{[ y nm , ]() t }
nm , Œ
is related to the FT of the dyadic wavelets as follows
ˆ
nm , ( w)= ¢( w) nm , ( ( w)
)
n
n n j2
mW
w
= 2 W¢( w) Y00
, 2 W( w)
e
Y W Y W
from which we see that warped wavelets are not simply generated
by dilating and translating a mother wavelet as in
dyadic wavelets (Eq. 3). Rather, the “translated wavelets” are
generated by generally non-rational allpass filtering e -j2n mW(w) .
Scaling depends on the warping map W(w) as well. However,
frequency warped wavelets have the remarkable property that
their frequency resolution, i.e., their essential frequency
support, may be arbitrarily assigned by proper choice of the
map W(w). Indeed, if the cut-off frequencies of dyadic
wavelets are fixed at 2 -n p, the cut-off frequencies of the
warped wavelets are
w
y nm , t
nm ,
nm , Œ
()∫ [ ]() t
n
- -n
= W 1 ( 2 p)
For our purposes we just need to add the remark that genuine
scale a wavelets, 0 < a < 1, may be generated by selecting as
warping map a function W(w) satisfying the following property
(a-homogeneity):
1
W( aw)= W( w)
2
( )
( ) - ( )
(3)
(4)
(5)
(6)

16 G. Evangelista
3
2.5
2
1.5
1
0.5
0
0 0.5 1 1.5 2 2.5 3
Fig. 1. Example of warping map for power of a wavelet cut-off
choice.
such as
reported in Figure 1. Repeated application of Equation 6
shows that
By deriving both sides of Equation 8 with respect to w one
obtains
By substituting Equations 8 and 9 in Equation 5 we
obtain
Yˆ
which shows that warped wavelets generated by an a-homogeneous
warping map are obtained by dilating the family of
mother wavelets Ŷ 0,m (w), m Œ , while wavelets at fixed scale
level n are allpass related:
Yˆ
W
W( w)=
pw
w
W( a -n
w)= 2
n W( w)
-n
n
¢( a w)= ( 2a) W¢( w)
-n
Yˆ
n
( w)= a
2 -
( a w)
nm , 0,
m
n
jmW
a w
( w)= Yˆ
( w) e
- ( -
)
nm , n,
0
-
2 log a
Without loss of generality we can assume that
W(p) = p
so that the cut-off frequencies of the scale a warped wavelets
are fixed at
w n = a n p
w
p
(7)
(8)
(9)
If a = 1 – 2 then the map (Eq. 7) reverts to the identical map
W(w) = w and the warped wavelets coincide with the
dyadic wavelets. If a > 1 – 2 then the warped wavelets achieve a
finer frequency resolution than the dyadic wavelets.
Discrete-time frequency warping and
Laguerre transform
The continuous-time warped wavelet expansion illustrated in
the previous section is applicable to continuous-time signals
and requires warping of the signal, a task that is computationally
difficult to perform. An important property of dyadic
wavelet expansions is that they have a fast algorithm to
iteratively compute the expansion coefficients even in the
continuous-time case. In this section we consider frequency
warping of discrete-time signals and relate this to an orthogonal
transform that can be computed by means of a chain of
rational allpass filters.
Since the Discrete-Time Fourier Transform (DTFT) of a
sequence x(n) is periodic, reversible frequency warping of
discrete-time signals requires an invertible frequency map of
the interval [-p, p] onto itself. As we did in the continuoustime
case, we will assume that q (w) is almost everywhere
differentiable, has odd parity and maps the point p into itself.
As discussed in the previous section, we are interested in procedures
for unwarping the signal, producing the signal xˆ(n)
whose DTFT is
It follows that the discrete-time unwarping operator -1 is
orthogonal and has kernel
-1
+ d
-
-1
1 p q
1
jn ( w- kq ( w)
)
W ( n,
k)=
Ú e dw
2p
-p
dw
1 + p
jk ( w- nq( w)
)
= Ú q¢( w)
e dw
2p
-p
The unwarped signal is
xn ˆ ()= [ - 1 -
x]()= n W 1 ( nk , ) xk ( )= l ( kxk ) ( )
where we defined
(10)
(11)
(12)
Notice that Equation 11 is in the form of a scalar product of
the signal with the sequence l n (k). By the orthogonality of
the operator
and the set {l n (k)} nΠis orthogonal and complete. In fact:
and
Â
k
-1
dq
-1
ˆX ( w)= X q ( w)
dw
Â
k
-1
l n ()∫ k W ( n,
k)
W - 1
( n, k )= W ( k,
n )
Â
( )
Â
-1
l () k l ()= k W ( n, k) W( k, n¢
)= d ,
n n¢
n n¢
k
k
n

Flexible wavelets for music signal processing 17
Â
n
Â
-1
l () k l ( k¢
)= W ( n, k) W( k¢
, n)=
d ,
n n k k¢
k
Thus, the discrete-time unwarped signal is given by the
sequence of expansion coefficients
xn ˆ ()= x,
ln
(13)
of the signal over the orthogonal basis {l n (k)} nΠand the
signal x(n) is recovered from xˆ(n) by the expansion series
Fig. 2. Structure for computing discrete-time signal unwarping by
means of the Laguerre transform.
xn ()= Â xk ˆ () l k () n
k
In turn, if the signal is causal, i.e., if x(k) = 0 for k < 0,
then the scalar product (Eq. 13) may be computed by convolving
the time-reversed signal y(-k) = x(k) by l n (k) and
sampling the result at n = 0:
•
•
x,ln = Â x( k) ln( k)= Â y( 0 -k) ln( k)
k = 0
k = 0
Furthermore, from Equations 12 and 2 it is easy to recognize
that the DTFT of l n (k) satisfies the following recurrence:
3
2.5
2
1.5
b =.9
with
Hence
L
n
( w)= L ( w)
e
n-1
L 0 ( w)= q¢( w)
- jqw
( )
(14)
1
0.5
b =-.9
L
n
jnqw
( w)= L ( w) e
- ( )
0
where the term e -jnq(w) can be assimilated to the response of
an allpass filter. However, for digital implementations one
needs to constrain -q(w) to be the phase of a causal and
stable, rational allpass filter. For reversible warping we need
to constrain q (w) to a one-to-one mapping of the normalized
frequency interval [-p, +p]. Furthermore, we want to map a
real signal into a real warped signal, thus q(w) must have
odd parity. It is easy to show that these conditions are verified
if and only if -q(w) is the phase of a first-order real
allpass filter with transfer function
-1
z - b
Az ()= , - 1< b < 1
-1
1 - bz
The corresponding basis set l n (k) is recognized to be the
discrete Laguerre basis (Broome, 1965) (Evangelista &
Cavaliere, 1998d) (Oppenheim et al., 1971), orthogonal and
complete over the set of finite energy causal signals, with
b sinw
qw ( )= w+
2 arctan , - 1< b < 1 (15)
1 - b cosw
This one-parameter family of warping curves (Eq. 15) is
shown in Figure 3. In this family, for any allowed value of
the parameter b, the curves corresponding to b and to -b are
inverse of each other. The curve for b = 0 is the identical map
q(w) = w. Furthermore,
0
0 0.5 1 1.5 2 2.5 3
Fig. 3. Family of warping curves associated to the Laguerre
transform.
2
1 - b
q¢( w)=
1- 2bcosw
+ b
In computing the Laguerre transform of a causal signal, the
function in Equation 14 can be replaced by a causal realization
of q¢( w)
, namely
1 - b
1 - be j
2
L 0 ( w)=
- w
Laguerre warping is the unique one-to-one warping that can
be numerically computed by means of rational filters. A
digital structure implementing Laguerre warping is shown in
Figure 2. There, the input signal is time reversed, filtered by
L 0 (w) and fed to a dispersive delay line consisting of a chain
of allpass filters The warped signal is obtained by collecting
the samples present at the output of each filter at time n = 0.
Although the computational structure in principle requires
an infinite number of filter sections, by a simple argument
on the maximum group delay, one can show (Evangelista
& Cavaliere, 1998d) that, to a good approximation, the
2

18 G. Evangelista
3
2.5
2
Fig. 5.
Two-channel warped filter bank structure.
1.5
1
0.5
0
0 0.5 1 1.5 2 2.5 3
Fig. 6.
unwarped downsampler
Equivalent downsampled warped filtering structures.
Fig. 4. Frequency warping harmonic partials to obtain inharmonic
partials.
Laguerre transform of a finite-length N signal can be computed
by means of
1 + b
M ª N
1 - b
allpass sections. As already observed, the inverse Laguerre
transform can be computed using the same structure of
Figure 2 provided that we reverse the sign of the Laguerre
parameter b.
Frequency warping is per se an interesting effect from a
musical point of view. In fact, by frequency warping a periodic
signal one obtains an inharmonic distribution of the partials,
as shown in Figure 4, typical of bells, drums or piano
tones in the low-register. Therefore, the frequency spectrum
of an initial sound is morphed into another sound, possibly
belonging to a different instrument class. Time-varying
versions of the warping algorithm have been devised
(Evangelista, 2000b). Further approximations via overlapadd
of the Short-Time Laguerre Transform allow us to compute
the effect in real time (Evangelista, 2000b). Sound
examples obtained by frequency warping can be visited at
http://lcavwww.epfl.ch/~gianevan/research/fwex.html.
Warped filter banks and iterated
warped wavelets
Two-channel warped filter bank
In the classical construction of dyadic wavelets (Daubechies,
1992) a two-channel critically sampled filter bank serves as
building block. This is based on two quadrature mirror filters
(QMF) H(z), low-pass, and G(z), high-pass. These filters are
half-band with cut-off frequency w c = p – 2
. In (Evangelista &
Cavaliere, 1998d) the structure of Figure 5 was considered
as a building block. There, the signal is frequency unwarped
by means of Laguerre transform (LT) prior to filtering. The
operators [Ø] and [≠] denote, respectively, downsampling
(decimation) by 2 and upsampling (zero-insertion) by 2.
Unwarping the signal is equivalent to warping the filters and
to unwarping the downsampling operators (Evangelista &
Cavaliere, 1998d), as depicted in Figure 6. Thus, on one
hand the pass-band of the filters is altered and the cut-off
frequency is moved to w c = q -1 ( p – 2
), on the other hand the
equivalent unwarped downsampling operator, which embeds
unwarping and conventional downsampling, resets the band
of the filtered signal to half-band prior to downsampling. By
combining the Laguerre transform with an orthogonal filter
bank one obtains an orthogonal perfect reconstruction structure
whose bands can be arbitrarily allocated by fixing the
parameter b.
The structure of Figure 5 can be generalized to include an
ordinary discrete wavelet transform structure in place of the
two-channel filter bank. This provides a first form of discretetime
warped wavelets. However, the analysis bands cannot
be arbitrarily selected since they are determined by transforming
the cut-off frequencies 2 -n p according to a single
curve in the Laguerre family. As shown in the next section,
fully arbitrary bandwidth allocation is achieved by iterating
the full warped filter bank structure.
Frequency warping may be successfully employed for
transforming inharmonic sounds into harmonic ones or viceversa.
When used in conjunction with the PSWT one obtains
the Pitch-Synchronous Frequency Warped Wavelet Transform
(PSFWWT) useful for decomposing inharmonic instrument
sounds into regular pseudo-periodic components plus

Flexible wavelets for music signal processing 19
...
Fig. 9. Synthesis filter bank for computing the inverse
discrete-time frequency warped wavelet transform.
jmWn
Ynm , ( w)= Ln, ( Wn ( w)
)
Ê 1
G Wn( w)
ˆ
0 -1 Fn-10
, ( w)
e
Ë 2 ¯
where
n
1
Fn, 0( Ï
w)= Lk,
0( Wk 1( w)
)
Ê
H Wk( w)
ˆ¸
’ Ì
-
Ó
Ë 2 ¯
˝
˛
k = 1
is the DTFT of the warped scaling sequence,
- ( w )
(16)
(17)
Fig. 7. Frequency domain characteristics of pitchsynchronous frequency
warped wavelets.
Fig. 8. Analysis filter bank for computing the discrete-time frequency
warped wavelet transform.
fluctuations. The magnitude FT of the basis elements is
reported in Figure 7. The PSFWWT decomposition allows,
e.g., to resolve the hammer noise in a piano tone. An account
of this method is given in (Evangelista & Cavaliere, 1998c)
(Testa et al., 1997), where the Laguerre warping family is seen
to close match the dispersion curves provided by both experimental
data and physical model based on a 4th order PDE.
Discrete-time warped wavelets with arbitrary
band allocation
A discrete-time warped wavelet decomposition may be
obtained by iterating the two-channel warped filter bank.
Similarly to the ordinary wavelet transform, identical warped
filter bank structures are nested in the low-pass branch. The
corresponding analysis and synthesis structures are reported
in Figures 8 and 9, respectively. Each warping stage is characterized
by a different value b k of the Laguerre parameter.
The warped wavelets are the impulse responses of the lower
branches of the synthesis structure for any value of the shift.
One can show (Evangelista & Cavaliere, 1998c) that the
DTFT of the level n warped wavelet has the form
...
2
1-
bk
L k ,0( w)=
- jw
1-
be k
is a causal realization of qk¢( w)
and
W k ( w)= J k oJ k - o...
1 oJ1( w)
(18)
with
Jk( w)= 2qk( w)
Notice that, due to the presence of the frequency dependent
delay term e -jmW n(w)
in Equation 16, wavelets pertaining to the
same level n are obtained from each other by an allpass filtering
operation, rather than by elementary time shift as in
ordinary dyadic wavelets. The cut-off frequency of the level
n warped scaling sequence is given by
-
wn
= W 1
n ( p)
which is the cut-off frequency of the narrowest band filter
H( – 1 2 W n (w)) in the product (Eq. 17).
Given a choice of the cut-off frequencies w n < w n-1 < ...
< w 1 , there exists a unique solution for the Laguerre parameters
b k , k = 1,..., n, obtained by the following recurrence
Ê p w1
ˆ
b1
= tan -
Ë 4 2 ¯
Ê p Wk-1( wk)
ˆ
b k = tan -
(19)
Ë 4 2 ¯
Consequently, to any choice of the cut-off frequencies is
uniquely associated a corresponding set of parameters. The
iterated warped filter bank is a perfect reconstruction orthogonal
structure equivalently described by a discrete-time
warped wavelet basis. Cut-off frequencies may be arranged
on a perceptual scale, such as Bark scale, leading to perceptually
organized orthogonal transforms (Evangelista &
Cavaliere, 1998a; 1998b). The example reported in Figure
10 displays the magnitude Fourier transform of frequency
warped wavelets and scaling sequence achieving 1/3 of
octave frequency resolution. Sound examples of ordinary

20 G. Evangelista
7.7
6.6
5.5
4.4
3.3
2.2
1.1
0
0 0.5 1 1.5 2 2.5 3
Fig. 10. Magnitude Fourier transform of 1/3-octave frequency
warped wavelets and scaling sequence obtained by letting w n = a n p,
with a = 2 - – 1 3
.
wavelet and frequency warped wavelet analysis can be found
at http://lcavwww.epfl.ch/~gianevan/research/fwex.html.
Continuous-time warped wavelets
Expansion coefficients over ordinary continuous-time
wavelet bases may be iteratively computed by means of an
infinite cascade of two-channel filter banks. The discretetime
frequency warped wavelets illustrated in the previous
section can be extended to continuous-time bases. This
extension requires generalization of many of the mathematical
results illustrated in Daubechies (1992) and Mallat
(1998) The scaling function of ordinary dyadic wavelets is
obtained by means of a limit process involving rescaling of
the discrete-time scaling function:
where F0,0(w) c.t. and Fn,0(w) d.t. respectively denote the Fourier
transform of the level 0 continuous-time scaling function and
the DTFT of the level n scaling sequence (discrete-time).
Since (Mallat 1998)
then
F
( w)=
lim
n-1
dt ..
F 00 , w ’
k = 0
k
( )= H( 2 w)
•
ct ..
F 00 , ( w)=
’
k = 1
1 dt .. Ê w ˆ
F , n
2 Ë 2 ¯
ct ..
00 ,
nƕ
n n 0
Ê w
H
Ë
k
2
2
ˆ
¯
The iterated frequency warped counterpart of this construction
is much more complex and requires a deep study of
properties the iterated map (Eq. 18) and of the convergence
of the parameter selection algorithm (Eq. 19). One can
show that, in spite of the intricate analytic form of the
iterates W k (w), enforcing the following exponential cut-off
condition:
w n = a n p, n = 1, 2, . . .
guarantees that the parameter sequence b k obtained by Equation
10, simply converges to
At the same time, we have
where G -1 (w) is an increasing and continuously differentiable
eigenfunction of the composition-by-J •
-1
operator with
eigenvalue a, i.e.,
G -1 ° J • -1 = aG -1 (20)
Since, under general assumptions, the eigenfunction of the
composition operator is unique up to a multiplicative constant
(Shapiro, 1993) (Evangelista, 2000a) and since G -1 (p)
= p, then
-1
G - 1 pg n ( w)
( w)=
lim
n Æ• -1
g ( p)
where
Thus, G -1 (w) is also proportional to the limit iterate of the
map J • -1 . Furthermore, from Equation 20 we obtain
and
-1
Wn
( w) -
lim
nƕ
n =
1
G ( w )
a
-1 -1 -1
g n ( w)= J•
oLoJ•
( w)
1442443
-
J 1 -
• ( w)= G aG
1 ( w)
-
g 1 n -
( w)= G a G
1 ( w)
n
1 a
b• = - 2
1+
2 a
.
n times
( )
( )
This allows us to define the Fourier transform of the
continuous-time frequency warped scaling function as
follows:
Ê 1
H g
Under rather general assumptions, this construction leads
to the definition of asymptotically scale a continuous-time
warped wavelets. The expansion coefficients over the resulting
basis can be iteratively computed by means of the struc-
-1
•
k
˜ ct ..
Ë 2
F 00 , ( w)=
’
k = 0 2
( w)
ˆ
¯
n

Flexible wavelets for music signal processing 21
3
2.5
2
1.5
1
0.5
0
0 20 40 60 80 100 120
Fig. 11.
Tiling the time-frequency plane by means of frequency warped wavelets.
ture in Figure 8. For more details on this extension, the
mathematically inclined reader is referred to (Evangelista,
2000a).
The warped wavelet decomposition leads to an unconventional
tiling of the time-frequency plane shown in Figure
11. Due to the frequency dependent delay of each basis
element, the time-frequency localization zones are characterized
by curved boundaries.
Conclusions
In this paper we explored the panorama of wavelet transforms
obtained by means of frequency warping techniques,
with particular attention to their realizability in computationally
efficient structures based on rational discrete-time
allpass filters and two-channel filter banks. The flexibility of
the transform is exploited for the representation of music
signals in perceptually or objectively meaningful components,
allowing for adaptation to perceptual scales or for
orthogonal separation of transients and noise in harmonic or
inharmonic sounds via pitch-synchronous methods. The
refinable frequency resolution of frequency warped wavelets
is a fundamental characteristic for the analysis of any musical
sound.
Frequency warping emerges as an appealing technique in
musical signal processing for the creation of new effects and
for the analysis and synthesis of features of sounds by means
of adapted representations.
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