Elimination of unwanted signals in audio
materials by using wavelet transform
Zoran Brajević,
Croatian Radio, Prisavlje 3
Zagreb, Croatia
zbrajevic@hrt.hr
Abstract - As a sound engineer, employed at Croatian
Radio, I realized that the tools for the reconstruction and
restoration of old recordings (or damaged recordings) are
insufficient and at least they have deficient analyzing method.
Orthogonal frequency-based systems, such as the DFT do not
offer a good insight into the temporal localization of unwanted
parts of the audio material such as pulse-formed signal (e.g.,
clicks). In this search for such a tool, I reached for the discrete
wavelet transformation (DWT) which is realized by double
decomposition in order to obtain wavelet coefficients and a
graphic depiction of the coefficients. Distortion is measured with
according mean square error (MSE) and it is compared with the
number of discarded wavelet coefficients. It is also done the
comparison between DWT and the results obtained with
frequency based systems based on discrete Fourier transform
(DFT). After being performed on an arbitrary mathematical
function, the wavelet transform is applied on a sound example.
(Wave 22.05 kHz, 8 bit). It is shown that DWT is dealing very
well with both, noise and discrete disturbance which are the most
common problems in the daily work with audio material.
Keywords – noise distrubances, audio material, Discrete Fourier
Transform, Wavelet transform,
I. INTRODUCTION
Clicks are short impulsive disturbances typical for old
records [1]. Beside clicks and crackles, historically, there
are broadband noise and wow & flutter, as typical
unwanted signal disturbance. Crackles are small clicks
while wow and flutter are disturbances that cause pitch
changes. Hence the need for developing tools which can
operate in the time domain and, modeled on the DFT-based
tools, had a graphic representation of unwanted events in
audio. Wavelet transform has found a wide application in
the field of elimination of interference and noise in audio
signals. One simple and effective method for improving of
the useful signal is suppression of those wavelet
coefficients which have small amplitude. Graphic
presentation (graphic depiction) will show us which
wavelet coefficients are redundant, and we expect them
fading away in places where observed function has a
derivation [2].
X(k)
s(n)+w(n)
DWT
Ẋ(k)
ŝ(n)
IDWT
Figure 1. Nonlinear processing of wavelet coefficients obtained by
discrete wavelet transformation.
II. FAST WAVELET TRANSFORM AND FILTER
BANKS
The recursive calculation of wavelet coefficients is
used in realization of analytic branches. At the same time is
noted that every subset of lower-level coefficients is
incorporated in a set of higher-level (or ranking)
coefficients. This means that the higher-level coefficients
can be expressed with lower-level (or ranked) coefficients
by a simple recursive operation.
Figure 2. Fast wavelet-transform
In this way we are coming to the analytical part of the
fast wavelet transform (Fig. 2). The starting point for
calculating the coefficients are the symmetric formulae:
um
+ 1
=
um , 2n + vm , 2n
+ 1
(2.1)
2
um
- 1
=
um , n - vm , n
(2.2)
2
Formulas 2.1 and 2.1 are translated into the language of
digital signal processing. First of all, the coefficients:
um + 1 = (um , n )n Î Z and vm + 1 = (vm , n )n Î Z should be
understood as a series of discrete signals. The next two
operations are introduced:
um
+ 1
= (¯ 2)A (um )
vm
+ 1
= (¯ 2)D (vm )
(2.3)
(2.4)
The function (↓) is called downsampling and in
this case we are performing the downsampling by factor 2.
This means that every second (consecutive) sample, within
the given series, is omitted. A(xm) and D(xm) are
representing the moving Average operation and the moving
Difference operation [1]. Moving average and moving
difference are a type of digital filter used to analyze a set of
data points by creating a series of averages of different
subsets of the full data set. The mentioned operations are
applied to get wavelet coefficients by using a recursive
method described before. Discrete signal x = (xm )m
Î Z
will be transformed according to formulae (2.5) and (2.6)
æ ö
A çççèx ø÷÷÷÷n =
xn + xn
æ ö
D çççèx ø÷÷÷÷n =
+ 1
, while the synthesis is shown in Figure 4;
(2.5)
2
xn - xn
+ 1
(2.6)
2
The procedure can be reversed and from the
higher-level coefficients (um and vm) get the lower-level (or
ranked) coefficients (um-1 and vm-1) and by that is the right
branch of dwt-coder (right branch on the Fig.1).
um
- 1
= Ă (­ 2)(um ) + Ď (­ 2)(vm )
(2.7)
Where are Ă (x ) m and Ď (x ) m given as follows:
æ ö
Ă ççèçx ø÷÷÷÷n =
æ
çç
çè
Ďx
ö
÷
÷
÷n
÷
ø
xn - xn -
1
(2.8)
xn + xn -
=
1
(2.9)
2
opposite operation than A (xm) and D (xm) ([1], [2]). It
comes from the conditions obtained by the z-transform:
æ
ö
This double branch of filter is called the Perfect
Reconstruction Filterbank. (PR-Filterbank). Coefficients
um,n are approximate coefficients and vm,n are details
coefficients ([1], [2]) and these coefficients are used in
detection of clicks in audio signal sample.
2
Operation (↑2) means upsampling by factor 2 and
performs inserting zero between two adjacent (consecutive)
members of a given input. At this point we have all
elements (formulae (2.1) - (2.9)) necessary for the
definition of fast wavelet transform (FWT). As we can see,
æ ö
æ ö
the functions Ă çççèx ø÷÷÷÷ and Ď çççèx ø÷÷÷÷ are calculating the exact
æ ö
Figure 4. Filter series for signal synthesis
æ
ö
æ
ö
ö
÷
÷
÷
÷
ø
Multiresolution analysis is a theory which puts
development of wavelet coefficients and digital signal
processing under common denominator. After the analysis
in the z-domain, the math shows us that is, basically,
enough to satisfy the following two equations which
connect the coefficients with the conditions of
orthogonality:
ψ :=
å
kÎ Z
Ă ççèçz ø÷÷÷÷ Ă ççèççz - 1 ø÷÷÷÷ + Ă èççç- z ø÷÷÷÷ Ă èçççç- z - 1 ø÷÷÷÷ = 2
æ
çç
çè
III. MULTIRESOLUTION ANALYSIS (MRA) AND
CONSTRUCTION OF WAVELET WITH
MRA
gk ×φ - 1, k with gk := (- 1)k hl -
k
(2.10)
æ
çç
çè
Where the relation between A z and Ă z
ö÷
÷
÷
ø÷
is given
(3.1)
2å
ψ(t) =
with:
k
(- 1) h
l - k
φ (2t - k)
kÎ Z
æ
æ ö
ö
A çççèz ÷÷÷÷ø = Ă çççèçz - 1 ÷÷÷÷ø
(2.11)
One more request which has to be fulfilled, considering
æ ö
Ă çççèz ø÷÷÷÷ filter, is symmetry;
æ ö
Ă çççèz ÷÷÷÷ø =
1
2
(1 + z )
-l
(2.12)
æ ö
, where l is odd. Namely, realization of Ă çççèz ø÷÷÷÷ in
practice will be low pass FIR (Finite Impulse Response)
filter.
Block-scheme of the analytical series of filters is shown
in Figure 3:
Formulae (3.1) showing us the mother-wavelet. We also
see that the moved copy (wavelet again) ψ0,n helps φ0,n to
became an orthonormal basis on V-1. If we look briefly at
1
Haar case, we notice that the coefficients h 0 = h 1 =
2
are also the coefficients of mother-wavelet of the Haar
function. All other coefficients are zero. (hk=0). If we
choose l=1, we get, from (3.1) : ψ(t ) = φ (2t ) - φ (2t - 1) .
And before the end of this brief mathematical
analysis, let us summarize: Arbitrary mathematical function
will be approximated with the wavelet coefficients as
follows:
The difference should
approximations ([3], [6]):
A mf =
å
be
noticed,
u m , n φm , n , where, um, n
=
φm,n,f
between
(3.2)
mÎ Z
, and the details:
Figure 3. Filter series for signal analysis
Dm f =
å
nÎ Z
vm , n ψm , n
, where, vm, n
=
ψm,n,f
(3.3)
m1
å
A m 0 f = A m 1f +
Dm 0 f
(3.4)
m = m 0+ 1
The function D m f is detail which appears on interval
with length 2m and stretch and shift along that interval.
In the similar way we can observe the details on A1,
A2, A3 …, i.e., D 1 f , D 2 f , D 3 f ….
Precisely this characteristic of wavelet transform we
will use later (in the next chapter) at the concrete
application of the noise elimination applied on the audio
signal. (Wave, 22.05 kHz, 8 bit) ([3],[6]).
IV. ANALYSIS AND RESULTS
First of all, we will apply the wavelet analysis and
Fourier transformation (DFT), (i.e. frequency based system
t
analysis), on two functions: f 1(t ) = sin(t ) ×rect ( ) ,
p
t
( rect ( ) is rectangle function of width π), with
p
discontinuities at –π/2 and π/2, and
f 2(t ) = u(t - 1) , ( u(t - 1) ) is unit step function shifted
by 1), with discontinuity at 1. Those functions are chosen
because they have obvious discontinuities and exactly these
kinds of discontinuities are very good approximation of
discrete disturbances in audio material. The distortion of
resulting function will be measured with MSE (Mean
Square Error) ([4], [5]):
MSE :=
1
N
N- 1
å
~2
fi - fi
Figure 9. Wavelet coefficients at discontinuities at f1(t)
Matlab function ‘kizo_skalendiagram’ is searching for
discontinuities on a test functions (Fig. 9 and 10) and on
the sound signal (Fig. 15). Values of coefficients are
represented with the level of grey (white is 0 and black is
255). Matlab function kizo_rad_1_ uses wavelet
coefficients to reconstruct the original test function (Figure
13.) and considered audio signal (Figure 16) to detect
impulse signals while Matlab function ‘kizoanafurCTFT’
is used for DFT synthesis of test function (Fig. 14). All
programs use mathematical principles described in
chapters’ I-III.
(4.1)
i= 0
, and, in the last example, SNR (Signal to Noise Ratio):
SNR = 10 log 10
1
N
æ f 2i ö
÷
çç
÷
ççMSE ÷
÷
ø
i= 0 è
N- 1
å
(4.2)
Where f is input function with N samples and f ~ is
output function also with N samples.
Figure 10. Wavelet coefficients at discontinuities at f2(t)
Figure 8. Test functions
t
f 1(t ) = sin(t ) ×rect ( ) and f 2(t ) = u(t - 1) (4.3)
p
Figure 11. DFT analysis of f1(t)
Figure 12. DFT analysis on f2(t)
Samplitude is high-definition digital audio workstation
(DAWs), specializing in recording, editing, mixing and
mastering and it is used only for graphic depiction of audio
signal (Fig. 15 and 16).
Figure 15. Wave ‘Klarinet.wav’ 22.050 Hz, 8 bit with click on 1. and
2. sec
Figure 16. Wave ‘Klarinet.wav’ 22.050 Hz, 8 bit with 75% reduced
click on 1. and 2. sec (MSE=0.0877, improvement of aprox.
SNR=3.2dB)
V. CONCLUSION
Wavelet transform is giving us quick and visible
analysis of the discrete disturbance of audio material, while
DFT analysis is not able to do. Furthermore, MSE of
wavelet reconstruction is approximately 3 times better with
25% less coefficients then DFT analysis. However, the
best solution would be cooperation between the DFT and
DWT analysis, where discrete disturbance would be solved
with DWT processing and noise with DFT processing.
REFERENCES
[1]
Figure 13. Reconstruction of f1(t) with 75 wavelet-coefficients
(MSE=0.0155)
[2]
[3]
[4]
[5]
Figure 14. Reconstruction of f1(t) with 100 Fourier-coefficients
(MSE=0.0454)
Christoph Musialik and Urlich Hatje “Frequency-domain
processors for efficient removal of noise and unwanted audio
events”, Algoritmix GmbH, AES 26th conference 2005
M. J. Roberts, ˝Signals and Systems“, McGraw-Hill, New
York, 2004.
Gerhard Doblinger, “Programierung in der digitalen
Signalverarbeitung”, Schlembach Fachverlag. 2001.
Yves Meyer, “Precursors and Development in Mathematics”,
Princeton University Press 2006.
Stephane G. Mallat, “Multiresolution Signal Decomposition”
IEEE Transactions on pattern analysis Vol. II 1989.