U. Glaeser

FIGURE 27.1 Two-band filter bank: (a) analysis bank, (b) synthesis bank.
FIGURE 27.2 Discrete wavelet transformation (DWT): (a) two-band analysis tree, (b) two-band synthesis tree.
number of parameters, description {b kl, a kl} is somehow more efficient because parameters a kl are less
important than b kl and consequently {b (k+1)n}. From Eqs. (27.11a) and (27.11b) we conclude that parameters
b kl and a kl result from a lowpass and a highpass filtering of parameters b (k+1)n, respectively, with a
two-band splitting filter bank of impulse responses h 0(−n) and h 1(−n), respectively, followed by a down
sampling with factor 2 (Fig. 27.1(a)). This procedure can be continued many times in order to obtain
even more efficient parametric representation. If only parameters b kl are split, which is the case in the
classical DWT, a kind of an octave signal analysis filter bank results (Figs. 27.2(a) and 27.3). If also
parameters a kl are split (wavelet packet) and/or a multiband splitting filter bank is used (multiband wavelet
system) [5], very flexible analysis filter banks can be realized (Fig. 27.4), e.g., those simulating along the
frequency axis the distribution of a set of nonoverlapping peripheral auditory filters (cf., section 27.4).
Another quite efficient approach for the parametric description of audio is the so-called sinusoidal
modeling often used for the analysis and synthesis of musical instrument sounds [41]. The audio signal
x(t) is modeled by a set of tones and noise.
© 2002 by CRC Press LLC
b ( k + 1)
n
b ( k −1)
j
↑2
a ( k −1)
j
↑2
b ( k + 1)
n
bkl
akl
h0( −n)
1( ) n h −
∑
0( ) l h
1( ) l h
↑2
↑2
↓2
↓2
h0( −n)
1( ) n h −
(a)
0( ) n h
1( ) n h
(b)
akl
+
(a)
akl
(b)
t
(27.12)
The tones have slowly varying parameters: amplitude a k(t) and frequency w k(t). Additionally, an
appropriate noise model has to be used. A perceptually acceptable noise model can be obtained by adding
↑2
↑2
↓2
↓2
h0( −l)
h1( −l)
bkl
akl
b ( k + 1)
n
+
0( ) n h
1( ) n h
↓2
↓2
b ( k −1)
j
a ( k −1)
j
b ( k + 1)
n
+
x() t =
ak() t sin⎛
ωk( τ)
dτ+
φ ⎞
⎝ k + noise
⎠
k
∫
τ=0