3.4 Wavelet Frames Based on Nonstationary Gabor Frames
3.4.2 Tight Wavelet Frames with Compactly Supported Functions
As a representative for all compactly supported functions, the construction of wavelet frames G.hl; al/is described with Hann windows. It is possible, however, to generalize the results of this section to any compactly supported window in the finite dimensional case.
In the following letbe the Hann window defined in (3.3.46). The warping function is given byW Dlogd, where > 0is a fixed constant. The resulting warped Hann window is then defined as
d;l;./DDd l ./D ( 1
pdl 1
2C12cos. .logd l//
dl 1 dlC1
0 else ;
(3.4.7) for; d 2RCandl 2Œl1; l2Dl1Cl; l3 Dl1C2l; : : : ; lnDl1C.n 1/l. Hence, the parameter serves for adjusting the bandwidthbwof the first warped Hann window.
If the first wavelet is supposed to be specified atfminit follows by a simple computation that
l1Dlogdfmin: (3.4.8)
For a given bandwidthbwthe parameter is computed by
D ln.d /
ln
1 2
bwd l1C q
bwd l12
C4
; (3.4.9)
Algorithm 3:Tight Wavelet Frames with Hann Windows Input :L- signal length
sf - sampling frequency d - scaling base
fmin- location of first window bw- bandwidth of first window bins- number of frequency bins Output:hOl - set of window functions
al - set of corresponding time sampling steps
1 compute initial parametersl1,andl according to Equations (3.4.8) - (3.4.10)
2 l l1
3 whiledlC1 < sf2 do
4 hOl d;l;./
5 hO l d;l;. /
6 a l;l L
dlC1 dl 1 1
7 l lCl
8 end
9 add two cover functions at zero and Nyquist frequency
which results from settingbwDdl1C1 dl1 1 and solving for. If the difference between two consecutivel’s satisfies
l D 2
bins; (3.4.10)
withbins> 2being an integer, the translated versions of the warped Hann windows will sum up to a constant, cf. (3.3.48). This procedure can be repeated, progressively covering the positive frequency axis until the support of a scaled window reaches the Nyquist frequency. The time sampling parameteral for each corresponding window hOl is chosen according to Corollary 3.3.4. Since it is assumed that all input signals f are real valued, it is sufficient to cover negative frequencies by mirroring corresponding wavelets, i.e., d;l;. /. Furthermore, two additional windows have to be added covering the zero and Nyquist frequency. Usually, wavelet coefficients resulting from these two windows do not contain any useful information and can therefore be neglected (Balazs et al., 2011). For frames, however, the windows are needed
(a)binsD4 (b)binsD7
Figure 3.5:Two examples of window functions with warped Hann windows, each constituting a tight wavelet frame with correspondingal, differing only in the number of bins.
to preserve the structure of the partition of unity. The construction scheme is summarized in Algorithm 3.
Two examples of tight wavelet frames based on the warped Hann window can be seen in Figure 3.5. Signal length isLD1800at a sampling frequency ofsf D1800Hz. Furthermore, d D2,fminD150Hz andbwD150Hz. Figure 3.5a and 3.5b show windowshOl forbins D4 andbinsD7, resulting in tight wavelet framesG.hl; al/with boundsAD1:5andAD2:625 and redundancies 2.76 and 4.18, respectively. For convenient plotting, each window is normed such that
hOl
2 D1. The frequency axis in Fig. 3.5 is adapted to match the frequency domain representation of MATLAB’s fft-routine, where the second half with frequencies larger than the Nyquist frequency corresponds to negative frequencies.
3.4.3 Approximately Tight Wavelet Frames with Non-Compactly Supported Functions
General Construction
In this section a general framework to construct wavelet frames from non-compactly supported mother wavelets is presented. In order to be consistent with the construction of frames with compactly supported functions, the same characteristic parametersfmin,bwandbinsare used.
The position of the first wavelet fminand the number of bins can be adapted without further consideration. The bandwidthbwof the first wavelet, however, needs to be defined for functions
without compact support. A common approach would be to define the bandwidth as the width where the function drops below a certain point, e.g. the 3 dB bandwidth. Similar to this approach, the bandwidthbwcan be defined by utilizing the essential support in Definition 3.3.11. Hence, bwDN2"bw N1"bw; (3.4.11) where the bandwidth depends on the chosen"bw.
As a starting point of the construction scheme, letbe a continuous square integrable function, which attains its maximum at zero. Similar to the painless construction in the previous section, the warping function is given byW Dlogd for some parameter > 0. The scaled versions of can be written as
d;l; DDd l ./D.logd l/; (3.4.12)
whered; > 0andl 2Œl1; l2 Dl1Cl; l3Dl1C2l; : : : ; lnDl1C.n 1/l.
Initial parametersl1, andl can then be computed in the following manner. The position of the first wavelet immediately yields
l1Dlogdfmin; (3.4.13)
since the maximum value of is located at zero. The parameter can be computed by considering the bandwidth of the first wavelet
logd l12U"bw (3.4.14)
2
"
dl1C
N"bw 1
; dl1C
N"bw 2
#
; (3.4.15)
for some"bw. Hence
bwDdl1C
N"bw 2
dl1C
N"bw 1
: (3.4.16)
The nonlinear expression in (3.4.16) is guaranteed to have a solution for 2.0;1/, which can be seen from the following Lemma.
Lemma 3.4.2. Letbw; l1 > 0andd > 1, be positive real numbers. Further let,N1; N2 2RC such thatN2> N1. Then, the nonlinear expression . /
. /DdN2 dN1 d l1bw; (3.4.17)
has exactly one zero in the interval.0;1/.
Proof. Obviously, lim!1 . /D d l1bw< 0. On the other hand,dN2 > dN1sinceN2>
N1and therefore . /is strictly monotonic decreasing with lim!0 . /D C1> 0. The parameter can then be estimated using numerical methods for finding real roots, e.g., as proposed by Forsythe et al. (1977, p. 161). After computing the parameters for the initial wavelet, dilates can be constructed with
l D 1
bins: (3.4.18)
By subsequently increasinglwithlthe functions d;l;cover the positive frequency axis up to the Nyquist frequency. Similar to the painless construction, negative frequencies are covered by using mirrored versions of the corresponding windows. Again, two additional windows are needed to cover zero and Nyquist frequency explicitly. The time sampling parameter al is computed as described in Section 3.3.2: for each scaled wavelethOl the time sampling stepal is chosen such that
al L N2"a
l N1"a
l
1
; (3.4.19)
where N1=2"a
l denotes the bounds of the essential support of corresponding wavelets hOl based on parameter "a. Note the difference between "a and "bw: the former is a measure responsible for the redundancy of the frame, the latter only affecting the bandwidth of the first wavelet. According to the results in Section 3.3.2, the corresponding wavelet frame operator is numerically diagonal for suitable small "a and the canonical dual frame of the painless approximation. The general construction scheme is summarized in Algorithm 4.
Examples of Tight Wavelet Frames
In the following finite dimensional examples letL D1800and the sampling frequencysf D 1800Hz be fixed. Further, the first wavelet should be located atfminD150Hz with a bandwidth bw D 150Hz for"bw D 1e 6. Different generating functions might lead to different total numbers of windows for the samebinsparameter, depending on the decay behavior. Thus, the number of bins is adjusted for each function such that the total number of waveletsK D38is constant. In the case of tight frame constructions the function
ZD
K
X
lD1
ˇ ˇ ˇhOl
ˇ ˇ ˇ
2
; (3.4.20)
Algorithm 4:Wavelet Frames with Non-Compactly Supported Functions Input : - generating wavelet
L- signal length
sf - sampling frequency d - scaling base
fmin- location of first window bw- bandwidth of first window
"bw- bandwidth parameter for essential support bins- number of frequency bins
"a- parameter for choosing time sampling steps Output:hOl - set of window functions
al - set of corresponding time sampling steps
1 compute initial parametersl1,(based on"bw) andl according to Equations (3.4.8), (3.4.16) and (3.4.18)
2 l l1
3 whiledlCN2 < sf2 do
4 hOl d;l;
5 hO l mirrored version of d;l;
6 a l;l L N2"a N1"a 1
7 l lCl
8 end
9 add the two cover functions at zero and Nyquist frequency
should be, at least within numerical precision, a constant. As shown in Section 3.3.3, this depends on the chosen number of bins.
First, consider to be the Gaussian. The resulting set of warped Gaussian wavelets hOl is illustrated in Figure 3.6a. Figure 3.6b illustrates the set of wavelets, whereis chosen such that is the Equalizer withD25, i.e.,./D Eq.d/and D0for negative frequencies. This substitution is justified since the domain of the Equalizer is restricted toRC. The singularity at D 0 introduced by setting function values to zero for negative frequencies is in a finite dimensional setting close to machine precision. Corresponding valuesZ for the partition of unity are given byZD3:18with absolute numerical error max.Z/ min.Z/D3:6e 15for
(a)Warped Gaussian (b)Equalizer
Figure 3.6:Examples of non-compactly supported wavelet functions forming a partition of unity.
the warped Gaussian andZD3:25with max.Z/ min.Z/D3:1e 15for the Equalizer. Each set of wavelet functions would constitute a tight frame, at least within numerical precision, if parameters"aare chosen sufficiently small. A more detailed evaluation follows in the subsequent section.