Diagonality of the Nonstationary Gabor Frame Operator with Non-

Corollary 3.3.4(Painless Nonstationary Expansions in Frequency Domain (Balazs et al., 2011, Cor. 2)). Leth_l 2W .R/be such thathO_lis bandlimited on the intervalŒp_l; q_lfor alll 2Z. The corresponding nonstationary Gabor system is the setG.h_l; al/D fTa_lkhl W k; l 2Zg. If time shift parametersa_l are chosen such thata_{l ql}¹_pl, the corresponding nonstationary Gabor frame operatorS_h;his the convolution operator

S_h;hf DF ¹ X

l2Z

1 a_l

ˇ ˇ ˇhO_l

ˇ ˇ ˇ

2

Of

!

; f 2L².R/: (3.3.8)

Thus,G.h_l; a_l/is frame forL².R/if and only if there exist constantsA; B > 0satisfying the inequality

0 < AX

l2Z

1 a_l

ˇ ˇ ˇhO_lˇ

ˇ ˇ

2

B <1; (3.3.9)

almost everywhere.

Similar to Corollary 3.3.3, inverting the frame operator gives canonical dual windowsO_l^ıin frequency domain

O

_l^ıD hO_l P

l2Z 1 al

ˇ ˇ ˇhO_l

ˇ ˇ ˇ

2; 8l 2Z: (3.3.10)

Corresponding analysis coefficients are then defined by c_k;l D˝

f; T_alkh_l˛ DD

f ; MO _alkhO_lE

k; l2Z; (3.3.11)

showing that the results from Theorem 3.3.2 and above Corollary 3.3.4 are essentially the same up to a Fourier transform. Hence,S_h;halso admits a Walnut representation.

Obviously, such painless constructions only work for windows with compact support, either time- or bandlimited. More general windows which are neither time- nor bandlimited are considered in the following.

3.3.2 Diagonality of the Nonstationary Gabor Frame Operator with

time-sampling pointsa_k.

Definition 3.3.5(ı-separated Set). A set of sampling pointsfa_k 2RW k2Zgisı-separated if there existsı > 0, such that

ja_k amj> ı; 8k¤m: (3.3.12) Dörfler and Matusiak then show that the nonstationary Gabor frame operatorSg;gis bounded from above and below if for everyk,g_k 2W .R/and the set of windowsfg_kgk2Zsatisfies the following two conditions:

for two constantsA; B > 0

0 < AX

k2Z

jg_k.x/j² B <1; (3.3.13) for allk 2Z.

for constants Ck > 0, windows gk have polynomial decay around a ı-separated set fa_k 2RWk2Zgsuch that

jg_k.x/j C_k.1C jx a_kj/ ^k; 8x 2R; (3.3.14) wherek > 2for allk2Z.

With these two conditions, Theorem 3.4 by Dörfler and Matusiak (2014) can be concisely summarized as follows. If windowsgkhave polynomial decay and sufficient overlap, a sequence fb_k⁰gk2Z exists such that forb_k b⁰_k for all k 2 Z, the system G.g_k; b_k/constitutes a frame for L².R/. Thus, nonstationary Gabor frames can always be constructed for functions with sufficient decay properties by choosing sufficiently dense frequency sampling parametersbk. For the sake of completeness, an equivalent result holds for windowsh_l which decay polynomially in frequency domain: with sufficient overlap of windowshO_l, there exists a sequencefa_l⁰gl2Z

such that for allal a⁰_l the systemG.hl; al/forms a frame forL².R/.

In the regular Gabor case the frame operator commutes with time-frequency shifts. For nonstationary Gabor frames the frame operator might not commute with modulations, i.e., S ¹.Mb_klgk/¤Mb_klS ¹gk. Holighaus (2014, Thm. 3) proved, that under certain conditions the inverse frame operator of nonstationary Gabor frames possesses a similar structure as the original frame operator. Further, these conditions also guarantee that the canonical dual frame ofG.g_k; bk/ is again a nonstationary Gabor frame with the same modulation parameters bk

(Holighaus, 2014, Cor. 5). Unfortunately, this only holds for compactly supported windows.

For functions which are neither time- nor bandlimited Dörfler and Matusiak (2015) propo-sed construction schemes for approximately dual frames, where approximate dual frames are characterized by the following definition adapted from Christensen and Laugesen (2010).

Definition 3.3.6(Approximate Dual Frame). Two framesG.g_k; b_k/andG._k; b_k/are said to be approximate dual frames, whenever

I Sg;

2< 1; (3.3.15)

for any frame operatorS with windowsgand.

As stated by Christensen (2016, Ch. 6.5), an upper bound of 1 might not be sufficient to guarantee that a functionf is approximately close toSg;f. In a finite dimensional setting this would contradict the perfect reconstruction constraint. A more reasonable approach is given by

I Sg;

₂"r (3.3.16)

for some positive"r 1. With the following lemmata it can be shown, that under certain assumptions, this approximation can be made arbitrarily small, similar to Theorem 3.2.5 for the regular Gabor case. This extends the results by Dörfler and Matusiak (2015) as they only considered approximate dual frames as defined in (3.3.15). Starting point is a brief lemma about an upper bound involvingı-separated sets, taken from Dörfler and Matusiak (2014).

Lemma 3.3.7(Dörfler and Matusiak (2014, Lem. 2.2b)). Letfak 2RWk2Zgbe aı-separated set. For > 1the following inequality holds

ess sup

x2R

X

k2Z

.1C jx a_kj/ 2 1C.1Cı/ .ı ¹C/. 1/ ¹

: (3.3.17) The corresponding proof can be found in (Dörfler and Matusiak, 2014). The following lemma characterizes the side diagonals of the nonstationary Gabor frame operator with respect to frequency sampling parametersb_k.

Lemma 3.3.8. Letgk have polynomial decay around aı-separated setfak 2R W k 2Zgfor allk2Z, i.e., there exist constantsC_k 2ŒCL; CUand_k 2ŒL; U, where the setsŒCL; CU andŒL; Uare positive andL> 2, such that

jgk.x/j Ck.1C jx akj/ ^k; 8x2R; (3.3.18)

for all windowsg_k. Then

blim_k!0

X

l2Znf0g

G_l^g;g

1 D0; 8k2Z; (3.3.19)

whereG_l^g;gis defined according to(3.3.4).

Proof. The general idea of this proof is based on the proof of Theorem 3.4 by Dörfler and Matusiak (2014) with minor modifications. First, let0 < < L 2. Using the polynomial decay ofg_k gives

ˇ ˇ ˇ ˇ

1 bk

g_k

x l

bk

g_k.x/

ˇ ˇ ˇ ˇD 1

bk

ˇ ˇ ˇ ˇ

g_k

x l

bk

ˇ ˇ ˇ

ˇjg_k.x/j (3.3.20)

C_k² b_k

1C

ˇ ˇ ˇ ˇ

x a_k l b_k

ˇ ˇ ˇ ˇ

k

.1C jx a_kj/ ^k (3.3.21)

C_k² bk

1C

ˇ ˇ ˇ ˇ

x a_k l bk

ˇ ˇ ˇ ˇ

_kC.1C/

.1C jx a_kj/ ^k (3.3.22)

C_k²

b_k .1C jx akj/ ^.1C/

1C ˇ ˇ ˇ ˇ l b_k

ˇ ˇ ˇ ˇ

kC.1C/

(3.3.23) C_k².1C jx a_kj/ ^.1C/jlj ^kC.1C/b_k^{k .2C/}; (3.3.24) where the estimate from (3.3.22) to (3.3.23) results from the inequality

.1C jxCyj/ .1C jxj/.1C jyj/ ; (3.3.25) for x; y 2 R and 0 (Dörfler and Matusiak, 2014). Now, choose " < CL and set b_k D "

Ck

1=k

. Thus,

G_l^g;g

1 Dess sup

x2R

ˇ ˇ ˇ ˇ ˇ

X

k2Z

1 b_kg_k

x l

b_k

g_k.x/

ˇ ˇ ˇ ˇ ˇ

(3.3.26)

ess sup

x2R

X

k2Z

C_k^1C.2C/=kjlj ^kC1C"^{1 .2C/=k}.1C jx a_kj/ ^.1C/

max

k2ZRkjlj ^LC1C"^{1 .2C/=L}ess sup

x2R

X

k2Z

.1C jx akj/ ^.1C/; (3.3.27)

whereR_k DC_k^1C.2C/=k. Using Lemma 3.3.7, the essential supremum on the right hand side of (3.3.27) can then be bounded by

G_l^g;g

1max

k2ZRkjlj ^{.L 1 /}"^{1 .2C/=L}2

1C.1Cı/ ^.1C/.ı ¹C1C/ ¹

: (3.3.28) Hence, withD2

1C.1Cı/ ^.1C/.ı ¹C1C/ ¹ X

l2Znf0g

G_l^g;g

1 max

k2ZR_k"^{1 .2C/=L} X

l2Znf0g

jlj ^{.L 1 /} (3.3.29) The summation of (3.3.29) over all l 2 Znf0g reduces to the Riemann zeta function and is convergent as per definitionL 1 > 1. Further, Equation (3.3.29) tends to 0whenever

"! 0since the exponent of"satisfies1 .2C/=L > 0and the claim in (3.3.19) follows readily.

Now, analogously to Theorem 3.2.5 for Gabor frames, a similar result can be derived for nonstationary Gabor frames. It is an adaption of Proposition 4.1 by Dörfler and Matusiak (2015).

Theorem 3.3.9(Diagonality of the Nonstationary Gabor Frame Operator). For everyk2Z, let g_k have polynomial decay around aı-separated set and let there exist positive constantsA; B such that

0 < AG₀^g;g DX

k2Z

1

b_k jg_k.x/j² B <1; 8x2R: (3.3.30) Define dual windows by

k D G₀^g;g 1

gk; (3.3.31)

for allk2Z. With identity operatorI, the nonstationary Gabor frame operator satisfies

I Sg;

₂

P

l2Znf0g

G_l^g;g 1

ess infG₀^g;g : (3.3.32)

Furthermore, ifb_k !0for everyk2Z,

blimk!0

I Sg;

₂D0; (3.3.33)

whereSg; depends onbk, see(3.3.3).

Proof. The proof for estimating the upper bound in (3.3.32) follows similar arguments as in Equations (3.2.14) to (3.2.18). Equation (3.3.33) follows then directly from Lemma 3.3.8 and the lower bound ofG₀^g;g.

This result shows that, quite similar to the Gabor case in a finite dimensional setting, the non-stationary Gabor frame operator can be made diagonal within machine precision for sufficiently small frequency shift parametersbk. Similarly, by duality of (3.3.11) and (3.3.1), all of the above assumptions are also valid for functionshO_l with sufficient decay properties in frequency domain.

This can be summarized in the following Corollary, showing that the nonstationary Gabor frame operator tends to the identity, whenever time sampling parametersa_l tend pointwise to zero.

Corollary 3.3.10. For every l 2 Z, let h_l be such thathO_l has polynomial decay around a separated set and let there exist positive constantsA; Bsuch that

0 < AG₀^h;hDX

l2Z

1 al

ˇ ˇ ˇhO_l

ˇ ˇ ˇ

2

B <1: (3.3.34)

With identity operatorI, the nonstationary Gabor frame operator satisfies

I FS_h;

₂ P

k2Znf0g

G_k^h;h

1

ess infG₀^h;h

; (3.3.35)

where dual windows are defined by O _l D

G₀^h;h 1

hO_l (3.3.36)

for alll 2Z. Furthermore, ifa_l !0for everyl 2Z,

alim_l!0

I FS_h;

₂D0; (3.3.37)

whereFS_h; depends ona_l.

In order to put the results from Theorem 3.3.9 and Corollary 3.3.10 into more practical terms, consider the following definition, similar to the"-concentration proposed by Donoho and Stark (1989):

Definition 3.3.11(Essential Support). Letf 2L².R/be a function withkfk2 D1. For any

", with0 < " < 1, there exists an intervalŒN1; N2RwithN1< N2, such that

f f ŒN₁;N₂

2 2D

Z

R

ˇ

ˇf .x/ f .x/ŒN₁;N₂

ˇ ˇ

2dx < "; (3.3.38) whereis the characteristic function of the specified set. The essential support off is then denoted by the intervalU^" DŒN1; N2.

With this definition, so called almost painless nonstationary Gabor frames can be constructed, similar to (Dörfler and Matusiak, 2015, Def. 4). For some", let

zk Dgk_U^"

k; (3.3.39)

be the painless approximation of windowsg_k without compact support. Whenever frequency-shift parametersb_kare chosen such thatG.z_k; b_k/satisfies Theorem 3.3.2,G.g_k; b_k/is called an almost painless nonstationary Gabor frame. Obviously, the smaller"the smaller the difference between painless and almost painless nonstationary Gabor frame. This implies the following discrete construction scheme for nonstationary Gabor framesG.g_k; b_k/with windowsg_kwithout compact support:

1. Choose arbitraryg_k with sufficient decay properties (e.g., Gaussians with different vari-ances) localized at different time locations such that Equations (3.3.13) and (3.3.14) are satisfied.

2. Chooseb_k by considering the corresponding painless approximation framez_k Dg_kU^"

k

for a specific", such thatG.z_k; b_k/is a frame according to Theorem 3.3.2.

The resulting nonstationary Gabor frameG.g_k; b_k/with non-compactly supportedg_kcan then be related to Theorem 3.3.9 as follows. With decreasing"the frequency shift parameterbk will also decrease according to Theorem 3.3.2 and therefore, the limiting case of" ! 0 implies b_k !0for allk2Z. Theorem 3.3.9 then states that the nonstationary Gabor frame operator is the identity, if dual frames are chosen according to (3.3.31). In a finite dimensional setting the resulting frame operator is diagonal within machine precision for sufficiently small choices of

". Hence, no computational expensive inverting of the frame operator is required, resulting in fast algorithms to compute dual frames.

Im Dokument The Affine Uncertainty Principle, Associated Frames and Applications in Signal Processing (Seite 37-44)