Abstract
Recently, a Monte Carlo approach was proposed for processing highly redundant continuous frames. In this paper, we present and analyze applications of this new theory. The computational complexity of the Monte Carlo method relies on the continuous frame being so-called linear volume discretizable (LVD). The LVD property means that the number of samples in the coefficient space required by the Monte Carlo method is proportional to the resolution of the discrete signal. We show in this paper that the continuous wavelet transform (CWT) and the localizing time-frequency transform (LTFT) are LVD. The LTFT is a time-frequency representation based on a 3D time-frequency space with a richer class of time-frequency atoms than classical time-frequency transforms like the short time Fourier transform (STFT) and the CWT. Our analysis proves that performing signal processing with the LTFT has the same asymptotic complexity as signal processing with the STFT and CWT (based on FFT), even though the coefficient space of the LTFT is higher dimensional.
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Ali, S., Antoine, J., Gazeau, J.: Continuous frames in Hilbert space. Ann. Phys. 222(1), 1–37 (1993)
Aliprantis, C.D.: An Invitation to Operator Theory. American Mathematical Society, Rhode Island (2002)
Balazs, P.: Basic definition and properties of Bessel multipliers. J. Math. Anal. Appl. 325(1), 571–585 (2007)
Balazs, P., Bayer, D., Rahimi, A.: Multipliers for continuous frames in Hilbert spaces. Journal of Physics A: Mathematical and Theoretical 45(24) (2012)
Balazs, P., Gröchenig, K., Speckbacher, M.: Kernel theorems in coorbit theory. arXiv:1903.02961, [math.FA] (2019)
Balazs, P., Laback, B., Eckel, G., Deutsch, W.A.: Time-frequency sparsity by removing perceptually irrelevant components using a simple model of simultaneous masking. Trans. Audio Speech Lang. Proc. 18(1), 34–49 (2010). https://doi.org/10.1109/TASL.2009.2023164
Bass, R.F., Gröchenig, K.: Relevant sampling of band-limited functions. Ill. J. Math. 57 (1), 43–58 (2013). https://doi.org/10.1215/ijm/1403534485
Candès, J., Donoho, D.L.: Continuous curvelet transform: II. discretization and frames. Appl. Comput. Harmon. Anal. 19(2), 198–222 (2005)
Christensen, O.: An Introduction to Frames and Riesz. Birkhäuser, Bases (2002)
Crochiere, R.: A weighted overlap-add method of short-time fourier analysis/synthesis. IEEE Trans. Acoust. Speech Sig. Process. 28(1), 99–102 (1980). https://doi.org/10.1109/TASSP.1980.1163353
Dahlke, S., Kutyniok, G., Maass, P., Sagiv, C., Stark, H.G., Teschke, G.: The uncertainty principle associated with the continuous shearlet transform. International Journal of Wavelets Multiresolution and Information Processing 06(02), 157–181 (2008)
Daubechies, I.: Ten lectures on wavelets SIAM: Society for industrial and applied mathematics (1992)
Donoho, D., Johnstone, J.: Ideal spatial adaptation by wavelet shrinkage. Biometrika 81(3), 425–455 (1994). https://doi.org/10.1093/biomet/81.3.425
Donoho, D.L., Johnstone, I.M., Kerkyacharian, G., Picard, D.: Wavelet shrinkage: Asymptopia. J. R. Stat. Soc. Series B (Methodological) 57 (2), 301–369 (1995)
Driedger, J., Müller, M.: A review of time-scale modification of music signals. Applied Sciences 12(2) (2016)
Duflo, M., Moore, C.: On the regular representation of a nonunimodular locally compact group. J. Funct. Anal. 21, 209–243 (1976)
Evangelista, G., Dörfler, M., Matusiak, E.: Arbitrary phase vocoders by means of warping. Musica/Tecnologia 7(0) (2013)
Feichtinger, H.G.: Un espace de banach de distributions tempérées sur les groupes localement compacts abéliens. C. R. Acad. Sci. Paris S er 290 (17), A791–A794 (1980)
Führ, H., Xian, J.: Relevant sampling in finitely generated shift-invariant spaces. J. Approx. Theory 240, 1–15 (2019). https://doi.org/10.1016/j.jat.2018.09.009
Folland, G.B.: Harmonic analysis in phase space. (AM-122). Princeton University Press, Princeton (1989)
Führ, H.: Abstract Harmonic Analysis of Continuous Wavelet Transforms. Springer, Berlin (2005)
Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Basel (2001)
Grossmann, A., Morlet, J.: Decomposition of Hardy functions into square integrable wavelets of constant shape. SIAM J. Math. Anal. 15(4), 723–736 (1984)
Grossmann, A., Morlet, J., Paul, T.: Transforms associated with square integrable group representations i. general results. J. Math. Phys. 26 (10), 2473–2479 (1985)
Guo, Q., Yu, S., Chen, X., Liu, C., Wei, W.: Shearlet-based image denoising using bivariate shrinkage with intra-band and opposite orientation dependencies. In: 2009 International Joint Conference on Computational Sciences and Optimization. https://doi.org/10.1109/CSO.2009.218, vol. 1, pp 863–866 (2009)
Hagen, R., Roch, S., Silbermann, B.: C*-algebras and Numerical Analysis. CRC Press, Boca Raton (2001)
Holighaus, N., Wiesmeyr, C., Balazs, P.: Continuous warped time-frequency representations—coorbit spaces and discretization. Appl. Comput. Harmon. Anal. 47(3), 975–1013 (2019)
Hörmander, L.: The Analysis of Linear Partial Differential Operators, vol. I. Springer, Berlin (1983)
Kalisa, C., Torrésani, B.: N-dimensional affine Weyl-Heisenberg wave. Annales de l’Institut Henri Poincare Physique Theorique 59, 201–236 (1993)
Laroche, J., Dolson, M.: Phase-vocoder: about this phasiness business. In: Proceedings of 1997 Workshop on Applications of Signal Processing to Audio and Acoustics, pp. 4 pp.– (1997)
Laroche, J., Dolson, M.: Improved phase vocoder time-scale modification of audio. Trans. Speech Audio Process. 7(3), 323–332 (1999)
Levie, R., Avron, H.: Randomized signal processing with continuous frames. arXiv:1808.08810 [math.NA] (2018)
Liuni, M., Roebel, A.: Phase vocoder and beyond. Music/Technology 7, 73–89 (2013)
Majdak, P., Balázs, P., Kreuzer, W., Dörfler, M.: A time-frequency method for increasing the signal-to-noise ratio in system identification with exponential sweeps. 2011 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 3812–3815 (2011)
Matz, G., Hlawatsch, F. In: Boashas, B. (ed.) : Time-Frequency Transfer Function Calculus Of Linear Time-Varying Systems, chapter 4.7 in Time-Frequency Signal Analysis And Processing: A Comprehensive Reference, pp 135–144. Elsevie, Oxford (2003)
Olivero, A., Torrésani, B., Kronland-Martinet, R.: A class of algorithms for time-frequency multiplier estimation. IEEE Transactions on AudioSpeech, and Language Processing 21, 1550–1559 (2013)
Ottosen, E.S., Dörfler, M.: A phase vocoder based on nonstationary Gabor frames. IEEE/ACM Transactions on Audio Speech, and Language Processing 25, 2199–2208 (2017)
Patel, D., Sampath, S.: Random sampling in reproducing kernel subspaces of lp(rn). J. Math. Anal. Appl. 491(1), 124270 (2020). https://doi.org/10.1016/j.jmaa.2020.124270
Pettis, B.J.: On integration in vector spaces. Trans. Am. Math. Soc. 44(2), 277–304 (1938)
Portnoff, M.: Implementation of the digital phase vocoder using the fast Fourier transform. IEEE Transactions on Acoustics Speech, and Signal Processing 24(3), 243–248 (1976). https://doi.org/10.1109/TASSP.1976.1162810
Průša, Z., Holighaus, N.: Phase vocoder done right. In: Proceedings of 25th European Signal Processing Conference (EUSIPCO-2017), pp. 1006–1010. Kos (2017)
Rahimi, A., Najati, A., Dehghan, Y.N.: Continuous frame in Hilbert spaces. Methods Funct. Anal. Topology 12(2), 170–182 (2006)
Smith, J.O.: Mathematics of the Discrete Fourier Transform (DFT) W3K Publishing (2007)
Stoeva, D., Balazs, P.: Invertibility of multipliers. Appl. Comput. Harmon. Anal. 33(2), 292–299 (2012)
Torrésani, B.: Wavelets associated with representations of the affine Weyl–Heisenberg group. J. Math. Phys. 32(5), 1273–1279 (1991). https://doi.org/10.1063/1.529325
Torrésani, B.: Time-frequency representations: wavelet packets and optimal decomposition. Annales de l’Institut Henri Poincare Physique Theorique 56(2), 215–234 (1992). https://hal.archives-ouvertes.fr/hal-01280027
Velasco, G.A.: Relevant sampling of the short-time fourier transform of time-frequency localized functions. arXiv:1707.09634 [math.FA] (2017)
Vretbladr, A.: Fourier Analysis and its Applications. Springer, Berlin (2000)
Zolzer, U.: DAFX: Digital Audio Effects, 2nd edn. Wiley, Hoboken (2011)
Acknowledgements
Ron Levie was supported by the DFG Grant DFG SPP 1798 “Compressed Sensing in Information Processing.” Haim Avron was partially supported by BSF grant 2017698.
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Appendices
Appendix 1: Pseudo inverse of the analysis operator
For an injective linear operator with close range \(B:\mathcal {V}\rightarrow \mathcal {W}\) between the Hilbert spaces \(\mathcal {V}\) and \(\mathcal {W}\), we define the pseudo inverse [26, Section 2.1.2]
where \(R_{B\mathcal {V}}:\mathcal {W}\rightarrow B\mathcal {V}\) is the surjective operator given by the orthogonal projection from \(\mathcal {W}\) to \(B\mathcal {V}\) and restriction of the image space to the range \(B\mathcal {V}\), and \(B\big |_{\mathcal {V}\rightarrow B\mathcal {V}}\) is the restriction of the image space of B to its range \(B\mathcal {V}\), in which it is invertible. Note that \(R_{B\mathcal {V}}^{*}\) is the operator that takes a vector from \(B\mathcal {V}\) and canonically embeds it in \(\mathcal {W}\), and \(P_{B\mathcal {V}}=R_{B\mathcal {V}}^{*}R_{B\mathcal {V}}:\mathcal {W}\rightarrow \mathcal {W}\) is the orthogonal projection upon \(B\mathcal {V}\). Note that \(V_{f}^{+}\) exists. Indeed, by the frame inequality (7), Vf is bounded both from above and below, so it must be injective with closed range [2, Chapter 2]. In the following, we collect basic properties from frame analysis (see, e.g., [42], [9, Section 5.6] and [26, Section 2.1.2])
Lemma 25
Let \(f:G\rightarrow {\mathscr{H}}\) be a continuous frame with frame bounds A, B. Then the following properties hold. Let \(s\in {\mathscr{H}}\).
-
1.
The operator \(V_{f}^{*}\) is the pseudo inverse of \(V_{f}^{+*}\), and \(V_{f}^{+*}[{\mathscr{H}}]=V_{f}[{\mathscr{H}}]\).
-
2.
\(V_{f}^{+*}V_{f}^{*}=V_{f}V_{f}^{+}=P_{V_{f}[{\mathscr{H}}]}\).
-
3.
\(S_{f}^{-1}=(V_{f}^{*}V_{f})^{-1}= V_{f}^{+}V_{f}^{+*}\).
-
4.
\(V_{f}^{+*}[s]=V_{S_{f}^{-1}f}[s]=V_{f}[S_{f}^{-1}s]\).
-
5.
\(V_{f}^{+}=S_{f}^{-1}V_{f}^{*}\).
-
6.
\(\left \|V_{f}^{+}\right \|{~}_{2}\geq A^{-1/2}\).
Proof
We prove 5, and note that the rest are basic properties of dual frames and pseudo inverse. See, e.g., [42] and [9, Section 5.6] for dual frames, and [26, Section 2.1.2] for pseudo inverse.
□
Appendix 2: Proofs
1.1 2.1 Proofs of Section 4
The mapping (x, ω, τ)↦fx, ω, τ is continuous as a mapping \([\tau _{1},\tau _{2}]\times \mathbb {R}^{2}\rightarrow L^{2}(\mathbb {R})\), since \(\mathcal {T}(x),{\mathscr{M}}(\omega ),\mathcal {D}(\gamma )\) are continuous, in x, ω, and γ respectively, in the strong topology [22, Section 9.2]. Hence, \(V_{f}[s]:G\rightarrow \mathbb {C}\) is a continuous function for every \(s\in L^{2}(\mathbb {R})\), and thus measurable. To show that f is continuous frame, it is left to show the existence of frame bounds \(0<A\leq B<\infty \) satisfying (7). Equivalently we show that Vf is injective and
where \(V_{f}^{+}:L^{2}(G)\rightarrow L^{2}(\mathbb {R})\) is the pseudo inverse of Vf as defined in Section 3.1.
We start by deriving a formula for the LTFT atoms \(\hat {f}_{x,\omega ,\tau }\) in the frequency domain.
Lemma 26
LTFT atoms take the following form in the frequency domain.
For fτ(t) = τ− 1/2f(τ− 1t)e2πit we have
and another formula in case \(a_{\tau }\leq \left |\omega \right |\leq b_{\tau }\) is
Proof
Equation (57) is a direct result of Lemma 4 and (30).
We can write \(f_{\tau } = {\mathscr{M}}(1) \mathcal {D}(\tau )h\). Another formula in case \( a_{\tau }\leq \left |{\omega }\right |\leq b_{\tau } \) is
so
We can also write
□
For convenience, we repeat here Definition 13. The sub-frame filter kernels \(\hat {S}_f^{\text {low}}, \hat {S}_f^{\text {mid}}\) and \(\hat {S}_f^{\text {high}}\) are the functions \(\mathbb {R}\rightarrow \mathbb {C}\) defined by
The frame filter kernel \(\hat {S}_f:\mathbb {R}\rightarrow \mathbb {C}\) is defined by
Let band ∈{low,mid,high}. For convenience, we recall (34)
Thus, by (57),
for the band corresponding to ω.
Proof of Theorem 12
Denote by \(V_f^{\text {band}}[s]\) the restriction of Vf[s] to (x, ω, τ) satisfying \(\omega \in \mathcal {J}_{\tau }^{\text {band}}\). We offer the following informal computation.
Here, δ is the delta functional, and the informal computation with the delta functional can be formulated appropriately similarly to the usual Calderón’s reproducing formula in continuous wavelet analysis (see, e.g., [12, Proposition 2.4.1 and 2.4.1], [8, Theorem 1], and [11, Theorem 2.5]).
Now note that by the fact the three \(\mathcal {J}^{\text {band}}_{\tau }\) domains are disjoint, so
Our goal now is to show that \(\hat {S}_{f}(z)\) is bounded from below by some A > 0 and from above by some B > 0 for every \(z\in \mathbb {R}\). The constants A, B are the frame bounds. In the following, we construct implicit upper and lower bounds for \(\hat {S}_{f}(z)\), without any effort to make these bound realistic estimates of \(\left \|V_{f}\right \|{~}^{2}\) and \(\left \|V_{f}^{+}\right \|{~}^{2}\). The goal is to prove that f is a continuous frame, rather than to obtain good frame bounds. In Section 4.2, we explain separately that numerically estimating \(\hat {S}_{f}\) and \(\hat {S}_{f}^{-1}\) give good estimates for the frame bounds.
Next, we show that there is some A > 0 such that for every \(z\in \mathbb {R} \hat {S}_{f}(z)\geq A\). For simplicity, we consider the case where aτ = a and bτ = b are constants. The general case is shown similarly with the appropriate modifications. Let z ≥ 0, and note that the case z ≤ 0 is shown symmetrically. By the fact that f is a non-negative function,
Since f is compactly supported, \(\hat {f}\) is smooth, so there is some ν > 0 such that for every z ∈ (−ν, ν)
We now distinguish between three cases: \(z\in [0,a], z\in \left .\left (a,b\right .\right ]\), and \(z\in (b,\infty )\).
In case z ∈ [0, a],
By lugging (64) in (65), for any ω satisfying
we have
Let \(\mathcal {I}_{z}\) denote the interval \((z-\nu \frac {a}{\tau _{2}}, z+\nu \frac {a}{\tau _{2}})\cap (-a,a)\), and note that the length of \(\mathcal {I}_{z}\) is bounded from below by
where μ is the standard Lebesgue measure or \(\mathbb {R}\). Thus, by the fact that the integrand of (35) is non-negative, the fact that μτ([τ1, τs]) = 1, and by (66) and (67),
If \(z\in \left .\left (a,b\right .\right ]\),
By (64), for any ω satisfying
we have
Let \(\mathcal {I}_{z}\) denote the interval \((z-\nu \frac {a}{\tau _{2}}, z+\nu \frac {a}{\tau _{2}})\cap (a,b)\), and note that the length of \(\mathcal {I}_{z}\) is bounded from below by
Thus, by the fact that the integrand of (35) is non-negative, by μτ([τ1, τs]) = 1, (69) and (70),
Last, if \(z\in (b,\infty )\),
By (64), for any ω satisfying
we have
Let \(\mathcal {I}_{z}\) denote the interval \((z-\nu \frac {b}{\tau _{2}}, z+\nu \frac {b}{\tau _{2}})\cap (b,\infty )\), and note that the length of \(\mathcal {I}_{z}\) is bounded from below by
Thus, by the fact that the integrand of (35) is non-negative, by μτ([τ1, τs]) = 1, (72) and (73),
By taking \(A=\min \limits \{C_{1},C_{2},C_{3}\}\), for every z ≥ 0
and thus
Next, we bound \(\left \|V_{f}\right \|{~}^{2}\) from above. Note that
where \(V_{f^{\text {band}}}\) now denotes Vf restricted to (x, ω, τ) with \(\omega \in \mathcal {J}^{\text {band}}_{\tau }\), for any band ∈{low, mid, high}. The systems flow and fhigh are both STFT systems restricted in phase space to a sub-domain of frequencies, and integrated along τ ∈ (τ1, τ2). By extending flow and fhigh to the whole frequency axis \(\mathbb {R}\), we increase \(\left \|{V_{f^{\text {low}}}}\right \|\) and \(\left \|{V_{f^{\text {high}}}}\right \|\) to the frame bound of the STFT which is 1. This shows that
It is left to bound \(\left \|V_{f^{\text {mid}}}\right \|{~}^{2}\) from above. Note that fmid cannot be extended to a CWT frame, since this system is not based on an admissible wavelet.
Recall the pseudo mother wavelet defined in (31)
In the following, we use the bound
By (62)
and by Lemma 26,
Let us change variable in (76) and consider the interval ω ∈ [a, b]. By ω− 1z = y, we have ω = zy− 1, dω = −zy− 2dy, and
Thus
Therefore, by (75),
To conclude,
□
Next, we prove Proposition 14, that states that for any \(s\in L^{2}(\mathbb {R})\), the frame operator of the LTFT is given in the frequency domain by
Proof of Proposition 14
For any band ∈{low,mid,high}, define the operator \(S_{f}^{\text {band}}\) by
and observe that
We show that for any band ∈{low,mid,high}, and any \(s\in L^{2}(\mathbb {R})\),
We offer the following informal computation, analogues to the proof of Theorem 12.
Here, δ is the delta functional, and the formal computation with the delta functional can be formulated appropriately as explained in the proof of Theorem 12. □
1.2 2.2 Proofs of Section 5.1
Recall that the sequence of spaces \(\{V_{M}\cap \mathcal {R}_{C}\}_{M\in \mathbb {N}}\), from Definition 15 and (40), is the discretization on which we prove the LVD property of the CWT. Recall that ξ is supported in (− 1/2, 1/2), non-negative, and continuously differentiable, and obtains zero only outside (− 1/2, 1/2). Recall that the class \(\mathcal {R}_{C}\) is the set of all signals q ∈ L2[− 1/2, 1/2] such that
and
Moreover, recall that
Next, we prove Proposition 17, which states that \(\{V_{M}\cap \mathcal {R}_{C}\}_{M\in \mathbb {N}}\) is a discretization of \(\mathcal {R}_{C}\).
Proof of Proposition 17
Let δ. Given \(q\in \mathcal {R}_{C}\), we can approximate q by a smooth function \(p\in L^{2}(-1/2,1/2)\cap L^{\infty }(-1/2,1/2)\) that vanishes in a neighborhood of − 1/2 and 1/2, up to the small errors
Since p and ξ− 1p have continuously differentiable periodic extensions, their Fourier series converge to p and ξ− 1p respectively in both L2(− 1/2, 1/2) and \(L^{\infty }(-1/2,1/2)\) [48, Section 4.4]. We denote by vM the truncation of the Fourier series of ξ− 1p up to the frequency M, multiplied by ξ. Namely,
There is thus a large enough M, such that vM ∈ VM satisfies
Given any \(\delta ^{\prime }>0\), by choosing δ small enough, and M large enough, (81) and (82) guarantee that \(v_{M}\in V_{M}\cap \mathcal {R}_{C}\) and \(\left \|v_{M}-q\right \|{~}_{2}<\delta ^{\prime }\). □
We prove Proposition 18 in a sequence of claims. Consider the phase space domain GM of (41). Recall that [−S, S] is the support of the mother wavelet f. Thus, [−S/ω, Sω] is the support of the dilated wavelet ω1/2f(ωz). Since the signal q is supported in time in [− 1/2, 1/2], Vf[q](x, ω) is zero for any x∉(− 1/2 − S/ω, 1/2 + S/ω). As a result, restricting Vf[q] to the phase space domain
is equivalent to restricting Vf[q] to GM. We thus consider without loss of generality the domain \(G_{M}^{\prime }\) instead of GM in this section.
We define the following inner product that corresponds to enveloping by ξ.
Definition 27 (Weighted Lebesgue space)
For any two measurable \(q,p:[-1/2,1/2]\rightarrow \mathbb {C}\)
where \(\left \langle \xi ^{-1}q,\xi ^{-1}p\right \rangle \) is the L2[− 1/2, 1/2] inner product. Denote
Denote by L2;ξ(− 1/2, 1/2) the Hilbert space of signals with \(\left \|q\right \|{~}_{\xi }<\infty \).
The following lemma characterizes the behavior of admissible wavelets about the zero frequency.
Lemma 28
Let \(f\in L_{2}(\mathbb {R})\) be an admissible wavelet supported in [−S, S]. Then
Proof
First note that by the fact that f is supported in (−S, S), it is also in \(L^{1}(\mathbb {R})\) by the Cauchy–Schwarz inequality, with
By the wavelet admissibility condition \(\hat {f}(0)=0\). Moreover, since f is compactly supported, \(\hat {f}\) is smooth. Thus, for z > 0,
As a result
where \(\left \|\hat {f}^{\prime }\right \|{~}_{\infty } \leq \left \|2\pi yf(y)\right \|{~}_{1}\) since \((\mathcal {F}\hat {f}^{\prime })(y)=2\pi i yf(y)\), and
For z < 0 the proof is similar. □
Note that the basis \(\{e^{2\pi i m x}\xi (x)\}_{m=-M}^{M}\) of VM is an orthonormal system in L2;ξ(− 1/2, 1/2), since \(\{e^{2\pi i m x}\}_{m=-M}^{M}\) are orthonormal in L2(− 1/2, 1/2). By Parseval’s identity, for q ∈ VM satisfying
we have
We prove Proposition 18 by embedding the signal class \(\mathcal {R}_{C}\) in a richer space, and proving linear volume discretization for the richer space. Consider the signal space \(\mathcal {S}_{E}\) of signals q ∈ L2(− 1/2, 1/2) satisfying
for some fixed E > 0.
Lemma 29
\(\mathcal {R}_{C}\subset \mathcal {S}_{E}\) for any E ≥ C2.
Proof
Let \(q=\xi s \in \mathcal {R}_{C}\). By Cauchy-Schwartz inequality and by (37) and (38)
so
□
Proposition 30
Under the above construction, with ξ satisfying (44) with k > 2, we have μ(GM) ≤ 5WM, and for every \(q_{M}\in V_{M}\cap \mathcal {S}_{E}\) we have
where the O notation O(W− 1) is with respect to \(W\rightarrow \infty \), and oM(1) is a function that goes to zero as \(M\rightarrow \infty \).
Next, we prove Proposition 18, which we copy here for the convenience of the reader.
Proposition 18 Consider a smooth enough ξ in the sense
for some k > 2 and D > 0, and the corresponding discrete spaces \(\{V_{M}\}_{M\in \mathbb {N}}\) of (40). The continuous wavelet transform with a compactly supported mother wavelet \(f\in L_{2}(\mathbb {R})\) is linear volume discretizable with respect to the class \(\mathcal {R}_{C}\) and the discretization \(\{V_{M}\cap \mathcal {R}_{C}\}_{M\in \mathbb {N}}\), with the envelopes ψM defined by (43) for large enough W that depends only on 𝜖 of Definition 6.
By Lemma 29, Proposition 18 is now a corollary of Proposition 30, where for every 𝜖 we choose W and M0 large enough, so that for every M > M0
Proof of Proposition 30
Since \(\{e^{2\pi i m x}\xi (x)\}_{m=-M}^{M}\) is an orthonormal basis of VM ⊂ L2;ξ(− 1/2, 1/2), in the frequency domain signals in VM are spanned by \(\{V_{m}(z)=\hat {\xi }(z-m)\}_{m=-M}^{M}\) (see Lemma 4). We bound Vn(z) by
For \(\hat {q} = {\sum }_{m=-M}^{M} c_{m} \hat {b}_{m}\), with \(\left \|q\right \|{~}_{\xi }=\left \|\{c_{m}\}_{m=1}^{N}\right \|{~}_{2}\), we have \(\left |c_{m}\right |\leq \left \|q\right \|{~}_{\xi }\) for all m. Thus
For \(\left |z\right |\leq M+2\), (88) leads to the bound
and (89) is maximized by taking z = 0. We extend the sum to m between \(-\infty \) and \(\infty \), and without loss of generality increase the value of the sum by choosing z = 0. Thus, since k > 2,
Now, for \(\left |z\right |>M+2\), without loss of generality consider z > 0. By (88) we have
Overall,
We consider the domain \(G_{M}^{\prime } = \{(x,\omega )\ |\ W^{-1}M^{-1} < \omega <AM \}\). Recall that restricting to \(G_{M}^{\prime }\) is equivalent to restricting to GM. Let ψM denote by abuse of notation the projection in phase space that restricts functions to the domain \(G_{M}^{\prime }\). Next, we bound the error \(\left \|(I-\psi _{M})V_{f}[q]\right \|{~}_{2}\) for signals in VM. Note that for every \(\omega \in \mathbb {R}\)
Indeed, by Lemma 4
so by Plancherel’s identity
Let us now bound the right-hand side of (92) for ω such that \((x,\omega )\notin G_{M}^{\prime }\). We then integrate the result for \(\omega \in (WM,\infty )\), and for ω ∈ (0, W− 1M− 1), to bound the error in phase space truncation by ψM. Negative ω are treated similarly.
We start with ω > WM. Here, we decompose the integral along z into two integrals with boundaries in 0, M + 2 + (0.5(ω − M − 2))1/2. For each segment of the integral along z, by additivity of the integral, we integrate ω along \((WM,\infty )\) and show that the resulting value is small. The first segment gives
By Lemma 28, the dilated mother wavelet satisfies
so
for G(ω) = (M + 2 + (0.5(ω − M − 2))1/2)2ω− 3. We study G(ω) for different values of ω. When ω > M2, the value (M + 2 + (0.5(ω − M − 2))1/2)2 is bounded by 4ω for large enough M, so
When ω < M2,(M + 2 + (0.5(ω − M − 2))1/2)2 is bounded by 4M2 for large enough M, so
For ω > M2, integrating via the bound (95) gives
where oM(1) is a function that converges to zero as \(M\rightarrow \infty \). For WM < ω < M2, integrating via the bound (96) gives
Next, we study \(z\in \Big (M+2 + \big (0.5(\omega -M-2)\big )^{1/2},\infty \Big )\). Here, we take the maximum of the signal squared, and take the 2 norm of the window. By (90) we obtain for large enough M
Integrating the bound (100) along \(w\in (WM,\infty )\) gives
since k > 2 and \(\left \|q\right \|{~}_{\xi }\leq E\left \|q\right \|{~}_{2}\).
Last, we integrate ω ∈ (0, W− 1M− 1). By (90)
Thus
Thus, the integration of the bound (102) for ω ∈ (0, W− 1M− 1) gives
Summarizing the estimates (99), (101), and (103), together with the analogue bounds for ω < 0, we obtain
Last, since by the frame assumption \(\left \|q\right \|{~}_{2}^{2} \leq A^{-1}\left \|V_{f}(q)\right \|{~}_{2}^{2}\), we have
This means that given 𝜖 > 0, we may choose W large enough to guarantee \(\mathcal {J}(W,M) < \epsilon \) up from some large enough M0, and also guarantee for every \(M\in \mathbb {N}\)
with C𝜖 = 3W by (42). □
1.3 2.3 Proofs of Section 5.2
Recall that every q ∈ VM, R is a trigonometric polynomials supported on L2(−R/2, R/2)
with \(\left \|\mathbf {c}\right \|{~}_{2}=\left \|q\right \|{~}_{2}\). Since the Fourier transform of the indicator function of [− 1/2, 1/2] is the sinc function \(\text {sinc}(z) := \frac {\sin \limits (\pi x)}{\pi x}\), the normalized indicator function of [−R/2, R/2] is given in the frequency domain by
Hence,
recall that (45) reads: for every z > Y or z < −Y
Recall that STFT envelope GM, R of (47) is defined by
In the proof of Proposition 21, we use the following simple fact that can be shown by a direct calculation.
Lemma 31
Let Vf be the STFT based on the window \(f\in L^{2}(\mathbb {R})\), and let \(q\in L^{2}(\mathbb {R})\) be a signal. Then
The following lemma will be used in the proofs of Propositions 21 and 22.
Lemma 32
Let \(f\in L^{2}(\mathbb {R})\) be supported in [−S, S] and satisfy (50). Let q ∈ VM, R, with R = O(M). Then for every W > 4,
where [−WM/R, WM/R]c is the set \(\{\omega \in \mathbb {R}\ |\ w\notin [-WM/R,WM/R]\}\), and oW(1) is a function that decays to zero as \(W\rightarrow \infty \).
Proof
We consider ω > 0 and z > 0, and note that the other cases are similar. For each value of ω > WM/R, we decompose the integral (107) along z into the two integrals in
For z ∈ (0, (W1/2M/R + ω)/2), since ω ≥ MW/R and z ≤ (W1/2M/R + ω)/2,
so \(\left |z-\omega \right |{~}^{-2\kappa }\) obtains its maximum at z = (W1/2M/R + ω)/2. Thus, by (105),
Integrating the bound (109) for \(\omega \in (WM/R,\infty )\) gives
Note that (M/R)1 − 2κ = O(1) since R = O(M) and κ > 1/2.
For \(z\in \big ((W^{1/2}M/R+\omega )/2,\infty \big ), \hat {q}\) decays like M1/2(z − M)− 1. Indeed, since z > M
Now, by (111),
Integrating the bound (112) for \(\omega \in (WM/R,\infty )\) gives
The bounds (110) and (113) give together (108). □
Proof of Proposition 21
Let q ∈ VM, R Lemmas 31 and 32, are combined to give \(\left \|{(I-\psi ^{W}_{M,R})V_{f}[q]}\right \|{~}_{2}= o_{W}(1)\left \|q\right \|{~}_{2}\), so by the frame inequality
This means that given 𝜖 > 0, we may choose W large enough to guarantee \(\frac {\left \|{(I-\psi ^{W}_{M,R})V_{f}[q]}\right \|{~}_{2}}{\left \|{V_{f}[q]}\right \|{~}_{2}} < \epsilon \), and also guarantee that for every \(R,M\in \mathbb {N}, \left \|\psi _{M,R}\right \|{~}_{1}\leq C_{\epsilon } M\), with C𝜖 = 2W(1 + 2S), by (106), for R = O(M). □
The proof of Proposition 22 is similar to that of Proposition 21. We start with an analogous lemma to Lemma 31, based on (30) and Lemma 4.
Lemma 33
Let Vf be the LTFT (Definition 11), and let \(q\in L^{2}(\mathbb {R})\) be a signal. Then, for any (ω, τ) such that \(\left |\omega \right |>b^{M,R}_{\tau }\).
Proof of Proposition 22
Let 𝜖 > 0. Note that since Vf[sM, R] is supported in the x direction in \([R/2- \tau _{2}/a^{M,R}_{\tau }, R/2+ \tau _{2}/a^{M,R}_{\tau }]\), it is enough to show that restricting the frequency direction of G to − WM/R < ω < WM/R results in an error less than 𝜖. Note that all of the truncated atoms, with \(\left |\omega \right |\geq WM/ R\), are high frequency STFT atoms.
By Lemma 33, we must study the error
Note that all atoms \([\mathcal {D}(b^{M,R}_{\tau }/\tau )\hat {f}]\) in (115) satisfy (105) with some constant \(Y^{\prime },C^{\prime \prime }\) instead of \(Y,C^{\prime }\). Hence, by Lemma 32, the error (115) is of order \(o_{W}(1)\left \|s_{M,R}\right \|{~}^{2}_{2}\). The rest of the proof is the same as the proof of Proposition 21. □
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Levie, R., Avron, H. Randomized continuous frames in time-frequency analysis. Adv Comput Math 48, 25 (2022). https://doi.org/10.1007/s10444-022-09941-7
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DOI: https://doi.org/10.1007/s10444-022-09941-7
Keywords
- Signal processing
- Continuous wavelet transform
- Stochastic methods
- Time-frequency analysis
- Localizing time-frequency transform
- Phase vocoder