Wave-Shape Function Analysis

Abstract

We propose to combine cepstrum and nonlinear time–frequency (TF) analysis to study multiple component oscillatory signals with time-varying frequency and amplitude and with time-varying non-sinusoidal oscillatory pattern. The concept of cepstrum is applied to eliminate the wave-shape function influence on the TF analysis, and we propose a new algorithm, named de-shape synchrosqueezing transform (de-shape SST). The mathematical model, adaptive non-harmonic model, is introduced and the de-shape SST algorithm is theoretically analyzed. In addition to simulated signals, several different physiological, musical and biological signals are analyzed to illustrate the proposed algorithm.

Appendices

Appendix

Proof of Theorem 3.4

In this section, we provide an analysis of STCT in Theorem 3.4 step by step

• first step: approximate the ANH function by a “harmonized” function by Taylor’s expansion and evaluate its STFT;

• second step: evaluate the $\gamma$ power of the absolute value of STFT. Since in general there will be more than one ANH component in the ANH function, we have to handle the possible interference between different ANH components. We will apply the Erdös–Turán inequality to control the interference;

• third step: find the Fourier transform of the $\gamma$ power of the absolute value of STFT and finish the proof.

We start from the first Lemma, which allows us to locally approximate an ANH function by a sinusoidal function.

Lemma 7.1

Take $\epsilon {>0}$, a sequence $c \in \ell ^1$, $N\in \mathbb {N}$ and $0< C<\infty$. For $f(t)=\frac{1}{2}B_0(t)+\sum _{\ell =1}^\infty \cos (2\pi \phi _\ell (t))\in \mathcal {D}^{c,C,N}_\epsilon$, for each $\ell \in \{0\}\cup \mathbb {N}$ we have

$$\begin{aligned} |B_\ell (t+s)-B_\ell (t)|\le&\epsilon c(\ell ) |s|(\phi '_1(t)+\frac{1}{2}\Vert \phi ''_{1}\Vert _{L^\infty }|s|),\end{aligned}$$

(48)

$$\begin{aligned} |\phi '_\ell (t+s)-\phi '_\ell (t)|\le&\epsilon \ell |s|(\phi '_1(t)+\frac{1}{2}\Vert \phi ''_{1}\Vert _{L^\infty }|s|). \end{aligned}$$

(49)

Proof

Assume that $s>0$. The proof for $s\le 0$ is the same. By the assumption of $B_\ell (t)$, we have

$$\begin{aligned} |B_\ell (t+s)-B_\ell (t)|&=\,\left| \int _0^sB_\ell '(t+u)d u\right| \\&{ \le \epsilon c(\ell ) \int _0^s\phi _1'(t+u) d u \quad \text{ by } \text{ the } \text{ slowly } \text{ varying } \text{ condition } \text{(16), }}\\&\le \,\epsilon c(\ell ) \int _0^s \left( \phi _1'(t)+\int _0^u \phi _1''({t+}y)dy\right) d u\\&\le \epsilon c(\ell ) \left( \phi _1'(t)s+\frac{1}{2}\Vert \phi ''_1\Vert _{L^\infty }s^2\right) . \end{aligned}$$

The proof of (49) follows by the same argument.

$$\begin{aligned} |\phi '_{\ell }(t+s) - \phi '_{\ell }(t)|&= \left| \int _0^s \phi ''_{\ell }(t+u) du \right| \\&\le \epsilon \ell \int _0^s \phi _1'(t+u)du \quad \text{ by } \text{ the } \text{ slowly } \text{ varying } \text{ condition } \text{(16), } \\&=\epsilon \ell \int _0^s \left( \phi _1'(t) + \int _0^u \phi _1''(t+y)dy\right) du\\&\le \epsilon \ell \left( \phi _1'(t) s + \frac{1}{2} \Vert \phi _1''\Vert _{L^{\infty }} s^2 \right) . \end{aligned}$$

$\square$

The following Lemma leads to the first part of the Theorem 3.4, regarding the STFT. In short, for the superposition of ANH functions in $\mathcal {D}_{\epsilon ,d}$, at each time t the function behaves like a sinusoidal function and the STFT could be approximately explicitly.

Lemma 7.2

Fix $\epsilon {>0}$ and $d>0$. Take $f(t)=\sum _{k=1}^Kf_k(t)\in \mathcal {D}_{\epsilon ,d}$. Then, the STFT of f at $t\in \mathbb {R}$ is

$$\begin{aligned} V^{(h)}_f(t,\xi )= \frac{1}{2}\sum _{k=1}^K\sum _{\ell =-N_k}^{N_k} B_{k,\ell }(t)\hat{h}(\xi -\phi '_{k,\ell }(t))e^{i2\pi \phi _{k,\ell }(t)}+\epsilon _0(t,\xi ), \end{aligned}$$

(50)

where $\xi \in \mathbb {R}$ and $\epsilon _0(t,\xi )$ is defined in (62). Furthermore, $|{\epsilon _0}(t,\xi )|$ is of order $\epsilon$ and decays at the rate of $|\xi |^{-1}$ as $|\xi |\rightarrow \infty$.

Proof

Since $f\in L^\infty \cap C^1\subset \mathcal {S}'$ and $h\in \mathcal {S}$, by the linearity of the STFT, we have

$$\begin{aligned} V^{(h)}_f(t,\xi )=\sum _{k=1}^K\sum _{\ell =0}^\infty V^{(h)}_{f_{k,\ell }}(t,\xi ), \end{aligned}$$

(51)

where $f_{k,0}=\frac{1}{2}B_{k,0}(\cdot {)}$ and $f_{k,\ell }(\cdot ):=B_{k,\ell }(\cdot )\cos (2\pi \phi _{k,\ell }(\cdot ))$ for $\ell =1,2,\ldots$. Denote

$$\begin{aligned}&\tilde{V}^{(h)}_{f_{k,\ell }}(t,\xi ) := \int B_{k,\ell }(t) \cos (2\pi (\phi _{k,\ell }(t) + \phi '_{k,\ell }(t)(x-t))) h(x-t) e^{-i2\pi \xi (x-t)}dx\nonumber \\&\tilde{V}_{f_{k,0}}^{(h)}(t,\xi )=\frac{1}{2}\int B_{k,0}(t)h(x-t)e^{-i2\pi \xi (x-t)}dx \end{aligned}$$

(52)

where $k=1,\cdots , K$ and $\ell = 1,\cdots , \infty$. Next, fix $k\in \{1,\ldots ,K\}$, we evaluate the difference between $V^{(h)}_{f_{k,\ell }}(t,\xi )$ and $\tilde{V}^{(h)}_{f_{k,\ell }}(t,\xi )$. For each $\ell \in \mathbb {N}\cup \{0\}$, denote

$$\begin{aligned} \epsilon _{k,\ell }(t,\xi ):=V^{(h)}_{f_{k,\ell }}(t,\xi )-{ \tilde{V}^{(h)}_{f_{k,\ell }}(t,\xi )}. \end{aligned}$$

(53)

We show that $|\epsilon _{k,\ell }(t,\xi )|$ is of order $\epsilon$ and linearly dependent on $c_{k}(\ell )$ for all $t,\xi \in \mathbb {R}$. First, note that

$$\begin{aligned} \left| \epsilon _{k,\ell }(t,\xi ) \right|&\le \int \left| B_{k,\ell }(x) - B_{k,\ell }(t)\right| \left| h(x-t) \right| dx \nonumber \\&\quad + B_{k,\ell }(t) \int \left| \cos (2\pi \phi _{k,\ell }(x)) - \cos ( 2\pi (\phi _{k,\ell }(t) + \phi '_{k,\ell }(t)(x-t) ))\right| \nonumber \\&\quad \times \left| h(x-t) \right| dx \end{aligned}$$

(54)

and that

$$\begin{aligned}&\left| \cos (2\pi \phi _{k,\ell }(x)) - \cos ( 2\pi (\phi _{k,\ell }(t) + \phi '_{k,\ell }(t)(x-t) )\right| \nonumber \\&\quad \le 2\pi \left| \phi _{k,\ell }(x)-\phi _{k,\ell }(t)-\phi _{k,\ell }'(t)(x-t) \right| \le 2\pi \int _0^{x-t} \left| \phi '_{k,\ell }(t+u)-\phi '_{k,\ell }(t)\right| du \end{aligned}$$

(55)

Denote

$$\begin{aligned} M_k:=\Vert \phi '_{k,1}\Vert _{L^\infty }. \end{aligned}$$

Clearly, $\Vert \phi _{k,1}''\Vert _{L^\infty }\le \epsilon M_k$. Combining the above inequalities and Lemma 7.1, we have

$$\begin{aligned} |\epsilon _{k,\ell }(t,\xi )|\le&\, \int |B_{k,\ell }(x)-B_{k,\ell }(t)||h(x-t)|dx\\&+2\pi B_{k,\ell }(t)\int \int _0^{x-t}|\phi '_{k,\ell }(t+u)-\phi '_{k,\ell }(t)|du |h(x-t)|dx\\ \le&\, \epsilon \big [c_{k}(\ell )\big (\phi _{k,1}'(t)I_1+\frac{1}{2} \epsilon M_kI_2\big )+\pi B_{k,\ell }(t)\ell (\phi _{k,1}'(t)I_2+\frac{1}{3}\epsilon M_kI_3)\big ] \end{aligned}$$

which is of order $\epsilon$ since $\phi _{k,1}'(t)$ and $B_{k,1}(t)$ are bounded. Note that $\epsilon _{k,0}(t,\xi )\le \epsilon c_{k}(\ell )(\phi '_{k,1}(t) + \epsilon M_kI_2/2)$ since the phase $\phi _{k,0}=0$. Furthermore, note that $|\epsilon _{k,\ell }(t,\xi )|$ decays at the rate of $|\xi |^{-1}$ as $|\xi |\rightarrow \infty$ since

$$\begin{aligned} B_{k,\ell }(x) \cos (2\pi \phi _{k,\ell }(x)) - B_{k,\ell }(t) \cos (2\pi ( \phi _{k,\ell }(t)) + \phi '_{k,\ell }(t)(x-t) ) \in C^1. \end{aligned}$$

(56)

Denote

$$\begin{aligned} E^{(1)}_{k}(t,\xi ):=\sum _{\ell =0}^\infty \epsilon _{k,\ell }(t,\xi ), \end{aligned}$$

which converges by (15) that $\sum _{\ell =1}^\infty \ell B_{k,\ell }(t)\le C_k \sqrt{\frac{1}{4}B^2_{k,0}(t)+\frac{1}{2}\sum _{\ell =1}^\infty B^2_{k,\ell }(t)}$, and hence

$$\begin{aligned} |E^{(1)}_{k}(t,\xi )|&\le \epsilon \Big (\Vert c_k\Vert _{\ell ^1}\big [\phi _{k,1}'(t)I_1+\frac{1}{2}\epsilon M_kI_2 {\big ]}\nonumber \\&\quad +\pi C_k {\sqrt{\frac{1}{4}B^2_{k,0}(t)+\frac{1}{2}\sum _{\ell =1}^\infty B^2_{k,\ell }(t)}} (\phi _{k,1}'(t)I_2+\frac{1}{3}\epsilon M_kI_3) \Big ). \end{aligned}$$

(57)

Thus, $E^{(1)}_{k}(t,\xi )$ is of order $\epsilon$.

Finally, for each $k\in \{1,\ldots ,K\}$, denote

$$\begin{aligned} E^{(2)}_{k}(t,\xi ):= { \sum _{\ell =N_k +1}^{\infty } \tilde{V}^{(h)}_{f_{k,\ell }}(t,\xi ).} \end{aligned}$$

By the Plancherel identity, we have

$$\begin{aligned} \tilde{V}^{(h)}_{f_{k,\ell }}(t,\xi ) =\frac{1}{2}B_{k,\ell }(t)\Big [\hat{h}(\xi -\phi '_{k,\ell }(t))e^{i2\pi \phi _{k,\ell }(t)}+\hat{h}(\xi +\phi '_{k,\ell }(t))e^{-i2\pi \phi _{k,\ell }(t)}\Big ]. \end{aligned}$$

(58)

Thus, by the assumption that (14) that $\sum _{\ell =N_k+1}^\infty B_{k,\ell }(t)\le \epsilon \sqrt{\frac{1}{4}B^2_{k,0}(t)+\frac{1}{2}\sum _{\ell =1}^\infty B^2_{k,\ell }(t)}$, we have

$$\begin{aligned} \Big |E^{(2)}_{k}(t,\xi )\Big |\le & {} \frac{1}{2}\sum _{\ell \in \mathbb {Z}\backslash {\{ -N_k, \cdots , N_k\} }} B_{k,\ell }(t)|\hat{h}(\xi -\phi '_{k,\ell }(t))|\nonumber \\\le & {} \epsilon I_0 {\sqrt{\frac{1}{4}B^2_{k,0}(t)+\frac{1}{2}\sum _{\ell =1}^\infty B^2_{k,\ell }(t)}}, \end{aligned}$$

(59)

where the last inequality holds since $\Vert \hat{h}\Vert _{L^\infty }\le I_0$ by a direct bound. Thus, we have

$$\begin{aligned} {\sum _{k=1}^K \sum _{\ell =0}^{\infty } \tilde{V}^{(h)}_{f_{k,\ell }}}(t,\xi )=\frac{1}{2}\sum _{k=1}^K\sum _{\ell =-N_k}^{N_k} B_{k,\ell }(t)\hat{h}(\xi -\phi '_{k,\ell }(t))e^{i2\pi \phi _{k,\ell }(t)}+\sum _{k=1}^KE^{(2)}_{k}(t,\xi ),\nonumber \\ \end{aligned}$$

(60)

where $|E^{(2)}_{k}(t,\xi )|$ is of order $\epsilon$. Furthermore, $|E^{(2)}_{k}(t,\xi )|$ decays faster than $|\xi |^{-1}$ as $|\xi |\rightarrow \infty$ since $\sum _{\ell =1}^\infty B_{k,\ell }(t)<\infty$ and $\sum _{k=1}^K \sum _{\ell =0}^{N_k} \tilde{V}^{(h)}_{f_{k,\ell }}(t,\xi )$ decays faster than $|\xi |^{-1}$ as $|\xi |\rightarrow \infty$.

We thus have

$$\begin{aligned} \sum _{k=1}^K\sum _{\ell =0}^\infty \tilde{V}^{(h)}_{f_{k,l}}(t,\xi )=\frac{1}{2}\sum _{k=1}^K\sum _{\ell \in \mathbb {Z}} B_{k,\ell }(t)\hat{h}(\xi -\phi '_{k,\ell }(t))e^{i2\pi \phi _{k,\ell }(t)}.\nonumber \\ \end{aligned}$$

(61)

Putting (53) and (60) together, we have

$$\begin{aligned} V^{(h)}_f(t,\xi )&\, =\sum _{k=1}^K\sum _{\ell =0}^\infty V^{(h)}_{f_{k,\ell }}(t,\xi ) =\sum _{k=1}^K\sum _{\ell =0}^\infty [{\tilde{V}^{(h)}_{f_{k,\ell }}}(t,\xi )+\epsilon _{k,\ell }(t,\xi )]\\&\,=\sum _{k=1}^K[\frac{1}{2}\sum _{\ell \in \mathbb {Z}} B_{k,\ell }(t)\hat{h}(\xi -\phi '_{k,\ell }(t))e^{i2\pi \phi _{k,\ell }(t)}+E^{(1)}_{k}(t,\xi )],\\&\,=\sum _{k=1}^K\big [\frac{1}{2}\sum _{\ell =-N_k}^{N_k} B_{k,\ell }(t)\hat{h}(\xi -\phi '_{k,\ell }(t))e^{i2\pi \phi _{k,\ell }(t)}+E^{(1)}_{k}(t,\xi )+E^{(2)}_{k}(t,\xi )\big ]. \end{aligned}$$

Denote

$$\begin{aligned} \epsilon _0(t,\xi ):=\sum _{k=1}^K[E^{(1)}_{k}(t,\xi )+E^{(2)}_{k}(t,\xi )], \end{aligned}$$

(62)

which is of order $\epsilon$ and $|\epsilon _0(t,\xi )|$ decays at the rate of $|\xi |^{-1}$ as $|\xi |\rightarrow \infty$. We thus have the proof. $\square$

Lemma 7.3

Fix $\epsilon {>0}$ and $d>0$. Take $f(t)=\sum _{k=1}^Kf_k(t)\in \mathcal {D}_{\epsilon ,d}$. Fix a window function $h\in \mathcal {S}$. For each $t\in \mathbb {R}$ and $\xi \in \mathbb {R}$, we have

$$\begin{aligned}&\sum _{k=1}^K\sum _{\ell =-N_k}^{N_k} B_{k,\ell }(t)\hat{h}(\xi -\phi '_{k,\ell }(t))e^{i2\pi \phi _{k,\ell }(t)}\\ =\,&\sum _{k=1}^K\sum _{\ell =-N_k}^{N_k} B_{k,\ell }(t)\hat{h}(\xi -\ell \phi '_{k,1}(t))e^{i2\pi \phi _{k,\ell }(t)}+\epsilon _1(t,\xi ),\nonumber \end{aligned}$$

(63)

where $\epsilon _1(t,\xi )$ is defined in (67) satisfying

$$\begin{aligned} |\epsilon _1(t,\xi )|\le \epsilon 2\pi I_1 \sum _{k=1}^K \phi _{k,1}'(t)\sum _{\ell =-N_k}^{N_k} B_{k,\ell }(t)\chi _{\tilde{Z}_{k,\ell }}(\xi ), \end{aligned}$$

(64)

where $\tilde{Z}_{k,\ell }(t) := [(\ell - \epsilon ) \phi '_{k,1}(t) - \Delta , (\ell + \epsilon ) \phi '_{k,1} + \Delta ]$. Note that the support of $\epsilon _1(t,\xi )$ is inside $[-\max _{k}((N_k+\epsilon )\phi _{k,1}'(t))-\Delta ,\,\max _{k}((N_k+\epsilon )\phi _{k,1}'(t))+\Delta ]$. In particular, we have

$$\begin{aligned} V^{(h)}_f(t,\xi )= \frac{1}{2}\sum _{k=1}^K\sum _{\ell =-N_k}^{N_k} B_{k,\ell }(t)\hat{h}(\xi -\ell \phi '_{k,1}(t))e^{i2\pi \phi _{k,\ell }(t)}+\epsilon _2(t,\xi ), \end{aligned}$$

(65)

where $\epsilon _2(t,\xi )=\epsilon _0(t,\xi )+\epsilon _1(t,\xi )$, which is of order $\epsilon$ and $|\epsilon _2(t,\xi )|$ decays at the rate of $|\xi |^{-1}$ as $|\xi |\rightarrow \infty$.

Proof

The proof is straightforward by the smoothness assumption of h and Taylor’s expansion. Indeed, by the assumption that $\left| \frac{\phi '_{k,\ell }(t)}{\phi '_{k,1}(t)}-\ell \right| \le \epsilon$, we know that $|\phi '_{k,\ell }(t)-\ell \phi '_{k,1}(t)|\le \epsilon \phi _{k,1}'(t)$ for all $\ell =1,\ldots$. Thus, since $\hat{h}$ is compactly supported on $[-\Delta ,\Delta ]$, we have that for $\xi \in \tilde{Z}_{k,\ell }$,

$$\begin{aligned} |\hat{h}(\xi -\phi '_{k,\ell }(t))-\hat{h}(\xi -\ell \phi '_{k,1}(t))|\le \epsilon \phi _{k,1}'(t)\Vert \hat{h}'\Vert _{L^\infty }\le 2\pi \epsilon \phi _{k,1}'(t)I_1, \end{aligned}$$

(66)

where we use the bound $\Vert \hat{h}'\Vert _{L^\infty }\le 2\pi I_1$; for $\xi \notin \tilde{Z}_{k,\ell }$,

$$\begin{aligned} |\hat{h}(\xi -\phi '_{k,\ell }(t))-\hat{h}(\xi -\ell \phi '_{k,1}(t))|=0. \end{aligned}$$

Denote

$$\begin{aligned} \epsilon _1(t,\xi ):=\sum _{k=1}^K\sum _{\ell =-N_k}^{N_k} B_{k,\ell }(t)(\hat{h}(\xi -\phi '_{k,\ell }(t))-\hat{h}(\xi -\ell \phi '_{k,1}(t)))e^{i2\pi \phi _{k,\ell }(t)}. \end{aligned}$$

(67)

By a direct bound, we have

$$\begin{aligned} |\epsilon _1(t,\xi )|=&\left| \sum _{k=1}^K\sum _{\ell =-N_k}^{N_k} B_{k,\ell }(t)(\hat{h}(\xi -\phi '_{k,\ell }(t))-\hat{h}(\xi -\ell \phi '_{k,1}(t)))e^{i2\pi \phi _{k,\ell }(t)}\right| \\ \le&\, \sum _{k=1}^K\sum _{\ell =-N_k}^{N_k} B_{k,\ell }(t)\left| \hat{h}(\xi -\phi '_{k,\ell }(t))-\hat{h}(\xi -\ell \phi '_{k,1}(t))\right| \nonumber \\ \le \,&\epsilon 2\pi I_1 \sum _{k=1}^K \phi _{k,1}'(t)\sum _{\ell =-N_k}^{N_k} B_{k,\ell }(t)\chi _{\tilde{Z}_{k,\ell }},\nonumber \end{aligned}$$

(68)

which leads to the claim. The proof of (65) comes from a direct combination of (50) and (63). $\square$

By the assumption that $0<\Delta \le \phi _{1,1}'(t)/4$, we know that for a fixed $k\in \{1,\ldots ,K\}$, $Z_{k,i}(t)\cap Z_{k,j}(t)=\emptyset$ for all $i\ne j$, where $Z_{k,\ell }$ is defined in (34). Thus, when $K=1$, we know that for any $\gamma >0$, the $\gamma$ power of the absolute value of the major term in (65) becomes

$$\begin{aligned} \left| \sum _{\ell =-N_1}^{N_1} B_{1,\ell }(t)\hat{h}(\xi -\phi '_{1,\ell }(t))e^{i2\pi \phi _{1,\ell }(t)}\right| ^\gamma =\sum _{\ell =-N_1}^{N_1} B^\gamma _{1,\ell }(t)|\hat{h}(\xi -\phi '_{1,\ell }(t))|^\gamma \end{aligned}$$

since the supports of $\hat{h}(\xi -\phi '_{1,i}(t))$ and $\hat{h}(\xi -\phi '_{1,j}(t))$ do not overlap, when $i\ne j$. However, when $K>1$, although $Z_{k,1}(t)\cap Z_{\ell ,1}(t)=\emptyset$ when $k\ne \ell$ since $\Delta <d/4$, there is no guarantee that $Z_{k,i}(t)\cap Z_{\ell ,j}(t)=\emptyset$ when $k\ne \ell$ and $i\ne j$. So, when $K>1$, we need to be careful when we take the power.

Definition 7.4

Fix $\epsilon { >0}$ and $d>0$. Take $f(t)=\sum _{k=1}^Kf_k(t)\in \mathcal {D}_{\epsilon ,d}$. Define ${S_1(t)}=\emptyset$, and for each $k\in \{{2},\ldots ,K\}$, define

$$\begin{aligned} S_{k}(t):= & {} \{i,-i|\, 1\le i\le N_k,\,Z_{k,i}(t)\nonumber \\ {}&\cap&Z_{\ell ,j}(t)\ne \emptyset ,\,j\in { \{1,\ldots ,N_\ell \}}\backslash S_\ell (t),\,\ell =1,\ldots ,k-1 \} { \cup \{0\}}. \end{aligned}$$

(69)

Furthermore, define

$$\begin{aligned} Y_{\text {no-OL}}(t)&:=\cup _{k=1}^K\cup _{i\in { \{0, \pm 1,\ldots ,\pm N_k\}} \backslash S_k}{Z}_{k,i}(t)\subset \mathbb {R}\\ Y_{\text {with-OL}}(t)&:=\cup _{k=1}^K\cup _{i\in S_k} {Z}_{k,i}(t)\subset \mathbb {R}.\nonumber \end{aligned}$$

(70)

The set $S_k(t)$ indicates the multiples of the kth ANH function that have the danger of overlapping with the other ANH functions. To be more precise, for $k\in \{2,\ldots ,K\}$ and $\ell \in \{1,\ldots ,k-1\}$, the supports of $\hat{h}(\xi -i\phi _{k}'(t))$ and $\hat{h}(\xi -j \phi _{\ell }'(t))$, where $i\in {\{0,\pm 1,\ldots ,\pm N_k\}}\backslash S_k$ and $j\in { \{0,\pm 1,\ldots ,\pm N_\ell \}} \backslash S_\ell$ do not overlap. The sets $Y_{\text {no-OL}}(t)$ and $Y_{\text {with-OL}}(t)$ are used to control the overlapping of multiples associated with different ANH components. Note that the supports of all summands in $\sum _{k=1}^K\sum _{\ell \in \{{0,}\pm 1,\ldots ,\pm N_k\}\backslash S_k}B_{k,\ell }(t)|\hat{h}(\xi -\ell \phi '_{k,1}(t))|$ do not overlap.

To evaluate $|V^{(h)}_f(t,\xi )|^\gamma$, we need the following bounds to control the influence of taking the $\gamma$ power.

Lemma 7.5

Suppose $x\ge y\ge 0$. For $0<\gamma \le 1$, we have

$$\begin{aligned} (x+y)^\gamma \le x^\gamma +\gamma y^\gamma . \end{aligned}$$

(71)

Proof

When $x=y=0$, this is the trivial case. Suppose $x\ge y> 0$ or $x>y\ge 0$. By Taylor’s expansion, we have

$$\begin{aligned} (x+y)^\gamma =x^\gamma (1+\frac{y}{x})^\gamma \le x^\gamma +\gamma \frac{y}{x}x^\gamma =x^\gamma +\gamma \big (\frac{y}{x}\big )^{1-\gamma }y^\gamma . \end{aligned}$$

(72)

Since $y/x\le 1$, we obtain the bound. $\square$

Lemma 7.6

Suppose Assumption 3.2 holds and take $0<\gamma \le 1$. Then we have

$$\begin{aligned} |V^{(h)}_f(t,\xi )|^\gamma&\,=\frac{1}{2^\gamma }\sum _{k=1}^K\sum _{\ell =-N_k}^{N_k}B^\gamma _{k,\ell }(t)|\hat{h}(\xi -\ell \phi '_{k,1}(t))|^\gamma +\delta _3(t,\xi )+{\epsilon _3}(t,\xi ), \end{aligned}$$

(73)

where $\delta _3(t,\xi )$ is defined in (74) and $\epsilon _3(t,\xi )$ is defined in (75). Moreover, $\delta _3(t,\xi )=0$ when $K=1$. When $K>1$, $\delta _3(t,\xi )$ is supported on $Y_{\text {with-OL}}(t)$ and is bounded by $\frac{I_0^\gamma }{2^\gamma }\sum _{k=2}^K\sum _{\ell \in S_k}B^\gamma _{k,\ell }(t)\chi _{Z_{k,\ell }}(\xi )$. $\epsilon _3(t,\xi )$ satisfies $|{\epsilon _3}(t,\xi )|\le |\epsilon _{2}(t,\xi )|^\gamma$.

Proof

Let $\delta _3(t,\xi )$ and ${\epsilon _3}(t,\xi )$ be defined as

$$\begin{aligned} \delta _3(t,\xi ):= & {} \Big |\frac{1}{2} \sum _{k=1}^K \sum _{\ell = -N_k}^{N_k} B_{k,\ell }(t) \hat{h}(\xi - \ell \phi '_{k,1}(t))e^{i2\pi \phi _{k,\ell }(t)}\Big |^{\gamma } \nonumber \\&- \frac{1}{2^{\gamma }} \sum _{k=1}^K \sum _{\ell = -N_k}^{N_k} B_{k,\ell }^{\gamma }(t) |\hat{h}(\xi - \ell \phi '_{k,1}(t))|^{\gamma } \end{aligned}$$

(74)

and

$$\begin{aligned} {\epsilon _3}(t,\xi ) := |V^{(h)}_f(t,\xi )|^\gamma -\Big |\frac{1}{2} \sum _{k=1}^K \sum _{\ell = -N_k}^{N_k} B_{k,\ell }(t) \hat{h}(\xi - \ell \phi '_{k,1}(t))e^{i2\pi \phi _{k,\ell }(t)}\Big |^{\gamma }. \end{aligned}$$

(75)

That is,

$$\begin{aligned} |V^{(h)}_f(t,\xi )|^\gamma = \frac{1}{2^{\gamma }} \sum _{k=1}^K \sum _{\ell = -N_k}^{N_k} B_{k,\ell }^{\gamma }(t) |\hat{h}(\xi - \ell \phi '_{k,1}(t))|^{\gamma } + \delta _3(t,\xi ) + {\epsilon _3}(t,\xi ).\nonumber \\ \end{aligned}$$

(76)

According to Lemmas 7.5 and 7.3, when $\epsilon$ is small enough, by the triangular inequality that $\big ||V^{(h)}_f(t,\xi )|- |\frac{1}{2} \sum _{k=1}^K \sum _{\ell = -N_k}^{N_k} B_{k,\ell }(t) \hat{h}(\xi - \ell \phi '_{k,1}(t))e^{i2\pi \phi _{k,\ell }(t)}|\big |\le |\epsilon _2(t,\xi )|$, we have

$$\begin{aligned} |{\epsilon _3}(t,\xi )| \le |\epsilon _2(t,\xi )|^{\gamma }. \end{aligned}$$

(77)

Note that when $\xi \in Y_{\text {no-OL}}(t)$, $\delta _3(t,\xi ) =0$ since the supports of all summands in $\sum _{k=1}^K\sum _{\ell =-N_k}^{N_k}B_{k,\ell }(t)|\hat{h}(\xi -\ell \phi '_{k,1}(t))|$ do not overlap for each $\xi \in Y_{\text {no-OL}}(t)$. Therefore, we have

$$\begin{aligned} \delta _3(t,\xi ) = \frac{1}{2^\gamma } \left( \Big |\sum _{k=2}^K\sum _{\ell \in S_k} B_{k,\ell }(t) \hat{h}(\xi - \ell \phi '_{k,1}(t))e^{i2\pi \phi _{k,\ell }(t)}\Big |^{\gamma }\right. \nonumber \\ - \left. \sum _{k=2}^K\sum _{\ell \in S_k}B_{k,\ell }^{\gamma }(t) |\hat{h}(\xi - \ell \phi '_{k,1}(t))|^{\gamma }\right) . \end{aligned}$$

(78)

Hence,

$$\begin{aligned} |\delta _3(t,\xi ) |&= \left| \Big |\frac{1}{2} \sum _{k=1}^K \sum _{\ell \in S_k(t)} B_{k,\ell }(t) \hat{h}(\xi - \ell \phi '_{k,1}(t))e^{i2\pi \phi _{k,\ell }(t)}\Big |^{\gamma }\right. \\&\left. \quad - \frac{1}{2^{\gamma }} \sum _{k=1}^K \sum _{\ell \in S_k(t)} B_{k,\ell }^{\gamma }(t) |\hat{h}(\xi - \ell \phi '_{k,1}(t))|^{\gamma } \right| \\&= \frac{1}{2^{\gamma }} \sum _{k=1}^K \sum _{\ell \in S_k(t)} B_{k,\ell }^{\gamma }(t) |\hat{h}(\xi - \ell \phi '_{k,1}(t))|^{\gamma }\\ {}&\quad - \left| \frac{1}{2} \sum _{k=1}^K \sum _{\ell \in S_k(t)} B_{k,\ell }(t) \hat{h}(\xi - \ell \phi '_{k,1}(t))e^{i2\pi \phi _{k,\ell }(t)}\right| ^{\gamma }, \end{aligned}$$

since $\big |\frac{1}{2} \sum _{k=1}^K \sum _{\ell \in S_k(t) } B_{k,\ell }(t) \hat{h}(\xi - \ell \phi '_{k,1}(t))e^{i2\pi \phi _{k,\ell }(t)}\big |^{\gamma } \le \frac{1}{2^{\gamma }} \sum _{k=1}^K \sum _{\ell \in S_k(t) } B_{k,\ell }^{\gamma }(t) |\hat{h}(\xi - \ell \phi '_{k,1}(t))|^{\gamma }$ by Lemma 7.5. Note that when $K=1$, $S_1(t) = \emptyset$. Putting these together, we have

$$\begin{aligned} |\delta _3(t,\xi )| \le \frac{1}{2^\gamma }\sum _{k=2}^K\sum _{\ell \in S_k}B^\gamma _{k,\ell }(t)\Vert \hat{h}\Vert ^\gamma _{L^\infty }\chi _{Z_{k,\ell }}(\xi )\le \frac{I^\gamma _0}{2^\gamma } \sum _{k=2}^K\sum _{\ell \in S_k}B^\gamma _{k,\ell }(t)\chi _{Z_{k,\ell }}(\xi )\nonumber \\ \end{aligned}$$

(79)

which completes the proof. $\square$

Before finishing the proof, we need to control the error introduced by $\delta _3(t,\xi )$ in Lemma 7.6 when $K\ge 2$. Note that $\delta _3(t,\xi )$ is supported on $Y_{\text {with-OL}}(t)$. We now control this set.

Lemma 7.7

Suppose Assumption 3.2 holds and $K>1$. For each $t\in \mathbb {R}$, we have for each $k\in \{2,\ldots ,K\}$ the following bound:

$$\begin{aligned} \frac{\#S_{k}(t)}{N_k}\le { \sum _{\ell =1}^{k-1}}\Big [\frac{4\Delta }{\phi '_{\ell ,1}(t)} +E^{(\ell )}(N_k)\Big ], \end{aligned}$$

(80)

where $\#S_{k}(t)$ is the cardinal number of the set $S_{k}(t)$ and $E^{(\ell )}(N_k)\ge 0$ is defined in (85). Clearly $\frac{\#S_{1}(t)}{N_1}=0$.

This Lemma gives a bound of the set $S_k(t)$, which indicates that only a small fraction of the multiples of the kth ANH function has the danger of overlapping with other ANH function.

Proof

Fix $k\in \{2,3,\ldots ,K\}$ and $\ell \in \{1,\ldots ,k-1\}$. Define a set

$$\begin{aligned} S_{k,\ell }(t):=\{m|\, { m\in \mathbb {N}\cup \{0\}},\,Z_{k,m}(t)\cap Z_{\ell ,j}\ne \emptyset ,\,j\in { \mathbb {N}\cup \{0\}}\}, \end{aligned}$$

(81)

which is the set of multiples of $\phi '_{k,1}(t)$ that overlap some multiples of $\phi '_{\ell ,1}(t)$. Clearly, $S_{k}(t) \subset \cup _{\ell =1}^{k-1}S_{k,\ell }(t)$ and $S_{k,\ell _1}(t)$ and $S_{k,\ell _2}(t)$ might overlap when $\ell _1\ne \ell _2$. Thus, $\#S_{k}(t)\le { \sum _{\ell =1}^{k-1}}\#S_{k,\ell }(t)$. To evaluate the cardinality of the set $S_{k,\ell }(t)$, denote a sequence $s_{k,\ell }(m)$, $m\in \mathbb {N}$, so that

$$\begin{aligned} s_{k,\ell }(m)=m\phi '_{k,1}(t)\,\, (\text {mod }\phi '_{\ell ,1}(t)). \end{aligned}$$

(82)

By the compactly supported assumption of $\hat{h}$, when $s_{k,\ell }(m)$ lands in

$$\begin{aligned} \mathcal {Z}_{k,\ell }:=[0,2\Delta ]\cup [\phi '_{\ell ,1}(t)-2\Delta ,\phi '_{\ell ,1}(t)), \end{aligned}$$

we know that $Z_{k,m}(t)\cap Z_{\ell ,j}\ne \emptyset$ for some j; that is,

$$\begin{aligned} S_{k,\ell }(t)=\{ {0} \le m \le N_k \, | \,s_{k,\ell }(m)\in \mathcal {Z}_{k,\ell } \}. \end{aligned}$$

When $\phi '_{k,1}(t)/\phi '_{\ell ,1}(t)$ is a rational number, that is, $\phi '_{k,1}(t)/\phi '_{\ell ,1}(t)=a/b$, where $a,b\in \mathbb {N}$ and are co-prime numbers, then the sequence $\{s_{k,\ell }(m)\}_{m\in \mathbb {N}}$ only lands on $\{0,\phi '_{\ell ,1}(t)/b,\ldots ,(b-1)\phi '_{\ell ,1}(t)/b\}$ uniformly on $[0,\phi '_{\ell ,1}(t))$ since the integer a has a multiplicative inverse modulo b; that is, there exists $n_0$ such that $an_0\,\, (\text {mod } b)=1$. Thus the claim holds with the worst bound

$$\begin{aligned} \frac{\#S_{k}(t)}{N_k}\le \sum _{\ell =1}^{k-1}\frac{4\Delta }{\phi '_{\ell ,1}(t)}. \end{aligned}$$

(83)

When $\phi '_{k,1}(t)/\phi '_{\ell ,1}(t)$ is an irrational number, the sequence $\{s_{k,\ell }(m)\}$ is equidistributed on $[0,\phi '_{\ell ,1}(t)]$ by Weyl’s criterion. We apply the following well-known Erdös–Turán inequality [41, Corollary 1.1] to bound $\frac{\#S_{k,\ell }(t)}{N_k}$:

$$\begin{aligned}&\Big | \frac{\#S_{k,\ell }(t)}{N_k} - \frac{4\Delta }{\phi '_{\ell ,1}(t)} \Big | \le \frac{1}{J+1} + \frac{3}{N_k}\sum _{n=1}^{J} \frac{1}{n} \left| \sum _{m={0}}^{N_k} e^{i2 \pi n s_{k,\ell }(m) } \right| \end{aligned}$$

(84)

for all positive J. Denote $E^{(\ell )}_J(N_k)$ to be the right hand side of (84). Then the best upper bound we could obtain from Erdös–Turán inequality is

$$\begin{aligned} E^{(\ell )}(N_k):=\min _{J\in \mathbb {N}}E^{(\ell )}_J(N_k), \end{aligned}$$

(85)

which goes to zero when $N_k\rightarrow \infty$; that is, when $N_k\rightarrow \infty$, the chance that $s_{k,\ell }(m)$ would land in $\mathcal {Z}_{k,\ell }$ is $\frac{4\Delta }{\phi '_{\ell ,1}(t)}$. Thus, in general we know that for the pair $(k,\ell )$, we have

$$\begin{aligned} \frac{\#S_{k,\ell }(t)}{N_k}\le \frac{4\Delta }{\phi '_{\ell ,1}(t)}+E^{(\ell )}(N_k) \end{aligned}$$

and hence

$$\begin{aligned} \#S_{k,\ell }(t)\le N_k\Big [\frac{4\Delta }{\phi '_{\ell ,1}(t)}+E^{(\ell )}(N_k)\Big ], \end{aligned}$$

(86)

which is the number of multiples of $\phi '_{k,1}(t)$ that are close to some multiples of $\phi '_{\ell ,1}(t)$. In conclusion, we have

$$\begin{aligned} \frac{\#S_{k}(t)}{N_k}\le \sum _{\ell =1}^{k-1}\Big [\frac{4\Delta }{\phi '_{\ell ,1}(t)} +E^{(\ell )}(N_k)\Big ]. \end{aligned}$$

(87)

$\square$

By putting the above Lemmas together, we can prove Theorem 3.4, which shows that the STCT does provide the necessary information for the fundamental IF of the ANH function, even when there are more than one component.

Proof of Theorem 3.4

Note that in general $|V^{(h)}_f(t,\cdot )|^\gamma$ is a tempered distribution, so we can define the Fourier transform in the distribution sense. Define a $\ell ^1$ sequence $b_k$, where $b_k(\ell )=B^\gamma _{k,\ell }(t)$ for all $\ell \in \{0,\ldots ,N_k\}$, $b_k(\ell )=0$ for all $\ell >N_k$, and $b_k(-\ell )=b_k(\ell )$ for all $\ell \in \mathbb {N}\cup \{0\}$. By a direct calculation, for $q>0$, we have

$$\begin{aligned}&\mathcal {F}(\sum _{\ell =-N_k}^{N_k}B^\gamma _{k,\ell }(t)\delta _{\ell \phi _{k,1}'(t)}\star |\hat{h}|^\gamma )(q)=\widehat{|\hat{h}|^\gamma }(q)\sum _{\ell =-N_k}^{N_k}b_k(\ell )e^{i2\pi \ell \phi _{k,1}'(t)q}\nonumber \\ {}&\quad =\widehat{|\hat{h}|^\gamma }(q)\sum _{\ell =-\infty }^{\infty }b_k(\ell )e^{i2\pi \ell \phi _{k,1}'(t)q}=\widehat{|\hat{h}|^\gamma }(q){\hat{b}_k(q)}, \end{aligned}$$

(88)

where $\hat{b}_k$ is the discrete-time Fourier transform of the $\ell ^1$ sequence $b_k$, which is a continuous and real.

For the term $\delta _3$, since $\delta _3(t,\cdot )$ is compactly supported, continuous by (78) and is bounded by (79), $\delta _3(t,\cdot )\in L^1$ and its Fourier transform could be well defined as a function. Since the support of $\delta _3$, which is determined by the overlapped multiples of different ANH functions, could not be controlled, we apply the Riemann-Lebesgue theorem to evaluate a simple bound:

$$\begin{aligned}&|\int \delta _3(t,\xi )e^{-i2\pi \xi q}d \xi |\le { \frac{I_0^{\gamma }}{2^{\gamma }}}\sum _{k=2}^K \sum _{\ell \in S_k(t)}B^\gamma _{k,\ell }(t)\int \chi _{{Z}_{k,\ell }}(\xi ) d\xi \nonumber \\ \le \,&{ 2\Delta I_0^{\gamma } \sum _{k=2}^K \sum _{\ell \in S_k(t)}B^\gamma _{k,\ell }(t) \le 2 \Delta I_0^{\gamma }\sum _{k=2}^K B^\gamma _{k,1}(t)\sum _{\ell \in S_k(t)}c^\gamma _k(\ell ) } \end{aligned}$$

(89)

since $|Z_{k,\ell }| =2\Delta$. To control $\sum _{\ell \in S_k(t)}c_k^\gamma (\ell )$, we apply the simple bound $c_k(\ell )\le \Vert c_k\Vert _{\ell ^\infty }$ for all $\ell =0,1,\ldots ,N_k$. This leads to

$$\begin{aligned} \sum _{\ell \in S_k(t)}c^\gamma _k(\ell )&\le \#S_{k}(t)\Vert c_k^\gamma \Vert _{\ell ^\infty }\le \Vert c_k^\gamma \Vert _{\ell ^\infty }N_k\sum _{\ell =1}^{k-1}\Big [\frac{4\Delta }{\phi '_{\ell ,1}(t)}+E^{(\ell )}(N_k)\Big ], \end{aligned}$$

where the last inequality holds by Lemma 7.7. Thus, the first term

$$\begin{aligned} E_1:=\mathcal {F}[\delta _3(t,\cdot )] \end{aligned}$$

(90)

is bounded by

$$\begin{aligned} |E_1|\le 2\Delta I_0^\gamma \sum _{k=2}^K B^\gamma _{k,1}(t)\Vert c_k^\gamma \Vert _{\ell ^\infty }N_k\sum _{\ell =1}^{k-1}\Big [\frac{4\Delta }{\phi '_{\ell ,1}(t)}+E^{(\ell )}(N_k)\Big ]. \end{aligned}$$

Note that $K=1$, since $\delta _3(t,\xi )=0$, we know that $E_1=0$ and the bound holds trivially.

The error term ${\epsilon _3}(t,\xi )$ is of order $\epsilon ^\gamma$ but in general it decays at the rate of $|\xi |^{-\gamma }$ as $|\xi |\rightarrow \infty$, so its Fourier transform is evaluated in the distribution sense. Denote $E_2:=\mathcal {F}[{\epsilon _3}(t,\cdot )]$. We have

$$\begin{aligned} |E_2(\psi )|=\big |\int {\epsilon _3}(t,\xi ) \hat{\psi }(\xi )d\xi \big |\le \Vert {\epsilon _3}(t,\cdot )\Vert _{L^\infty } \Vert \hat{\psi }\Vert _{L^1} \end{aligned}$$

(91)

for all $\psi \in \mathcal {S}$. We have thus obtained the claim. $\square$

Remark 1

Note that the bound for $E_1$, which is the Fourier transform of $\delta _3$, is the worst bound, since we could not control the locations of the overlaps between those multiples of different ANH components in the STFT. The problem we encounter could be simplified to the following analytic number theory problem: given an irrational number $\alpha$. Denote $\beta _n=n\alpha -[n\alpha ]$, where $n\in \mathbb {N}\cup \{0\}$ and [x] means the integer part of x. Denote the set $I=\{n,-n| n\in \mathbb {N}\cup \{0\},\,0\le \beta _n<\zeta \}\cup \{n,-n|\beta _n>1-\zeta \}$, where $\zeta >0$ is a small number. Then, what is the spectral distribution of $\sum _{n\in I}\delta _n\star g$, where g is a smooth and compact function supported on $[-\zeta /2,\zeta /2]$?

Proof of Corollary 3.5

By (37), $b_k(\ell )$ is non-zero for $\ell \in \{-N_k,\ldots ,0,\ldots ,N_k\}$. Thus $\hat{b}_k$ is a continuous, real, and periodic function with the period equal to $1/\phi _{k,1}'(t)$. By (57), (59), and (64), $\epsilon _2(t,\xi )$ is bounded by $Q\epsilon$, where

$$\begin{aligned} Q&:=\sum _{k=1}^K\Big [ \big (\Vert c_k\Vert _{\ell ^1}[\phi _{k,1}'(t)I_1+\frac{1}{2}\epsilon M_kI_2]\\&\quad +\pi C_k {\sqrt{\frac{1}{4}B^2_{k,0}(t)+\frac{1}{2}\sum _{\ell =1}^\infty B^2_{k,\ell }(t)}} (\phi _{k,1}'(t)I_2+\frac{1}{3}\epsilon M_kI_3) \big ) \\&\quad + I_0\sqrt{\frac{1}{4}B^2_{k,0}(t)+\frac{1}{2}\sum _{\ell =1}^\infty B^2_{k,\ell }(t)} + 2\pi I_1 \phi _{k,1}'(t)\sum _{\ell =-N_k}^{N_k} B_{k,\ell }(t)\chi _{\tilde{Z}_{k,\ell }}\Big ]. \end{aligned}$$

Thus, when $\sqrt{\frac{1}{4}B^2_{k,0}(t)+\frac{1}{2}\sum _{\ell =1}^\infty B^2_{k,\ell }(t)}$ is sufficiently large and $\epsilon$ is sufficiently small, $\frac{1}{2^{\gamma }} \sum _{k=1}^K \sum _{\ell = -N_k}^{N_k} B_{k,\ell }^{\gamma }(t) |\hat{h}(\xi - \ell \phi '_{k,1}(t))|^{\gamma }$ dominantes $|\epsilon _2(t,\xi )|^\gamma$, since $B^\gamma _{k,\ell }(t)>\epsilon ^{\gamma /2}\big (\frac{1}{4}B^2_{k,0}(t)+\frac{1}{2}\sum _{\ell =1}^\infty B^2_{k,\ell }(t)\big )^{\gamma /2}$ and $\epsilon _3(t,\xi )$ is bounded by $Q^\gamma \epsilon ^\gamma$. Moreover, when $\Delta N_k$ is sufficiently small, $\frac{1}{2^{\gamma }} \sum _{k=1}^K \sum _{\ell = -N_k}^{N_k} B_{k,\ell }^{\gamma }(t) |\hat{h}(\xi - \ell \phi '_{k,1}(t))|^{\gamma }$ also dominates $\delta _3(t,\xi )$, and hence we finish the proof. $\square$