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Approximating Sampled Sinusoids and Multiband Signals Using Multiband Modulated DPSS Dictionaries

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Abstract

Many signal processing problems—such as analysis, compression, denoising, and reconstruction—can be facilitated by expressing the signal as a linear combination of atoms from a well-chosen dictionary. In this paper, we study possible dictionaries for representing the discrete vector one obtains when collecting a finite set of uniform samples from a multiband analog signal. By analyzing the spectrum of combined time- and multiband-limiting operations in the discrete-time domain, we conclude that the information level of the sampled multiband vectors is essentially equal to the time–frequency area. For representing these vectors, we consider a dictionary formed by concatenating a collection of modulated discrete prolate spheroidal sequences (DPSS’s). We study the angle between the subspaces spanned by this dictionary and an optimal dictionary, and we conclude that the multiband modulated DPSS dictionary—which is simple to construct and more flexible than the optimal dictionary in practical applications—is nearly optimal for representing multiband sample vectors. We also show that the multiband modulated DPSS dictionary not only provides a very high degree of approximation accuracy in an MSE sense for multiband sample vectors (using a number of atoms comparable to the information level), but also that it can provide high-quality approximations of all sampled sinusoids within the bands of interest.

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Notes

  1. By equivalent, we mean that \(\varvec{B}_{N,\mathbb {W}}\varvec{x}\) = \(\mathcal {I}_N(\mathcal {B}_\mathbb {W}(\mathcal {I}^*_N(\varvec{x})))\) for any \(\varvec{x}\in \mathbb {C}^N\).

  2. By “bands of interest,” we mean the union of intervals \(\mathbb {F}\) for continuous-time signals and \(\mathbb {W}\) for discrete-time signals. We assume these bands are known and are used to construct the multiband modulated DPSS dictionary. The results in this paper, however, can also have application in the problem of detecting the active bands from a set of possible candidates, as was studied in [13].

  3. For convenience, we use \(\varvec{B}_{N,W}\) instead of \(\varvec{B}_{N,[-W,W]}\) to denote the matrix which is equivalent to the operator \(\mathcal {I}_N\mathcal {B}_{[-W,W]}\mathcal {I}^*_N\). This is also the reason that we use \(\lambda _{N,W}\), \(s_{N,W}\) and \(\varvec{s}_{N,W}\) (which will be defined later) instead of \(\lambda _{N,[-W,W]}\), \(s_{N,[-W,W]}\) and \(\varvec{s}_{N,[-W,W]}\).

  4. Though a small \(\epsilon \) may require N large enough such that our results hold, \(\frac{\sum _i \lfloor 2NW_i(1-\epsilon )\rfloor }{\sum _i \lceil 2NW_i(1+\epsilon )\rceil }\) (the ratio between the sizes of the two dictionaries) may become close to 1.

  5. Note that \(\varvec{X}\) has the eigen-decomposition \(\varvec{X} = \varvec{V}\varvec{D}\varvec{V}^H\) where \(\varvec{V}\) is an orthonormal matrix and \(\varvec{D}\) is a diagonal matrix whose diagonal elements are non-negative, giving the square root \(\varvec{X}^{1/2} = \varvec{V} \varvec{D}^{1/2}\varvec{V}^H\).

  6. This can be verified as \(3N C_5(\mathbb {W},\epsilon )e^{-\frac{\widetilde{C}_2(\mathbb {W},\epsilon )N}{2}} = 3 C_5(\mathbb {W},\epsilon )e^{-N(\frac{\widetilde{C}_2(\mathbb {W},\epsilon )}{2}-\frac{\log N}{N})}\le 3 C_5(\mathbb {W},\epsilon )e^{-N\frac{\widetilde{C}_2\mathbb {W},\epsilon )}{4}}\le 1\) for all \(N\ge \max \{(\frac{4}{C_2(\mathbb {W},\epsilon )})^2,~\frac{4}{C_2(\mathbb {W},\epsilon )}\log (3C_5(\mathbb {W},\epsilon ))\}\). Here the first inequality follows because \(\frac{\log N}{N}\le \frac{1}{N^{1/2}}\le \frac{C_2(\mathbb {W},\epsilon )}{4}\) for all \(N\ge (\frac{4}{C_2(\mathbb {W},\epsilon )})^2\).

  7. Hogan and Lakey [23] considered the scaled and shifted Prolate Spheroidal Wave Fuctions (PSWF’s) and provided conditions on a shift parameter such that the scaled and shifted PSWF’s form a frame or a Riesz basis for the Paley–Wiener space.

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Acknowledgments

We gratefully acknowledge Mark Davenport, Armin Eftekhari, and Justin Romberg for valuable discussions and insightful comments; and the anonymous reviewers for their constructive comments.

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Authors and Affiliations

Authors

Corresponding author

Correspondence to Michael B. Wakin.

Additional information

Communicated by Hans G. Feichtinger.

This work was supported by NSF grant CCF-1409261.

Appendices

Appendix 1: Proof of Lemma 3.1

Proof

Let \(\varvec{y}\in \mathbb {C}^N,\varvec{y}\ne \varvec{0}\) be an arbitrary vector. Then

$$\begin{aligned}&\langle \mathcal {I}_N(\mathcal {B}_{\mathbb {W}}(\mathcal {I}^*_N(\varvec{y}))),\varvec{y}\rangle \\&\quad =\sum _{m=0}^{N-1}\mathcal {I}_N(\mathcal {B}_{\mathbb {W}}(\mathcal {I}^*_N(\varvec{y})))[m]\overline{\varvec{y}}[m]=\sum _{m=0}^{N-1}\left( \sum _{n=0}^{N-1}\int _{\mathbb {W}}e^{j2\pi f(m-n)}df\varvec{y}[n]\right) \overline{\varvec{y}}[m]\\&\quad =\int _{\mathbb {W}}\left( \sum _{m=0}^{N-1}e^{j2\pi f m}\overline{\varvec{y}}[m]\right) \left( \sum _{n=0}^{N-1}e^{-j2\pi f n}\varvec{y}[n]\right) df\\&\quad =\int _{\mathbb {W}}\left| \sum _{n=0}^{N-1}\varvec{y}[n]e^{-j2\pi f n}\right| ^2df>0, \end{aligned}$$

where \(\overline{\varvec{y}}\) is the complex-conjugate of the vector \(\varvec{y}\), \(\sum _{n=0}^{N-1}\varvec{y}[n]e^{-j2\pi f n}\) is the DTFT of \(\mathcal {I}^*_N(\varvec{y})\), and the last inequality is derived from the fact that compactly supported signals cannot have perfectly flat magnitude response.

By Parsevel’s Theorem, we know \(\int _{-1/2}^{1/2}|\sum _{n=0}^{N-1}\varvec{y}[n]e^{-j2\pi f n}|^2d f=||\varvec{y}||_2^2\). Therefore

$$\begin{aligned} \langle \mathcal {I}_N(\mathcal {B}_{\mathbb {W}}(\mathcal {I}^*_N(\varvec{y}))),\varvec{y}\rangle =\int _{\mathbb {W}}\left| \sum _{n=0}^{N-1}\varvec{y}[n]e^{-j2\pi f n}\right| ^2d f<||\varvec{y}||_2^2. \end{aligned}$$

Thus, we have

$$\begin{aligned} 0<\min _{\varvec{y}\in \mathbb {C}^N} \frac{\langle \mathcal {I}_N(\mathcal {B}_{\mathbb {W}}(\mathcal {I}^*_N(\varvec{y}))),\varvec{y}\rangle }{||\varvec{y}||_2^2}\le \lambda _{N,\mathbb {W}}^{(l)}\le \max _{\varvec{y}\in \mathbb {C}^N} \frac{\langle \mathcal {I}_N(\mathcal {B}_{\mathbb {W}}(\mathcal {I}^*_N(\varvec{y}))),\varvec{y}\rangle }{||\varvec{y}||_2^2}<1 \end{aligned}$$

for all \(l\in [N]\).

By noting that \(\mathcal {I}_N\mathcal {B}_{\mathbb {W}}\mathcal {I}^*_N\) is equivalent to \(\varvec{B}_{N,\mathbb {W}}\), we have

$$\begin{aligned} \sum _{l=0}^{N-1}\lambda _{N,\mathbb {W}}^{(l)} = \text {trace}(\varvec{B}_{N,\mathbb {W}})=\sum _{n=0}^{N-1}\varvec{B}_{N,\mathbb {W}}[n,n]=\sum _{n=0}^{N-1}\int _{\mathbb {W}}e^{j2\pi f 0}df=N|\mathbb {W}|. ~~~ \end{aligned}$$

\(\square \)

Appendix 2: Proof of Theorem 3.2

Proof

First we state a useful inequality about the Frobenius norm of positive semi-definite matrices. Suppose \(\varvec{X}\in \mathbb {C}^{N\times N}\) and \(\varvec{Y}\in \mathbb {C}^{N\times N}\) are two arbitrary positive semi-definite matrices. Then

$$\begin{aligned} ||\varvec{X}+\varvec{Y}||_F^2= & {} trace {\left( (\varvec{X}+\varvec{Y})^H(\varvec{X}+\varvec{Y})\right) }\\= & {} ||\varvec{X}||_F^2+||\varvec{Y}||_F^2 +2 trace {(\varvec{X}^H\varvec{Y})}\\&\ge ||\varvec{X}||_F^2+||\varvec{Y}||_F^2, \end{aligned}$$

where the last inequality is derived from the fact that \(trace (\varvec{X}^H\varvec{Y})\) is nonnegative, which can be showed as follows. By the hypothesis that \(\varvec{X}\) and \(\varvec{Y}\) are positive semi-definite matrices, we have the factorization \(\varvec{X}^H = \varvec{X} = \varvec{X}^{1/2}\varvec{X}^{1/2}\), where \(\varvec{X}^{1/2}\) is also a positive semi-definite matrix.Footnote 5 Then we conclude that \(\text {trace}{(\varvec{X}^H\varvec{Y})} = \text {trace}{(\varvec{X}^{1/2}\varvec{X}^{1/2}\varvec{Y})} = \text {trace}{(\varvec{X}^{1/2}\varvec{Y}\varvec{X}^{1/2})}\ge 0\), since \(\varvec{X}^{1/2}\varvec{Y}\varvec{X}^{1/2}\) is also a positive semi-definite matrix.

We next bound the Frobenius norm of \(\varvec{B}_{N,W_i}\) by

$$\begin{aligned}&||\varvec{B}_{N,W_i}||_F^2 = N(2W_i)^2+\underset{m\ne n}{\sum \sum }\left( \frac{\sin \left( 2\pi W_i(m-n)\right) }{\pi (m-n)}\right) ^2\\&\quad =4NW_i^2+2\sum _{n=1}^{N-1}(N-n)\left( \frac{\sin \left( 2\pi W_in\right) }{\pi n}\right) ^2\\&\quad =4NW_i^2+2N\sum _{n=1}^{N-1}\left( \frac{\sin \left( 2\pi W_in\right) }{\pi n}\right) ^2-2\sum _{n=1}^{N-1}n\left( \frac{\sin \left( 2\pi W_in\right) }{\pi n}\right) ^2\\&\quad = 4NW_i^2{+}2N\left( W_i-2W_i^2{-}\sum _{n=N}^{\infty }\left( \frac{\sin \left( 2\pi W_in\right) }{\pi n}\right) ^2\right) \!-\!2\sum _{n=1}^{N-1}n\left( \frac{\sin \left( 2\pi W_in\right) }{\pi n}\right) ^2\\&\quad \ge 4NW_i^2+2N\left( W_i-2W_i^2-\frac{1}{\pi ^2}\int _{N-1}^\infty \frac{1}{x^2}dx\right) -2\frac{1}{\pi ^2}\left( \int _{1}^{N-1}\frac{1}{x}dx+1\right) \\&\quad =2NW_i-\frac{2}{\pi ^2}\frac{2N-1}{N-1}-\frac{2}{\pi ^2}\log (N-1), \end{aligned}$$

where the fourth line follows from Parseval’s theorem \(\sum _{n=-\infty }^{\infty }\left( \frac{\sin \left( 2\pi W_in\right) }{\pi n}\right) ^2 = \int _{-W_i}^{W_i}df = 2W_i\), which indicates that \(\sum _{n=1}^{\infty }\left( \frac{\sin \left( 2\pi W_in\right) }{\pi n}\right) ^2 = W_i-2W_i^2\).

Now applying the above results yields

$$\begin{aligned} ||\varvec{B}_{N,\mathbb {W}}||_F^2= & {} ||\sum _{i=0}^{J-1}\varvec{E}_{f_i}\varvec{B}_{N,W_i}\varvec{E}_{f_i}^H||_F^2\\\ge & {} \sum _{i=0}^{J-1}||\varvec{B}_{N,W_i}||_F^2\\\ge & {} \sum _{i=0}^{J-1}\left( 2NW_i-\frac{2}{\pi ^2}\frac{2N-1}{N-1}-\frac{2}{\pi ^2}\log (N-1)\right) \\= & {} N|\mathbb {W}|-J\left( \frac{2}{\pi ^2}\frac{2N-1}{N-1}+\frac{2}{\pi ^2} \log (N-1)\right) , \end{aligned}$$

where the second line follows since \(\varvec{E}_{f_i}\varvec{B}_{N,W_i}\varvec{E}_{f_i}^H\) is positive semi-definite. Recalling the result stated in Lemma 3.1 that \(\sum _{l=0}^{N-1}\lambda _{N,\mathbb {W}}^{(l)} =\text {trace}(\varvec{B}_{N,\mathbb {W}}) = N|\mathbb {W}|\), we get

$$\begin{aligned}&\sum _{l=0}^{N-1}\lambda _{N,\mathbb {W}}^{(l)}(1-\lambda _{N,\mathbb {W}}^{(l)})\\&\quad = \text {trace}(\varvec{B}_{N,\mathbb {W}}) - ||\varvec{B}_{N,\mathbb {W}}||_F^2\le J\left( \frac{2}{\pi ^2}\frac{2N-1}{N-1}+\frac{2}{\pi ^2}\log (N-1)\right) . \end{aligned}$$

Thus, equation (13) follows by noting that for any \(\varepsilon \in (0,\frac{1}{2})\) one has

$$\begin{aligned}&\sum _{l=0}^{N-1}\lambda _{N,\mathbb {W}}^{(l)}(1-\lambda _{N,\mathbb {W}}^{(l)})\\&\quad \ge \sum _{\{l:\varepsilon \le \lambda _{N,\mathbb {W}}^{(l)}\le 1-\varepsilon \}}\lambda _{N,\mathbb {W}}^{(l)}(1-\lambda _{N,\mathbb {W}}^{(l)})\ge \varepsilon (1-\varepsilon )\#\{l:\varepsilon \le \lambda _{N,\mathbb {W}}^{(l)}\le 1-\varepsilon \}. \end{aligned}$$

\(\square \)

Appendix 3: Proof of Theorem 3.3

Proof

A precise proof of a similar result for time- and band-limiting operators in the continuous domain was first given in [28]. Izu and Lakey [25] extend the result to multiple intervals in the frequency domain or time domain. Their work forms the foundation of the following analysis.

As we have noted, the two operators \(\mathcal {T}_N\mathcal {B}_{\mathbb {W}}\mathcal {T}_N\) and \(\mathcal {I}_N\mathcal {B}_{\mathbb {W}}\mathcal {I}^*_N\) have the same eigenvalues. We work with \(\mathcal {T}_N\mathcal {B}_{\mathbb {W}}\mathcal {T}_N\) to prove Theorem 3.3. For convenience, we also use \(\lambda _{N,\mathbb {W}}^{(0)},\lambda _{N,\mathbb {W}}^{(1)},\ldots ,\lambda _{N,\mathbb {W}}^{(N-1)}\) to denote the decreasing eigenvalues for the operator \(\mathcal {T}_N\mathcal {B}_{\mathbb {W}}\mathcal {T}_N\). We let S([N]) denote the subspace of all finite-energy sequences supported only on the index set [N], that is

$$\begin{aligned} S([N]) = \{y: y\in \ell _2(\mathbb {Z}), \mathcal {T}_N(y) = y\}. \end{aligned}$$

First, for all integers \(l\in [N]\), the Weyl–Courant minimax representation of the eigenvalues can be stated as

$$\begin{aligned} \lambda _{N,\mathbb {W}}^{(l)}= & {} \left\{ \begin{array}{c} \min _{S_l}\max _{y\in \ell _2(\mathbb {Z}),y\perp S_l}\frac{\langle \mathcal {T}_N(\mathcal {B}_{\mathbb {W}}(\mathcal {T}_N(y))),y\rangle }{\langle y,y\rangle }, \\ \max _{S_{l+1}}\min _{y\in \ell _2(\mathbb {Z}),y\in S_{l+1}}\frac{\langle \mathcal {T}_N(\mathcal {B}_{\mathbb {W}}(\mathcal {T}_N(y))), y\rangle }{\langle y,y\rangle }, \end{array} \right. \nonumber \\= & {} \left\{ \begin{array}{c} \min _{S_l}\max _{y\in S([N]),y\perp S_l}\frac{\langle \mathcal {T}_N(\mathcal {B}_{\mathbb {W}}(\mathcal {T}_N(y))),y\rangle }{\langle y,y\rangle },\nonumber \\ \max _{S_{l+1}}\min _{y\in S([N]),y\in S_{l+1}}\frac{\langle \mathcal {T}_N(\mathcal {B}_{\mathbb {W}}(\mathcal {T}_N(y))),y\rangle }{\langle y,y\rangle }, \end{array} \right. \nonumber \\= & {} \left\{ \begin{array}{c} \min _{S_l}\max _{y\in S([N]),y\perp S_l}\frac{\int _{\mathbb {W}}|\widetilde{y}(f)|^2df}{||y||_2^2},\\ \max _{S_{l+1}}\min _{y\in S([N]),y\in S_{l+1}}\frac{\int _{\mathbb {W}}|\widetilde{y}(f)|^2df}{||y||_2^2}, \end{array}\right. \end{aligned}$$
(23)

where \(S_l\) is an l-dimensional subspace of \(\ell _2(\mathbb {Z})\), and \(\widetilde{y}(f)\) is the DTFT of the sequence y. Noting that all the eigenvectors of \(\mathcal {T}_N\mathcal {B}_{\mathbb {W}}\mathcal {T}_N\) belong to S([N]), we restrict to \(y\in S([N])\) in the second line. \(\square \)

Lemma 4.1

Consider the bandlimited sequence \(g\in \ell _2(\mathbb {Z})\) whose DTFT is given by

$$\begin{aligned} \widetilde{g}(f)=\left\{ \begin{array}{cl} \sqrt{2N}\cos (N\pi f)e^{-j2\pi f\lfloor \frac{N}{2}\rfloor }, &{}\quad |f|\le \frac{1}{2N},\\ 0, &{}\quad \frac{1}{2N}<|f|\le \frac{1}{2}. \end{array} \right. \end{aligned}$$
(24)

Then \(||g||_2^2 = 1\) and \(g[n]\ge \frac{1}{\sqrt{2N}}\) for all \(n\in [N]\).

Proof (of Lemma 4.1)

First it is easy to check that \(||g||_2^2 = \int _{-\frac{1}{2}}^{\frac{1}{2}}|\widetilde{g}(f)|^2df = 1\). Then computing the inverse DTFT directly yields

$$\begin{aligned} g[n] = \frac{1}{\sqrt{2N}}\text {sinc}\left( \frac{n-\lfloor \frac{N}{2}\rfloor }{N}-\frac{1}{2}\right) + \frac{1}{\sqrt{2N}}\text {sinc}\left( \frac{n-\lfloor \frac{N}{2}\rfloor }{N}+\frac{1}{2}\right) . \end{aligned}$$

Let \(\xi (t) = \text {sinc}(t-\frac{1}{2})+\text {sinc}(t+\frac{1}{2})\). Taking the directive of \(\xi (t)\), we would find on \([-\frac{1}{2},\frac{1}{2}]\) that \(\xi (t)\) achieves its minimum value of 1 at the points \(t=\pm \frac{1}{2}\). Therefore, \(g[n]\ge \frac{1}{\sqrt{2N}}\) since \(|\frac{n-\lfloor \frac{N}{2}\rfloor }{N}|\le \frac{1}{2}\) for all \(n\in [N]\). \(\square \)

1.1 Upper Bound

From equation (23), we know that

$$\begin{aligned} \lambda _{N,\mathbb {W}}^{(l)}=\min _{S_l}\max _{y\in S([N]),y\perp S_l}\frac{\int _{\mathbb {W}}|\widetilde{y}(f)|^2df}{||y||_2^2}. \end{aligned}$$

Therefore, in order to bound the eigenvalues from above, it suffices to pick an appropriate l-dimensional subspace \(S_l\subset \ell _2(\mathbb {Z})\) and then find a uniform upper bound for the quantity above for all time-limited sequences \(y\in S([N])\) orthogonal to \(S_l\).

Consider the bandlimited sequence \(g\in \ell _2(\mathbb {Z})\) defined in (24). Let \(\mathcal {E}_{f_0}:\ell _2(\mathbb {Z})\rightarrow \ell _2(\mathbb {Z})\) denote a modulating operator with \(\mathcal {E}_{f_0}(y)[n]:=e^{j2\pi f_0n}y[n]\) for all \(n\in \mathbb {Z}\) and \(f_0\in [-\frac{1}{2},\frac{1}{2}]\). Set

$$\begin{aligned} L_+ = \left\{ n'\in \mathbb {Z}:-\left\lfloor \frac{N}{2}\right\rfloor \le n'\le \left\lfloor \frac{N-1}{2}\right\rfloor ,~\left( \frac{n'}{N}-\frac{1}{2N},\frac{n'}{N}+\frac{1}{2N}\right) \cap \mathbb {W}\ne \emptyset \right\} \end{aligned}$$

and hence \(\iota _+ = \#L_+\). Let \(S_{\iota _+}\) be the \(\iota _+\)-dimensional subspace of \(\ell _2(\mathbb {Z})\) spanned by the functions \(\mathcal {E}_{\frac{n'}{N}}g, n'\in L_+\), that is,

$$\begin{aligned} S_{\iota _+}:= \text {span}\left( \{\mathcal {E}_{\frac{n'}{N}}g\}_{ n'\in L_+}\right) . \end{aligned}$$

If the time-limited sequence \(y\in S([N])\) is orthogonal to \(S_{\iota _+}\), then

$$\begin{aligned} 0=\langle y,\mathcal {E}_{\frac{n'}{N}}g\rangle =\left\langle \widetilde{y},\widetilde{g}\left( \cdot -\frac{n'}{N}\right) \right\rangle =\left( \widetilde{y}\star \widetilde{\overline{g}}\right) \left( \frac{n'}{N}\right) =:g_y[n'], ~n'\in L_+, \end{aligned}$$

where \(\overline{g}: = g^*\) is the complex-conjugate of the sequence g and \(\widetilde{\overline{g}}\) is the DTFT of \(\overline{g}\).

Now it follows that

$$\begin{aligned} \sum _{n'=-\lfloor \frac{N}{2}\rfloor }^{\lfloor \frac{N-1}{2}\rfloor }|g_y[n']|^2&=\sum _{n'\in L_+^C}|g_y[n']|^2 \nonumber \\&=\sum _{n'\in L_+^C}\left| \int _{\frac{n'-1/2}{N}}^{\frac{n'+1/2}{N}}\widetilde{y}(f)\widetilde{\overline{g}}(\frac{n'}{N}-f)df\right| ^2\nonumber \\&\le \sum _{n'\in L_+^C}\left( ||g||_2^2\int _{\frac{n'-1/2}{N}}^{\frac{n'+1/2}{N}}|\widetilde{y}(f)|^2df\right) \nonumber \\&\le \int _{f\notin \mathbb {W}}|\widetilde{y}(f)|^2df \nonumber \\&=||y||_2^2-\int _{f\in \mathbb {W}}|\widetilde{y}(f)|^2df, \end{aligned}$$
(25)

where \(L_+^C\) is defined as \(L_+^C := \{n'\in \mathbb {Z}:-\lfloor \frac{N}{2}\rfloor \le n'\le \lfloor \frac{N-1}{2}\rfloor ,~n'\notin L_+\}\), the second line holds because g is bandlimited to \([-\frac{1}{2N},\frac{1}{2N}]\), the third line follows from the Cauchy–Schwarz inequality, and the fourth line holds because \(||g||_2=1\) and by construction, the set \(\cup _{n'\in L_+}[\frac{n'}{N}-\frac{1}{2N},\frac{n'}{N}+\frac{1}{2N}]\) covers the intervals \(\mathbb {W}\) completely. On the other hand, let \(y\odot \overline{g}\) denote the pointwise product between y and \(\overline{g}\), that is \((y\odot \overline{g})[n] = y[n]\overline{g}[n]\). Note that \(y\odot \overline{g}\) has the same support in time as y, namely [N], and \(\{\frac{1}{\sqrt{N}}{\varvec{e}}_{\frac{n'}{N}},-\lfloor \frac{N}{2}\rfloor \le n'\le \lfloor \frac{N-1}{2}\rfloor \}\) forms an orthobasis (normalized DFT basis) for \(\mathbb {C}^N\). We can rewrite \(g_y[n'] = \varvec{e}_{\frac{n'}{N}}^H\left( y\odot \overline{g}\right) \), which can be viewed as the DFT of \(y\odot \overline{g}\). Therefore, using Parseval’s theorem, we acquire

$$\begin{aligned} \sum _{n'=-\lfloor \frac{N}{2}\rfloor }^{\lfloor \frac{N-1}{2}\rfloor }|g_y[n']|^2 = N||y\odot \overline{g}||_2^2\ge \frac{1}{2}||y||_2^2 \end{aligned}$$

since by hypothesis, \(g[n]\ge \frac{1}{\sqrt{2N}}\) for all \(n\in [N]\). Now, combining the above lower bound on the energy of the sequence \(g_y\) and the upper bound in (25), we observe that

$$\begin{aligned} \frac{1}{2}||y||_2^2\le ||y||_2^2-\int _{f\in \mathbb {W}}|\widetilde{y}(f)|^2df, \end{aligned}$$

and therefore,

$$\begin{aligned} \lambda _{N,\mathbb {W}}^{(\iota _+)}\le \frac{\int _{\mathbb {W}}|\widetilde{y}(f)|^2df}{||y||_2^2}\le \frac{1}{2}. \end{aligned}$$

1.2 Lower Bound

In the other direction, consider the minimax representation

$$\begin{aligned} \lambda _{N,\mathbb {W}}^{(l)} = \max _{S_{l+1}}~\min _{y\in S([N]),y\in S_{l+1}}\frac{\int _{\mathbb {W}}|\widetilde{y}(f)|^2df}{||y||_2^2}. \end{aligned}$$

In order to find a lower bound for the eigenvalues, it suffices to pick an appropriate \((l+1)\)-dimensional subspace \(S_{l+1}\subset \ell _2(\mathbb {Z})\) and then find a uniform lower bound for the quantity above for all time-limited sequences \(y\in S([N])\) inside \(S_{l+1}\). With g as defined in (24), let the time-limited sequence \(h\in \ell _2([N])\) be such that \(h[n] = 1/\overline{g}[n]\) for all \(n\in [N]\). We set

$$\begin{aligned} L_- := \left\{ n'\in \mathbb {Z}:-\left\lfloor \frac{N}{2}\right\rfloor \le n'\le \left\lfloor \frac{N-1}{2}\right\rfloor ,~\left( \frac{n'}{N}-\frac{1}{2N},\frac{n'}{N}+\frac{1}{2N}\right) \subset \mathbb {W}\right\} , \end{aligned}$$

and hence \(\iota _- = \#L_-\). Let \(S_{\iota _-}\) be the \(\iota _-\)-dimensional subspace of \(\ell _2(\mathbb {Z})\) spanned by the functions \(\mathcal {E}_{\frac{n'}{N}}h, n'\in L_-\), that is,

$$\begin{aligned} S_{\iota _-}:= \text {span}\left( \{\mathcal {E}_{\frac{n'}{N}}h\}_{ n'\in L_-}\right) . \end{aligned}$$

Suppose \(y\in S_{\iota _-}\) (and hence \(y\in \ell _2([N])\)). Then we may write

$$\begin{aligned} y=\sum _{n'\in L_-}b_{n'}\mathcal {E}_{\frac{n'}{N}}h \end{aligned}$$

for some coefficients \(b_{n'}\). Moreover,

$$\begin{aligned} y\odot \overline{g} = \sum _{n'\in L_-}b_{n'}{\varvec{e}}_{\frac{n'}{N}}. \end{aligned}$$

Noting that \(\{\frac{1}{\sqrt{N}}{\varvec{e}}_{\frac{n'}{N}},-\lfloor \frac{N}{2}\rfloor \le n'\le \lfloor \frac{N-1}{2}\rfloor \}\) forms an orthobasis for \(\mathbb {C}^N\), we obtain

$$\begin{aligned} \sum _{n'\in L_-}|b_{n'}|^2 = N||y\odot \overline{g}||_2^2 = N\sum _{n=0}^{N-1}|y[n]\odot \overline{g}[n]|^2\ge \frac{1}{2} \sum _{n=0}^{N-1}|y[n]|^2=\frac{1}{2}||y||_2^2 \end{aligned}$$

since by definition, \(g[n]\ge \frac{1}{\sqrt{2N}}\) for all \(n\in [N]\). On the other hand,

$$\begin{aligned} b_{n'} = \sum _{n=0}^{N-1}\overline{g}[n]y[n]e^{-j2\pi \frac{n'}{N}n} = \langle y, \mathcal {E}_{\frac{n'}{N}} g\rangle . \end{aligned}$$

Now using the same procedure as in (25), one has

$$\begin{aligned} \sum _{n'\in L_-}|b_{n'}|^2= & {} \sum _{n'\in L_-}|\langle y, \mathcal {E}_{\frac{n'}{N}} g\rangle |^2 \\= & {} \sum _{n'\in L_-}|\int _{\frac{n'-1/2}{N}}^{\frac{n'+1/2}{N}} \widetilde{y}(f)\widetilde{\overline{g}}(\frac{n'}{N}-f)df|^2\\\le & {} \sum _{n'\in L_-}\left( ||g||_2^2\int _{\frac{n'-1/2}{N}}^{ \frac{n'+1/2}{N}}|\widetilde{y}(f)|^2df\right) \\\le & {} \int _{f\in \mathbb {W}}|\widetilde{y}(f)|^2df, \end{aligned}$$

where the last line holds since by construction, the set \(\cup _{n'\in L_i}[\frac{n'}{N}-\frac{1}{2N},\frac{n'}{N}+\frac{1}{2N}]\) is a subset of the intervals \(\mathbb {W}\). Altogether, we then conclude that for any \(y\in S_{\iota _-}\) (and hence \(y\in S([N])\)),

$$\begin{aligned}\frac{1}{2}||y||_2^2\le \int _{f\in \mathbb {W}}|\widetilde{y}(f)|^2df. \end{aligned}$$

And hence

$$\begin{aligned} \lambda _{N,\mathbb {W}}^{(\iota _--1)}\ge \frac{\int _{f\in \mathbb {W}}|\widetilde{y}(f)|^2df}{||y||_2^2}\ge \frac{1}{2}. \end{aligned}$$

\(\square \)

Appendix 4: Proof of Theorem 3.4

1.1 Proof of Eigenvalues that Cluster Near Zero

Proof

Since \(\varvec{B}_{N,\mathbb {W}}=\sum _{i=0}^{J-1}\varvec{E}_{f_i}\varvec{B}_{N,W_i}\varvec{E}_{f_i}^H\) , according to [24] (see pp. 181), the following holds

$$\begin{aligned} \lambda _{N,\mathbb {W}}^{(l)} \le \sum _{i=0}^{J-1} \lambda _{N,W_i}^{(l_i)} \end{aligned}$$

for all \(l_i\in [N], i \in [J]\) and \(l=\sum _{i=0}^{J-1} l_i\in [N]\).

Fix \(\epsilon \in (0,\frac{1}{|\mathbb {W}|}-1)\). For each \(i\in [J]\), let \(N_{1}(W_i,\epsilon )\), \(C_3(W_i,\epsilon )\) and \(C_4(W_i,\epsilon )\) be the constants specified in Lemma 2.3 with respect to \(W_i\) and \(\epsilon \). If we let \(\overline{N}_1(\mathbb {W},\epsilon ) = \max {\{N_{1}(W_i,\epsilon ), ~\forall ~ i \in [J]\}}\), then we have

$$\begin{aligned} \lambda _{N,W_i}^{(l_i)}\le C_{3}(W_i,\epsilon )e^{-C_{4}(W_i,\epsilon )N},\quad \forall ~ l_i\ge \lceil 2NW_i(1+\epsilon )\rceil , i\in [J] \end{aligned}$$

for all \(N \ge \overline{N}_1(\mathbb {W},\epsilon )\). Hence, by choosing \(l_i\ge \lceil 2NW_i(1+\epsilon )\rceil , ~\forall ~ i\in [J]\), we have

$$\begin{aligned} \lambda _{N,\mathbb {W}}^{(l)} \le \sum _{i=0}^{J-1} C_{3}(W_i,\epsilon )e^{-C_{4}(W_i,\epsilon )N} \le \overline{C}_{3}(\mathbb {W},\epsilon )e^{-\overline{C}_4(\mathbb {W},\epsilon )N}, \end{aligned}$$

for all \(N \ge \overline{N}_1(\mathbb {W},\epsilon )\) and \(l\ge \sum _i \lceil 2NW_i(1+\epsilon )\rceil \),where \(\overline{C}_3(\mathbb {W},\epsilon ) = J\max {\{C_{3}(W_i,\epsilon ), ~\forall ~ i \in [J]\}}\) and \(\overline{C}_4(\mathbb {W},\epsilon ) = \min {\{C_{4}(W_i,\epsilon ), ~\forall ~ i \in [J]\}}\). \(\square \)

1.2 \(\varepsilon \)-Pseudo Eigenvalue and Eigenvectors

Definition 4.2

(\(\varepsilon \)-pseudo eigenvalue and eigenvector [34]) Let \(\varvec{X}\in \mathbb {C}^{N\times N}\) be any matrix and \(\varvec{u}\in \mathbb {C}^N\) be any vector with unit \(l_2\)-norm. Given \(\varepsilon >0\), the number \(\lambda \in \mathbb {C}\) and vector \(\varvec{u}\in \mathbb {C}^N\) are an \(\varepsilon \)-pseudo eigenpair of \(\varvec{X}\) if the following condition is satisfied:

$$\begin{aligned} ||(\varvec{X}-\lambda \varvec{I})\varvec{u}||_2^2\le \varepsilon . \end{aligned}$$

Lemma 4.3

Suppose \(\mathbb {W}\) is a fixed finite union of J pairwise disjoint intervals as defined in (5). Fix \(\epsilon \in (0,1)\). For each \(i\in [J]\), let \(N_{0}(W_i,\epsilon )\) be the constant specified in Lemma 2.3 with respect to \(W_i\) and \(\epsilon \) and let \(\widetilde{N}_0(\mathbb {W},\epsilon ) = \max {\{N_{0}(W_i,\epsilon ), ~\forall ~ i \in [J]\}}\). Then for all \(l_i\le 2NW_i(1-\epsilon ), i \in [J]\) and \( N>\widetilde{N}_0(\mathbb {W},\epsilon )\), (\(\lambda _{N,W_i}^{(l_i)}\), \(\varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)}\)) is an \(\varepsilon \)-pseudo eigenpair of \(\mathcal {I}_N\mathcal {B}_{\mathbb {W}}\mathcal {I}^*_N\) with \(\varepsilon \le 2C_{1}(W_i,\epsilon )e^{-C_{2}(W_i,\epsilon )N}\), or in detail

$$\begin{aligned} \mathcal {I}_N(\mathcal {B}_{\mathbb {W}}(\mathcal {I}^*_N(\varvec{E}_{f_i} \varvec{s}_{N,W_i}^{(l_i)})))=\lambda _{N,W_i}^{(l_i)}\varvec{E}_{f_i} \varvec{s}_{N,W_i}^{(l_i)}+\varvec{o}_i^{(l_i)}, \end{aligned}$$

where \(\varvec{o}_i^{(l_i)} = \mathcal {I}_N(\mathcal {B}_{\mathbb {W}\setminus [f_i-W_i,f_i+W_i]}(\mathcal {I}^*_N(\varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)})))\) and \(||\varvec{o}_i^{(l_i)}||_2^2\le 2C_{1}(W_i,\epsilon )e^{-C_{2}(W_i,\epsilon )N}\). Here \(\mathbb {W}\setminus [f_i-W_i,f_i+W_i] = \bigcup \limits _{i'\ne i}[f_{i'}-W_{i'},f_{i'}+W_{i'}]\) means the set difference between \(\mathbb {W}\) and \([f_i-W_i,f_i+W_i]\), and \(C_{1}(W_i,\epsilon )\) and \(C_{2}(W_i,\epsilon )\) are the constants specified in Lemma 2.3 corresponding to \(W_i\) and \(\epsilon \) for all \(i\in [J]\).

Proof (of Lemma 4.3)

According to the definition of the operator \(\mathcal {I}_N\mathcal {B}_{\mathbb {W}}\mathcal {I}^*_N\),

$$\begin{aligned}&\left( \mathcal {I}_N(\mathcal {B}_{\mathbb {W}}(\mathcal {I}^*_N(\varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)})))\right) [m]\\&\quad =\sum _{n=0}^{N-1}\sum _{i'=0}^{J-1}e^{j2\pi f_{i'}(m-n)}\frac{\sin (2\pi W_{i'}(m-n))}{\pi (m-n)}e^{j2\pi f_in}\varvec{s}_{N,W_i}^{(l_i)}[n]\\&\quad =e^{j2\pi f_im}\lambda ^{(l_i)}_{N,W_i}\varvec{s}_{N,W_i}^{(l_i)}[m]+\sum _{n=0}^{N-1}\sum _{i'=0,i'\ne i}^{J-1} e^{j2\pi f_{i'}(m-n)}\\&\qquad \times \,\frac{\sin (2\pi W_{i'}(m-n))}{\pi (m-n)}e^{j2\pi f_in}\varvec{s}_{N,W_i}^{(l_i)}[n]\\&\quad =e^{j2\pi f_im}\lambda ^{(l_i)}_{N,W_i}\varvec{s}_{N,W_i}^{(l_i)}[m]+\mathcal {I}_N(\mathcal {B}_{\mathbb {W}\setminus [f_i-W_i,f_i+W_i]}(\mathcal {I}^*_N(\varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)})))[m]. \end{aligned}$$

In what follows, we will bound the energy of \(\varvec{o}_i^{(l_i)} = \mathcal {I}_N(\mathcal {B}_{\mathbb {W}\setminus [f_i-W_i,f_i+W_i]}(\mathcal {I}^*_N(\varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)})))\) as

$$\begin{aligned} ||\varvec{o}_i^{(l_i)}||_2^2= & {} ||\mathcal {I}_N(\mathcal {B}_{\mathbb {W}\setminus [f_i-W_i,f_i+W_i]}(\mathcal {I}^*_N(\varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)})))||_2^2\\\le & {} ||\mathcal {B}_{\mathbb {W}\setminus [f_i-W_i,f_i+W_i]}(\mathcal {I}^*_N(\varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)}))||_2^2\\\le & {} ||\mathcal {B}_{[-\frac{1}{2},\frac{1}{2}]\setminus [f_i-W_i,f_i+W_i]}(\mathcal {I}^*_N(\varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)}))||_2^2\\= & {} ||\varvec{s}_{N,W_i}^{(l_i)}||_2^2-||\mathcal {B}_{[f_i-W_i,f_i+W_i]}(\mathcal {I}^*_N(\varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)}))||_2^2\\\le & {} ||\varvec{s}_{N,W_i}^{(l_i)}||_2^2-||\mathcal {I}_N(\mathcal {B}_{[f_i-W_i,f_i+W_i]}(\mathcal {I}^*_N(\varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)})))||_2^2\\\le & {} 1-(\lambda ^{(l_i)}_{N,W_i})^2\le 1-(1-C_{1}(W_i,\epsilon )e^{-C_{2}(W_i,\epsilon )N})^2\\= & {} 2C_{1}(W_i,\epsilon )e^{-C_{2}(W_i,\epsilon )N}-(C_1(W_i,\epsilon )e^{-C_2(W_i,\epsilon )N})^2\\\le & {} 2C_{1}(W_i,\epsilon )e^{-C_{2}(W_i,\epsilon )N} \end{aligned}$$

for all \(l_i\le \lfloor 2NW_i(1-\epsilon )\rfloor , i\in [J]\) and \(N\ge \widetilde{N}_0(\mathbb {W},\epsilon )\). Here the second inequality in the sixth line follows simply from Lemma 2.3 since \(\widetilde{N}_0(\mathbb {W},\epsilon )\ge N_0(W_i,\epsilon )\).\(\square \)

Using this result, we now show the first \(\approx N|\mathbb {W}|\) eigenvalues of \(\mathcal {I}_N\mathcal {B}_{\mathbb {W}}\mathcal {I}^*_N\) are close to 1.

1.3 Proof of Eigenvalues that Cluster Near One

The main idea is to guarantee that the sum of the first \(\approx N\left| \mathbb {W}\right| \) eigenvalues is sufficiently close \(N|\mathbb {W}|\). Then we conclude that the first \(\approx N|\mathbb {W}|\) eigenvalues cluster near one by applying the fact that the eigenvalues are upper bounded by 1. First we state the following useful results.

Lemma 4.4

([13] Lemma 5.1) Fix \(\epsilon \in (0,1)\). Let \(k_i = \lfloor 2NW_i(1-\epsilon )\rfloor , ~\forall ~i \in [J]\), and let \(\varvec{\Psi }\) be the dictionary as defined in (12). Then for any pair of distinct columns \(\varvec{\psi }_1\) and \(\varvec{\psi }_2\) in \(\varvec{\Psi }\), we have

$$\begin{aligned} \left| \langle \varvec{\psi }_1, \varvec{\psi }_2\rangle \right| \le 3\sqrt{\widetilde{C}_1(\mathbb {W},\epsilon )}e^{-\frac{\widetilde{C}_2(\mathbb {W},\epsilon )}{2}N} \end{aligned}$$
(26)

and

$$\begin{aligned} \left\| \varvec{\Psi }^H\varvec{\Psi }\right\| _2 \le 1+3N \sqrt{\widetilde{C}_1(\mathbb {W},\epsilon )}e^{-\frac{\widetilde{C}_2(\mathbb {W},\epsilon )N}{2}} \end{aligned}$$

if \(N\ge \widetilde{N}_0(\mathbb {W},\epsilon )\), where \(\widetilde{C}_1(\mathbb {W},\epsilon ) = \max {\{C_{1}(W_i,\epsilon ), ~\forall ~ i \in [J]\}}\) and \(\widetilde{C}_2(\mathbb {W},\epsilon ) = \min {\{C_{2}(W_i,\epsilon ),~\forall ~i \in [J]\}}\). Here \(||\varvec{\Psi }^H\varvec{\Psi }||_2\) is the spectral norm (or largest singular value) of \(\varvec{\Psi }^H\varvec{\Psi }\).

Lemma 4.5

([24]) Let \(\varvec{X}\in \mathbb {C}^{N\times N}\) be a Hermitian matrix, and let \(\lambda _0(\varvec{X}),\lambda _1(\varvec{X}),\ldots ,\lambda _{N-1}(\varvec{X})\) be its eigenvalues arranged in decreasing order. Then,

$$\begin{aligned} \lambda _0(\varvec{X})+\lambda _1(\varvec{X})+\cdots +\lambda _{r-1}(\varvec{X}) = \max _{\varvec{U}\in \mathbb {C}^{N\times r},\varvec{U}^H\varvec{U}=\varvec{I}_r} trace (\varvec{U}^H\varvec{X}\varvec{U}), \end{aligned}$$

where \(\varvec{I}_r\) is the \(r\times r\) identity matrix with \(1\le r\le N\).

Based on this result, we propose the following generalized result concerning the sum of the first r eigenvalues.

Lemma 4.6

Let \(\varvec{X}\in \mathbb {C}^{N\times N}\) be a positive-semidefinite (PSD) matrix, and let \(\lambda _0(\varvec{X}),\lambda _1(\varvec{X}),\ldots , \lambda _{N-1}(\varvec{X})\) be its eigenvalues arranged in decreasing order. Then, for any matrix \(\varvec{M}\in \varvec{C}^{N\times r}, 1\le r\le N\), the following inequality holds

$$\begin{aligned} \lambda _0(\varvec{X})+\lambda _1(\varvec{X})+\cdots +\lambda _{r-1}(\varvec{X}) \ge trace (\varvec{M}^H\varvec{X}\varvec{M})/\Vert \varvec{M}^H\varvec{M}\Vert _2. \end{aligned}$$

Proof (of Lemma 4.6)

Let \(\sigma _0(\varvec{M}),\ldots ,\sigma _{r-1}(\varvec{M})\) denote the decreasing singular values of the matrix \(\varvec{M}\). Denote \(\varvec{M} = \varvec{U}_r\varvec{\Sigma }_r\varvec{V}^H_r\) as the truncated SVD of \(\varvec{M}\), where \(\varvec{\Sigma }_r\) is an \(r\times r\) diagonal matrix with \(\sigma _0(\varvec{M}),\ldots ,\sigma _{r-1}(\varvec{M})\) along its diagonal.

Now applying Lemma 4.5, we obtain

$$\begin{aligned} \sum _{l = 0}^{r-1} \lambda _l(\varvec{X})&\ge \text {trace}(\varvec{U}_r^H\varvec{X}\varvec{U}_r)\\&\ge \text {trace}(\varvec{\Sigma }_r\varvec{U}_r^H\varvec{X}\varvec{U}_r\varvec{\Sigma }_r)/(\sigma _0(\varvec{M}))^2\\&=\text {trace}(\varvec{V_r}\varvec{\Sigma _r}\varvec{U}_r^H\varvec{X}\varvec{U}_r\varvec{\Sigma }_r\varvec{V}^H_r)/\Vert \varvec{M}^H\varvec{M}\Vert _2\\&=\text {trace}(\varvec{M}^H\varvec{X}\varvec{M})/\Vert \varvec{M}^H\varvec{M}\Vert _2, \end{aligned}$$

where the first line follows directly from Lemma 4.5, the second line is obtained because \(\varvec{U}_r^H\varvec{X}\varvec{U}_r\) is PSD and hence its main diagonal elements are non-negative, and the third line follows because \(\varvec{V}_r\) is an orthobasis and \((\sigma _0(\varvec{M}))^2 = \Vert \varvec{M}^H\varvec{M}\Vert _2\). \(\square \)

We are now ready to prove the main part. Fix \(\epsilon \in (0,1)\). Let \(k_i = \lfloor 2NW_i(1-\epsilon )\rfloor , \forall i\in [J]\), and let \(\varvec{\Psi }\) be the dictionary as defined in (12). We have

$$\begin{aligned}&\sum _{l=0}^{J-1+\sum _i \lfloor 2NW_i(1-\epsilon )\rfloor }\lambda _{N,\mathbb {W}}^{(l)} \ge \text {trace}\left( \varvec{\Psi }^H\varvec{B}_{N,\mathbb {W}}\varvec{\Psi }\right) \Big /\left\| \varvec{\Psi }^H\varvec{\Psi }\right\| _2\\&= \left( \sum _{i=0}^{J-1}\sum _{l_i= 0}^{\lfloor 2NW_i(1-\epsilon )\rfloor }\left( (\varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)})^H\mathcal {I}_N(\mathcal {B}_{\mathbb {W}}(\mathcal {I}^*_N(\varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)})))\right) \right) \Big /\left\| \varvec{\Psi }^H\varvec{\Psi }\right\| _2\\&= \left( \sum _{i=0}^{J-1}\sum _{l_i= 0}^{\lfloor 2NW_i(1-\epsilon )\rfloor }\left( \left( \varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)}\right) ^H\left( \lambda _{N,W_i}^{(l_i)}\varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)}+\varvec{o}_i^{(l_i)}\right) \right) \right) \Big /\left\| \varvec{\Psi }^H\varvec{\Psi }\right\| _2\\&\ge \left( \sum _{i=0}^{J-1}\sum _{l_i= 0}^{\lfloor 2NW_i(1-\epsilon )\rfloor }\left( \lambda _{N,W_i}^{(l_i)}-\Vert \varvec{o}_{i}^{(l_i)}\Vert _2\right) \right) \Big /\left\| \varvec{\Psi }^H\varvec{\Psi }\right\| _2\\&\ge \frac{\left( \sum _{i=0}^{J-1}\sum _{l_i= 0}^{\lfloor 2NW_i(1-\epsilon )\rfloor }\left( 1-C_1(W_i,\epsilon )e^{-C_2(W_i,\epsilon )N}-\sqrt{2}\sqrt{C_1(W_i,\epsilon )}e^{-\frac{C_2(W_i,\epsilon )}{2}N}\right) \right) }{\left( 1+3N \sqrt{\widetilde{C}_1(\mathbb {W},\epsilon )}e^{-\frac{\widetilde{C}_2(\mathbb {W},\epsilon )N}{2}}\right) }\\&\ge \frac{\left( \sum _{i=0}^{J-1}\sum _{l_i= 0}^{\lfloor 2NW_i(1-\epsilon )\rfloor }\left( 1-\widetilde{C}_1(\mathbb {W},\epsilon )e^{-\widetilde{C}_2(\mathbb {W},\epsilon )N}-\sqrt{2}\sqrt{\widetilde{C}_1(\mathbb {W},\epsilon )}e^{-\frac{\widetilde{C}_2(\mathbb {W},\epsilon )}{2}N}\right) \right) }{\left( 1+3N \sqrt{\widetilde{C}_1(\mathbb {W},\epsilon )}e^{-\frac{\widetilde{C}_2(\mathbb {W},\epsilon )N}{2}}\right) }\\&\ge \frac{J+\sum _i \lfloor 2NW_i(1-\epsilon )\rfloor -3NC_5(\mathbb {W},\epsilon )e^{-\frac{\widetilde{C}_2(\mathbb {W},\epsilon )}{2}N}}{1+3N C_5(\mathbb {W},\epsilon )e^{-\frac{\widetilde{C}_2(\mathbb {W},\epsilon )N}{2}}}\\&=\frac{\left( J+\sum _i \lfloor 2NW_i(1-\epsilon )\rfloor -3NC_5(\mathbb {W},\epsilon )e^{-\frac{\widetilde{C}_2(\mathbb {W},\epsilon )}{2}N}\right) \left( 1-3N C_5(\mathbb {W},\epsilon )e^{-\frac{\widetilde{C}_2(\mathbb {W},\epsilon )N}{2}}\right) }{\left( 1+3N C_5(\mathbb {W},\epsilon )e^{-\frac{\widetilde{C}_2(\mathbb {W},\epsilon )N}{2}}\right) \left( 1-3N C_5(\mathbb {W},\epsilon )e^{-\frac{\widetilde{C}_2(\mathbb {W},\epsilon )N}{2}}\right) }\\&\ge \frac{J+\sum _i \lfloor 2NW_i(1-\epsilon )\rfloor -6N^2C_5(\mathbb {W},\epsilon )e^{-\frac{\widetilde{C}_2(\mathbb {W},\epsilon )}{2}N} +\left( 3NC_5(\mathbb {W},\epsilon )e^{-\frac{\widetilde{C}_2(\mathbb {W},\epsilon )}{2}N}\right) ^2}{1-\left( 3N C_5(\mathbb {W},\epsilon )e^{-\frac{\widetilde{C}_2(\mathbb {W},\epsilon )N}{2}}\right) ^2}\\&\ge J+\sum _i \lfloor 2NW_i(1-\epsilon )\rfloor - 6N^2C_5(\mathbb {W},\epsilon )e^{-\frac{\widetilde{C}_2(\mathbb {W},\epsilon )}{2}N} \end{aligned}$$

for all \(N\ge \max \{\widetilde{N}_0(\mathbb {W},\epsilon ),N'(\mathbb {W},\epsilon )\}\), where \(N'(\mathbb {W},\epsilon ) =\max \{(\frac{4}{C_2(\mathbb {W},\epsilon )})^2,~\frac{4}{C_2(\mathbb {W},\epsilon )}\log (3C_5(\mathbb {W},\epsilon ))\}\) is the constant such that \(3N C_5(\mathbb {W},\epsilon )e^{-\frac{\widetilde{C}_2(\mathbb {W},\epsilon )N}{2}}<1\) for all \(N\ge N'(\mathbb {W},\epsilon )\).Footnote 6 Here the first line follows directly from Lemma 4.6, the second line follows because \(\text {trace}\left( \varvec{\Psi }^H\varvec{B}_{N,\mathbb {W}}\varvec{\Psi }\right) = \text {trace}\left( \sum _{i=0}^{J-1}\varvec{\Psi }_i^H\varvec{B}_{N,\mathbb {W}}\varvec{\Psi }_i\right) \) and \(\varvec{B}_{N,\mathbb {W}}\) is equivalent to \(\mathcal {I}_N\mathcal {B}_{\mathbb {W}}\mathcal {I}^*_N\), the third line follows from Lemma 4.3, the fourth line follows from the Cauchy–Schwarz inequality which indicates that \(|(\varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)})^H\varvec{o}_{i}^{(l_i)}|\le ||\varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)}||_2||\varvec{o}_{i}^{(l_i)}||_2 = ||\varvec{o}_{i}^{(l_i)}||_2\), the fifth line follows from Lemmas 2.3, 4.3 and 4.4, the seventh line follows by setting \(C_5(\mathbb {W},\epsilon ) = \max \{\widetilde{C}_1(\mathbb {W},\epsilon ),\sqrt{\widetilde{C}_1(\mathbb {W},\epsilon )}\}\), the ninth line follows because \(J+\sum _i \lfloor 2NW(1-\epsilon )\rfloor \le N\), and the last line follows because by assumption \(3N C_5(\mathbb {W},\epsilon )e^{-\frac{\widetilde{C}_2(\mathbb {W},\epsilon )N}{2}}<1\).

By noting that \(0<\lambda _{N,\mathbb {W}}^{(N-1)}\le \lambda _{N,\mathbb {W}}^{(0)}<1\) from Lemma 3.1, we acquire

$$\begin{aligned} \lambda _{N,\mathbb {W}}^{(l)}= & {} \left( \sum _{l'=0}^{J-1+\sum _i \lfloor 2NW_i(1-\epsilon )\rfloor }\lambda _{N,\mathbb {W}}^{(l')}\right) - \left( \sum _{l'=0,l'\ne l}^{J-1+\sum _i \lfloor 2NW_i(1-\epsilon )\rfloor }\lambda _{N,\mathbb {W}}^{(l')}\right) \\\ge & {} \left( \sum _{l'=0}^{J-1+\sum _i \lfloor 2NW_i(1-\epsilon )\rfloor }\lambda _{N,\mathbb {W}}^{(l')}\right) - \left( J-1+\sum _i\lfloor 2NW_i(1-\epsilon )\rfloor \right) \\\ge & {} 1-6N^2C_5(\mathbb {W},\epsilon )e^{-\frac{\widetilde{C}_2(\mathbb {W},\epsilon )}{2}N} \end{aligned}$$

for all \(l\le J-1+\sum _i \lfloor 2NW_i(1-\epsilon )\rfloor \), where the second line follows by setting \(\lambda _{N,\mathbb {W}}^{(l')},~l'\ne l\) to 1. Fix \(\mathbb {W}\) and \(\epsilon \). It is always possible to find a constant \(N'\) such that \(3N C_5(\mathbb {W},\epsilon )e^{-\frac{\widetilde{C}_2(\mathbb {W},\epsilon )N}{2}}<1\) for all \(N\ge N'\). Now, for convenience, we set \(\overline{C}_1(\mathbb {W},\epsilon ) = 6C_5(\mathbb {W},\epsilon )\), \(\overline{C}_2(\mathbb {W},\epsilon ) = \frac{\widetilde{C}_2(\mathbb {W},\epsilon )}{2}\), and \(\overline{N}_0(\mathbb {W},\epsilon ) = \max \{\widetilde{N}_0(\mathbb {W},\epsilon ),N'\}\). This completes the proof of Theorem 3.4.    \(\square \)

Appendix 5: Proof of Theorem 3.6

Proof

First denote the eigen-decomposition of \(\varvec{B}_{N,\mathbb {W}}\) as

$$\begin{aligned} \varvec{B}_{N,\mathbb {W}} = \varvec{U}_{N,\mathbb {W}}\varvec{\Lambda }_{N,\mathbb {W}}\varvec{U}_{N,\mathbb {W}}^H, \end{aligned}$$

where \(\varvec{\Lambda }_{N,\mathbb {W}}\) is an \(N\times N\) diagonal matrix whose diagonal elements are the eigenvalues \(\lambda _{N,\mathbb {W}}^{(0)},\lambda _{N,\mathbb {W}}^{(1)},\ldots ,\lambda _{N,\mathbb {W}}^{(N-1)}\) and \(\varvec{U}_{N,\mathbb {W}}\) is a square (\(N\times N\)) matrix defined by

$$\begin{aligned} \varvec{U}_{N,\mathbb {W}} :=[\varvec{u}_{N,\mathbb {W}}^{(0)} ~ \varvec{u}_{N,\mathbb {W}}^{(1)} ~ \ldots ~ \varvec{u}_{N,\mathbb {W}}^{(N-1)}]. \end{aligned}$$

Also let \(\varvec{a} = \varvec{U}_{N,\mathbb {W}}^H\varvec{\psi }\) be the coefficients of \(\varvec{\psi }\) represented by \(\varvec{U}_{N,\mathbb {W}}\).

Fix \(\epsilon \in (0,\min \{1,\frac{1}{|\mathbb {W}|}-1\})\). Suppose \(\varvec{\psi }\) is a column of \(\varvec{\Psi }_i\) for some particular \(i\in [J]\). Now from Lemma 4.3, we have

$$\begin{aligned} \varvec{B}_{N,\mathbb {W}}\varvec{\psi }=\lambda _{N,W_i}^{(l_i)} \varvec{\psi }+\varvec{o}_i^{(l_i)} \end{aligned}$$

for some \(l_i\le \lfloor 2NW_i(1-\epsilon )\rfloor \).

Plugging the eigen-decomposition of the matrix \(\varvec{U}_{N,\mathbb {W}}\) into the above equation, we require

$$\begin{aligned} \varvec{\Lambda }_{N,\mathbb {W}}\varvec{a} = \lambda _{N,W_i}^{(l_i)}\varvec{a}+\widehat{\varvec{o}}_i^{(l_i)}, \end{aligned}$$

where \(\widehat{\varvec{o}}_i^{(l_i)} = \varvec{U}_{N,\mathbb {W}}^H\varvec{o}_i^{(l_i)}\). The elementary form of the above equation is

$$\begin{aligned} \lambda _{N,\mathbb {W}}^{(m)}\varvec{a}[m] = \lambda _{N,W_i}^{(l_i)}\varvec{a}[m]+ \widehat{\varvec{o}}_i^{(l_i)}[m] \end{aligned}$$

for all \(m\in [N]\).

Now we have

$$\begin{aligned} ||\varvec{\psi }-\varvec{P}_{{\varvec{\Phi }}}\varvec{\psi }||_2^2&=\sum _{m=\sum _i \lceil 2NW_i(1+\epsilon )\rceil }^{N-1}|\varvec{a}[m]|^2 = \sum _{m=\sum _i \lceil 2NW_i(1+\epsilon )\rceil }^{N-1}\frac{\left| \widehat{\varvec{o}}_i^{(l_i)}[m]\right| ^2}{\left| \lambda _{N,W_i}^{(l_i)}-\lambda _{N,\mathbb {W}}^{(m)}\right| ^2} \nonumber \\&\le \frac{\sum _{m=\sum _i \lceil 2NW_i(1+\epsilon )\rceil }^{N-1}\left| \widehat{\varvec{o}}_i^{(l_i)}[m]\right| ^2}{\left( 1-\widetilde{C}_1(\mathbb {W},\epsilon )e^{-\widetilde{C}_2(\mathbb {W},\epsilon )N}-\overline{C}_3(\mathbb {W},\epsilon )e^{-\overline{C}_4(\mathbb {W},\epsilon )N}\right) ^2}\nonumber \\&\le \frac{||\varvec{o}_i^{(l_i)}||^2}{\left( 1-\widetilde{C}_1(\mathbb {W},\epsilon )e^{-\widetilde{C}_2(\mathbb {W},\epsilon )N}-\overline{C}_3(\mathbb {W},\epsilon )e^{-\overline{C}_4(\mathbb {W},\epsilon )N}\right) ^2}\nonumber \\&\le \frac{2\widetilde{C}_1(\mathbb {W},\epsilon )e^{-\widetilde{C}_2(\mathbb {W},\epsilon )N}}{\left( 1-\widetilde{C}_1(\mathbb {W},\epsilon )e^{-\widetilde{C}_2(\mathbb {W},\epsilon )N}-\overline{C}_3(\mathbb {W},\epsilon )e^{-\overline{C}_4(\mathbb {W},\epsilon )N}\right) ^2} \end{aligned}$$
(27)

for all \(N\ge \max \{\overline{N}_0(\mathbb {W},\epsilon ),\overline{N}_1(\mathbb {W},\epsilon )\}\), where the second line follows by bounding the \(\lambda _{N,W_i}^{(l_i)}\) term using \(1-C_1(W_i,\epsilon )e^{-C_2(W_i,\epsilon )N}\) (which is not less than \(1-\widetilde{C}_1(\mathbb {W},\epsilon )e^{-\widetilde{C}_2(\mathbb {W},\epsilon )N}\)) from Lemma 2.3 and bounding the \(\lambda _{N,\mathbb {W}}^{(m)}\) terms using Theorem 3.4, and the fourth line follows because \(||\varvec{o}_i^{(l_i)}||^2\le 2C_1(W_i,\epsilon )e^{-C_2(W_i,\epsilon )N}\le 2\widetilde{C}_1(\mathbb {W},\epsilon )e^{-\widetilde{C}_2(\mathbb {W},\epsilon )N}\).

The following general result will help in extending (27) to an angle between the subspaces.

Lemma 4.7

Let \(\mathcal {S}_{\varvec{U}}\) and \(\mathcal {S}_{\varvec{V}}\) be the subspaces spanned by the columns of the matrices \(\varvec{U}\in \mathbb {C}^{N\times q}\) and \(\varvec{V}\in \mathbb {C}^{N\times r}\), respectively. Here \(r \le q \le N\). Suppose each column of \(\varvec{V}\) is normalized so that \(\Vert \varvec{v}_l\Vert _2 = 1\) and is close to \(\mathcal {S}_{\varvec{U}}\) such that for some \(\delta _1\), \(\Vert \varvec{v}_l - \varvec{P}_{\varvec{U}}\varvec{v}_l\Vert _2^2\le \delta _1\) for all \(l\in [r]\). Furthermore, suppose the columns of \(\varvec{V}\) are approximately orthogonal to each other such that for some \(\delta _2\), \(\left| \langle \varvec{v}_k, \varvec{v}_l\rangle \right| \le \delta _2\) for all \(k\ne l\). Then we have

$$\begin{aligned} \cos (\Theta _{\mathcal {S}_{\varvec{U}}\mathcal {S}_{\varvec{V}}})\ge \sqrt{\frac{1 - \delta _1- N\left( \delta _2+\sqrt{\delta _1}\right) }{1 + N \delta _2}}. \end{aligned}$$

Proof (of Lemma 4.7)

Any \(\varvec{v}\in \mathcal {S}_{\varvec{V}}\) can be written as a linear combination of \(\varvec{v}_l\) in the form \(\varvec{v} = \sum _l \alpha _l \varvec{v}_l\). We first bound the \(l_2\) norm of \(\varvec{v}\) by

$$\begin{aligned} \Vert \varvec{v}\Vert _2^2&= \Vert \sum _{l=0}^{r-1} \alpha _l \varvec{v}_l\Vert _2^2\\&= \sum _{l=0}^{r-1} |\alpha _l|^2\Vert \varvec{v}_l\Vert _2^2 + \sum _{l=0}^{r-1}\sum _{k=0, k\ne l}^{r-1}\langle \alpha _l\varvec{v}_l,\alpha _k \varvec{v}_k \rangle \\&\le \sum _{l=0}^{r-1} |\alpha _l|^2 + \sum _{l=0}^{r-1}\sum _{k=0, k\ne l}^{r-1}|\alpha _l||\alpha _k|\delta _2\\&\le \sum _{l=0}^{r-1} |\alpha _l|^2 + \sum _{l=0}^{r-1}\sum _{k=0, k\ne l}^{r-1}\frac{|\alpha _l|^2+|\alpha _k|^2}{2}\delta _2\\&= \left( \sum _{l=0}^{r-1} |\alpha _l|^2\right) \left( 1 + (r-1)\delta _2\right) \le \left( \sum _{l=0}^{r-1} |\alpha _l|^2\right) \left( 1 + N\delta _2\right) , \end{aligned}$$

where the third line follows from the hypothesis that \(\left| \langle \varvec{v}_k, \varvec{v}_l\rangle \right| \le \delta _2\) for all \(k\ne l\). Similarly,

$$\begin{aligned} \Vert \varvec{P}_{\varvec{U}}\varvec{v}\Vert _2^2&= \Vert \sum _{l=0}^{r-1} \varvec{P}_{\varvec{U}}\left( \alpha _l \varvec{v}_l\right) \Vert _2^2\\&= \sum _{l=0}^{r-1} |\alpha _l|^2\Vert \varvec{P}_{\varvec{U}}\varvec{v}_l\Vert _2^2 + \sum _{l=0}^{r-1}\sum _{k=0, k\ne l}^{r-1}\left\langle \alpha _l\varvec{P}_{\varvec{U}}\varvec{v}_l,\alpha _k \varvec{P}_{\varvec{U}}\varvec{v}_k \right\rangle \\&= \sum _{l=0}^{r-1} |\alpha _l|^2\Vert \varvec{P}_{\varvec{U}}\varvec{v}_l\Vert _2^2 + \sum _{l=0}^{r-1}\sum _{k=0, k\ne l}^{r-1}\left\langle \alpha _l \varvec{v}_l,\alpha _k \left( \varvec{v}_k - (\varvec{v}_k -\varvec{P}_{\varvec{U}}\varvec{v}_k)\right) \right\rangle \\&\ge \sum _{l=0}^{r-1} |\alpha _l|^2\left( 1-\delta _1\right) - \sum _{l=0}^{r-1}\sum _{k=0, k\ne l}^{r-1}|\alpha _l||\alpha _k|\left( \delta _2+\sqrt{\delta _1}\right) \\&= \left( \sum _{l=0}^{r-1} |\alpha _l|^2\right) \left( 1 - \delta _1- (r-1)\left( \delta _2+\sqrt{\delta _1}\right) \right) \\&\ge \left( \sum _{l=0}^{r-1} |\alpha _l|^2\right) \left( 1 - \delta _1- N\left( \delta _2+\sqrt{\delta _1}\right) \right) , \end{aligned}$$

where the fourth line follows because \(\langle \varvec{v}_l, \varvec{v}_k -\varvec{P}_{\varvec{U}}\varvec{v}_k\rangle \le \Vert \varvec{v}_l\Vert _2\Vert \varvec{v}_k - \varvec{P}_{\varvec{U}}\varvec{v}_k\Vert _2\le \sqrt{\delta _1}\) and \(\left| \langle \varvec{v}_k, \varvec{v}_l\rangle \right| \le \delta _2\) for all \(k\ne l\).

Therefore, for any non-zero vector \(\varvec{v}\in \mathcal {S}_{\varvec{V}}\) we have

$$\begin{aligned} \frac{\Vert \varvec{P}_{\varvec{U}}\varvec{v}\Vert _2^2}{\Vert \varvec{v}\Vert _2^2}\ge \frac{1 - \delta _1- N\left( \delta _2+\sqrt{\delta _1}\right) }{1 + N \delta _2}. \end{aligned}$$

\(\square \)

Finally, (15) follows from Lemma 4.7 by replacing \(\varvec{U}\) with \(\varvec{\Phi }\) and \(\varvec{V}\) with \(\varvec{\Psi }\), and assigning \(\delta _1\) with the upper bound in (27) and \(\delta _2\) with the upper bound in (26). \(\square \)

Appendix 6: Proof of Theorem 3.7

Proof

For each \(i\in [J]\), define \(\overline{\varvec{\Psi }}_i = [\varvec{E}_{f_i}\varvec{S}_{N,W_i}\sqrt{\varvec{\Lambda }_{N,W_i}}]_{k_i}\) for some given \(k_i\in \{1,2,\ldots ,N\}\). We construct the scaled multiband modulated DPSS matrix \(\overline{\varvec{\Psi }}\) byFootnote 7

$$\begin{aligned} \overline{\varvec{\Psi }}:=[\overline{\varvec{\Psi }}_0 ~ \overline{\varvec{\Psi }}_1 ~ \cdots ~ \overline{\varvec{\Psi }}_{J-1}]. \end{aligned}$$
(28)

The main idea is to bound \(\left\| \varvec{P}_{\varvec{\Psi }} \varvec{u}_{N,\mathbb {W}}^{(l)}\right\| _2\) using \(\left\| \overline{\varvec{\Psi }}~\overline{\varvec{\Psi }}^H\varvec{u}_{N,\mathbb {W}}^{(l)}\right\| _2\). In order to use this argument, we first give out some useful results. \(\square \)

Lemma 4.8

Suppose \(\overline{\varvec{\Psi }}\) is the matrix defined in (28) with some given \(k_i\in \{1,2,\ldots ,N\}, \forall i\in [J]\). Then

$$\begin{aligned} \left\| \overline{\varvec{\Psi }}\right\| _2\le 1. \end{aligned}$$

Proof (of Lemma 4.8)

Let \(\varvec{y}\in \mathbb {C}^N\). Then

$$\begin{aligned} \left\| \overline{\varvec{\Psi }}^H\varvec{y}\right\| _2^2&=\sum _{i=0}^{J-1}\sum _{l_i=0}^{k_i-1}|\langle \varvec{y},\varvec{E}_{f_i}\sqrt{\lambda _{N,W_i}^{(l_i)}}\varvec{s}_{N,W_i}^{(l_i)}\rangle |^2\\&=\sum _{i=0}^{J-1}\sum _{l_i=0}^{k_i-1}\langle \varvec{y},\varvec{E}_{f_i}\sqrt{\lambda _{N,W_i}^{(l_i)}}\varvec{s}_{N,W_i}^{(l_i)}\rangle \langle \varvec{E}_{f_i}\sqrt{\lambda _{N,W_i}^{(l_i)}}\varvec{s}_{N,W_i}^{(l_i)}, \varvec{y}\rangle \\&=\sum _{i=0}^{J-1}\sum _{l_i=0}^{k_i-1} \varvec{y}^H\varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)}\lambda _{N,W_i}^{(l_i)}(\varvec{s}_{N,W_i}^{(l_i)})^H \varvec{E}_{f_i}^H\varvec{y}\\&\le \sum _{i=0}^{J-1}\sum _{l_i=0}^{N-1} \varvec{y}^H\varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)}\lambda _{N,W_i}^{(l_i)}(\varvec{s}_{N,W_i}^{(l_i)})^H \varvec{E}_{f_i}^H\varvec{y}\\&=\sum _{i=0}^{J-1} \varvec{y}^H\varvec{E}_{f_l}\mathcal {I}_N(\mathcal {B}_{W_i}(\mathcal {I}^*_N( \varvec{E}_{f_i}^H\varvec{y})))=\sum _{i=0}^{J-1} \langle \mathcal {I}_N(\mathcal {B}_{W_i}(\mathcal {I}^*_N( \varvec{E}_{f_i}^H\varvec{y}))),\varvec{E}_{f_i}^H\varvec{y}\rangle \\&=\sum _{i=0}^{J-1} \langle \mathcal {B}_{W_i}(\mathcal {I}^*_N( \varvec{E}_{f_i}^H\varvec{y})),\mathcal {I}^*_N(\varvec{E}_{f_i}^H\varvec{y})\rangle \\&=\sum _{i=0}^{J-1} \langle \mathcal {B}_{W_i}(\mathcal {I}^*_N( \varvec{E}_{f_i}^H\varvec{y})),\mathcal {B}_{W_i}(\mathcal {I}^*_N(\varvec{E}_{f_l}^H\varvec{y}))\rangle =\sum _{i=0}^{J-1} ||\mathcal {B}_{W_i}(\mathcal {I}^*_N( \varvec{E}_{f_i}^H\varvec{y}))||_2^2\\&=\sum _{i=0}^{J-1}\int _{f_i-W_i}^{f_i+W_i}|\widetilde{\varvec{y}}(f)|^2df=\int _{-1/2}^{1/2}\left( \sum _{i=0}^{J-1}\mathbbm {1}_{[f_i-W_i,f_i+W_i)}(f)\right) |\widetilde{\varvec{y}}(f)|^2df ,\nonumber \end{aligned}$$

where the fourth line follows because \(\varvec{y}^H\varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)}\lambda _{N,W_i}^{(l_i)}(\varvec{s}_{N,W_i}^{(l_i)})^H \varvec{E}_{f_i}^H\varvec{y} = ||\sqrt{\lambda _{N,W_i}^{(l_i)}}(\varvec{s}_{N,W_i}^{(l_i)})^H \varvec{E}_{f_i}^H\varvec{y}||_2^2\ge 0\), the fifth line follows because \(\sum _{l_i=0}^{N-1}\varvec{s}_{N,W_i}^{(l_i)}\lambda _{N,W_i}^{(l_i)}(\varvec{s}_{N,W_i}^{(l_i)})^H \varvec{x} =\mathcal {I}_N(\mathcal {B}_{W_i}(\mathcal {I}^*_N(\varvec{x})))\), and we use \(\widetilde{\varvec{y}}(f) = \sum _{n=0}^{N-1}\varvec{y}[n]e^{-j2\pi fn}\) as the DTFT of \(\mathcal {I}^*_N(\varvec{y})\) in the last two equations.

Noting that \(\sum _{i=0}^{J-1}\mathbbm {1}_{[f_i-W_i,W_i+f_i)}(f)\le 1\) for all \(f\in [-\frac{1}{2},\frac{1}{2}]\) since we assume there is no overlap between each interval \([f_i-W_i,W_i+f_i)\), we conclude

$$\begin{aligned} ||\overline{\varvec{\Psi }}^H\varvec{y}||_2^2\le \int _{-1/2}^{1/2}|\widetilde{\varvec{y}}(f)|^2df = ||\varvec{y}||_2^2 \end{aligned}$$

and

$$\begin{aligned} ||\overline{\varvec{\Psi }}||_2\le 1. \end{aligned}$$

\(\square \)

Lemma 4.9

For any \(k_i\in \{1,2,\ldots ,N\}, i\in [J]\), let \(\varvec{\Psi }\) and \(\overline{\varvec{\Psi }}\) be the matrices defined in (12) and (28) respectively. Then for any \(\varvec{y}\in \mathbb {C}^{N\times 1}\),

$$\begin{aligned} ||\varvec{P}_{\varvec{\Psi }}\varvec{y}||_2\ge ||\overline{\varvec{\Psi }}~\overline{\varvec{\Psi }}^H\varvec{y}||_2. \end{aligned}$$
(29)

Proof (of Lemma 4.9)

Let \(\overline{\varvec{\Psi }} = \varvec{U}_{\overline{\varvec{\Psi }}}\Sigma _{\overline{\varvec{\Psi }}}\varvec{V}_{\overline{\varvec{\Psi }}}^H\) be a reduced SVD of \(\overline{\varvec{\Psi }}\), where both \(\varvec{U}_{\overline{\varvec{\Psi }}}\) and \(\varvec{V}_{\overline{\varvec{\Psi }}}\) are orthonormal matrices of the proper dimension, and \(\varvec{\Sigma }_{\overline{\varvec{\Psi }}}\) is a diagonal matrix whose diagonal elements are the non-zero singular values of \(\overline{\varvec{\Psi }}\). We have

$$\begin{aligned} ||\overline{\varvec{\Psi }}~\overline{\varvec{\Psi }}^H\varvec{y}||_2&= ||\varvec{U}_{\overline{\varvec{\Psi }}}\Sigma _{\overline{\varvec{\Psi }}}^2\varvec{U}_{\overline{\varvec{\Psi }}}^H\varvec{y}||_2\\&\le ||\varvec{U}_{\overline{\varvec{\Psi }}}^H\varvec{y}||_2\\&= ||\varvec{U}_{\overline{\varvec{\Psi }}}\varvec{U}_{\overline{\varvec{\Psi }}}^H\varvec{y}||_2\\&= ||\varvec{P}_{\varvec{\Psi }}\varvec{y}||_2 \nonumber \end{aligned}$$

where the second lines follows because \(||\overline{\varvec{\Psi }}||_2\le 1\) and hence the diagonal elements \(\varvec{\Sigma }_{\overline{\varvec{\Psi }}}\) are bounded above by 1, and the fourth line follows because each column in \(\overline{\varvec{\Psi }}\) is in also \(\varvec{\Psi }\) and hence \(||\varvec{P}_{\varvec{\Psi }}\varvec{y}||_2=||\varvec{P}_{\varvec{U}_{\overline{\Psi }}}\varvec{y}||_2\). \(\square \)

Now we turn to prove Theorem 3.7. By (29), we observe that

$$\begin{aligned} ||\varvec{P}_{\varvec{\Psi }}\varvec{u}_{N,\mathbb {W}}^{(l)}||_2&\ge ||\overline{\varvec{\Psi }}~\overline{\varvec{\Psi }}^H\varvec{u}_{N,\mathbb {W}}^{(l)}||_2\\&= ||\sum _{i=0}^{J-1}\sum _{l_i=0}^{k_i-1} \varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)}\lambda _{N,W_i}^{(l_i)}(\varvec{s}_{N,W_i}^{(l_i)})^H \varvec{E}_{f_i}^H\varvec{u}_{N,\mathbb {W}}^{(l)}||_2\\&= ||\varvec{B}_{N,\mathbb {W}}\varvec{u}_{N,\mathbb {W}}^{(l)}- \sum _{i=0}^{J-1}\sum _{l_i=k_i}^{N-1} \varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)}\lambda _{N,W_i}^{(l_i)}(\varvec{s}_{N,W_i}^{(l_i)})^H \varvec{E}_{f_i}^H\varvec{u}_{N,\mathbb {W}}^{(l)}||_2\\&\ge ||\varvec{B}_{N,\mathbb {W}}\varvec{u}_{N,\mathbb {W}}^{(l)}||_2 - \sum _{i=0}^{J-1}\sum _{l_i=k_i}^{N-1}|| \varvec{E}_{f_i}\varvec{s}_{N,W_i}^{(l_i)}\lambda _{N,W_i}^{(l_i)}(\varvec{s}_{N,W_i}^{(l_i)})^H \varvec{E}_{f_i}^H\varvec{u}_{N,\mathbb {W}}^{(l)}||_2\\&\ge \lambda _{N,\mathbb {W}}^{(l)} - \sum _{i=0}^{J-1}\sum _{l_i=k_i}^{N-1}\lambda _{N,W_i}^{(l_i)}. \end{aligned}$$

\(\square \)

Appendix 7: Proof of Corollary 3.8

Proof

It follows from Theorem 3.7 that

$$\begin{aligned} ||\varvec{P}_{\varvec{\Psi }}\varvec{u}_{N,\mathbb {W}}^{(l)}||_2&\ge \lambda _{N,\mathbb {W}}^{(l)} - \sum _{i=0}^{J-1}\sum _{l_i=k_i}^{N-1}\lambda _{N,W_i}^{(l_i)}\\&\ge 1 - \overline{C}_1(\mathbb {W},\epsilon )N^2e^{-\overline{C}_2(\mathbb {W},\epsilon )N} - \sum _{i=0}^{J-1}\sum _{l_i=k_i}^{N-1}C_3(W_i,\epsilon )e^{-C_4(W_i,\epsilon )N}\\&\ge 1 - \overline{C}_1(\mathbb {W},\epsilon )N^2e^{-\overline{C}_2(\mathbb {W},\epsilon )N} - \sum _{i=0}^{J-1}\sum _{l_i=k_i}^{N-1}\frac{1}{J}\overline{C}_3(\mathbb {W},\epsilon )e^{-\overline{C}_4(\mathbb {W},\epsilon )N}\\&\ge 1 - \overline{C}_1(\mathbb {W},\epsilon )N^2e^{-\overline{C}_2(\mathbb {W},\epsilon )N} - N\overline{C}_3(\mathbb {W},\epsilon )e^{-\overline{C}_4(\mathbb {W},\epsilon )N} \nonumber \end{aligned}$$

for all \(N\ge \max \{\overline{N}_0(\mathbb {W},\epsilon ),\overline{N}_1(\mathbb {W},\epsilon )\}\), where the second line follows by bounding the \(\lambda _{N,\mathbb {W}}^{(l)}\) term using Theorem 3.4 and by bounding the \(\lambda _{N,W_i}^{(l_i)}\) terms using Lemma 2.3, and the third line follows because \(\overline{C}_3(\mathbb {W},\epsilon ) = J\max {\{C_{3}(W_i,\epsilon ), ~\forall ~ i \in [J]\}}\) and \(\overline{C}_4(\mathbb {W},\epsilon ) = \min {\{C_{4}(W_i,\epsilon ), ~\forall ~ i \in [J]\}}\).

Let \(\kappa _2(N,\mathbb W,\epsilon ) =\overline{C}_1(\mathbb {W},\epsilon )N^2e^{-\overline{C}_2(\mathbb {W},\epsilon )N}+ N\overline{C}_3(\mathbb {W},\epsilon )e^{-\overline{C}_4(\mathbb {W},\epsilon )N}\). Then \(||\varvec{u}_{N,\mathbb {W}}^{(l)} - \varvec{P}_{\varvec{\Psi }}\varvec{u}_{N,\mathbb {W}}^{(l)}||_2^2\le 2\kappa _2(N,\mathbb W,\epsilon ) - \kappa _2^2(N,\mathbb W,\epsilon )\). Noting also that \(\langle \varvec{u}_{N,\mathbb {W}}^{(l)}, \varvec{u}_{N,\mathbb {W}}^{(k)}\rangle = 0\) for all \(k\ne l\), (16) follows directly from Lemma 4.7.     \(\square \)

Appendix 8: DTFT of DPSS Vectors

The results presented in this appendix are useful in Appendix 9, where we analyze the performance of the DPSS vectors for representing sampled pure tones inside the band of interest. Let \(\widetilde{\varvec{s}}_{N,W}^{(l)}(f)\) denote the DTFT of the sequence \(\mathcal {T}_N(s_{N,W}^{(l)})\), i.e., \(\widetilde{\varvec{s}}_{N,W}^{(l)}(f) = \sum _{n=0}^{N-1}s_{N,W}^{(l)}[n]e^{-j2\pi fn}\). Figure 1 shows \(\widetilde{\varvec{s}}_{N,W}^{(l)}(f)\) for all \(l\in [N]\) with \(N=1024\) and \(W = \frac{1}{4}\). We observe that the first \(\approx 2NW\) DPSS vectors have their spectrum mostly concentrated in \([-W,W]\), only a small fraction of DPSS vectors whose indices are near 2NW have a relatively flat spectrum over \([-\frac{1}{2},\frac{1}{2}]\), and the remaining DPSS vectors have their spectrum mostly concentrated outside of the band \([-W,W]\). This phenomenon is captured formally in the asymptotic expressions for \(\lambda _{N,W}^{(l)}\) and \(\widetilde{\varvec{s}}^{(l)}_{N,W}(f)\) from [37].

Fig. 1
figure 1

Illustration of \(\left| \widetilde{\varvec{s}}_{N,W}^{(l)}(f)\right| ^2\), or the energy in \(\{\varvec{e}_f\}\) captured by each DPSS vector. The horizontal axis stands for the digital frequency f, which ranges over \([-\frac{1}{2},\frac{1}{2}]\), while the vertical axis stands for the index \(l\in [N]\). The l-th horizontal line shows \(10\log _{10}\left| \widetilde{\varvec{s}}_{N,W}^{(l)}(f)\right| ^2\). Here \(N = 1024\) and \(W= \frac{1}{4}\).

Lemma 4.10

([37]) Fix \(W\in (0,\frac{1}{2})\) and \(\epsilon \in (0,1)\). Let \(\alpha := 1-A = 1-\cos 2\pi W\).

  1. 1.

    For fixed l, as \(N \rightarrow \infty \), we have

    $$\begin{aligned} 1-\lambda _{N,W}^{(l)}\sim c_5^2/\left( 2\sqrt{2\alpha }\right) \end{aligned}$$

    and

    $$\begin{aligned} \widetilde{\varvec{s}}_{N,W}^{(l)}(f)\sim \left\{ \begin{array}{ll}c_3f_4(f),&{}\quad W\le |f|\le \arccos (A-N^{-3/2})/2\pi ,\\ c_5f_5(f),&{}\quad \arccos (A-N^{-3/2})/2\pi \le |f|\le 1/2. \end{array}\right. \end{aligned}$$

    Here

    $$\begin{aligned} c_5&= (l!)^{-1/2}\pi ^{1/4}2^{(14l+15)/8}\alpha ^{(2l+3)/8}N^{(2l+1)/4}\\&(\sqrt{2}+\sqrt{\alpha })^{-N}(2-\alpha )^{(N-l-1/2)/2}\\&= (l!)^{-1/2}\pi ^{1/4}2^{(14l+15)/8}\alpha ^{(2l+3)/8}N^{(2l+1)/4}(2-\alpha )^{-(l+1/2)/2}e^{-\frac{\gamma }{2} N},\\ c_3&= \pi ^{1/2}2^{-1/2}\alpha ^{-1/4}[2-\alpha ]^{-1/4}N^{1/2}c_5 = O(N^{1/2}) c_5, \\ \gamma&= \log \left( 1+\frac{2\sqrt{\alpha }}{\sqrt{2}-\sqrt{\alpha }}\right) ,\\ f_4(f)&= J_0\left( \frac{N}{\sqrt{2-\alpha }}\sqrt{A-\cos \left( 2\pi f\right) }\right) ,\\ f_5(f)&=\frac{\cos \left( \frac{N}{2}\arcsin \left( \theta (f)\right) +\frac{1}{2}(l+\frac{1}{2})\arcsin \left( \phi (f)\right) +(l-N)\frac{\pi }{4}+\frac{3\pi }{8}\right) }{\left( (A-\cos \left( 2\pi f\right) )(1-\cos \left( 2\pi f\right) )\right) ^{1/4}},\\ \theta (f)&= \frac{\alpha +2\cos \left( 2\pi f\right) }{2-\alpha }, ~\phi (f) = \frac{(2-3\alpha )-(2+\alpha )\cos \left( 2\pi f\right) }{(2-\alpha )(1-\cos \left( 2\pi f\right) )}, \end{aligned}$$

    where \(J_0\) is the Bessel function of the first kind.

  2. 2.

    As \(N \rightarrow \infty \) and with \(l=\lfloor 2NW(1-\epsilon ')\rfloor \) for any \(\epsilon '\in (0,\epsilon ]\), we have

    $$\begin{aligned} 1-\lambda _{N,W}^{(l)}\sim 2\pi L_2^{-1}d_6^2 \end{aligned}$$

    and

    $$\begin{aligned} \widetilde{\varvec{s}}_{N,W}^{(l)}(f)\sim \left\{ \begin{array}{ll}d_4g_5(f),&{}\quad W\le |f|\le \arccos (A-N^{-1})/2\pi ,\\ d_6g_6(f), &{}\quad \arccos (A-N^{-1})/2\pi \le |f|\le 1/2. \end{array}\right. \end{aligned}$$

    Here

    $$\begin{aligned} d_6= & {} (L_2)^{-1/2}\pi ^{1/2}2^{1/2}e^{-CL_4/4}e^{-NL_3/2}, \\ d_4= & {} (L_2)^{-1/2}\pi (1-A^2)^{-1/4}e^{-CL_4/4}e^{-NL_3/2}N^{1/2},\\ g_5(f)= & {} J_0\left( N\sqrt{\frac{B-A}{1-A^2}\left( \cos (2\pi f)-A\right) }\right) ,\\ g_6(f)= & {} R(f)\cos \left( \pi N\int _{f}^{1/2}\sqrt{\frac{B-\cos (2\pi t)}{A- \cos (2\pi t)}}dt \right. \\&\left. +\, \frac{\pi C}{2}\int _f^{1/2}\frac{dt}{\sqrt{\left( B - \cos (2\pi t)\right) \left( A - \cos (2\pi t)\right) }} + \theta \right) ,\\ R(f)= & {} \left| \left( B - \cos (2\pi f)\right) \left( A - \cos (2\pi f) \right) \right| ^{-1/4}, \\ C= & {} \frac{1}{L_2}\mod \left( \frac{N}{2}L_1 + \left( 2+(-1)^l\right) \frac{\pi }{4},2\pi \right) ,\\ \theta= & {} \mod \left( \frac{\pi }{4}-\frac{N}{2}L_5-\frac{C}{4}L_6,2\pi \right) ,\\ L_1= & {} \int _B^1 P(\xi )d\xi ,~L_2 = \int _B^1 Q(\xi )d\xi ,~L_3 = \int _A^B P(\xi )d\xi ,\\L_4= & {} \int _A^B Q(\xi )d\xi ,~L_5 = \int _{-1}^A P(\xi )d\xi ,~L_6 = L_2,\\ P(\xi )= & {} \left| \frac{\xi - B}{\left( \xi - A\right) \left( 1-\xi ^2\right) } \right| ^{1/2},~Q(\xi ) = \left| \left( \xi - B\right) \left( \xi - A\right) \left( 1-\xi ^2\right) \right| ^{-1/2}, \end{aligned}$$

    where B is determined so that \(\int _B^1\sqrt{\frac{\xi - B}{\left( \xi - A\right) \left( 1 - \xi ^2\right) }}d\xi = \frac{l}{N}\pi \) and \(\mod (y,2\pi )\) returns the remainder after division of y by \(2\pi \).

Appendix 9: Proof of Theorem 3.9

Noting that \(\varvec{S}_{N,W}\) forms an orthobasis for \(\mathbb {C}^{N\times N}\), the main idea is to show that the DPSS vectors \(\varvec{s}_{N,W}^{(2NW(1+\epsilon ))}, \varvec{s}_{N,W}^{(2NW(1+\epsilon )+1)}, \ldots ,\varvec{s}_{N,W}^{(N-1)}\) have their spectrum most concentrated outside of the band \([-W,W]\).

Since the sequence \(s_{N,W}^{(l)}\) is exactly bandlimited to the frequency range \(|f|\le W\), we know that its DTFT \(\widetilde{s}_{N,W}^{(l)}(f):=\sum _{n=-\infty }^\infty s_{N,W}^{(l)}[n]e^{j2\pi fn}\) vanishes for all \(W<|f|<\frac{1}{2}\). By noting that the first \(\approx 2NW\) DPSS’s are also approximately time-limited to the index range \(n=0,1,\ldots ,N-1\), we may expect that \(\widetilde{\varvec{s}}_{N,W}^{(l)}(f):=\sum _{n=0}^{N-1} \varvec{s}_{N,W}^{(l)}[n]e^{j2\pi fn}\) is also approximately 0 for all \(W<|f|<\frac{1}{2}\) and \(l\le 2NW(1-\epsilon )\). This illustrates informally why the DTFT of the first \(\approx 2NW\) DPSS vectors is concentrated inside the band \([-W,W]\). By employing the antisymmetric property [37] which states that \(|\widetilde{\varvec{s}}_{N,W}^{(l)}(f)| = |\widetilde{\varvec{s}}_{N,\frac{1}{2}-W}^{(N-1-l)}(\frac{1}{2}-f)|\), we then have that the DPSS vectors \(\varvec{s}_{N,W}^{(2NW(1+\epsilon ))}\), \(\varvec{s}_{N,W}^{(2NW(1+\epsilon )+1)}\), \(\ldots ,\varvec{s}_{N,W}^{(N-1)}\) are almost orthogonal to any sinusoid with frequency inside the band \([-W,W]\).

Recall that \(\widetilde{\varvec{s}}_{N,W}^{(l)}(f)\) is the DTFT of the sequence \(\mathcal {T}_N(s_{N,W}^{(l)})\), i.e., \(\widetilde{\varvec{s}}_{N,W}^{(l)}(f) = \sum _{n=0}^{N-1}s_{N,W}^{(l)}[n]e^{-j2\pi fn}\). We have

$$\begin{aligned} \langle \varvec{s}_{N,W}^{(l)}, \varvec{e}_{f}\rangle = \widetilde{\varvec{s}}_{N,W}^{(l)}(f), \end{aligned}$$

for all \(l\in [N]\). As we have observed in Fig. 1, the spectrum of the first \(\approx 2NW\) DPSS vectors is approximately concentrated on the frequency interval \([-W,W]\). This behavior is captured formally in the following results.

Corollary 4.11

Let \(A = \cos 2\pi W\). For fixed \(W\in (0,\frac{1}{2})\) and \(\epsilon \in (0,\min (\frac{1}{2W}-1,1))\), there exists a constant \(C_6(W,\epsilon )\) (which may depend on W and \(\epsilon \)) such that

$$\begin{aligned} |\widetilde{\varvec{s}}_{N,W}^{(l)}(f)|\le C_6(W,\epsilon )N^{3/4}e^{-\frac{C_2(W,\epsilon )}{2}N},~~ W\le |f|\le 1/2 \end{aligned}$$

for all \(N\ge N_0(W,\epsilon )\) and \(l\le 2NW(1-\epsilon )\). Here \(C_2(W,\epsilon )\) and \(N_0(N,\epsilon )\) are constants specified in Lemma 2.3.

Proof (of Corollary 4.11)

The main approach is to bound \(\widetilde{\varvec{s}}_{N,W}^{(l)}(f),~ W\le |f|\le 1/2\) with the expressions presented in Lemma 4.10. Suppose \(\epsilon \in (0,1)\) is fixed.

  1. 1.

    For fixed l and large N: In order to quantify the decay rate of \(|\widetilde{\varvec{s}}_{N,W}^{(l)}(f)|\), we exploit some results concerning of \(f_4(f)\) from [32] and \(f_5(f)\) as follows:

    $$\begin{aligned} |J_0(x)|\le 1, ~\forall ~ x\ge 0, \end{aligned}$$
    (30)

    and for any \(\frac{\arccos (A-N^{-3/2})}{2\pi }\le |f|\le 1/2\), one may verify that

    $$\begin{aligned} |f_5(f)|&\le \frac{1}{\left( (A-\cos \left( 2\pi f\right) )(1-\cos \left( 2\pi f\right) )\right) ^{1/4}}\\&\le \frac{1}{\left( (A-\left( A-N^{-3/2})\right) (1-\left( A-N^{-3/2})\right) \right) ^{1/4}}\\&\le \frac{1}{\left( (N^{-3/2}))(N^{-3/2}))\right) ^{1/4}} = N^{3/4}, \end{aligned}$$

    where the last line follows because \(1-A\ge 0\). Recall that \(c_3 {=} \pi ^{1/2}2^{-1/2}\alpha ^{-1/4}\left( 2{-}\alpha \right) ^{-1/4}N^{1/2}c_5\) and \(c_5 {\sim } \sqrt{2\sqrt{2\alpha }\left( 1-\lambda _{N,W}^{(l)}\right) }\). Plugging these into Lemma 4.10 and utilizing Lemma 2.3, we get the exponential decay of \(|\widetilde{\varvec{s}}_{N,W}^{(l)}(f)|\), \(|f|\ge W\) as

    $$\begin{aligned} |\widetilde{\varvec{s}}_{N,W}^{(l)}(f)|\le \left\{ \begin{array}{ll}C_7'(W,\epsilon )N^{1/2}e^{-\frac{C_2}{2}N},&{}\quad W\le |f|\le \arccos \left( A-N^{-3/2}\right) /2\pi ,\\ C_8'(W,\epsilon )N^{3/4}e^{-\frac{C_2}{2}N}, &{}\quad \arccos \left( A-N^{-3/2}\right) /2\pi \le |f|\le 1/2, \end{array}\right. \end{aligned}$$

    for fixed l and \(N\ge N_0(W,\epsilon )\). Here \(C_7'(W,\epsilon ) {=} \pi ^{1/2}2^{1/4}\left( 2-\alpha \right) ^{-1/4}\sqrt{C_1(W,\epsilon )}\), \(C_8'(W,\epsilon ) = (2\sqrt{2\alpha }C_1(W,\epsilon ))^{1/2}\), and \(N_0(W,\epsilon )\), \(C_1(W,\epsilon )\) and \(C_2(W,\epsilon )\) are constants as specified in Lemma 2.3.

  2. 2.

    For large N and \(l=\lfloor 2NW(1-\epsilon ')\rfloor , ~\forall ~\epsilon '\in (0,\epsilon ]\): Note that \(\int _B^1\sqrt{\frac{\xi - B}{\left( \xi - A\right) \left( 1 - \xi ^2\right) }}d\xi \) is a decreasing function of B and \(\int _A^1\sqrt{\frac{\xi - A}{\left( \xi - A\right) \left( 1 - \xi ^2\right) }}d\xi = 2W\pi >\frac{l}{N}\pi \). Hence \(1>B>A\). Now we have

    $$\begin{aligned} |g_6(f)|\le |R(f)|\le \frac{1}{\left( A-\cos (2\pi f)\right) ^{1/2}}\le \frac{1}{\left( A-(A-N^{-1})\right) ^{1/2}}\le N^{1/2} \end{aligned}$$

    for all \(\arccos (A-N^{-1})/2\pi \le |f|\le 1/2\). Recall that \(\left| g_5(f)\right| \le = 1\) from (30), \(d_4 = \pi ^{1/2}(1-A^2)^{-1/4}2^{-1/2}N^{1/2}d_6\) and \(d_6 \sim \sqrt{\frac{1-\lambda _{N,W}^{(l)}}{2\pi }}\). Plugging these into Lemma 4.10 and utilizing the bound on \(\lambda _{N,W}^{(l)}\) in Lemma 2.3, we get the exponential decay of \(|\widetilde{\varvec{s}}_{N,W}^{(l)}(f)|\), \(|f|\ge W\) as

    $$\begin{aligned} |\widetilde{\varvec{s}}_{N,W}^{(l)}(f)|\le \left\{ \begin{array}{ll}C_7''(W,\epsilon )N^{1/2}e^{-\frac{C_2}{2}N},&{} W\le |f|\le \arccos [A-N^{-1}]/2\pi ,\\ C_8''(W,\epsilon )N^{1/2}e^{-\frac{C_2}{2}N}, &{} \arccos [A-N^{-1}]/2\pi \le |f|\le 1/2, \end{array}\right. \end{aligned}$$

    for all \(l=\lfloor 2NW(1-\epsilon ')\rfloor , ~\forall ~\epsilon '\in (0,\epsilon ]\) and \(N\ge N_0(W,\epsilon )\). Here \(C_8''(W,\epsilon ) = \sqrt{C_1(W,\epsilon )/2\pi }\), \(C_7''(W,\epsilon ) = 2^{-1}(1-A^2)^{-1/4}\sqrt{C_1(W,\epsilon )}\), and \(N_0(W,\epsilon )\), \(C_1(W,\epsilon )\) and \(C_2(W,\epsilon )\) are constants as specified in Lemma 2.3.

Set

$$\begin{aligned}&C_6(W,\epsilon ) = \max \left\{ C_7'(W,\epsilon ),C_8'(W,\epsilon ),C_7''(W,\epsilon ),C_8''(W,\epsilon )\right\} \\&\quad = \max \left\{ \pi ^{1/2}\left( \frac{2}{2-\alpha }\right) ^{1/4},2^{-1}(1-A^2)^{-1/4}\right\} \sqrt{C_1(W,\epsilon )}. \end{aligned}$$

This completes the proof of Corollary 4.11.  \(\square \)

Lemma 4.12

([37]) For fixed \(W\in (0,\frac{1}{2})\) and \(\epsilon \in (0,\frac{1}{2W}-1)\), \(\widetilde{\varvec{s}}_{N,W}^{(l)}(f)\) and \(\widetilde{\varvec{s}}_{N,\frac{1}{2}-W}^{(N-1-l)}(f)\) satisfy

$$\begin{aligned} \left| \widetilde{\varvec{s}}_{N,W}^{(l)}(f)\right| = \left| \widetilde{\varvec{s}}_{N,\frac{1}{2}-W}^{(N-1-l)}\left( \frac{1}{2}-f\right) \right| \end{aligned}$$

for all \(l\ge 2NW(1+\epsilon )\).

Now we can conclude that \(\langle \varvec{e}_{f}, \varvec{s}_{N,W}^{(l)} \rangle \) decays exponentially in N for all \(l\ge 2NW(1+\epsilon )\) and \(|f|\le W\) by combining the above results.

Corollary 4.13

Fix \(W\in (0,\frac{1}{2})\) and \(\epsilon \in (0,\frac{1}{2W}-1)\). Let \(W' = \frac{1}{2} - W\) and \(\epsilon ' = \frac{W}{\frac{1}{2}-W}\epsilon \). Then

$$\begin{aligned} |\langle \varvec{e}_{f}, \varvec{s}_{N,W}^{(l)} \rangle | = |\widetilde{\varvec{s}}_{N,W}^{(l)}(f)|\le C_6(W',\epsilon ')N^{3/4}e^{-\frac{C_2(W',\epsilon ')}{2}N},\quad \forall |f|\le W~ \end{aligned}$$

for all \(N\ge N_0(W',\epsilon ')\) and all \(l\ge 2NW(1+\epsilon )\). Here, \(C_2(W',\epsilon ')\) and \(N_0(W',\epsilon ')\) are constants specified in Lemma 2.3 with respect to \(W'\) and \(\epsilon '\), and \(C_6(W',\epsilon ')\) is the constant specified in Corollary 4.11 with respect to \(W'\) and \(\epsilon '\).

Proof of Corollary 4.13

Let \(l' = N - 1 - l\). For all \(l\ge 2NW(1+\epsilon )\), we have

$$\begin{aligned} l' = N-1-l \le N-2NW(1+\epsilon ) = 2N\left( \frac{1}{2}-W\right) \left( 1-\frac{W}{\frac{1}{2}-W}\epsilon \right) . \end{aligned}$$

Let \(W' = \frac{1}{2} - W\) and \(\epsilon ' = \frac{W}{\frac{1}{2}-W}\epsilon \in (0,1)\). It follows from from Corollary 4.11 and Lemma 4.12 that

$$\begin{aligned} |\langle \varvec{e}_{f}, \varvec{s}_{N,W}^{(l)} \rangle | = |\langle \varvec{e}_{\frac{1}{2}-f}, \varvec{s}_{N,W'}^{(l')} \rangle |\le C_6(W',\epsilon ')N^{3/4}e^{-\frac{C_2(W',\epsilon ')}{2}N},\quad \forall ~|f|\le W \end{aligned}$$

for all \(N\ge N_0(W',\epsilon ')\). \(\square \)

Recall that \(C_6(W',\epsilon ') = \max \left\{ \pi ^{1/2}\left( \frac{2}{\alpha }\right) ^{1/4},2^{-1}(1-A^2)^{-1/4}\right\} \sqrt{C_1(W',\epsilon ')}\) with \(A = \cos (2\pi W)\) and \(\alpha = 1-A\). As W gets closer to 0 or \(\frac{1}{2}\), the variable \((1-A^2)^{-1/4}\) becomes larger, and we have \((1-A^2)^{-1/4} \rightarrow 1/\sqrt{2\pi W}\) as \(W \rightarrow 0\). Also we have \(\left( \frac{2}{\alpha }\right) ^{1/4} \rightarrow 1/\sqrt{\pi W}\) as \(W \rightarrow 0\). Therefore, for any non-negligible bandwidth which is the main assumption in this paper, the variable \(\max \left\{ \pi ^{1/2}\left( \frac{2}{\alpha }\right) ^{1/4},2^{-1}(1-A^2)^{-1/4}\right\} \sqrt{C_1(W',\epsilon ')}\) would not be too large.

Now, for fixed \(W\in (0,\frac{1}{2})\) and \(\epsilon \in (0,\frac{1}{2W}-1)\), we have

$$\begin{aligned} ||\varvec{e}_f - \varvec{P}_{[\varvec{S}_{N,W}]_k}\varvec{e}_f ||_2^2&= \sum _{l=2NW(1+\epsilon )}^{N-1} |\langle \varvec{e}_{f}, \varvec{s}_{N,W}^{(l)} \rangle |^2 \\&\le \sum _{l=2NW(1+\epsilon )}^{N-1} C_6^2(W',\epsilon ')N^{3/2}e^{-C_2(W',\epsilon ')N}\\&\le C_9(W',\epsilon ')N^{5/2}e^{-C_2(W',\epsilon ')N} \end{aligned}$$

for all \(|f|\le W\) and \(N\ge N_0(W',\epsilon ')\), where \(C_9(W',\epsilon ') = C_6^2(W',\epsilon ')\). \(\square \)

Appendix 10: Proof of Corollary 3.10

Proof

Suppose \(f\in [f_i - W_i, f_i + W_i]\) for some particular \(i\in [J]\). Let \(C_{10}(\mathbb {W},\epsilon ) = \max \{C_{9}(W_i',\epsilon '),\forall i\in [J]\}\) and \(C_{11}(\mathbb {W},\epsilon ) = \min \{C_{2}(W_i',\epsilon '),\forall i\in [J]\}\). It follows from Theorem 3.9 that

$$\begin{aligned} ||\varvec{e}_f - \varvec{P}_{\varvec{\Psi }}\varvec{e}_f||_2^2&\le ||\varvec{e}_f - \varvec{P}_{[\varvec{E}_{f_i}\varvec{S}_{N,W_i}]_{2NW_i(1+\epsilon )}}\varvec{e}_f||_2^2\\&=||\varvec{e}_{f-f_i} - \varvec{P}_{[\varvec{S}_{N,W_i}]_{2NW_i(1+\epsilon )}}\varvec{e}_{f-f_i}||_2^2\\&\le C_9(W_i',\epsilon ')N^{5/2}e^{-C_2(W_i',\epsilon ')N}\le C_{10}(\mathbb {W},\epsilon )N^{5/2}e^{-C_{11}(\mathbb {W},\epsilon )N} \end{aligned}$$

for all \(N\ge N_0(W_i',\epsilon ')\). We complete the proof by setting \(N_2(\mathbb {W},\epsilon ) = \max \{N_0(W_i',\epsilon '),\forall i\in [J]\}\). \(\square \)

Appendix 11: Proof of Theorem 3.11

Proof

Since \(\varvec{x}_0, \varvec{x}_1, \ldots , \varvec{x}_{J-1}\) are independent and zero-mean, we have

$$\begin{aligned}&\mathbb {E}\left[ \left\| \varvec{x}\right\| _2^2\right] =\sum _{n=0}^{N-1}\mathbb {E}\left[ \left| \varvec{x}[n]\right| ^2\right] =\sum _{n=0}^{N-1}\sum _{0\le i,i'\le J-1}\mathbb {E}\left[ \varvec{x}_i[n]\overline{\varvec{x}_i'[n]}\right] \\&\quad =\sum _{n=0}^{N-1}\sum _{i=0}^{J-1}\mathbb {E}\left[ \left| \varvec{x}_i[n]\right| ^2\right] =N\sum _{i=0}^{J-1} \frac{1}{J}=N. \end{aligned}$$

Applying Theorem 2.4, we acquire

$$\begin{aligned} \mathbb {E}\left[ \left\| \varvec{x}_i-\varvec{P}_{\left[ \varvec{E}_{f_i}\varvec{S}_{N,W_i}\right] _{k_i}}\varvec{x}\right\| _2^2\right] = \frac{1}{|\mathbb {W}|}\sum _{l=k_i}^{N-1}\lambda _{N,W_i}^{(l)}. \end{aligned}$$

Note that the power spectrum \(P_{x_i}(F)\) assumed in (18) results in the constant \(\frac{1}{|\mathbb {W}|}\) instead of \(\frac{1}{2W_i}\).

Now, we have

$$\begin{aligned}&\mathbb {E}\left[ \left\| \varvec{x}-\varvec{P}_{\varvec{\Psi }}\varvec{x}\right\| _2^2\right] \\&\quad =\mathbb {E}\left[ \left\| \sum _{i=0}^{J-1}\varvec{x}_i-\varvec{P}_{\varvec{\Psi }}(\sum _{i=0}^{J-1}\varvec{x}_i)\right\| _2^2\right] =\mathbb {E}\left[ \left\| \sum _{i=0}^{J-1}\left( \varvec{x}_i-\varvec{P}_{\varvec{\Psi }}\varvec{x}_i\right) \right\| _2^2\right] \\&\quad =\mathbb {E}\left[ \left( \sum _{i=0}^{J-1}\left( \varvec{x}_i-\varvec{P}_{\varvec{\Psi }}\varvec{x}_i\right) ^H\right) \left( \sum _{i=0}^{J-1}\left( \varvec{x}_i-\varvec{P}_{\varvec{\Psi }}\varvec{x}_i\right) \right) \right] \\&\quad =\mathbb {E}\left[ \sum _{i=0}^{J-1}\left\| \varvec{x}_i-\varvec{P}_{\varvec{\Psi }}\varvec{x}_i\right\| _2^2+\sum _{i=0}^{J-1}\sum _{i'=0,i'\ne i}^{J-1}\left( \varvec{x}_i-\varvec{P}_{\varvec{\Psi }}\varvec{x}_i\right) ^H\left( \varvec{x}_{i'}-\varvec{P}_{\varvec{\Psi }}\varvec{x}_{i'}\right) \right] \\&\quad = \sum _{i=0}^{J-1} \mathbb {E}\left[ \left\| \varvec{x}_i-\varvec{P}_{\varvec{\Psi }}\varvec{x}_i\right\| _2^2\right] + \sum _{i=0}^{J-1}\sum _{i'=0,i'\ne i}^{J-1}\mathbb {E} \left[ \left( \varvec{x}_i-\varvec{P}_{\varvec{\Psi }}\varvec{x}_i\right) ^H\left( \varvec{x}_{i'}-\varvec{P}_{\varvec{\Psi }}\varvec{x}_{i'}\right) \right] \\&\quad = \sum _{i=0}^{J-1} \mathbb {E}\left[ \left\| \varvec{x}_i-\varvec{P}_{\varvec{\Psi }}\varvec{x}_i\right\| _2^2\right] + \sum _{i=0}^{J-1}\sum _{i'=0,i'\ne i}^J\mathbb {E}\left[ \varvec{x}_i^H\varvec{x}_{i'}-\varvec{x}_i^H\varvec{P}_{\varvec{\Psi }}\varvec{x}_{i'}\right] \\&\quad =\sum _{i=0}^{J-1} \mathbb {E}\left[ \left\| \varvec{x}_i-\varvec{P}_{\varvec{\Psi }}\varvec{x}_i\right\| _2^2\right] \le \sum _{i=0}^{J-1} \mathbb {E}\left[ \left\| \varvec{x}_i-\varvec{P}_{[\varvec{E}_{f_i}\varvec{S}_{N,W_i}]_{k_i}}\varvec{x}_i\right\| _2^2\right] \\&\quad = \sum _{i=0}^{J-1}\frac{1}{|\mathbb {W}|}\sum _{l=k_i}^{N-1}\lambda _{N,W_i}^{(l)} \end{aligned}$$

where the equality in the seventh line follows because \(\mathbb {E}\left[ \varvec{x}_{i'}^H\varvec{x}_i\right] =\left( \mathbb {E}\left[ \varvec{x}_{i'}\right] \right) ^H\left( \mathbb {E}\left[ \varvec{x}_i\right] \right) =0\) and \(\mathbb {E}\left[ \varvec{x}_{i'}^H\varvec{P}_{\varvec{\Psi }}\varvec{x}_i\right] =\left( \mathbb {E}\left[ \varvec{x}_{i'}\right] \right) ^H\left( \mathbb {E}\left[ \varvec{P}_{\varvec{\Psi }}\varvec{x}_i\right] \right) =0\) for all \(i',i \in [J], i'\ne i\), and the inequality in the seventh line follows because the column space of \([\varvec{E}_{f_i}\varvec{S}_{N,W_i}]_{k_i}\) is inside the column space of \(\varvec{\Psi }\) for all \(i\in [J]\). \(\square \)

Appendix 12: Proof of Corollary 3.12

Proof

It is useful to express the sampled bandpass signal \(\varvec{x}\) as

$$\begin{aligned} \varvec{x} = \int _{\mathbb {W}}\widetilde{x}(f)\varvec{e}_fdf, \end{aligned}$$
(31)

where we recall that \(\widetilde{x}(f)\) denotes the DTFT of x[n], which is the infinite-length sequence that one obtains by uniformly sampling x(t) with sampling rate \(T_s\).

Now it follows from (31) that

$$\begin{aligned} \left\| \varvec{x}-\varvec{P}_{\varvec{\Psi }}\varvec{x}\right\| _2^2&= \left\| \int _{\mathbb {W}}\widetilde{x}(f)\varvec{e}_fdf - \int _{\mathbb {W}}\widetilde{x}(f)\varvec{P}_{\varvec{\Psi }}\varvec{e}_fdf\right\| _2^2\\&=\left\| \int _{\mathbb {W}}\widetilde{x}(f)(\varvec{e}_{f} - \varvec{P}_{\varvec{\Psi }}\varvec{e}_{f})df\right\| _2^2\\&\le \int _{\mathbb {W}}|\widetilde{x}(f)|^2df \cdot \int _{\mathbb {W}}\Vert \varvec{e}_{f} - \varvec{P}_{\varvec{\Psi }}\varvec{e}_{f}\Vert _2^2df\\&\le \int _{\mathbb {W}}|\widetilde{x}(f)|^2df \cdot C_{10}(\mathbb {W},\epsilon )N^{5/2}e^{-{C}_{11}(\mathbb {W},\epsilon )N}, \end{aligned}$$

where the third line follows from the Cauchy–Schwarz inequality and the last line follows from (17) and the fact that \(\int _{\mathbb {W}}\Vert \varvec{e}_{f} - \varvec{P}_{\varvec{\Psi }}\varvec{e}_{f}\Vert _2^2df\le |\mathbb {W}|\sup _{f\in \mathbb {W}} \Vert \varvec{e}_{f} - \varvec{P}_{\varvec{\Psi }}\varvec{e}_{f}\Vert _2^2\le \sup _{f\in \mathbb {W}} \Vert \varvec{e}_{f} - \varvec{P}_{\varvec{\Psi }}\varvec{e}_{f}\Vert _2^2\). \(\square \)

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Zhu, Z., Wakin, M.B. Approximating Sampled Sinusoids and Multiband Signals Using Multiband Modulated DPSS Dictionaries. J Fourier Anal Appl 23, 1263–1310 (2017). https://doi.org/10.1007/s00041-016-9498-2

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