Toeplitz-Structured Xampling System for Multipulse Signal

Abstract

We consider the Xampling system, under the framework of analog compressive sensing for the multipulse signal with unknown shapes. Several Xampling systems have paved the way for sub-Nyquist sampling of multipulse signal. Unfortunately, for practical application, existing architectures still have some drawbacks such as high complexity, low sampling efficiency and imperfect reconstruction. To the best of our knowledge, a simple and effective Xampling system for multipulse signal is still lacking. In this paper, we propose the Toeplitz-structured Xampling system for the multipulse signal to achieve the simplified system construction and improved system performance. Under the proposed scheme, Toeplitz-structured weight coefficients are used to construct the mixed function. Then, the system filter can be obtained with simplified form and low randomness. It is verified that with this filter, the proposed Xampling system shows low complexity, improved sampling efficiency and good reconstruction effect. In addition, considering both the noise and mismatch, we analyze the reconstruction error of the proposed Xampling system. It is shown that our scheme has bounded error and good robustness.

Appendices

Appendix A: Proof of Proposition III

For the output in (13), we have

$$ y_{l} (m) = (\text{e}^{ - 2\pi iblt} x(t) * s( - t))[WKm] = \int_{ - \infty }^{ + \infty } {\text{e}^{ - 2\pi iblt} x(t)} s(t - WKm) $$

(61)

Since x(t) is supported on [− T/2, T/2] and the shifted versions of s_m(t) are not overlapping between each other in (12), then

$$ \int_{ - \infty }^{ + \infty } {e^{ - 2\pi iblt} x(t)} s(t - WKm) = \int_{{ - \frac{T}{2}}}^{{\frac{T}{2}}} {e^{ - 2\pi iblt} x(t)} s(t - WKm) = \int_{{ - \frac{T}{2}}}^{{\frac{T}{2}}} {e^{ - 2\pi iblt} x(t)} s_{m} (t) $$

(62)

For the structured Xampling system, s_m(t) satisfies (35), then with the m ≤ M

$$ \left\{ {\begin{array}{*{20}l} {s_{m} (t)\; = s_{0} (t - am)\quad \text{if}\quad \left( { - K_{0} + m - 1} \right)a + {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}} \le t \le (K_{0} + 1)a - {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \hfill \\ {s_{m} (t)\; \ne s_{0} (t - am)\quad \text{if}\quad (K_{0} + 1)a - {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}} < t \le K_{0} a + {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}{\text{or}} - K_{0} a} \hfill \\ {\quad - {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}} \le t < \left( { - K_{0} + \Delta m - 1} \right)a + {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \hfill \\ \end{array} } \right. $$

(63)

For $- K_{0} a - {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}} \le t < \left( { - K_{0} + m - 1} \right)a + {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}$, we have

$$ \left\{ {\begin{array}{*{20}l} {s_{m} (t)\; = s_{M} (t + a(M - m))\quad \text{if}\quad \left( { - K_{0} - 1} \right)a + {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}} \le t \le \left( { - K_{0} + m - 1} \right)a + {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \hfill \\ {s_{m} (t)\; \ne s_{M} (t + a(M - m))\quad \text{if}\quad - K_{0} a - {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}} \le t < \left( { - K_{0} - 1} \right)a + {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}\quad } \hfill \\ \end{array} } \right. $$

(64)

Then,

$$ \left\{ {\begin{array}{*{20}l} {s_{m} (t)\; \ne s_{M} (t + a(M - m))} \hfill & {\text{if}\quad - K_{0} a - {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}} \le t < \left( { - K_{0} - 1} \right)a + {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \hfill \\ {s_{m} (t)\; = s_{M} (t + a(M - m))} \hfill & {\text{if}\quad \left( { - K_{0} - 1} \right)a + {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}} \le t \le \left( { - K_{0} + m - 1} \right)a + {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \hfill \\ {s_{m} (t)\; = s_{0} (t - am)} \hfill & {\text{if}\quad \left( { - K_{0} + m - 1} \right)a + {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}} \le t \le (K_{0} + 1)a - {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \hfill \\ {s_{m} (t)\; \ne s_{0} (t - am)} \hfill & {\text{if}\quad (K_{0} + 1)a - {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}} < t \le K_{0} a + {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \hfill \\ \end{array} } \right. $$

(65)

It is known that K₀ is chosen to satisfy the condition

$$ - \frac{{W_{g} }}{2} + (K_{0} + 1)a \ge \frac{T}{2}\quad \text{and}\quad ( - K_{0} - 1)a + \frac{{W_{g} }}{2} \le - \frac{T}{2} $$

(66)

And considering that x(t) is supported on [-T/2, T/2], the values of the s_m(t) on the interval $(K_{0} + 1)a - {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}} < t \le K_{0} a + {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}$ and $- K_{0} a - {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}} \le t < \left( { - K_{0} - 1} \right)a + {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}$ will not influence the output in (62).

Assume that

$$ \left\{ \begin{aligned} \begin{array}{l} s'_{m} (t)\; = s_{0} (t - am)\quad \text{if}\quad \left( { - K_{0} + m - 1} \right)a + {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}} \le t \le K_{0} a + {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}} \hfill \\ s'_{m} (t)\; = s_{M} (t + aM - am)\quad \text{if}\quad - K_{0} a - {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}} \le t \le \left( { - K_{0} + m - 1} \right)a + {\raise0.7ex\hbox{${W_{g} }$} \!\mathord{\left/ {\vphantom {{W_{g} } 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}} \hfill \\ \end{array}\end{aligned} \right. $$

(67)

Then,

$$ \begin{aligned} \int_{{ - \frac{T}{2}}}^{{\frac{T}{2}}} {e^{ - 2\pi iblt} x(t)} s_{m} (t) & = \int_{{ - \frac{T}{2}}}^{{\frac{T}{2}}} {e^{ - 2\pi iblt} x(t)} s'_{m} (t) = \int_{ - \infty }^{ + \infty } {e^{ - 2\pi iblt} x(t)} s'(t - am) \\ & = (e^{ - 2\pi iblt} x(t) * s'( - t))[am] = y'_{l} (m) \\ \end{aligned} $$

(68)

So Proposition III is proven.

Appendix B: Proof of Corollary I

Given the dual window g^*(t) and corresponding bounds A^* and B^*, where A^* is lower bound and B^* is upper bound, we have

$$ A^{*} \left\| {\left| x \right|} \right\|_{2}^{2} \le \sum\limits_{k} {\left| {\left\langle {x,g_{k}^{*} } \right\rangle } \right|}_{2}^{2} \le B^{*} \left\| {\left| x \right|} \right\|_{2}^{2} \quad \text{and}\quad \left\| {\sum\limits_{k} {c_{k} g_{k}^{*} } } \right\|_{2}^{2} \le B^{*} \sum\limits_{k} {\left| {c_{k} } \right|^{2} } $$

(69)

As g′(t) is the approximation of g(t), then

$$ \begin{aligned} & \left\| {x_{c} - \sum\limits_{{k = - K_{0} }}^{{K_{0} }} {\sum\limits_{{l = - L_{0} }}^{{L_{0} }} {\left\langle {x_{c} (t),M_{bl} T_{ak} g^{\prime } (t)} \right\rangle } M_{bl} T_{ak} g^{*} (t)} } \right\|_{2}^{2} \\ & \quad = \left\| {\sum\limits_{{k = - K_{0} }}^{{K_{0} }} {\sum\limits_{{l = - L_{0} }}^{{L_{0} }} {\left\langle {x_{c} (t),M_{bl} T_{ak} g(t)} \right\rangle } M_{bl} T_{ak} g^{*} (t)} - \sum\limits_{{k = - K_{0} }}^{{K_{0} }} {\sum\limits_{{l = - L_{0} }}^{{L_{0} }} {\left\langle {x_{c} (t),M_{bl} T_{ak} g^{\prime } (t)} \right\rangle } M_{bl} T_{ak} g^{*} (t)} } \right\|_{2}^{2} \\ & \quad = \left\| {\sum\limits_{{k = - K_{0} }}^{{K_{0} }} {\sum\limits_{{l = - L_{0} }}^{{L_{0} }} {\left\langle {x_{c} (t),M_{bl} T_{ak} (g(t) - g^{\prime } (t))} \right\rangle } M_{bl} T_{ak} g^{*} (t)} } \right\|_{2}^{2} \\ & \quad \le B^{*} \sum\limits_{{k = - K_{0} }}^{{K_{0} }} {\sum\limits_{{l = - L_{0} }}^{{L_{0} }} {\left| {\left\langle {x_{c} (t),M_{bl} T_{ak} (g(t) - g^{\prime } (t))} \right\rangle } \right|^{2} } } \\ \end{aligned} $$

(70)

Since

$$ \left( {\sum\limits_{{k = - K_{0} }}^{{K_{0} }} {\sum\limits_{{l = - L_{0} }}^{{L_{0} }} {\left| {\left\langle {x_{c} (t),M_{bl} T_{ak} (g(t) - g^{\prime } (t))} \right\rangle } \right|}^{2} } } \right)^{1/2} \le \mu_{1} \left\| {x_{c} (t)} \right\|_{2} $$

(71)

Then we have

$$ \left\| {x_{c} - \sum\limits_{{k = - K_{0} }}^{{K_{0} }} {\sum\limits_{{l = - L_{0} }}^{{L_{0} }} {\left\langle {x_{c} (t),M_{bl} T_{ak} g'(t)} \right\rangle } M_{bl} T_{ak} g^{*} (t)} } \right\|_{2}^{{}} \le \mu_{1} \sqrt {B^{*} } \left\| {x_{c} (t)} \right\|_{{_{2} }}^{{}} $$

(72)

So Corollary I is proven.

Appendix C: Proof of Corollary II

As $\hat{z}_{k,l}^{\prime }$ is the reconstructed Gabor coefficient, the total deviation in reconstruction can be given by:

$$ \begin{aligned} & \left\| {x - \sum\limits_{{k = - K_{0} }}^{{K_{0} }} {\sum\limits_{{l = - L_{0} }}^{{L_{0} }} {\hat{z}'_{k,l} } M_{bl} T_{ak} g^{*} (t)} } \right\|_{2}^{{}} \\ & = \left\| {x - \sum\limits_{{k = - K_{0} }}^{{K_{0} }} {\sum\limits_{{l = - L_{0} }}^{{L_{0} }} {z'_{k,l} } M_{bl} T_{ak} g^{*} (t)} + \sum\limits_{{k = - K_{0} }}^{{K_{0} }} {\sum\limits_{{l = - L_{0} }}^{{L_{0} }} {z'_{k,l} } M_{bl} T_{ak} g^{*} (t)} - \sum\limits_{{k = - K_{0} }}^{{K_{0} }} {\sum\limits_{{l = - L_{0} }}^{{L_{0} }} {\hat{z}'_{k,l} } M_{bl} T_{ak} g^{*} (t)} } \right\|_{2}^{{}} \\ & = \left\| {x - \sum\limits_{{k = - K_{0} }}^{{K_{0} }} {\sum\limits_{{l = - L_{0} }}^{{L_{0} }} {\left\langle {x_{c} (t),M_{bl} T_{ak} g'(t)} \right\rangle } M_{bl} T_{ak} g^{*} (t)} + \sum\limits_{{k = - K_{0} }}^{{K_{0} }} {\sum\limits_{{l = - L_{0} }}^{{L_{0} }} {(z'_{k,l} - \hat{z}'_{k,l} )} M_{bl} T_{ak} g^{*} (t)} } \right\|_{2}^{{}} \\ \end{aligned} $$

(73)

In Eq. (58), we have

$$ \begin{aligned} & \left\| {x - \sum\limits_{{k = - K_{0} }}^{{K_{0} }} {\sum\limits_{{l = - L_{0} }}^{{L_{0} }} {\left\langle {x_{c} (t),M_{bl} T_{ak} g'(t)} \right\rangle } M_{bl} T_{ak} g^{*} (t)} + \sum\limits_{{k = - K_{0} }}^{{K_{0} }} {\sum\limits_{{l = - L_{0} }}^{{L_{0} }} {(z'_{k,l} - \hat{z}'_{k,l} )} M_{bl} T_{ak} g^{*} (t)} } \right\|_{2}^{{}} \\ & \quad \le \left\| {x - \sum\limits_{{k = - K_{0} }}^{{K_{0} }} {\sum\limits_{{l = - L_{0} }}^{{L_{0} }} {\left\langle {x_{c} (t),M_{bl} T_{ak} g'(t)} \right\rangle } M_{bl} T_{ak} g^{*} (t)} } \right\|_{2}^{{}} + \mu_{2} \sigma_{N} \sqrt {B^{*} } \\ \end{aligned} $$

(74)

In Eq. (49) and Corollary I, we have

$$ \begin{aligned} & \left\| {x - \sum\limits_{{k = - K_{0} }}^{{K_{0} }} {\sum\limits_{{l = - L_{0} }}^{{L_{0} }} {\left\langle {x_{c} (t),M_{bl} T_{ak} g'(t)} \right\rangle } M_{bl} T_{ak} g^{*} (t)} } \right\|_{2}^{{}} \\ & \quad = \left\| {x - x_{c} + x_{c} - \sum\limits_{{k = - K_{0} }}^{{K_{0} }} {\sum\limits_{{l = - L_{0} }}^{{L_{0} }} {\left\langle {x_{c} (t),M_{bl} T_{ak} g'(t)} \right\rangle } M_{bl} T_{ak} g^{*} (t)} } \right\|_{2}^{{}} \\ & \quad \le \left\| {x - x_{c} } \right\|_{2}^{{}} + \mu_{1} \sqrt {B^{*} } \left\| {x_{c} (t)} \right\|_{{_{2} }}^{{}} \\ & \quad \le \sqrt {\varepsilon_{\varOmega } } \left\| {x(t)} \right\|_{2}^{{}} + \mu_{1} \sqrt {B^{*} } \left\| {x_{c} (t)} \right\|_{{_{2} }}^{{}} \\ \end{aligned} $$

(75)

Then, the total deviation can be expressed as:

$$ \left\| {x - \sum\limits_{{k = - K_{0} }}^{{K_{0} }} {\sum\limits_{{l = - L_{0} }}^{{L_{0} }} {\hat{z}'_{k,l} } M_{bl} T_{ak} g^{*} (t)} } \right\|_{2}^{{}} \le \sqrt {\varepsilon_{\varOmega } } \left\| {x(t)} \right\|_{2}^{{}} + \mu_{1} \sqrt {B^{*} } \left\| {x_{c} (t)} \right\|_{{_{2} }}^{{}} + \mu_{2} \sigma_{N} \sqrt {B^{*} } $$

(76)

So Corollary II is proven.