Contents I. Introduction
In many digital signal processing applications, it is usually of interest to speed up the sampling rate of analog/digital (A/D) converter. One of methods to achieve this purpose is to use the two-channel sampling methods such as interlaced sampling method, derivative sampling method and Hilbert transform sampling method [1]–[3]. Fig. 1 shows the block diagram of Hilbert transform sampling method, where is called the sampling period. The ideal frequency response of continuous-time Hilbert transformer (CTHT) is given by \begin{equation*} H_{c} (\omega )={{\begin{cases} {-j} & {\omega >0} \\ j & {\omega <0} \\ \end{cases}}} \tag{1}\end{equation*}
Clearly, the phase in positive frequency band is shifted by and the phase in negative frequency band is shifted by . Let the continuous-time Fourier transform pairs of be defined by
\begin{align*} X(\omega )&= \int _{-\infty }^\infty {x(t)e^{-j\omega t}} dt\tag {2a}\\ x(t)&= \frac {1}{2\pi }\int _{-\infty }^\infty {X(\omega )e^{j\omega t}} d\omega \tag {2b}\end{align*}
then the conventional Hilbert transform sampling theorem in [1] [3] is described below: If the is band-limited to ; that is, for , then the can be completely reconstructed from and its Hilbert transform samples in terms of
\begin{align*} x(t)\;{=}\;& \sum \limits _{n=-\infty }^\infty \left \{{{ {\begin{array}{c} x(nT_{s})\cos \left ({{{\frac {\sigma }{2}(t-nT_{s})}}}\right ) \\[2pt] \,\,-x_{H} (nT_{s})\sin \left ({{{\frac {\sigma }{2}(t-nT_{s})}}}\right ) \\[2pt] \end{array}}}}\right \} \\[2pt]& \times \text {sinc}\left ({{{\frac {\sigma (t-nT_{s})}{2\pi }}}}\right ) \tag {3}\end{align*}
where and for , 1 for . Defining two continuous-time functions as
\begin{align*} b_{0} (t)=\text {sinc}\left ({{{\frac {\sigma t}{2\pi }}}}\right )\cos \frac {\sigma t}{2}\quad b_{1} (t)=-\text {sinc}\left ({{{\frac {\sigma t}{2\pi }}}}\right )\sin \frac {\sigma t}{2} \\[2pt] \tag {4}\end{align*}
then equation (3) can be rewritten as
\begin{equation*} x(t)\!=\!\sum \limits _{n=-\infty }^\infty {x(nT_{s})b_{0} (t-nT_{s})} \!+\!x_{H} (nT_{s})b_{1} (t-nT_{s})\tag {5}\end{equation*}
The reconstruction method 1 in Fig. 1 can be realized by using (5). On the other hand, some researchers have invested significant efforts to develop fractional order signals and systems. The research topics include fractional order transform, fractional order differentiator and integrator, fractional delay filter and fractional Hilbert transformer [4]–[27]. The sampling and reconstruction of signals for the fractional Fourier transform and linear canonical transform have been investigated in [8]–[10]. Moreover, the fractional Hilbert transformer (FHT) is a generalized version of the conventional Hilbert transformer by introducing a fractional order to control the phase response [11]–[21]. So far, several typical methods have been presented to design FHTs including the analytical closed-form FIR and all-pass filter methods [13]–[17], Peano kernel method [20], and non-causal IIR FHT design with equiripple or flat phase response [21]. These methods have their unique features. The optical and micro-wave implementation of FHT can be found in [22] [23]. Due to the progress of the FHT design, it is interesting to study the research topic of fractional Hilbert transform sampling method whose block diagram is shown in Fig. 2. The ideal frequency responses of continuous-time fractional Hilbert transformers are given by
\begin{equation*} H_{\ell } (\omega )={{\begin{cases} {e^{-j\phi _{\ell }}} & {\omega >0} \\ {e^{j\phi _{\ell }}} & {\omega <0} \\ \end{cases}}} \tag {6}\end{equation*}
Clearly, the phase in positive frequency band is shifted by and the phase in negative frequency band is shifted by . If and are chosen, then and . In this case, fractional Hilbert transform sampling in Fig. 2 reduces to the Hilbert transform sampling in Fig. 1. Therefore, the sampling method in Fig. 2 is a generalization of the method in Fig. 1. Now, the reconstruction problem of fractional Hilbert transform sampling is described below. Given two discrete-time fractional Hilbert transform samples and in Fig. 2, how can the original continuous-time signal be recovered completely from these given samples? This problem will be studied in this paper and its filter bank implementation will be also presented. This paper is organized as follows. In Section II, a frequency-domain analysis method is applied to derive the fractional Hilbert transform sampling theorem such that the reconstruction method 2 in Fig. 2 can be obtained. In Section III, an analog filter bank method is presented to recover the original continuous-time signal from the discrete-time sampled signals. Because the analog filter bank is not easy to be implemented, a digital filter bank method is proposed to solve the reconstruction problem. To reduce the arithmetic implementation complexity, a sparse design of digital reconstruction filters is also studied. Finally, a conclusion is drawn.
(a) The two-channel Hilbert transform sampling method, where is an analog Hilbert transformer. (b) The reconstruction method of Hilbert transform sampling.
(a) The two-channel fractional Hilbert transform sampling method, where and are analog fractional Hilbert transformers with phase angles and . (b) The reconstruction method of fractional Hilbert transform sampling.
