Asymptotically perfect reconstruction in hybrid filter banks

The generalized sampling theorem of Papoulis [1], states that a deterministic continuous-time bandlimited signal x t ( ) is uniquely determined from the outputs of M linear systems, sampled at 1 M the Nyquist rate. The concept of multi-channel sampling of Brown, formulates the same property in terms of M analysis and M synthesis analog filters constituting an analog filter bank [2]. In this formulation, the signal band is divided into M equal subbands. Then, the transfer function of the synthesis filters in each subband is obtained as a function of the analysis filters by solving a linear system of equations. The generalized sampling theorem has been also extended to twodimensional signals [3] and linear T-periodic timevarying systems [4]. The generalized sampling theorem has been believed ill-posed because if one of the synthesis filters has infinite energy and the input signal is corrupted by additive noise, the output noise energy will not be bounded [5]. Nevertheless, it has been shown that if all of the synthesis filters have finite energy, the generalized theorem is well-posed, e.g.; the output noise energy is bounded [6]. This property has been further used to optimize the analysis filters for the reconstruction of oversampled noisy signals [7]. The generalized sampling theorem has been also demonstrated using a digital approach [8] and a relation between digital and analog filter banks has been established [9]. Digital filter banks have been studied extensively [10] and several design techniques with different objectives have been proposed. Here, we study the signal reconstruction problem in hybrid filter banks composed of analog and digital filters. Because of flexibility and simplicity of design of digital filter banks, we will study hybrid filter banks by referring them to the digital filter of Fig.1. In all of the following sections, the quantization noise is neglected.