RECONSTRUCTION OF MULTIBAND SIGNALS FROM NON-INVERTIBLE UNIFORM AND PERIODIC NONUNIFORM SAMPLES USING AN ITERATIVE METHOD Arash Amini and Farokh Marvasti Advanced Communications Research Institute (ACRI) & EE Department, Sharif University of Technology, Tehran, Iran arashsil@ee.sharif.edu, marvasti@sharif.edu 1. EXTENDED ABSTRACT One of the proposed methods for recovery of a band-limited signal from its samples, whether uniform or nonuniform, is the so called frame Method. In this method the original signal is reconstructed by iterative use of sampling-filtering blocks. Convergence of this method for linear invertible op- erators has been previously proved. In this paper we show that this method for non-invertible periodic nonuniform sam- plings as well as non-invertible uniform samples of Bandpass (or multi-band) signals will lead to the Pseudo-Inverse solu- tion. Convergence conditions in case of additive noise will also be discussed. Although uniform sampling at the Nyquist rate for low pass signals is quite straitforward, the extensions to band- pass and multi-band signals are not trivial. The bandwidth for such signals consists of separate positive frequency inter- vals with the overall length of B (thus the Nyquist rate is 2B). Therefore, uniform sampling at the rate 2B is likely to lead aliasing. The minimum alias-free uniform sampling rate for bandpass and multiband signals are not necessarily 2B [1]. One of the old proposed methods for reducing the sampling rate is the second order sampling [2]. In this method the classical uniform sampling points are substituted by two in- terleaved uniform sampling sets. The average sampling rate of 2B can be reached by proper selection of the interleaving parameter. Generalization to N th order samplings (or peri- odic nonuniform sampling) for bandpass and multi-band sig- nals have been studied by [3, 4, 5, 6]. Implementations of periodic nonuniform samplings are usually fulfilled by pass- ing the signal through different delaying filters and then uni- formly sampling each filter output. The idea of using a gen- eral filter bank (instead of delay filters) has been developed by [7, 8]. The proposed reconstruction methods are mainly based on interpolating functions. Since these functions are ban- dlimited, they cannot be timelimited. For practical imple- mentations these functions should be truncated. In many cases these truncations result in considerable errors. The al- ternative methods are iterative approaches. In these methods, by repeated use of a simple but not a perfect reconstruction method, the output gradually converges to the perfect solu- tion [9]. Here we consider the iterative method depicted in Fig. 1. As it is shown, to recover the original signal (x) from its distorted version (y), we repeatedly apply lowpass filtering and sampling denoted by the operator(D). The mathematical formulation of this method is as follows Figure 1: Block diagram of the iterative method where the operator D represents the lowpass filtering and periodic nonuniform sampling operations and the relaxation parame- ter λ 1. x k 1 λ y Dx k x k y Dx x 0 0 (1) For Optimizing the iterative method with respect to the convergence rate, accelerated methods has been proposed [10]. One of the advantages of the iterative method is that when the sampling scheme is non-invertible, it still converges while the interpolating methods diverge. We will show that the converging signal is the pseudo-inverse solution. We show that the sampling-filtering block can be modeled with a Hermitian matrix, thus by considering the (D) operator as a Hermitian matrix and the input and output as vectors, we can rewrite (1) as: k 1 λ k k 0 0 (2) where is the input noise vector. We can summarize the results of the convergence analysis in the following theorem: Theorem 1 The iterative method described in (eq.2) (for a Hermitian distorting matrix D) converges if and only if: 0 λ 2 d max (3) and we have: lim k ∞ k (4) where d max is the maximum eigen value of the matrix( ) and is its pseudo-inverse. For invertible ’s we have 1 . REFERENCES [1] N. L. Scott R. G. Vaughan and D. R. White. The theory of bandpass sampling. IEEE Trans. on Signal Proc., 39(9):1973–1984, 1991.