Parameter Selection in Periodic Nonuniform Sampling of Multiband Signals

📝 Abstract

Periodic nonuniform sampling has been considered in literature as an effective approach to reduce the sampling rate far below the Nyquist rate for sparse spectrum multiband signals. In the presence of non-ideality the sampling parameters play an important role on the quality of reconstructed signal. Also the average sampling ratio is directly dependent on the sampling parameters that they should be chosen for a minimum rate and complexity. In this paper we consider the effect of sampling parameters on the reconstruction error and the sampling ratio and suggest feasible approaches for achieving an optimal sampling and reconstruction.

💡 Analysis

Periodic nonuniform sampling has been considered in literature as an effective approach to reduce the sampling rate far below the Nyquist rate for sparse spectrum multiband signals. In the presence of non-ideality the sampling parameters play an important role on the quality of reconstructed signal. Also the average sampling ratio is directly dependent on the sampling parameters that they should be chosen for a minimum rate and complexity. In this paper we consider the effect of sampling parameters on the reconstruction error and the sampling ratio and suggest feasible approaches for achieving an optimal sampling and reconstruction.

📄 Content

Abstract—Periodic nonuniform sampling has been considered in literature as an effective approach to reduce the sampling rate far below the Nyquist rate for sparse spectrum multiband signals. In the presence of non-ideality the sampling parameters play an important role on the quality of reconstructed signal. Also the average sampling ratio is directly dependent on the sampling parameters that they should be chosen for a minimum rate and complexity. In this paper we consider the effect of sampling parameters on the reconstruction error and the sampling ratio and suggest feasible approaches for achieving an optimal sampling and reconstruction.

Index Terms — Condition number, nonuniform sampling, multiband signals, sample pattern
I. INTRODUCTION Periodic nonuniform sampling is proposed for sampling of multiband signals and described in the articles [1]-[6]. It has been shown that for sparse spectrum signals the sampling rate can be reduced much lower than Nyquist rate, while retain sufficient information. The average sample ratio is specified by sampling parameters that can be approached to Landau’s lower bound with proper selection of sampling parameters. However, the reduced sampling rates afforded by nonuniform scheme can be accompanied by increased error sensitivity. This paper is going to consider the sampling parameters and their effect on the sample ratio and reconstructed error.
The outline of the paper is as follows. In section II we provide a review of the sampling model and related definitions. Section III discusses the selection of sampling parameters L and p and finding the spectral index set from the band locations. Section IV shows the effect of sample pattern on the reconstructed signal in the presence of noise and provides a feasible algorithm to select a suitable sample pattern. A summary is given in section V. II. SAMPLING MODEL We consider the class of continuous complex-valued multi-band signals of finite energy and maximum frequency of fmax, with band locations that are specified by a subset
ࡲൌڂ ሾܽ௜, ܾ௜ሻ ே ௜ୀଵ of the real line [4][6]. Assume the signal x(t) of this class, is sampled nonuniformly at times
ݐൌሺ݊ܮ൅ܿ௜ሻܶ, ݊א Ժ, 1 ൑݅൑݌, the samples then are categorized into p sequences such that [4],[5] ݔ௜ሾ݊ሿൌ൜ݔሾ݊ܶሿ, ݊ൌ݉ܮ൅ܿ௜ , ݉א Ժ 0, ݋ݐ݄݁ݎݓ݅ݏ݁ (1)

where, ܶ is the base sample time, L is the period of pattern or block length, p is the number of samples are kept in each block L and ࡯ൌሼܿଵ, ܿଶ, … ܿ௣ሽ is the sample pattern [1].
The average sampling rate with choosing T=1/fmax is [5]

                                   ܦ⎯ൌ ቀ

௣ ௅ቁ ݂௠௔௫ (2)

After taking DFT from both sides of (1) and represent in matrix form, the model of sampled signal in the frequency domain and in the interval ܨ଴ൌሾ0, ௙೘ೌೣ ௅ሿ is given by [3]

                                ࢟ሺ݂ሻൌ ۯ ࢠሺ݂ሻ,    ׊ ݂א ܨ଴       (3) 

where y(f) is the known vector of observations as

                            ࢟ሺ݂ሻ௣ൈଵൌ

ۏ ێ ێ ێ ۍܺଵሺ݂ሻ ܺଶሺ݂ሻ ڭ ܺ௣ሺ݂ሻے ۑ ۑ ۑ ې , ׊ ݂א ܨ଴ (4)

that ܺ௜ሺ݂ሻ is the DFT of sequence xi[n], and ࢠሺ݂ሻ is the unknown vector of signal spectrum as

                  ࢠሺ݂ሻ௤ൈଵൌ

ۏ ێ ێ ێ ێ ۍܺሺ݂൅ ௞భ ௅்ሻ ܺሺ݂൅ ௞మ ௅்ሻ ڭ ܺሺ݂൅ ௞೜ ௅்ሻے ۑ ۑ ۑ ۑ ې , ׊ ݂א ܨ଴ (5)

that ܺሺ݂ሻ is the Fourier transform of x(t) and A is the known modulation matrix as

  ۯሺ݅, ݈ሻൌ

ଵ ௅்exp ቀ ௝ଶగ௖೔௞೗ ௅ ቁ, 1 ൑݅൑݌ , 1 ൑݈൑ݍ (6)

The equation (3) relates the unknown spectrum of signal with the sampled data via modulation matrix A.
If the spectrum of signal, ܺሺ݂ሻ, is sliced into L slots and indexed from 0 to L-1 , then the slots that are occupied by
Parameter Selection in Periodic Nonuniform Sampling of Multiband Signals

Moslem Rashidi, Sara Mansouri
Dep. of Signal Systems, Chalmers University of Technology, Goteborg, Sweden
e-mail: moslem@student.chalemrs.se,masara@student.chalmers.se

Fig. 1: Spectrum of multi-band signal is sliced into L=10 slots, the number of active slots is q=4 and the spectral index set is k={2,3,8,9} [6],[3]

part of the signal, are called active slots and their indices collected into a set, k, named spectral index set with length of q such as [6],[3],[4]:

                      ࢑ൌ൛݇ଵ, ݇ଶ, … . ݇௤ൟ,    ݍൌ|࢑|                    (7) 

Fig.1 shows the spectrum of a multi-band signal, sliced into L=10 slots , the number of active slots are q=4 and the spectral index set is k={2,3,8,9}. The unique solution of (3) can be obtained using a left inverse-e.g. the pseudo-inverse of A as [3],[4]

                       ࢠሺ݂ሻൌۯற ࢟ሺ݂ሻ    , ׊ ݂ א ܨ଴                   (8) 

provided that A is full rank and p ≥ q, hence the reconstruction can be obtained in time or frequency domain. III. SAMPLING PARAMETERS (L, p) Selection of sampling parameters L and p can be considered

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