^{1, 2}

^{1}

^{1}

^{1}

^{1}

^{2}

We consider the multigenerator system

For

Gabor analysis is a pervasive signal processing method for decomposing and reconstructing signals from their time-frequency (TF) projections, and Gabor representation is used in many applications ranging from speech processing and texture segmentation to pattern and object recognition, among others. However, as it is widely recognized, a single-windowed Gabor expansion is not enough to analyze the dynamic TF contents of signals that contain a wide range of spatial and frequency components, the resolution of which is normally very poor. Therefore, if one could incorporate a set of multiple windows of various TF localizations in a frame system, the representation of signals of multiple and/or time-varying frequencies would have their corresponding windowing templates and resolutions to relate to. To this purpose, one of the best choices may be the multigenerator Gabor system.

Multigenerator Gabor system is firstly presented by Zibulski and Zeevi in [

Note that

Motivated by [

In this section, we present some notations and lemmas, which will be needed in the rest of the paper. Let

Define

Note that

Let

Given a frame

For fixed positive integer

The following lemma follows from general characterizations of shift-invariant frames, see [

Let

In this section, we provide some sufficient and necessary conditions for a class of the multigenerator Gabor frame system to be a frame for

Firstly, we obtain the following theorem for the multigenerator Gabor system with the parameters

Let

If the Gabor system

If the Gabor system

The part (I) follows from the fact that

Next, we prove the second part. Suppose that the Gabor system

Moreover, we have the following sufficient condition for the multigenerator Gabor system with the parameters

Let

Define

Theorem

The following theorem gives necessary condition for the system

Let

Firstly, note that

The rest part of the proof is by contradiction. Assume that the upper condition in (

Now consider the function

In applications of frames, it is inconvenient that the frame decomposition, stated in [

Let

Let

Define

Next, we prove (

To proceed further, we need use the following symbols. For

Let

For fixed

Note that

Let

If

This work was supported by the National Natural Science Foundation of China (Grant no. 11071152) and the Natural Science Foundation of Guangdong Province (Grant nos. 10151503101000025 and S2011010004511); this research was also partially supported by the Opening Project of Guangdong Province Key Laboratory of Computational Science at the Sun Yat-sen University (Grant no. 201206012).