This note discusses the recovery of signals from undersampled data in the situation that such signals are nearly block sparse in terms of an overcomplete and coherent tight frame
Compressed sensing (CS) [
There are two common ways to recover
An alternative to
There are a growing number of practical scenarios in which the transform coefficient
Let
Note that we only require the
In the next section, we give two key lemmas. In Section
We begin with introducing the definition of block
The block
For convenience, in the remainder of this note, we use
From the parallelogram and the definition of block
In order to derive the main results of this note, we will introduce two useful inequalities related to
For all nonnegative integers
Since
For any
The proof is similar to the procedure of the proof of Lemma 1 in [
Note that Lemma
In this section, we establish several block
Let
The proof will follow the ideas from [
Since
Note that
Let
For
Different choices of
Different sufficient conditions.


Recovery condition 



















Though we have only considered bounded noise in Theorem
If the matrix
In Theorem
When the block size
When no noise is present and
This note has presented some conditions of the measurement matrix
This work was supported in part by National 973 Project of China under Grant no. 2013CB329404, Natural Science Foundation of China under Grant nos. 61273020 and 61075054 and Fundamental Research Funds for the Central Universities under Grant no. XDJK2010B005.