  The analysis sparsity model is a very effective approach in modern Compressed
Sensing applications. Specifically, redundant analysis operators can lead to
fewer measurements needed for reconstruction when employing the analysis
$l_1$-minimization in Compressed Sensing. In this paper, we pick an eigenvector
of the Zauner unitary matrix and -- under certain assumptions on the ambient
dimension -- we build a spark deficient Gabor frame. The analysis operator
associated with such a spark deficient Gabor frame, is a new (highly) redundant
Gabor transform, which we use as a sparsifying transform in Compressed Sensing.
We conduct computational experiments -- on both synthetic and real-world data
-- solving the analysis $l_1$-minimization problem of Compressed Sensing, with
four different choices of analysis operators, including our Gabor analysis
operator. The results show that our proposed redundant Gabor transform
outperforms -- in all cases -- Gabor transforms compared to state-of-the-art
window vectors of time-frequency analysis.
