  The autocorrelation of a Zadoff-Chu (ZC) sequence with a non-zero cyclically
shifted version of itself is zero. Due to the interesting property, ZC
sequences are widely used in the LTE air interface in the primary
synchronization signal (PSS), random access preamble (PRACH), uplink control
channel (PUCCH) etc. However, this interesting property of ZC sequence is not
useful in the random access initial uplink synchronization problem due to some
specific structures of the underlying problem. In particular, the state of the
art uplink synchronization algorithms do not perform equally for all ZC
sequences. In this work, we show a systematic procedure to choose the ZC
sequences that yield the optimum performance of the uplink synchronization
algorithms. At first, we show that the uplink synchronization is a sparse
signal recovery problem on an overcomplete basis. Next, we use the theory of
sparse recovery algorithms and identify a factor that controls performance of
the algorithms. We then suggest a ZC sequence design procedure to optimally
choose this factor. The simulation results show that the performance of most of
the state of the art uplink synchronization algorithms improve significantly
when the ZC sequences are chosen by using the proposed technique.
