Zadoff–Chu sequenceFrom Wikipedia, the free encyclopedia

Plot of a Zadoff-Chu sequence for u=7, N=353

A Zadoff–Chu sequence is a complex-valued mathematical sequence which, when applied to radiosignals,

gives rise to an electromagnetic signal of constant amplitude, whereby cyclically shifted versions of the

sequence imposed on a signal result in zero correlation with one another at the receiver. A generated

Zadoff–Chu sequence that has not been shifted is known as a "root sequence".

These sequences exhibits the useful property that cyclically shifted versions of itself are orthogonal to one

another, provided, that is, that each cyclic shift, when viewed within the time domain of the signal, is greater

than the combined propagation delay and multi-path delay-spread of that signal between the transmitter

and receiver.

The complex value at each position n of each root Zadoff–Chu sequence parametrised by u is given by

where

Zadoff–Chu sequences are CAZAC sequences (constant amplitude zero

autocorrelation waveform).

They're named after Solomon A. Zadoff and D. C. Chu. Note that the special

case   results in a Chu sequence.

Properties of Zadoff-Chu sequences[edit]

1. They are periodic with period   if   is odd.

2. If   is prime, Discrete Fourier Transform of Zadoff–Chu sequence is

another Zadoff–Chu sequence conjugated, scaled and time scaled.

 where   is the multiplicative inverse of u modulo  .

3. The auto correlation of a prime length Zadoff–Chu sequence with a

cyclically shifted version of itself is zero, i.e., it is non-zero only at one

instant which corresponds to the cyclic shift.

4. The cross correlation between two prime length Zadoff–Chu sequences,

i.e. different values of  , is constant  ,

provided that   is relative prime to  [1]

Usages[edit]

Zadoff–Chu sequences are used in the 3GPP LTE Long Term Evolution air

interface in the Primary Synchronization Signal (PSS), random access

preamble (PRACH), uplink control channel (PUCCH), uplink traffic channel

(PUSCH) and sounding reference signals (SRS). By

assigning orthogonal Zadoff–Chu sequences to each LTE eNodeB and

multiplying their transmissions by their respective codes, the cross-

correlation of simultaneous eNodeB transmissions is reduced, thus

reducing inter-cell interference and uniquely

identifying eNodeB transmissions. Zadoff–Chu sequence improve over

the Walsh–Hadamard codes used in UMTS because they result in a

constant-amplitude output signal, reducing the cost and complexity of the

radio's power amplifier.[2]

References[edit]

1. Jump up^ Branislav M, Popovic, “Generalized Chirp-Like polyphase

sequences with optimum correlaiton properties”, IEEE tran. Infom.

Theory, vol. 38, no. 4, 1992

2. Jump up^ Evolved Cellular Network Planning and Optimization for

UMTS and LTE, Lingyang Song and Jia Shen, CRC Press, 2011, New

York

http://www.quintillion.co.jp/3GPP/Specs/

S. Beyme and C. Leung (2009). "Efficient computation of DFT of

Zadoff-Chu sequences". Electron. Lett. 45 (9): 461–

463. doi:10.1049/el.2009.3330.

Zadoff Chu (ZC) Sequences