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[[File:Zadoff Chu 7 353.png|thumb|500px|Plot of a Zadoff-Chu sequence for u=7, N=353]]


A '''Zadoff–Chu sequence''' is a [[Complex number|complex-valued]] mathematical [[sequence]] which, when applied to [[radio]] signals, 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 [[Orthogonality|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.
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The complex value at each position ''n'' of each root Zadoff–Chu sequence parametrised by ''u'' is given by
 
: <math> x_u(n)=e^{-j\frac{\pi un(n+1+2q)}{N_\text{ZC}}}, \, </math>
 
where
 
: <math> 0 \le n < N_\text{ZC}-1, \,</math>
 
: <math> 0 < u < N_\text{ZC}\,\and \, \text{gcd}(N_\text{ZC},u)=1, </math>
 
: <math> q \in \Z, </math>
 
: <math> N_\text{ZC} =  \text{length of sequence.}\,</math>
 
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 <math> q = 0 </math> results in a [[Chu sequence]].
 
== Properties of Zadoff-Chu sequences ==
1. They are [[Periodic function|periodic]] with period <math>N_\text{ZC} </math> if <math>N_\text{ZC} </math> is odd.
 
: <math> x_u ( n + N_{\text{ZC}} )= x_u(n) </math>
 
2. If <math>N_\text{ZC} </math> is prime, [[Discrete Fourier Transform]] of Zadoff–Chu sequence is another Zadoff–Chu sequence conjugated, scaled and time scaled.
 
:  <math> X_{u}[k]= x_{u}^{*}(\tilde{u}k) X_{u}[0]</math> where <math>\tilde{u}</math> is the multiplicative inverse of u modulo <math>N_\text{ZC} </math>.
<!--
proof :
 
<math> x_{u}^{*}(\tilde{u}k) x_{u} (\tilde{u}k) = 1 </math>
 
By definition of multiplicative inverse, <math> u \tilde{u} = l N_\text{ZC} + 1  </math> where l is an integer.
 
Therefore <math> e^{-j\pi\frac{u \tilde{u} 2nk }{N_\text{ZC}}} = e^{-j\pi\frac{(lN_\text{ZC} +1) 2nk }{N_\text{ZC}}} = e^{-j\pi\frac{2nk }{N_\text{ZC}}} </math>
 
<math> \quad X_{u}[k] = \sum_{n=0}^{N_\text{ZC}-1} x_u ( n ) e^{-j\frac{2\pi nk}{N_\text{ZC}}} </math>
 
<math> \quad \quad    = \sum_{n=0}^{N_\text{ZC}-1}
                        e^{-j\pi\frac{un(n+1)}{N_\text{ZC}}}
                        e^{-j\pi\frac{2nk}{N_\text{ZC}}} </math>
 
Expanding and inserting <math> u \tilde{u} </math>
 
<math> \quad \quad    = \sum_{n=0}^{N_\text{ZC}-1}
                        e^{-j\pi\frac{un(n+1)}{N_\text{ZC}}}
                        e^{-j\pi\frac{u \tilde{u}2nk }{N_\text{ZC}}} </math>
 
<math> \quad \quad    = \sum_{n=0}^{N_\text{ZC}-1}
                        e^{-j\pi\frac{un(n+1)+u \tilde{u}2nk}{N_\text{ZC}}}  </math>
 
<math> \quad \quad    = \sum_{n=0}^{N_\text{ZC}-1}
                        e^{-j \pi u \frac{n^2 + 2n \tilde{u} k + n}{N_\text{ZC}}}  </math>
 
<math> \quad \quad    = \sum_{n=0}^{N_\text{ZC}-1}
                        e^{-j \pi u \frac{ (n + \tilde{u} k) (n + \tilde{u} k + 1) - \tilde{u} k (\tilde{u} k +1)}{N_\text{ZC}}}  </math>
 
<math> \quad \quad    = x_{u}^{*}(\tilde{u}k)  \sum_{n=0}^{N_\text{ZC}-1} x_{u} (\tilde{u}k+n) </math>
 
Since the summation over a period of a periodic signal is its DC term <math> X_{u}[0]  </math>
 
<math> X_{u}[k]= x_{u}^{*}(\tilde{u}k) X_{u}[0]</math>
-->
 
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[[cross correlation| correlation]] between two prime length Zadoff–Chu sequences, i.e. different values of <math>u, u=u_1, u=u_2 </math>, is constant <math> \frac{1}{\sqrt{N_\text{ZC}}} </math>, provided that <math> u_1 - u_2 </math> is relative prime to <math> N_\text{ZC} </math> <ref>Branislav M, Popovic, “Generalized Chirp-Like polyphase sequences with optimum correlaiton properties”, IEEE tran. Infom. Theory, vol. 38, no. 4, 1992</ref>
 
== Usages ==
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 [[3GPP Long Term Evolution|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 code]]s used in [[UMTS]] because they result in a constant-amplitude output signal, reducing the cost and complexity of the radio's power amplifier.<ref>Evolved Cellular Network Planning and Optimization for UMTS and LTE, Lingyang Song and Jia Shen, CRC Press, 2011, New York</ref>
 
== References ==
{{reflist}}
*http://www.quintillion.co.jp/3GPP/Specs/
*{{cite journal
| author = S. Beyme and C. Leung
| title = Efficient computation of DFT of Zadoff-Chu sequences
| journal = Electron. Lett.
| volume = 45 | issue = 9 | pages = 461–463 | year = 2009
| doi = 10.1049/el.2009.3330
}}
*[http://www.globalspec.com/reference/68075/203279/9-2-Zadoff-Chu-ZC-Sequences Zadoff Chu (ZC) Sequences]
 
{{DEFAULTSORT:Zadoff-Chu Sequence}}
[[Category:Radio communications]]

Latest revision as of 21:32, 12 January 2015


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