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Revision as of 03:20, 2 October 2013

A Zadoff–Chu sequence is a 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 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

x_{u}(n)=e^{-j{\frac {\pi un(n+1+2q)}{N_{\text{ZC}}}}},\,

where

0\leq n<N_{\text{ZC}}-1,\,

0<u<N_{\text{ZC}}\,\land \,{\text{gcd}}(N_{\text{ZC}},u)=1,

q\in \mathbb {Z} ,

N_{\text{ZC}}={\text{length of sequence.}}\,

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 $q=0$ results in a Chu sequence.

Properties of Zadoff-Chu sequences

1. They are periodic with period $N_{\text{ZC}}$ if $N_{\text{ZC}}$ is odd.

x_{u}(n+N_{\text{ZC}})=x_{u}(n)

2. If $N_{\text{ZC}}$ is prime, Discrete Fourier Transform of Zadoff–Chu sequence is another Zadoff–Chu sequence conjugated, scaled and time scaled.

X_{u}[k]=x_{u}^{*}({\tilde {u}}k)X_{u}[0]

where

{\tilde {u}}

is the multiplicative inverse of u modulo

N_{\text{ZC}}

.

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 $u,u=u_{1},u=u_{2}$ , is constant ${\frac {1}{\sqrt {N_{\text{ZC}}}}}$ , provided that $u_{1}-u_{2}$ is relative prime to $N_{\text{ZC}}$ ^[1]

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 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

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http://www.quintillion.co.jp/3GPP/Specs/
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Zadoff Chu (ZC) Sequences

↑ Branislav M, Popovic, “Generalized Chirp-Like polyphase sequences with optimum correlaiton properties”, IEEE tran. Infom. Theory, vol. 38, no. 4, 1992
↑ Evolved Cellular Network Planning and Optimization for UMTS and LTE, Lingyang Song and Jia Shen, CRC Press, 2011, New York

[1] Branislav M, Popovic, “Generalized Chirp-Like polyphase sequences with optimum correlaiton properties”, IEEE tran. Infom. Theory, vol. 38, no. 4, 1992

[2] Evolved Cellular Network Planning and Optimization for UMTS and LTE, Lingyang Song and Jia Shen, CRC Press, 2011, New York

[1]

[2]

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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.
+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 ==
+. 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>
+. 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>
+-->
+. 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.
+. 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]]

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