Zadoff Chu episode

Plot of a Zadoff-Chu sequence for u = 7, N = 353 (real part above, imaginary part below)

A Zadoff-Chu sequence (ZC), also Chu sequence or Frank-Zadoff-Chu sequence (FCZ), is a complex-valued mathematical sequence that has a constant amplitude and whose cyclic autocorrelation results in zero. Because of these properties, Zadoff-Chu sequences are also referred to as CAZAC sequences ( English constant amplitude zero autocorrelation waveform ). Applied to the propagation of an electromagnetic signal, no interference occurs even with multipath propagation, since the delayed ZC sequences arriving at the receiver are orthogonal to one another. They are named after Solomon A. Zadoff and David C. Chu.

Calculation of the terms of the sequence

The calculation of the kth term of the Zadoff-Chu sequence with parameter M (often also referred to as the root) depends on whether the length of the sequence is even or odd. The following applies to straight lengths: ${\displaystyle N}$

${\displaystyle x_{M}(k)={\text{exp}}\left(j{\frac {\pi Mk^{2}}{N}}\right),\qquad k=0,1,\dotsb ,N-1}$

for odd lengths:

${\displaystyle x_{M}(k)={\text{exp}}\left(j{\frac {\pi Mk(k+1)}{N}}\right),\qquad k=0,1,\dotsb ,N-1}$

the parameter M must be selected so that M and N are relatively prime . A new Zadoff-Chu sequence is thereby created for each parameter M. Through a conjunction of the entire sequence, a shift in the sequence or a phase change in the form

${\displaystyle {\text{exp}}\left(j{\frac {2\pi qk}{N}}\right),\qquad q\in \mathbb {Z} }$

However, nothing changes in the correlation properties of the sequence. This can also be used to describe the consequences more generally in this form:

${\displaystyle x_{M}(k)={\text{exp}}\left(-j{\frac {\pi Mk(k+c_{\text{f}}+2q)}{N}}\right),\qquad c_{\text{f}}=N{\bmod {2}}}$

use

Zadoff-Chu sequences are used in the LTE mobile communications standard for the primary synchronization signal (PSS), random access preamble (PRACH), uplink control channel (PUCCH), uplink traffic channel (PUSCH) and sounding reference signals (SRS). By using orthogonal Zadoff-Chu sequences for each base station (eNodeB) , interference between the cells can be reduced. Zadoff-Chu sequences are more favorable compared to the Walsh-Hadamard codes in UMTS , since they have a constant amplitude and thus make lower demands on the linearity of the transmitter amplifier.

Individual evidence

1. ↑ Hans-Jurgen Zepernick, Adolf Finger: Pseudo Random Signal Processing: Theory and Application . John Wiley & Sons, July 17, 2013, ISBN 978-1-118-69121-2 .
2. ↑ Patent US3099796 : Phase coded communication system. Registered on November 27, 1957 . ‌
3. ↑ ^{a b} D. Chu: Polyphase codes with good periodic correlation properties (Corresp.) . In: IEEE Transactions on Information Theory . tape 18 , no. 4 , July 1972, p. 531-532 , doi : 10.1109 / TIT.1972.1054840 .
4. ↑ Lingyang Song, Jia Shen: Evolved Cellular Network Planning and Optimization for UMTS and LTE. CRC Press, New York 2011.