Constant amplitude zero autocorrelation waveform: Difference between revisions

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In [[signal processing]], a '''Constant Amplitude Zero AutoCorrelation waveform''' (abbreviated '''CAZAC''') is a periodic [[complex number|complex]]-valued [[signal (electrical engineering)|signal]] with modulus one and out-of-phase periodic (cyclic) [[autocorrelation]]s equal to zero. CAZAC sequences find application in wireless communication systems, for example in [[3GPP Long Term Evolution]] for synchronisationsynchronization of mobile phones with base stations. [[Zadoff–Chu sequence]]s are well -known CAZAC sequences with special properties.

{{technical|date=June 2012}}

{{Refimprove|date=March 2009}}

== Example CAZAC Sequence ==

In [[signal processing]], a '''Constant Amplitude Zero AutoCorrelation waveform''' (abbreviated CAZAC) is a periodic [[complex number|complex]]-valued [[signal (electrical engineering)|signal]] with modulus one and out-of-phase periodic (cyclic) [[autocorrelation]] equal to zero. CAZAC sequences find application in wireless communication systems, for example in [[3GPP Long Term Evolution]] for synchronisation of mobile phones with base stations. [[Zadoff–Chu sequence]]s are well known CAZAC sequences with special properties.

For a CAZAC sequence of length <math>N</math> where <math>M</math> is relatively prime to <math>N</math> the <math>k</math>th symbol <math>u_k</math> is given by:<ref>{{Cite journal|last=Chu|first=D.|date=July 1972|title=Polyphase codes with good periodic correlation properties (Corresp.)|journal=IEEE Transactions on Information Theory|volume=18|issue=4|pages=531–532|doi=10.1109/TIT.1972.1054840|issn=1557-9654}}</ref>

In wireless [[CDMA]] technology, the given data encoded and in transmitter side number users assign to unique codes with help comman codes ( ex.. PN sequences, [[pseudorandom noise]]) . In CDMA technology data transmitted under same range of frequency but the users are identified by the unique number, the codes are matched between the particular users than data will transmitted.

===Even N===

The matching between the users is called synchronous CDMA. The coded data sending with constant amplitude and zero auto correlation . The [[constant envelope | constant amplitude]] means the sending and receiving easily without any correction. Zero auto correlation means the correlative variance between the users is zero, so no interaction or interference between the users.

<math>u_k = \exp \left(j \frac{M \pi k^2}{N} \right)</math>

===Odd N===

<math>u_k = \exp \left(j \frac{M \pi k (k+1)}{N} \right)</math>

==Power Spectrum of CAZAC Sequence==

The power spectrum of a CAZAC sequence is flat.

If we have a CAZAC sequence the time ___domain autocorrelation is an impulse

: <math>r(\tau)=\delta(n)</math>

The discrete fourier transform of the autocorrelation is flat

: <math>R(f) = 1/N</math>

Power spectrum is related to autocorrelation by

: <math>R(f) = \left| X(f) \right|^2</math>

As a result the power spectrum is also flat.

: <math>\left| X(f) \right|^2 = 1/N</math>

==References==

{{Reflist}}

==External links==

* [httphttps://www.math.umd.edu/~jjb/cazac/ CAZAC Sequence Generator (Java applet)]

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