# Carrier interferometry

Carrier interferometry

Carrier Interferometry (CI) is a type of spread spectrum multiple access typically employed with Orthogonal frequency-division multiplexing (OFDM). CI spreading codes are commonly used to spread data symbols across multiple OFDM subcarriers for diversity benefits and to shape the resulting superposition of coded subcarriers for reducing the Peak-to-Average Power (PAPR) of the transmitted signal.

Mathematical Description

In wireless multicarrier spread-spectrum, spreading is performed across orthogonal subcarrier frequencies to produce a transmit signal expressed byx = F-1Sbwhere F-1 is an inverse DFT, S is a spread-OFDM code matrix, and b is the transmitted symbol vector. The inverse DFT typically employs an over-sampling factor, so its dimension is "KxN" (where "K" > "N" is the number of time-domain samples per OFDM symbol block), whereas the dimension of the spread-OFDM code matrix is "NxN".

At the receiver, the received spread-OFDM signal is expressed byr = HF-1Sb,where H represents a channel matrix. Since the use of a cyclic prefix in OFDM changes the Toeplitz-like channel matrix into a circulant matrix, the received signal may be represented by

r = F-1ΛHFF-1Sb

= F-1ΛHSb

where the relationship H = F-1ΛHF is from the definition of a circulant matrix, wherein ΛH is a diagonal matrix whose diagonal elements correspond to the first column of the circulant channel matrix H. The receiver employs a forward DFT to produce

y = ΛHSb.

In conventional multicarrier spread-spectrum, such as MC-CDMA and Spread-OFDM employing Hadamard-Walsh spreading codes, multipath distortions cause correlations between the spreading codes, resulting in loss of orthogonality in the code space. These correlations are substantially uniform across the entire code space, resulting in each code subspace contributing interference to all of the other code subspaces. Thus, a receiver that employs multi-user detection needs to estimate and cancel interference due to transmissions in all active code subspaces other than the code subspace for a signal of interest.

CI spreading codes can be used to localize code subspace correlations in the presence of multipath. Such codes can greatly simplify multi-user detection. In the trivial case, S = I, where I is the identity matrix, gives regular OFDM without spreading. However, it is advantageous to identify spreading code matrices S that commute with ΛH. The following relationship establishes the necessary conditions for S to commute with ΛH:

r = F-1ΛHFF-1(ΛCF)b,

where S = ΛCF, and C is a circulant matrix defined by C = F-1ΛCF, where ΛC is the circulant’s diagonal matrix. Thus, the received signal r can be written as

r = F-1ΛHΛCFb = F-1ΛCΛHFb,

and the despread signal, prior to equalization, is expressed by y = ΛC-1Fr.In a particularly simple case, the spreading matrix S = ΛCF may be implemented with ΛC = I, such that the spreading matrix is an "NxN" DFT matrix. This “Circulant-Identity” case is a simple form of CI. Since OFDM’s over-sampled DFT is "KxN", the basic CI spreading matrix may simply be a sinc pulse-shaping filter, which maps each data symbol to a cyclically shifted (orthogonally positioned) superposition of OFDM subcarriers. The actual cyclic CI spreading matrix is expressed by C = F-1ΛCF. Other versions of CI, including other pulse shapes, may be produced by selecting different diagonal matrices ΛC.

History

* 1999: US patent [5,955,992] [http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&p=1&u=%2Fnetahtml%2FPTO%2Fsearch-bool.html&r=12&f=G&l=50&co1=AND&d=PTXT&s1=shattil.INNM.&OS=IN/shattil&RS=IN/shattil] , "Frequency-shifted feedback cavity used as a phased array antenna controller and carrier interference multiple access spread-spectrum transmitter"
* 1999: "Introduction of carrier interference to spread spectrum multiple access," Wireless Communications and Systems, 1999 Emerging Technologies Symposium 12-13 April 1999 Page(s):4.1 - 4.5.
* 2001: Multi-Carrier Technologies for Wireless Communication [http://www.amazon.com/dp/0792376188] , Springer; 1st edition (November 1, 2001)

ee also

* Subcarrier multiplexing

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