Multi-Carrier and Spread Spectrum Systems: From OFDM and MC ...

MC-CDMA 59i.e. the spreading is performed over all sub-carriers [7]. Thus, the resulting scheme is asingle-carrier system with cyclic extension and frequency domain equalizer. This schemehas a dynamic range of single-carrier systems. The computational efficient implementationof the more general case where the FFT spreading is performed over groups of subcarriersthat are interleaved equidistantly is described in Reference [8]. A comparisonof the amplitude distributions between Hadamard codes and Fourier codes shows thatFourier codes result in an equal or lower peak-to-average power ratio [9]. Fourier codesare applied for spreading in DFT-spread OFDM introduced in Chapter 3 and applied inthe uplink of LTE.Pseudo Noise (PN) Spreading CodesThe property of a PN sequence is that the sequence appears to be noise-like if theconstruction is not known at the receiver. They are typically generated by using shiftregisters. Often used PN sequences are maximum-length shift register sequences, knownas m-sequences. A sequence has a length ofn = 2 m − 1 (2.17)bits and is generated by a shift register of length m with linear feedback [44]. The sequencehas a period length of n and each period contains 2 m−1 ones and 2 m−1 − 1 zeros; i.e. itis a balanced sequence.Gold CodesPN sequences with better cross-correlation properties than m-sequences are part of theso-called Gold sequences [44]. A set of n Gold sequences is derived from a preferred pairof m-sequences of length L = 2 n − 1 by taking the modulo-2 sum of the first preferredm-sequence with the n cyclically shifted versions of the second preferred m-sequence. Byincluding the two preferred m-sequences, a family of n + 2 Gold codes is obtained. Goldcodes have a three-valued cross-correlation function with values {−1, −t(m),t(m)− 2},where{2t(m) =(m+1)/2 + 1 for m odd2 (m+2)/2 + 1 for m even . (2.18)Golay CodesOrthogonal Golay complementary codes can recursively be obtained by[ ]CL/2 CC L =L/2, ∀L = 2 m , m ≥ 1, C 1 = 1, (2.19)C L/2 −C L/2where the complementary matrix C L is defined by reverting the original matrix C L .Ifand A L and B L are L × L/2 matrices, thenC L = [A L B L ], (2.20)C L = [A L − B L ]. (2.21)