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

80 MC-CDMA and MC-DS-CDMASince the variances are assigned to real-valued noise, w (k) is multiplied by a factor of 2.When applying Walsh–Hadamard codes as spreading codes, the property can be exploitedthat the product C (g)lC (k)∗l,l = 0,...,L− 1, in half of the cases equals −1 and in theother half equals +1 ifg ≠ k. Furthermore, when assuming that the realizations b (k) =+1and b (k) =−1 are equally probable, the LLR for MC-CDMA systems with single-userdetection results in [27, 29]∣ L−1∑ ∣∣∣∣2G∣ l,l H l,lƔ (k) l=0= ⎛∣(K − 1) ⎝ 1 L−1∑L−1|G l,l H l,l | 2 1 ∑ ∣∣∣∣2 ⎞ w (k) . (2.86)−GL∣ l,l H l,l⎠ + σ 2 L−1∑|G l,l | 2L2l=0l=0When MMSE equalization is used in MC-CDMA systems, Equation (2.86) can be approximatedby [27]Ɣ (k) = 4 ∑|H l,l |w (k) , (2.87)Lσ 2 L−1l=0since the variance of G l,l H l,l reduces such that only the noise remains relevant.The gain with soft decision decoding compared to hard decision decoding in MC-CDMA systems with single-user detection depends on the spreading code length and isin the order of 4 dB for small L (e.g. L = 8) and reduces to 3 dB with increasing L (e.g.L = 64). This shows the effect that the spreading averages the influence of the fadingon a data symbol. When using LLRs instead of the soft decided information w (k) ,theperformance further improves up to 1 dB [26].Maximum Likelihood DetectionThe LLR for coded MC-CDMA mobile radio systems with joint detection based onMLSSE is given by( p(r | bƔ (k) (k) )=+1)= lnp(r | b (k) (2.88)=−1)and is inherently delivered in the symbol-by-symbol estimation process presented inSection 2.1.5.2. The set of all possible transmitted data vectors d µ where the consideredcoded bit b (k) of user k is equal to +1 is denoted by D (k)+ . The set of all possibledata vectors where b (k) is equal to −1 is denoted by D (k)− . The LLR for MC-CDMAsystems with MLSSE results in [27, 29]Ɣ (k) = ln⎛⎜⎝∑∀d µ ∈D (k)+∑∀d µ ∈D (k)−l=0(exp − 1 ) ⎞σ 2 2 (d µ , r)(exp − 1 ), (2.89)⎟σ 2 2 (d µ , r) ⎠where 2 (d µ , r) is the squared Euclidean distance according to Equation (2.44).