System Level Modeling and Optimization of the LTE Downlink

3. Physical Layer Modeling and LTE System Level Simulation3.1.1.5. Shadow FadingShadow fading is modeled to represent the deviations from the average pathloss valuesdue to geographical features such as terrain changes or buildings. It is generallymodeled as a log-normal distribution with zero mean, which although could also betreated as a time-dependent process, is preferable to treat as position-dependent dueto the convenience of storing it in map form.A typical standardized cell layout sets a log-normal distribution with a standarddeviation of 10 dB, as well as an inter-site correlation of 0.5 [69]. Since the shadowfading is interpreted as geographical variations, as sectors share the same site (i.e.,geographical location), an inter-sector correlation factor of one is assumed.In order to introduce spatial correlation to the points on each map, a method basedon the Cholesky decomposition of the correlation matrix R is employed. Thismethods allows us to introduce spatial correlation to an uncorrelated log-normallydistributedvector.Given an initial vector a of length K with a correlation matrix R a = E { aa H}equal to the identity matrix of size K (I K ), a correlated vector s with a predefinedcorrelation matrix R s can be obtained by performings = L s a, (3.17)where L s is the lower-triangular Cholesky decomposition of R s and the correlationmatrix of s is E { ss H} = L s L H s = R s .The values of the correlation coefficients in R s follow an exponential model wherecorrelation diminishes with distance, expressed as r(x) = e −αx [84, 85], with thedistance x in meters, and a typical value for α of 1/20 [86].As with the pathloss map, the shadow fading map is stored as pixels, with eachpixel representing a square of p × p meters. Thus, for a map of size M × N pixels,a correlation matrix of size M · N × M · N is required. As an example, the pathlossmap in Figure 3.7 encompasses an area of 2 080 × 2 402 m, with a resolution of5 m/pixel, resulting in a 416 × 481 matrix. The resulting correlation matrix wouldbe 200 096 × 200 096, requiring around 300 GB of memory for the correlation matrixR s alone assuming double-precision storage (8 bytes/value).In order to reduce complexity, an extension of the method proposed in [87] is employed.In order to calculate the value of the space-correlated value s n , just a setof neighboring pixels is taken into account. Since the correlated pixels have to begenerated following a certain order, only the previously processed pixels are takeninto account for the calculation of s n . For a neighbor count of 12 and a row-wiseprocessing order, the set of neighboring pixels s n−1 . . . s n−12 is depicted in Figure 3.836