An Efficient Method for Avoiding Shadow Fading Maps in System Level Simulations

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1 An Efficient Method for Avoiding Shadow Fading Maps in System Leve Simuations Thomas Dittrich, Martin Taranetz, Markus Rupp Christian Dopper Laboratory for Dependabe Wireess Connectivity for the Society in Motion Institute of Teecommunications, Technische Universität Wien Gusshausstrasse 25/389, A-0 Vienna, Austria Emai: {thomas.dittrich, ÖBB Infrastruktur AG Praterstern 3, A-0 Vienna, Austria Abstract In system-eve simuations of wireess communication systems, shadow fading is commony modeed by spatia autocorreation. Currenty, shadow fading vaues are generated using a shadow fading map, which rasterizes the region of interest. This map is commony generated at its fu extent, athough ony a sma part of it may be needed. In this paper we propose an approach for generating shadow fading maps that is aso capabe of generating singe shadow fading vaues. This is achieved by generaizing the shadow fading map to an -dimensiona array of correated random variabes, which is generated by appying a cooring transformation, in form of inear fitering, to an array of uncorreated random variabes. Moreover, in the paper we wi aso discuss the feasibiity of this approach and its appication for generating correated variabes over a continuous curve in space. Index Terms System-Leve Simuations, Shadow Fading Maps, Cooring Transformation, Fourier Transform of Gaussian Arrays I. ITRODUCTIO With each new generation of mobie communication systems, the compexity of the system has increased. This trend is expected to continue in the fifth generation of mobie communication systems (5G). Consequenty, system-eve simuations have become an insurmountabe too for evauating the performance of new schemes before reaizing them. For such simuations it is important to have accurate yet efficient techniques to generate actua instances that represent the effects of the physica channe. These effects are commony modeed as random variabes (RVs). In the particuar case of shadow fading modes, the ognorma shadowing mode is one of the most we accepted modes. Further, there is a broad consensus within the scientific community, that these RVs shoud exhibit a certain spatia correation representing the correation of the bocking objects []. In [2] a mode for such a spatia correation was introduced, stating that the correation decreases exponentiay with distance. The authors verified this mode by measurements in an urban area at 700 MHz and a suburban area at 900 MHz. Such correated shadow fading vaues (SFVs) are frequenty generated by dividing the region of interest (ROI) into a grid, aso denoted as shadow fading map (SFM), with one og-norma SFV for each eement in the grid [3 6]. A straightforward approach to generate such an SFM is to appy a cooring transformation to a set of independent and identicay distributed (iid) Gaussian RVs, which can be obtained by means of the Choesky decomposition (CD). The probem of the CD is that it has a very high computationa compexity and is thus not suitabe for arge SFMs [3]. Therefore, further approaches have been reported in iterature, which aim at combating the high compexity. In [3], the authors introduced the Low-Compexity Approximation of the Choesky Decomposition (LCCD). This approach cacuates a map s of correated vaues by iterativey correating the vaues of an uncorreated map a for every position. The correated variabes are cacuated by taking a set of vaues from s that are adjacent to the currenty considered position with a fixed offset in both spatia directions. This set of neighboring correated vaues is then whitened and stacked with the vaue from a at the currenty considered position. The correated random vaue for this position is then cacuated by a cooring transformation. For both the whitening and the cooring transformation the CD is utiized. This approach is currenty appied, for exampe, in the Vienna LTE-Advanced Downink System-Leve Simuator [7]. In [8], the authors described the modeing of spatia correation for a map of infinite size and continuous space by means of inear fitering. This approach cacuates the spectrum of a inear fiter by taking the square root of the power spectra density (PSD), corresponding to the desired autocorreation function (ACF). The fiter is then appied to a white Gaussian random process (RP) to generate the SFM with the desired ACF. This paper proposes an extension to the Linear Fitering Approach (LFA) in [8] to an -dimensiona, discrete and periodic random process. Our approach yieds a nove way to efficienty generate the spectrum of the SFM without any oss of accuracy as ong as the truncated periodic repetition of the desired autocorreation sequence (ACS) is aso an ACS. We derive the properties of an -dimensiona array of iid Gaussian RVs. By doing so we generate an array of uncorreated RVs directy in the frequency domain and perform the inear fitering aso in the frequency domain. We

2 aso expain how this approach reduces the compexity to one -dimensiona fast Fourier transform (FFT) if ony a sma subset of the map is actuay needed, thus making it obsoete to cacuate a fu SFM. As the resuting map is Gaussian, aso different probems can be soved by the LFA, such as the generation of ine of sight (LOS) modes [9]. A. otation Throughout this paper we wi utiize the foowing notation: ) Vectors wi be bod face and its scaar eements are accessed by a subscripted index, which starts at one. E.g. the -dimensiona vector a C has the eements a i for i {,..., }. 2) For any function f D C with domain D R, the argument wi be encosed by parantheses, i.e. f(x) with x R. For D Z, the argument of the function wi be encosed by brackets, i.e. f[k] with k Z. The same rue appies for RPs over a continuous or discrete space, respectivey. 3) The operators and denote eement-wise mutipication and division. I.e., for -dimensiona vectors a, b C the i-th eement of the resuts are (a b) i a i b i and (a b) i a i /b i. 4) The symbo for the compex unit is j. 5) Rea and Imaginary parts of a compex variabe c a+jb, with a, b R, are denoted by R{c} a and I{c} b. 6) The compex conjugate of a compex variabe c a + jb, with a, b R, is denoted by a superscript. I.e., c a jb. 7) An inner product of two compex vectors a, b C is written as a, b b i a i 8) For the discrete Fourier transform (DFT) and the inverse discrete Fourier transform (idft) the twidde factor is denoted by ω k,n,f e j2π k n,f 9) The convoution of two functions or sequences is denoted by. 0) For some sets S i with i {, 2,..., }, S i denotes the -fod Cartesian product. II. LIEAR FILTERIG APPROACH From a high-eve perspective, the LFA works as foows. Starting with a wide sense stationary (WSS), white, Gaussian process a(x, y) with ACF r a ( x, y) σ 2 δ( x, y) and PSD S a (f x, f y ) σ 2, it is possibe to generate another WSS Gaussian process b(x, y) with ACF r b ( x, y) and PSD S b (f x, f y ). This can be achieved by fitering a(x, y) with a fiter that has the impuse response h(x, y) and frequency response H(f x, f y ). The process b wi then have the ACF r b ( x, y) σ 2 h(x, y) h ( x, y) and PSD S b (f x, f y ) σ 2 H(f x, f y ) 2. A. Theoretic Description The generation of shadow fading maps by means of the LFA was introduced in [8] where the authors modeed the ACF as r x (d) e d α, () with α d D / n(2) and the decorreation distance d D, which indicates the distance, for which the correation has aready decreased to 0.5. The frequency response of the fiter is Sb (f x, f y ) H(f x, f y ). (2) σ The appication of this approach for simuations on a discrete and finite space was expained in [0, Chapter 2] for a one dimensiona RP. In this paper, we extend it to RPs over an - dimensiona space, i.e., each of the RPs a(t) and b(t) takes one RV in C for every continuous position t R. For the generation of an -dimensiona array of correated RVs, the processes a and b are samped with a certain resoution r R +. Further, we ony aow an ROI of hyperrectanguar shape, so that its quantization is a vaid support for the DFT and the resuting array is of finite size. The size of such an ROI is s rn with n and thus the samped processes a[k] a(rk) and b[k] b(rk) with k Z can be modeed as periodic outside the region of interest with a[k+i n] a[k] and b[k+i n] b[k], k, i Z. Within the array, the desired ACS is the samped version of () and thus, for the RP b[k] it is the truncated and periodic repetition of Equation () r b [k] r x (r min i Z k + i n 2 ). (3) Throughout the paper we utiized the foowing definition of the DFT. For a different definition, e.g. the unitary DFT, an appropriate scaing factor has to be appied to the resuts. Definition (-dimensiona DFT). Let x[k], with k Z be a periodic sequence with periodicity n (with eements n i, i {,..., }) and the set of a vaues for k within one period K n the DFT of x[k] is { ni+, ni+ +,..., ni }. Then, X[f] F {x[k]} [f] k K n x[k]ω k,n,f. (4) The idft then formuates as x[f] F {X[f]} [k] X[f]ω f,n,k. (5) f K n n i For the cacuation of the fiter we need a definition of the -dimensiona PSD and a reation to the ACS, which is given in the foowing. Definition 2. Let x[k], k Z be a random process with periodicity n (i.e., x[k] x[k + i n], i Z ), X[f] be the DFT of x[k] and n j be the j-th component of n. The PSD of x[k] is then S x [f] E { X[f] 2 }. n j (6) j This definition is achieved by extending the steps of [0, Chapter 0] to -dimensiona discrete periodic RPs. Simiar

3 to the continuous case, this definition fufis an extension of the Wiener-Khintchine-Einstein Theorem [0, Chapter 0] to -dimensiona discrete periodic processes. Theorem (Extension of Wiener-Khintchine-Einstein to dimensions). Let the random process x[k] be WSS, then its PSD can be cacuated from its ACS r x [k] E {x[n]x [n k]} of x[k] as S x [f] F {r x [k]} [f]. (7) Proof. By inserting Definition into Definition 2 for X[f], we find S x [f] E {x[k]x []} ω k,n,f. k, K n n j (8) j In the case of a WSS, process (8) becomes S x [f] j j n j n j (mk ) r x [k ]ω k,n,f k, K n r x [m]ω m,n,f k,m K n m K n r x [m]ω m,n,f F {r x [m]} [f]. As we ony empoy periodic processes for the SFM, we can cacuate the desired PSD by an -dimensiona FFT S b [f], with f Z. Based on this PSD, we cacuate the frequency response of a fiter which transforms a white process into a coored process with the given PSD. In genera, the - dimensiona frequency response of this fiter is (9) H[f] S b [f]φ[f], (0) where φ[f] is a conjugate symmetric periodic sequence with ampitude one. As we just need one reaization for H we can set the phasor sequence φ to one, without oss of generaity. For the generation of a process with the PSD S b [f] we et a[k] be a WSS, rea and white Gaussian process with variance σ 2 and appy it to the input of the fiter H. The output b[k] of the fiter is again WSS and has a Gaussian distribution with autocorreation σ 2 r b [k]. For an impementation according to [8] we set parameter 2 in the LFA, transform the frequency response of the fiter to the impuse response by an inverse fast Fourier transform (ifft) and fiter the white Gaussian process with this impuse response. The fitering can, for exampe, be performed by a convoution, which is of compexity n 2 i. As the process a[k] is modeed being periodic, the convoution is indeed a circuar convoution and thus it is equivaent to a mutipication of the spectra. In the new impementation fitering is performed by means of a mutipication in the frequency domain, which ote that this discrete PSD is in genera not equa to a samped version of the PSD of the continuous process without periodicity is of compexity n i. Further, we generate the spectrum A[f] F {a[k]} [f] of the white Gaussian process a[k] directy. The whoe SFM is thus obtained as b[k] F { F {r b [k]} [f]a[f]} [k]. () The necessary properties, to generate the spectrum directy, are the distribution and whether there exists a correation between A[f ] and A[f 2 ] for f f 2. For simpifying the notation, the n-periodic repetition of the unit puse, i Z f i n δ n [f] (2) 0, otherwise is appied. The properties of the spectrum are then given by the foowing theorem. Theorem 2 (DFT of normay distributed RVs). Let X[f] be a periodic random process with the properties. ) X[f] is periodic with periodicity n and conjugate symmetric, i.e., X[i n f] X [j n + f], i, j Z. (3) 2) The distribution of X[f] is a compex norma distribution with E {X[f]} 0 E {X[f]X [f]} Γ X σ 2 n i E {X[f]X[f]} C X[f] δ n [2f]σ 2 n i. (4) 3) For a f, f 2 Z X[f] is uncorreated within one period E {X[f ]X [f 2 ]} δ n [f f 2 ]σ 2 n i. (5) Then, the idft x[k] F {X[f]} [k] of this process is a normay distributed random process with zero mean, periodicity n, uncorreatedness within one period and variance σ 2. Further, the DFT of random process with the properties of x[k] fufis properies -3. For the proof of Theorem 2 we need a property of the characteristic function (CF), which is shown in the foowing Lemma Lemma. The CF of the sum of severa random variabes is the product of the individua CF, i.e., for L compex random variabes x C, {,..., L} and weights w C, {,..., L} the CF weight of the inear combination ˆx L w x Φˆx (t) E {exp [jr {t L w x }]} L E {exp [jr {t w x }]} L Φ x (w t) (6)

4 is a scaar function of one compex scaar parameter t. See Appendix A for the proof of Theorem 2. B. Feasibiity Boundary The description of the LFA suggests that it can generate an SFM that perfecty fufis the desired ACS. This is true as ong as r b [k] is an ACS. From Definition 2 it can be concuded that the PSD must be non-negative. However, as ater shown in Section IV, S b [f] F {r b [k]} [f] becomes negative for some frequencies if the resoution r is beow a certain threshod, which impies that r b [k] is not an ACS and thus the LFA becomes infeasibe. In Section IV we find that this feasibiity boundary is decreasing with increasing map size s for square maps. Thus, there exist at east two approaches to tacke this probem of infeasibe combinations of r and s. ) We cacuate the DFT of the sequence, defined in (3) and, if this contains negative vaues, set those to zero, which decreases the accuracy of this approach. 2) We increase the size of the map during the cacuation, such that the resuting PSD is non-negative. Then, we take ony a smaer part of the resuting SFM, which is as big as the desired area. This approach wi then yied the desired ACS without oss of accuracy but increases computationa compexity. C. Pointwise Cacuation and Interpoation The cacuation of the whoe SFM can be omitted if ony a sma amount of SFVs is needed, e.g., for individua receiver ocations. This can be achieved by cacuating ony singe points of the idft in (), which is even more efficient than cacuating the ifft for the whoe map. ote that the number of evauated points has to be sma compared to the map size as the compexity of the FFT for the whoe map is n n 2 og(n n 2 ) and for the cacuation of one singe point it is n n 2. Thus, if the number of required points is arger than og(n n 2 ), it is better to cacuate the whoe map in order to save computation time. In some simuation scenarios it is required to cacuate a SFV at a position that is not a mutipe of the resoution and thus does not ie on the grid of quantized positions, e.g., for a moving mobie with arbitrary speed in a systemeve simuation. To obtain vaues for those points one can set the resoution to ow vaues, quantize the positions or interpoate between neighboring points. Lower vaues for the resoution are not practica since a very sowy moving mobie requires extremey ow vaues, which are not feasibe anymore (see Section II-B). Quantized positions are aso not practica because then a sow-moving mobie coud jump over a whoe pixe in one time instant, which can correspond to a movement of severa meters. Interpoation can, for exampe, be reaized by zero-padding in the frequency domain, which tackes a the probems that where previousy mentioned for the other two approaches. This interpoation by zero-padding is straightforward to impement by repacing the discrete index k in the idft by a continuous position divided by the resoution. III. IMPLEMETATIO For the impementation of the LFA and the LCCD, there are some issues which need some attention. For the LFA, the resuting ACS is periodic. For the LCCD it may happen that not a neighbouring variabes are within the map. In the foowing we expain how these probems are treated within our impementations. We denote ˆn [n, n 2 ] T Z 2 the size of the resuting SFM. A. Linear Fitering Approach Due to the periodicity of the ACS there wi not ony be a high correation between the SFVs b[0, 0] and b[, 0] but aso between b[0, 0] and b[n, 0] when n ˆn. evertheess, the resuting SFM shoud have no periodicity. A simpe way to achieve this goa is generating a map of size n 2ˆn and defining the SFM as being a smaer part of part of size ˆn. The correation within this resuting map can then be found, with m k and m 2 k 2 2, as E{b[k, k 2 ]b[, 2 ]} r b [k, k 2 2 ] r b (r min (m + 2i n ) 2 + (m 2 + 2i 2 n 2 ) 2 ) i,i 2 Z (7) r b (r m 2 + m2 2 ), because n m n and n 2 m 2 n 2 for 0 k, < n and 0 k 2, 2 < n 2. I.e., there is no periodicity within the map. B. Low-Compexity Approximation of the Choesky Decomposition For the cacuation of the correated SFM each position of the map has to be considered one after another. The cacuation of the correated SFV at the current position (x, y) is based on the desired correation matrix for this eement and its neighboring eements with fixed offsets. I.e., when m neighboring vaues are considered and the set of spatia offsets is {(x, y ),..., (x m, y m )}, the position of the neighboring variabes is (x+x i, y +y i ), with i {, 2,..., m}. Regardess of how the offsets are chosen, for certain positions (x, y) the neighbors are not within the map and thus the size of the correation matrix depends on the position. It is very inefficient to cacuate the CD of the correation matrix for every position individuay as it stays constant for a arge number of eements in the midde of the map. Therefore, we introduce one booean variabe for every offset in and set it true if the position with this offset to the current position is within the map. Then, we appy a one to one mapping from this set of booean variabes to an index and store the CD of every possibe correation matrix, so that it can ater be accessed with the cacuated index. E.g., in the case of considering eight neighbors, as proposed in [3], there exist tweve different correation matrices, incuding the case that none of the considered neighbors is within the map. The amount of memory needed is 3496 byte, given that the matrices are stored with doube precision.

5 0 3 s s 80 s MSE Fig. : For square maps of size s s a back point denotes that the resoution r is not feasibe. Back dots denote that a configuration is not feasibe. Simuations run for r {0.0, 0.02,..., 0} and s {2, 3,..., 0} r Fig. 2: Logarithmic MSE of the empirica correation from the point in the center to every other point of the map for three different but fixed map sizes, s {, 80, }. The resoution takes the vaues r {0.0, 0.23, 0.45, 0.67, 0.89,.2,.34,.56,.78, 2.00}. IV. RESULTS In this section we wi anayze the feasibiity of the LFA, as it was expained in Section II-B, for varying map size. Then we wi anayze the LFA in terms of accuracy and compare it to the LCCD in terms of performance. We set parameter α in (). A change in α woud be equa to a change in the resoution thus we can keep this parameter constant during one simuation run and vary ony the resoution without oosing any information. A. Feasibiity Boundary As mentioned in Section II-B the DFT of the desired ACS can become negative and thus the LFA is not feasibe anymore. Figure shows a map of configurations and indicates by back dots whether a configuration is infeasibe. Those configurations are aways square maps of size s s and the resoution is set to r. To generate this figure, s was swept from 2 to 0 in steps of and r was swept from 0.0 to 0 in steps of 0.0. This set of configurations covers the whoe range of s within the boundaries because s can take ony vaues from, but for r it is just a sma subset of the range of a possibe vaues, indeed r R. The three curves show configurations, which resut in maps of the same size in terms of meters. From the figure it can be observed that, at east on this grid of configurations, the feasibiity boundary is decreasing with increasing s, except for the step from s 3 to s 4. For those map sizes the feasibiity boundary increases, but both of which resut in maps that are too sma to be reay usefu and thus they can be negected. It aso suggests that the feasibiity boundary decreases sower than ŝ/s, for a fixed ŝ. An anaytica approach to the feasibiity boundary is subject to future work. B. Accuracy For the measurement of the accuracy, a set of 0 6 SFMs is generated and the correation of the center point and each other point is measured empiricay. In the case that the configuration is infeasibe, negative vaues in the DFT of the desired ACS are set to zero. Figure 2 shows the mean squared error (MSE) of this empiric ACS to the desired ACS. For high resoutions this MSE is ower bounded by noise (the number of repetitions is high, but not infinite) but as expected the error increases and shows a systematic structure when r <.85, r <.09 and r < 0.79 for s, s 80 and s respectivey, i.e., when the configurations are infeasibe. For the smaest resoution that was measured it shows an opposite behavior, which is that the error decreases significanty. This is because the ACS is aready cose to a constant over the whoe map, which means that amost the whoe power is centered at the frequency f f 2 0 and thus the approximation for negative vaues has amost no infuence on the resut. Of course this effect is stronger for sma map sizes because the fuctuations in the ACS are ess than for arger maps which shows up in Figure 2, as for the smaest simuated resoution r 0.0 the error is increasing with increasing map size. But sti a systematic error shows up in the empiricay measured correation of the center point, which is shown in Figure 3, whereas the random error that occurs for high resoution vaues is shown in Figure 4 for s and r 2. Both errors are shown in terms of the squared difference of the empiricay measured correation and the desired ACS. The accuracy of interpoation is measured in the foowing way. First, a square map with s and r 2.5 is generated, which corresponds to a absoute size of 00 m 00 m. And second, this map is interpoated to s 97. This size was chosen

6 SE k x 30 Fig. 3: Squared error of the empiricay measured correation from the center point to every other point of the map. The size of the map is s and the resoution is r 0.0. k y SE k x Fig. 5: Squared error of the empirica correation from the center point to every other point of the map. This map was generated by interpoating a map of size s to s 97, the resoution of the origina map is r k y SE k x 30 Fig. 4: Squared error of the empiricay measured correation from the center point to every other point of the map. The size of the map is s and the resoution is r 2. because it is a prime number and therefore the points of the interpoated map do not exist in the origina map. The MSE of this map is , which is approximatey of the same magnitude as in the infeasibe region without interpoation. The squared error of the empiricay measured ACS to the desired ACS is shown in Figure 5. A justification why such errors (both for the infeasibe region and interpoation) are acceptabe was aready given in [3]: The magnitude of these errors is beow the uncertainty of the exponentia correation mode and thus they can be negected. C. Computation Time A fair comparison of the LFA and the LCCD is not possibe because it coud be that there exists an even more efficient impementation of the LCCD than ours. Indeed, the generation k y of 0 5 maps of size pixe took t LFA s with the LFA (the fiter is cacuated again for every map and nothing is reused) and t LCCD s for an impementation of the LCCD as it was described in Section III-B. Both approaches were impemented in Matab and paraeized over 2 threads on a machine with an Inte Core i7-980 Processor and 24 GB of RAM. For a map of size n n 2 the compexity of the inear fitering approach is in the order of n n 2 og(n n 2 ) and for the LCCD it is in the order of n n 2 which woud suggest that the resut shoud be t LCCD t LFA but there are two reasons that sti justify this resut: ) The first approach does not need significanty more computation except one FFT and one ifft and the second approach needs approximatey n n 2 matrix-vector mutipications of size 8 8 if eight neighbors are considered. Here it is possibe that the time that is needed for those mutipications is higher than og(n n 2 ). 2) The second approach needs an iterative indexing of every eement in the resuting map so that every correated SFV can be cacuated based on the previousy correated SFVs. Such an indexing is inefficient in Matab. V. COCLUSIO This paper presents an efficient way of generating an - dimensiona array of correated random variabes with a given ACS. It directy generates the spectrum of an array of iid Gaussian RVs and performs the fitering in the frequency domain. The properties of such a spectrum are derived in this paper. By knowing the spectrum of the correated array it is then possibe to cacuate singe eements of the array which makes a fu SFM obsoete as ong as the number of cacuated points is ow compared to the ogarithm of the array-size. As the theory is derived for an -dimensiona array, it is possibe to extend the correation modes to aso incude

7 correated variations over time and aso correation among different base stations in system-eve simuations. ACKOWLEDGEMETS This work has been funded by the Christian Dopper Laboratory for Dependabe Wireess Connectivity for the Society in Motion. The financia support by the Austrian Federa Ministry of Science, Research and Economy and the ationa Foundation for Research, Technoogy and Deveopment is gratefuy acknowedged. A. Proof of Theorem 2 APPEDIX Proof. As X[f] is the DFT of x[k], they both must be n- periodic due to the periodicity of the DFT. The remaining proof is spit into two parts: ) Properties of X[f] impy properties of x[k]: Property 2 impies that Γ X C X[f] for δ n [2f], which shows that, in this case, X[f] is a zero mean normay distributed random variabe with variance Γ X. In a the other cases C X[n] is zero and thus X[f] is a zero mean, circuary symmetric, compex normay distributed random variabe with the same variance. For the foowing et K n (i) {f K n δ n [2f] i} with i 0, be a disjoint and exhaustive partitioning of K n, and et e denote the number of even eements in n. The cardinaity of those sets is C () K n () 2 e and C (0) K n (0) n i 2 e. Considering the two cases that e 0 and e > 0 shows that C (0) is aways even, because in the first case both terms of the difference are odd and in the second case both terms are even. This partitioning spits the idft up into two sums n i F {X[f]} [k] X[f]ω f,n,k + f K n () X[f]ω f,n,k, + f K (0) n (8) from which it foows that the first sum consists ony of uncorreated random variabes, whie in the second sum every eement occurs exacty once in its origina form and once as its compex conjugate. ow take two (i 0, ) finite sequences f (i) K (i) n with {0,..., C (i) }, C(i) and X [f (0) ] X [f (0) C (0) sums in (8) as Y (i) C () n i x[k] 0 C () 0 C () 0 X[f (i) 0 {f (i) } K (i) ] and write the eements of the ]ω f (i) Y () C (0) + 0 C (0) Y () C (0) Y () The idft becomes,n,k Y (0) Y (0) + Y (0) 2R {Y (0) }. n (9) A the twidde factors in the first sum are + or and so the distribution of Y () is a zero mean norma distribution with variance Γ X. As mentioned before, the variabes X[f (0) ] are circuary symmetric, so the mutipication with a twidde factor does not change their distribution and thus the variabes 2R {Y (0) } are zero mean and normay distributed with variance 2Γ X. The variabe x[k] is thus aso normay distributed with zero mean and variance σ 2 x n 2 i C () 0 Γ X + C (0) 2 0 2Γ X () Γ X (C () + C (0) ) σ 2 n 2 i The ony thing that is eft in this part is the uncorreatedness of x[k] within one period. Therefore we cacuate the covariance of x[k ] and x[k 2 ] for δ n [k k 2 ] 0. Incuding property 3 we find n 2 i E {x[k ]x [k 2 ]} E f K n σ 2 σ 2 X[f ]X [f 2 ]ω f,n,k ω f2,n,k 2 f 2 K n n i f K n n i δ n [f f 2 ]ω f,n,k ω f2,n,k 2 f 2 K n ω f,n,k k 2 σ 2 n 2 i δ n [k k 2 ] 0, f K n which competes the first part of the proof. (2) 2) Properties of x[k] impy properties of X[f]: The distribution of X[f] can be found by using the resut of [] for the characteristic function of a compex Gaussian RV and Lemma. With Γ E{x[k]x [k]} σ 2 and C E{x[k]x[k]} σ 2 we can mode the norma distribution of x[k] with mean zero and variance σ 2 as a specia form of the compex norma distribution. The CF of each eement of the random process x[k] is then according to [] Φ x[k] (t) exp [ 4 (Γ t 2 + R{Ct t })]. (22) With Lemma and ˆx k Kn x[k]ω k,n,f, the CF of the spectrum X[f] F {x[k]} [f] is Φ X[f] (t) Φˆx (t) k K n exp [ 4 (Γ t 2 + R{Cω 2 k,n,f t t })] exp [ 4 (Γ X t 2 + R{C X[f] t t })], (23) where Γ X Γ n i and C X[f] δ n [2f]C n i. The distribution of the spectrum is thus a norma distribution if δ n [2f] and a circuary symmetric compex norma distribution otherwise. The mean is aways µ X[f] 0 and the

8 variance σ 2 X[f] σ2 n i. For the covariance we find E {X[f ]X [f 2 ]} k K n E {x[k]x []} ω k,n,f ω,n, f2 K n σ 2 ω k,n,f f 2 δ n [f f 2 ]σ 2 n i. k K n (24) This shows that the vaues of X[f] are uncorreated within one period. As the process x[k] is rea, periodic and discrete, it has to fufi the property of being conjugate symmetric, periodic and discrete in the frequency domain, which is X[i n f] X [j n + f], i, j Z. (25) REFERECES [] S. S. Szyszkowicz, H. Yanikomerogu, and J. S. Thompson, On the feasibiity of wireess shadowing correation modes, IEEE Transactions on Vehicuar Technoogy, vo. 59, no. 9, pp , ov. 0. [2] M. Gudmundson, Correation mode for shadow fading in mobie radio systems, Eectronics Letters, vo. 27, no. 23, pp , ov. 99. [3] H. Caussen, Efficient modeing of channe maps with correated shadow fading in mobie radio systems, in 05 IEEE 6th Internationa Symposium on Persona, Indoor and Mobie Radio Communications, vo., Sep. 05, pp [4] M. Ding, M. Zhang, D. López-Pérez, and H. Caussen, Correated shadow fading for ceuar network system-eve simuations with wrap-around, in 5 IEEE Internationa Conference on Communications (ICC), June 5, pp [5] I. Forke, M. Schinnenburg, and M. Ang, Generation of twodimensiona correated shadowing for mobie radio network simuation, WPMC, Sep, vo. 2, p. 43, 04. [6] M. Luo, G. Viemaud, and J.-M. Gorce, Generation of 2D correated random shadowing based on the deterministic MR- FDPF mode, EURASIP Journa on Wireess Communications and etworking, vo. 5, no., pp. 3, 5. [Onine]. Avaiabe: [7] M. Rupp, S. Schwarz, and M. Taranetz, The Vienna LTE- Advanced Simuators: Up and Downink, Link and System Leve Simuation, st ed., ser. Signas and Communication Technoogy. Springer Singapore, 6. [8] R. Fraie, J. F. Monserrat, J. Gozvez, and. Cardona, Mobie radio bi-dimensiona arge-scae fading modeing with site-to-site cross-correation, European Transactions on Teecommunications, vo. 9, no., pp. 0 06, 08. [Onine]. Avaiabe: [9] S. Schwarz, I. Safiuin, T. Phiosof, and M. Rupp, Gaussian modeing of spatiay correated LOS/LOS maps for mobie communications, in IEEE 84th Vehicuar Technoogy Conference, Montrea, Canada, Sep. 6. [0] S. L. Mier and D. Chiders, Probabiity and Random Processes (Second Edition), 2nd ed. Boston: Academic Press, 2. [Onine]. Avaiabe: science/artice/pii/b [] B. Picinbono, Second-order compex random vectors and norma distributions, IEEE Transactions on Signa Processing, vo. 44, no. 0, pp , Oct. 996.