A Wideband Space-Time MIMO Channel Simulator Based on the Geometrical One-Ring Model

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1 A Wideband Space-Time MIMO Channel Simulator Based on the Geometrical One-Ring Model Matthias Pätzold and Bjørn Olav Hogstad Faculty of Engineering and Science Agder University College 4898 Grimstad, Norway {matthias.paetzold, Abstract In this paper, we extend the geometrical one-ring multi-input multi-output (MIMO) channel model with respect to frequency-selectivity. Our approach enables the design of efficient and accurate simulation models for wideband space-time MIMO channels under isotropic scattering conditions. Two methods will be provided to compute the parameters of the simulation model. Especially, the space, time, and frequency correlation properties of the proposed wideband space-time MIMO channel simulator are studied analytically. It is shown that any given discrete or continuous power delay profile (PDP) can be incorporated in the simulation model. The high accuracy of the simulation model is demonstrated by comparing its statistical properties with those of the underlying reference model. Our procedure provides an important framework for developers of future wideband mobile communication systems to test and to verify new high data rate transmission concepts employing, e.g., space-time coded MIMO orthogonal frequency division multiplexing (OFDM) techniques. I. INTRODUCTION Currently, a number of standardization bodies supported by industries and research institutes are trying to establish new system standards for future high-speed wireless local area networks (WLANs) and 4th generation wireless systems employing MIMO-OFDM techniques. One important characteristic feature of future mobile communication systems is that they have much larger bandwidths than today s systems. The channel models that have been developed for the nd and 3rd generation mobile systems may therefore not be applicable in 4th generation systems []. For the design, optimization, and test of future wideband mobile systems, new MIMO channel reference and simulation models are required that sufficiently approximate the temporal, spatial, and frequency correlation properties of realistic frequency-selective space-time MIMO channels. Geometrical channel models with a ring of scatterers around the user [], [3] or around both the transmitter and the receiver [4], [5] have received significant attention in the past due to their simplicity. However, these channel models are frequencynonselective, which limits their usefulness to narrowband MIMO systems. In this paper, we show how the well-known one-ring model [], [3] can be extended with respect to frequency-selectivity. The wideband extension of the narrowband one-ring model was also the topic in [6], where the utility of the circular ring model has been demonstrated by fitting the time of arrival distribution and the PDP to empirical data. In our paper, we propose a new wideband extension of the onering model, which is capable of incorporating any given specified or measured PDP. Two methods are presented for the computation of the model parameters of the resulting simulation model. Closed-form expressions will be provided for the temporal autocorrelation function (ACF), the twodimensional (D) space cross-correlation function (CCF), and the frequency correlation function (FCF) of the simulation model as well as the reference model. This allows us to access the performance of the simulation model analytically by comparing its correlation properties with those of the reference model. It is shown that the designed MIMO channel simulator matches the underlying reference model exactly with respect to the FCF and a nearly perfect fitting can be achieved with respect to the temporal ACF and the D space CCF. The paper is structured as follows. Section II is devoted to a brief review of a known frequency-nonselective MIMO channel model derived from the geometrical one-ring model. Its extension to frequency-selectivity is new and the main topic of Section III. In Section IV, we will present two methods for the computation of the model parameters. Section V studies the temporal, spatial, and frequency correlation properties and demonstrates the performance of the proposed MIMO channel simulator. Finally, Section VI draws the conclusion. II. REVIEW OF THE ONE-RING MODEL A MIMO narrowband (frequency-nonselective) channel based on the geometrical one-ring scattering model was first proposed in [] and has further been developed in [7] and [3]. In this paper, we show how this model can be extended with respect to frequency-selectivity. For ease of comprehension, we redraw the geometrical one-ring model in Fig.. It is assumed that the base station (BS) is the transmitter and the mobile station (MS) is the receiver. The BS and the MS are equipped with M BS transmit and M MS receive antennas, respectively. The antenna element spacings at the BS and the MS are designated by δ BS and δ MS, respectively, and the multielement antenna tilt angles are denoted by α BS and α MS. The angle of motion is α v and the quantity φ BS max designates one half of the maximum angle of departure seen at the BS. In [3], it was shown that the time-variant complex channel gain, denoted by g pq (t), which describes the link from the /6/$ (c) 6 IEEE

2 δ BS A BS BS A BS α BS BS φn D n D n BS φ max D S n D n A MS Dn y φn MS α A MS v α MS Fig.. Geometrical model (one-ring model) for an elementary MIMO channel with local scatterers around the MS. qth transmit antenna to the pth receive antenna of the underlying flat fading M BS M MS MIMO channel model, can be expressed as where g pq (t) = N N n= a n,q = e jπ(3 q) δ BS λ b n,p = e jπ(3 p) δ MS λ f n = f max cos ( n α v ) MS δ MS a n,q b n,p e j(πfnt+θn) () [ ] cos(α BS)+φ BS max sin(α BS )sin(φms n ) () cos(φms n α MS) (3) for p =,,...,M MS and q =,,...,M BS. In the above equations, N is referred to as the number of exponential functions, λ is the carrier s wavelength, and f max is called the denotes the angle of arrival (AoA) of the nth incoming wave seen at the MS. The phases θ n in () are constants, as they are considered as outcomes of independent and identically distributed (i.i.d.) random variables with a uniform distribution over [, π). maximum Doppler frequency. The quantity n III. EXTENSION OF THE ONE-RING MODEL WITH RESPECT TO FREQUENCY-SELECTIVITY In the model described by (), it was assumed that the propagation delays τ n of all N incoming waves are approximately equal and small in comparison to the data symbol duration T s, i.e., τ max = max{ τ n} N n= T s. However, in wideband transmission systems, T s is much smaller than in narrowband systems. In such cases, the propagation delay differences cannot be neglected in comparison to the data symbol duration T s, and, as a consequence, the channel becomes frequency selective. To describe a frequency-selective channel, we refer to the discrete PDP S τ (τ )= v R x (4) c l δ(τ τ l) (5) l= or, equivalently, its Fourier transform, which is known as the FCF r τ (υ )= c le jπτ l υ. (6) l= Here, c l and τ l are the (amplitude) attenuation factor and the propagation delay of the lth path, respectively, and L is the number of different propagation paths. We assume that a PDP of the form (5) is given according to a specification or obtained from measurement data. This allows us to consider S τ (τ ) and r τ (υ ) as the PDP and FCF of a reference model, respectively. The problem is now the extension of the flat fading MIMO channel model in () with respect to frequency-selectivity in such a way that the FCF of the extended model is sufficiently close to a reference FCF in (6). To solve this problem, we observe from Fig. that the minimum and maximum propagation delays are given by τ min = D/c and τ max = τ min +R/c, respectively, where c is the speed of light. Without loss of generality, we can set τ min to, since the minimum delay τ min is common to all paths and can thus be neglected. Hence, the (relative) minimum and (relative) maximum propagation delays of the one-ring model are given by τ min = (7) τ max = R/c (8) respectively. Obviously, the ring radius R can be chosen such that τ L = τ max =R/c holds. In order to extend the one-ring channel model to the general case of a frequency-selective channel model, we partition the ring of scatterers into L pairs of segments I l (l =,,...,L) limited by ϕ l and ϕ l as illustrated in Fig.. Note that Fig.. I I ϕ ϕ y ϕ L ϕ L I L ϕ L ϕ L I L Partition of a ring of scatterers into L segment pairs. the number of segment pairs equals the number of different propagation paths L. Each pair of segments will be assigned to a single discrete propagation delay according to a fixed rule. To establish this rule, we express the propagation delay τ n of I L I L x

3 the nth path as τ n = R D + D + R +DR cos( n ) R c [ +cos(φ MS n ) ] = τ max +cos(φms n ) (9) where we have used (8) and the inequality D R. Solving this equation for n gives ( n = arccos τ n ) τ max. () Using this result, we can now establish the relationship between the given delays τ l in (5) and the angles ϕ l limiting the segments I l [see Fig. ] as follows ( ϕ l = arccos τ l ) τ L. () Recall that the AOAs n can be computed by using a proper parameter computation method, e.g., such as those described in [3], whereas the propagation delays τ l are given according to a specification (or measurements). Next, if n is within the interval I l, i.e., ϕ l < n ϕ l, l =, 3,...,L () then we perform the following assignments: c τ n τ l (3) a n,q a n,q,l (4) b n,p b n,p,l (5) f n f n,l (6) θ n θ n,l (7) where n =,,...,N and l =,,...,L. This result can be interpreted as follows. All propagation delays τ n of the onering model, which are within the range τ l < τ n τ l,are gathered together and assigned to a single discrete propagation delay τ l for all l =,,...,L. Here, τ n (n =,,...,N) denotes the actual propagation delay given by the nth scatterer S n, whereas τ l (l =,,...,L) denotes the lth delay related to the specified discrete channel profile of the reference model. In this way, we partition the N local scatterers S n into L pairs of segments. Let us denote N l as the number of scatterers within the lth segment pair, then L l= N l = N must hold. Obviously, the lth segment pair is related to the attenuation factor c l of the discrete PDP. By partitioning the ring of N scatterers into L segment pairs, each consisting of N l scatterers, and performing the assignments listed in (3) (7), the complex channel gain g pq (t) [see ()] can be extended to the impulse response of a frequency-selective M BS M MS MIMO channel model as follows h pq (τ,t) = l= c l Nl N l n= a n,q,l b n,p,l e j(πf n,lt+θ n,l ) δ(τ τ l). (8) Another possibility to incorporate frequency-selectivity in the one-ring model is given when multiple rings of scatterers around the MS with different radii are assumed. This assumption is more realistic, especially in combination with clusters of scatterers. Using the techniques described above, the development of a multiple-ring model with clusters of scatterers is straightforward and easy to implement. IV. PARAMETER COMPUTATION METHODS The model parameters to be determined are the AOAs n. In the following, we present two methods, which can be used to compute n on the assumption of isotropic scattering. A. The Generalized Method of Exact Doppler Spread The method of exact Doppler spread (MEDS) [8] has recently been developed further in [9]. According to this socalled generalized MEDS (GMEDS q ), the model parameters n are given by n = qπ ( n ) +, n =,,...,N (9) N where q {,, 3, 4} and is called the angle of rotation. Here, we choose q = 4, so that the scatterers are located equally spaced on the ring around the MS. To maximize the performance of the GMEDS 4, we define the angle of rotation as := φms n n = π 4 N. () Note that the GMEDS 4 provides a simple closed-form solution to the parameter computation problem. Its excellent performance will be demonstrated in Section V. B. The L p -Norm Method The L p -norm method (LPNM) is described in detail in [8]. The application of this method to the determination of the model parameters n requires the minimization of the following error function (L -norm) E rgpq = τ max τ max r gpq (τ) r gpq (τ) dτ / () where r gpq (τ) and r gpq (τ) are the temporal ACF of the reference model and the simulation model, respectively, and τ max defines the upper limit of the domain [,τ max ] over which the approximation of r gpq (τ) is of interest. If all scatterers are located on a ring, then a proper value for τ max is given by N/(4f max ). The expression for the temporal ACFs r gpq (τ) and r gpq (τ) will be provided in Subsection V-A. A (local) minimum of the error function in () can be found by using numerical optimization techniques. The performance of the LPNM will be studied in the next section.

4 V. CORRELATION PROPERTIES In this section, we study the statistical properties of the proposed space-time MIMO channel simulator. Of special interest are the temporal ACF, the D space CCF, and the FCF. Since the proposed simulation model is deterministic, its correlation properties have to be determined by time averages instead of ensemble averages. A. The Temporal ACF The temporal ACF r gpq (τ) of the time-variant complex channel gain r gpq (t) is defined as r gpq (τ) :=< g pq (t + τ) g pq(t) > () where < > denotes the time average operator. Substituting () in () gives r gpq (τ) = N N e jπfnτ (3) n= where f n = f n ( n ) is given by (4). Note that the above result is independent of p {,,...,M MS } and q {,,...,M BS }, i.e., the channel gains g pq (t) of all links are described by the same temporal ACF. When using the GMEDS 4, one can show that in the limit N, the temporal ACF r gpq (τ) approaches lim r g pq (τ) =r gpq (τ) =J (πf max τ) (4) N where r gpq (τ) denotes the temporal ACF of the reference model and J ( ) is the zero-order Bessel function of the first kind. The performance of the GMEDS 4 and the LPNM using N = 8 can be studied in Fig. 3, where the resulting temporal ACFs r gpq (τ) of the simulation model are shown in comparison to the ACF r gpq (τ) of the reference model. Both methods enable an excellent fitting in the interval from to τ max = N/(4f max ). In our following studies, we will explicitly use the GMEDS 4, since this method provides a closed-form solution which is almost as good as the LPNM. B. The D Space CCF The D space CCF ρ(δ BS,δ MS ) of g (t) and g (t) is defined as ρ(δ BS,δ MS ):=< g (t) g (t) >. (5) Substituting () in (5) results in the closed-form expression ρ(δ BS,δ MS )= N N a n,(δ BS )b n,(δ MS ). (6) n= When using the GMEDS 4, it can be shown that ρ(δ BS,δ MS ) ρ(δ BS,δ MS ) as N, where ρ(δ BS,δ MS ) denotes the D The time average of a waveform x(t) is defined as < x(t) > := T lim T T x(t)dt. T Temporal autocorrelation function.5 Reference model Simulation model (GMEDS 4, N = 8) Simulation model (LPNM, N = 8) Normalized time lag, τ f max Fig. 3. The temporal ACFs r gpq (τ) (reference model) and r gpq (τ) (simulation model) for isotropic scattering environments (αv =). space CCF of the reference model for which the following equation can be derived ρ(δ BS,δ MS ) = e jπ δ BS λ cos(α BS ) J (π{(φ BS maxδ BS /λ) sin (δ BS )+φ BS maxδ BS δ MS /λ sin(α BS )sin(α MS )+(δ MS /λ) } / ). (7) The above result has been obtained by using [, Eq. ( )] and [, Eq. (9.6.3)]. If the transmit and receive antennas are perpendicular to the x-axis, i.e., α BS = α MS = π/, then the expression in (7) reduces to ρ(δ BS,δ MS )=J (π ( φ BS δ BS max λ + δ MS λ )). (8) For this special case, the plots of the D space CCFs ρ(δ BS,δ MS ) and ρ(δ BS,δ MS ) are illustrated in Figs. 4 and 5, respectively. Here, the simulation model has been designed by using the GMEDS 4 with N =8. Obviously, there is no visual difference between ρ(δ BS,δ MS ) and ρ(δ BS,δ MS ). C. The FCF The FCF r τ (υ ) of the wideband MIMO channel simulator is defined as r τ (υ ):=< H pq (f + υ,t) H pq(f,t) > (9) where H pq (f,t) denotes the simulation model s time-variant transfer function, which is the Fourier transform of the impulse response h pq (τ,t) with respect to the propagation delays τ. Taking the Fourier transform of h pq (τ,t) in (8) with respect to τ results in H pq (f c l N l,t)= a n,q,l b n,p,l e j[π(f n,lt τ l υ )+θ n,l ]. Nl l= n= (3) Now, substituting (3) in (9) and averaging over time gives r τ (υ )= c l e jπτ l υ. (3) l=

5 δm S /λ δbs /λ Propagation delay profile, Sτ (τ ) D space cross-correlation function Simulation model R e fe re n c e m o d e l S im u la tio n m o d e l Fig. 4. The D space CCF ρ (δbs, δms ) of the simulation model (GMEDS4, N = 8, αbs = αms = π/, φbs max = ) Fig. 6. The PDPs Sτ (τ ) (reference model) and S τ (τ ) (simulation model) according to the 8-path HIPERLAN/ model C []. Reference Model δm S /λ 3 δbs /λ Fig. 5. The D space CCF ρ(δbs, δms ) of the reference model (N, αbs = αms = π/, φbs max = ). Frequency correlation function, rτ (υ ) D space cross-correlation function 4 Propagation delay, τ (ns) R e fe re n c e m o d e l S im u la tio n m o d e l Frequency separation, υ (MHz) To obtain the above result, we have to impose on the simulation model that the identity fn, = fm,k holds iff n = m and = k. This boundary condition is always fulfilled when the GMEDS4 is used with the angle of rotation according to (). It is worth mentioning that the FCF of the simulation model equals that of the reference model. This statement follows directly from a comparison of (3) and (6). As a consequence, the simulation model has exactly the same discrete PDP as the reference model. An application of our approach is illustrated in Fig. 6, which shows the PDP of the 8-path HIPERLAN/ model C [] corresponding to typical large open areas. Figure 7 presents the absolute value of the resulting FCF of both the simulation model and the reference model. If the PDP of the reference model is given in a continuous form, then we first have to find the discrete PDP, e.g., by using one of the methods described in [3]. After the discrete PDP has been determined, one can proceed with the techniques described above. VI. C ONCLUSION The well-known narrowband one-ring MIMO channel model has been extended with respect to frequency-selectivity. Two methods (GMEDS4 and LPNM) have been proposed for the computation of the model parameters of the proposed Fig. 7. Absolute value of the FCFs rτ (υ ) (reference model) and r τ (υ ) (simulation model) according to the 8-path HIPERLAN/ model C []. wideband MIMO channel simulator. In case of isotropic scattering, the GMEDS4 provides a closed-form solution, whereas the LPNM requires numerical optimization techniques. Our investigations have shown that both methods have nearly the same performance. Closed-form solutions have also been presented for the temporal ACF, D space CCF, FCF, and PDP of the proposed simulation model as well as for the corresponding reference model, which is obtained when the number of scatterers approaches infinity. It was shown that the proposed wideband MIMO channel simulator can be designed in such a way that its PDP equals that of any specified discrete PDP. This fact allows us to conclude that the FCF of the simulation and reference model are also identical. In addition, the temporal ACF and the D space CCF of the simulation model are extremely close to the respective correlation functions of the reference model over a domain which increases linearly with the number of exponential functions (number of scatterers). The proposed channel simulator is quite useful for studying the performance of future wideband mobile communication systems employing MIMO techniques.

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