MIMO Signal Description for Spatial-Variant Filter Generation

Transcription

1 MIMO Signal Description for Spatial-Variant Filter Generation Nadja Lohse and Marcus Bronzel and Gerhard Fettweis Abstract: Channel modelling today has to address time-variant systems with multiple antennas at the transmitter and at the receiver, the so-called MIMO (multiple multiple ) systems. In order to describe this time-variant with ly distributed antennas by a simple space-time filter, system functions are defined, which are based on a space-time analogy of Bellos "physical-intuitive" systems functions. In addition, the paper in hand considers space-time correlation functions and introduces the corresponding space-time filter approach. Introduction The growing demand on MIMO radio simulations has driven the investigations on space-time descriptions. Even through a verity of trends of such investigations eists, a novel approach is chosen that is presented in the following. The approach is a etension of Bellos research [,2]. We based our investigations on this work, since it results in a simple usable stochastic model, the time-variant Doppler-filter, which is unfortunately confined to time domain. So, it seemed to be useful to utilise this simple filter structure in the domain as well. The result is a generally usable mathematical construct, which is utilizable for space-selective time-variant filter design. 2 Spatial Etension of Bellos Model A Bellos Description in Time Domain The radio can be considered as a mathematical operator, which transforms signals stimulated at a vector-transmitter into signals observed at a vectorreceiver as indicated in figure. The parameters of a stimulated signal are denoted by and its dual signal parameter, the parameters of a observed one by t and f. Using this signal knowledge at and, the can be described by a linear integral operator, the so called kernel function, see above in figure. Bello did not use kernel functions to describe the radio, what he has justified it in his fundamental paper "Characterization of Randomly Time-Variant Linear Channel []: From a strictly mathematical point of view, Bellos's signal description in time domain by kernel function Bellos's signal description in time domain by system function time delay τ related to t frequency shift ν related to f Figure : Temporal signal description by kernel (above) or system operator (below). Dresden University of Technology, lohse@ifn.et.tu-dresden.de

2 the kernel system functions are sufficient to describe the time-frequency, - relations for a time-variant linear. From a physical-intuitive or engineering point of view, they are not as satisfactory, since they do not readily allow one to grasp by inspection the way in which the time-variant filter affects signals to produce signals. That is why, the will be described by system functions, where the temporal stimulated signal parameters t and f are projected onto observed and therefore the new difference variables time delay τ = t t and the frequency shift ν = f f were introduced. This is illustrated in the lower scheme of figure. B Space-Time Analogy In order to transfer the principle of Bellos temporal signal description to the domain, the analogy t=t t=t 2 t r =0 r =λ/4 r =λ/2 Figure 2: Signal propagation shown by comple rotator in space (horizontal) and in time (vertical). r in and temporal wave propagation is applied. Electro-magnetic waves propagate over space and j( t kr ) time, what can be epressed by a comple rotator in space and time srt (,) = Ae ω. Figure 2 illustrates it: The rotator in time j t e ω is shown at two different moments t i (vertically plotted) and the rotator in space jkr e at three different places i, r (horizontally plotted). In time, the rotator rotates with the rate of angular frequency ω counter clockwise, in space it rotates with the rate of the clockwise. Therefore, stationary points of a propagation wave are related by ω t = kr, () where r in r -domain and k in the frequency k -domain are defined in spherical coordinates with φ or Φ as azimuth angle and θ or Θ as elevation angle by: cosφcosθ cosφcosθ r r sinφcosθ = k = k sinφcos Θ. (2) sinθ sin Θ C Description in Spatial Domain Applying the space-time analogy, the signal in domain can be described similar to the temporal domain. The temporal values and were replaced by the corresponding ones, and frequency k, respectively, what characterize the signal description in domain by kernel functions, shown in the upper scheme of figure 3. This description is usually signal description in domain by kernel function signal description in domain by system function distance r related to r frequency shift k related to k Figure 3: Spatial signal description by kernel (above) or system operator (below).

3 used today [3]. In order to describe the by system functions, the difference values distance ρ = r r and frequency shift κ = k k were defined, see below in figure 3. 3 Input-Output lation A Deterministic Channel The deterministic space-time is determined by one of the following system functions: impulse response h, spectral response H, scatterer function s, or transfer function T in space and time 2. The - relation in time domain is given by the convolution of one system function with a stimulated function z(t) or with the dual one in frequency domain Z(f), which results in the observed function w(t) or its dual one W(f): wt () = zt ( τ) ht (, τ) dτ W( f) = Z( f ν) H( f, ν) dν (3) Applying the introduced signal dimensions, results in: wr ( ) = zr ( ρ) hr (, ρ) dρ W( k) = Zk ( κ) Hk (, κ) dκ (4) The - relationship of a, which shall be described by scatterer function in time s(, τν) or by transfer function in time Tt (, f ) is deduced in [, 2, 4]. Analogous I/O-relations in space can be derived using s( ρκ, ) and T ( r, k ) : j2π ft wt () = Z( ftt ) (, f ) e df W( ) = zt ( ) s(, ) d ν τ ντ τ jkr wr ( ) = ZkTrke ( ) (, ) dk W( κ) = zr ( ρ)( s κ, ρ) dρ (5) With (3) to (5) the - relations are given separately for space and time, which can be used for separate time-variant and space-variant filter design. In order to built a combined filter, the - relationship for the space-time system function has to be considered: j(2 π ft+ kr ) wtr (, ) = zt ( τ, r ρ) ht (, τ, r, ρ) dτdρ wtr (, ) = Z( f, ktt ) (, f, rke, ) dfdk W( f, k) = Z( f ν, k κ) H( f, ν, k, κ) dνdκ W( νκ, ) = zt ( τ, r ρ) s( ντκ,,, ρ) dτdρ (6) B Random Channel For a random space-time variant, all the above mentioned functions, the observed and stimulated functions and the system functions become stochastic processes, completely described by its cor- 2 Since the functions h, H, s and T are general notations for the system operators, the same notation is used in space and time domain.

4 relation properties, specified by Rf (, ) E{ f ( ) f( )} =. Taking this into account, the relationship can be derived from (3) and (4) directly: R (, tt ) = z ( t τ) zt ( τ ) R (, tt, ττ, ) dτdτ w RW( f, f ) = Z ( f ν) Z( f ν ) RH( f, f, νν, ) dνdν Rw(, ) = z ( ρ) z ( ρ ) Rh(,, ρ, ρ ) dρdρ R ( kk, ) = Z ( k κ) Zk ( κ ) R ( k, k, κκ, ) dκdκ W h H (7) The other relations can be deduced from (5) or (6) in a similar way. 4 Space-Time System Functions A Assumptions Firstly, space-time WSSUS is assumed: In a multi-path propagation environment different signals arrive with uncorrelated time delays τ p, directional distances ρ p and velocity-dependent Doppler shifts ν p and frequency shifts κ p (uncorrelated scatterers). Then, because of the Fourier-Transformrelationship between the space-time dimensions, the can be described as wide sense stationary (WSS) process in, and temporal and and k. Secondly, we will constrain the following considerations to the one-dimensional case in the domain, which reduces the vectors r, k of equation (2) to r = rcosφ and k = kcos Φ. Finally, we use the directional angle Φ instead of frequency suggested for finite antenna arrays in [5]. k and the directional angle φ instead of the as B Definition The significant space-time system functions and their twofold Fourier-Transform relationships are shown in figure 4. They were defined as the combination of the and temporal system functions. Function ht (, τ, r, ρ ) is the space-time response at to an unit impulse τ seconds in the past and at point r based on an impulse originating at ρ : ht (, τ, r, ρ ) = γ δ( r, φ φ ) δ(, t τ τ ), (8) ρ ρ p where denotes the number of incoming paths. Function s( ντκ,,, ρ ) is the space-time scatterer function with separation of multi-path components with respect to a temporal delay τ, offset ρ, frequency shift ν, and wave number separation κ : ht (, τ, r, ρ ) F f, k Tt (, f, rk, ) F ν,? F ν,? s( ντκ,,, ρ ) F f, k H( ν, f, κ, k ) Figure 4: Fourier Transform relation ship between the space-time system functions.

5 (,,, ) ( ) ( ) ( ) ( ) s ντκ ρ = γ δν ν p δτ τ p δ Φκ Φκ δφ p ρ φρ Function H( ν, f, κ, k ) is the spectral space-time response at to an at a - ν and at directional wave number k to an κ directional wave numbers apart from k : H( ν, f, κ, k ) γ e e δ( ) δν ( ν ) jdkcosφ ρ j2πdf τp = p Φκ Φκ p p Function Tt (, f, r, k ) is the space-ransfer function with superposition of multi-path components observed at and and at and temporal : (9) (0) p p p p Tt (, f, r, k ) = γ e e e e p jf ( Φκ ) dr jdk f ( φρ ) j2πdfτ j2πν dt () C Correlation Since WSSUS is assumed, the space-time correlation function of the signal in space and time only depends on the differences t, f, r and k. Hence, a reduction of correlation dimension is achieved: Rh (, tt, ττ,, rr,, ρρ, ) = E{ h (, t τ, r, ρ) ht ( + t, τ, r + r, ρ) } (2) = (, t τ, r, ρδτ ) ( τδ ) ( ρ ρ) h Rs ( νν,, ττ,, κκ,, ρ, ρ ) = E{ s ( ντκ,,, ρ) s( ντκ,,, ρ) } (3) = ( ντκ,,, ρδν ) ( νδτ ) ( τδκ ) ( κδ ) ( ρ ρ) s RH ( νν,, f, f, κκ,, kk, ) = E H ( ν, f, κ, k) H( ν, f + f, κ, k + k) = ( ν, f, κ, k) δν ( νδκ ) ( κ) { } H RT (, tt, f, f, rr,, k, k ) = E T (, t f, rktt, ) ( + t, f + f, r + rk, + k) = R ( t, f, r, k) { } T (4) (5) 5 Space-Time Filter Functions The filter function limits the dimensions to that region, where the is defined. Here, in order to understand the filter principle, only a simple filter function will be considered. Besides, the used filter functions depends on the considered dimensions. For limitation of the transfer function T, which is a function in original dimensions [,, and temporal ], rectangular functions can be applied multiplicatively. This results in the generalized system function T (, t f, r, k ), which is adapted to the real situation: real

6 t f k T real(, t f, r, k) = rect( )rect( ) rect( ) T( t, f, r, k) (6) Tb Ta Bb Ba Kb Ka The rectangular limitation of the transfer function T with original dimensions leads to a convolution of sinc-functions with the dual scatterer function in dimensions temporal delay τ, offset ρ, frequency shift ν, and wave number separation κ : s real( ντκ,,, ρ) = Tb Ta sinc(( Tb Ta) ν) Rb Ra sinc(( Rb Ra) ρ) s( ντκ,,, ρ) (7) since the sinc-function is the Fourier transform of the rect-function: + / i rect( ) F sinc(( ) ) i = ib i a i b i y a i (8) i b ia where the original dimensions are denoted by i and the dual ones by y i. For both space-time system functions impulse response h and the spectral response H, respectively, which contains mied dimensions i and y i, the generalized system function is given by: func (,,,, y, y,, y ) = real 2 i 2 i i rect( ) rect( ) i i sinc(( i i ) yi) func(, 2,, i,, 2,, i) b a b a y y y b a i (9) b ia whereby func corresponds to h and H. Other filter functions need to be investigated, in order to better reflect real scenarios. 6 Acknowledgments I would like to thank Dr. Ralf Kattenbach [4] from the University of Kassel for fruitful discussions, which resulted in the presented signal idea. 7 Conclusions Based on Bellos temporal signal description a novel MIMO space-time signal description was introduced in which the is represented by a space-time system operator. Compared to the traditional MIMO signal description this one allows the modelling by simple filter structures like the known time-variant Doppler filter. But, a new view of the with new parameters distance and frequency shift is required. Starting with one dimension the space-time system functions and for the random the correlation function were defined under space-time WSSUS assumption. A filter principle for -selective, time-variant filter design was introduced, which integrates the eisting limitations of a real by rectangular windows. revious radio signal descriptions should be contained as special cases.

7 8 Literature [] Bello,. A.: Characterisation of Randomly Time-Variant Linear Channels. IEEE Transactions on Communications Systems., age(s): , December963 [2] Bello,. : Time-Frequency Duality. IEEE Transactions on Information Theory, age(s): 8-33, January 964. [3] Molisch, A. F. et al.: Measurements of the capacity of MIMO systems in frequency-selective s. Vehicular Technology Conference, 200. VTC 200 Spring. IEEE VTS 53rd, Volume:, age(s): , 200 [4] Kattenbach, Ralf: Charakterisierung zeitvarianter Indoor-Funkkanäle anhand ihrer System- und Korrelationsfunktionen. Shaker Verlag, Juni 997 [5] John D. Kraus: Antennas. McGraw-Hill, 2 rd edition, 988