MIMO-CAPACITIES FOR BROADBAND IN-ROOM QUASI-DETERMINISTIC LINE-OF-SIGHT RADIO CHANNELS DERIVED FROM MEASUREMENTS

Transcription

1 MIMO-CAPACITIES FOR BROADBAND IN-ROOM QUASI-DETERMINISTIC LINE-OF-SIGHT RADIO CHANNELS DERIVED FROM MEASUREMENTS Andreas Knopp, Mohamed Chouayakh, and Berthold Lankl Department of Electrical and Electronics Engineering, Institute for Communications Engineering University of the Bundeswehr Munich, 85579, Neubiberg, Germany ABSTRACT We present broadband capacity snapshots as well as grid measurements derived with a fast 5 5 MIMO channelsounder in a typical, large-scale, non-mobile, in-room office scenario where we focus on Line-of-Sight (LOS) transmission channels. The accessible MIMO capacity is compared to its theoretic counterpart which would have been obtained if only the LOS signal without any reflections was considered. Our premise is to enable the user of quantifying the LOS signal s impact on the overall channel capacity for the particular scenario and coinciding to demonstrate the LOS s beneficial effect for the capacity. For a huge amount the beneficial effect of the LOS signal has to be ascribed to its receive signal power increase, an advantage which can hardly be balanced by any measures in the Non-LOS case. The results generally indicate the overall capacity staying fairly high even for single channel matrix realizations and widely independent from the current geometric antenna assembly. Besides, strong variations over frequency are observed. These variations are shown to be slightly reducible by rising the number of antennas, but more efficient, larger bandwidths seem to be an appropriate measure for stabilizing the overall channel capacity per bandwidth unit. I. INTRODUCTION Multiple Input-Multiple Output (MIMO) transmission systems promise high channel capacity gains and reliability improvements for fixed bandwidth and transmit power [1],[2]. Not only in the context of indoor wireless local area networks (WLANs) it is inevitable to be able to predict the channel capacity s availability and reliability for different scenarios and antenna arrangements. Especially broadband indoor transmission channels and as a subgroup in-room channels 1 are totally different from common statistical modelling approaches due to their strong LOS component in coincidence with low mobility. This paper focusses on the investigation of such an in-room channel and its capacity in terms of MIMO. Hence, in section II. we will describe the channel and confine it from statistical modelling approaches. We will summarize the the MIMO channel capacity calculation. Afterwards we will present and evaluate the results derived from channel measurements and capacity calculations in section III. In this section also our assumptions on the signal to noise ratio as well as on the measured channel transfer matrices are explained, providing a convenient method in order to calculate channel capacities from measured data. Last, we conclude our work in section IV. 1 here transmitter as well as receiver are located within the same room II. CHANNEL DESCRIPTION AND CAPACITY A. General presumptions on the channel In the context of MIMO applications and their capacity the indoor WLAN channel is commonly statistically modelled, where the modelling approaches concentrate on the Non-LOS (NLOS) transmission paths forming numerous and variously reflected waves ( rich scattering ). To fulfill these assumptions the LOS component must be distinctly attenuated by shadowing effects or when assuming high reflection factors ξ 1 the path lengths of the reflected waves must be in the range of the LOS path length. In this case the propagation loss of direct and reflected signal components range in the same order of magnitude and hence the reflection s impact rises. Here for narrow bandwidths and high mobility the well known Rayleigh fading model [3] can be applied and the entries of the channel transfer matrix (CTM) are assumed to be uncorrelated, zero mean, i.i.d. complex Gaussian random variables, a case which describes a convenient transmission channel for MIMO applications as its ergodic capacity is high [1] as long as the receive signal correlation is kept low. According to [4] the latter condition can be achieved by antenna spacings of about half-wavelength (λ/2) what therefore has become a common design issue for MIMO antenna arrays. When the direct LOS signal component gets present this fact can be incorporated leading to the well known Rice propagation model [3]. These statistical narrowband approaches which require high mobility are widely used when discussing early indoor transmission systems and they are included in the popular, cluster based indoor channel model introduced by Saleh [5]. Contrarily, in the case of the in-room propagation scenario discussed in the sequel a dominant LOS signal component that carries most of the signal energy is always present which leads to correlated signals at the receiver inputs. Here the Rayleigh fading model is no longer adequate for several reasons. Of course also in indoor scenarios the LOS signal component ist sometimes obstructed and therefore diffracted by objects but it is known from the theory of diffraction that no significant losses arise as long as the obstructing obstacle s size stays below the first Fresnel zone s dimension. For a typical WLAN center frequency of 2.4 GHz and distances of 2 m and 3 m between the obstacle and the transmitter (Tx) and receiver (Rx), respectively, the size of the first Fresnel zone for example is about 80 cm in diameter which is large compared to typical furniture or persons. Besides, one set of antennas (Tx or Rx) is typically mounted at the ceiling which distinctly reduces the risk of obstruction. In these scenarios the simple approach of a λ/2 antenna spacing is no longer sufficient as it can not exploit /06/$20.00 c 2006 IEEE

2 the maximum MIMO capacity with respect to LOS. By geometrically optimizing the antenna positions and spacings at the Tx and the Rx resulting in proper phase angle relations among the entries of the CTM, it is possible to make accessible the maximum MIMO capacity even in the case of correlated channels. The approach of [6] is a powerful, geometric optimization as it shows a strategy to construct antenna arrangements providing capacities close to the maximum even for arbitrary numbers of antennas, although some typical drawbacks imposing practical restrictions could not be mitigated, e.g. a lack of mobility or the need of unpractically large antenna spacings especially in the case of growing Tx-Rx distances. More important, it is inevitable that the LOS signal component never exists without reflections which destroy the carefully constructed phase angle relations within the channel matrix entries. Here the necessity of the optimum LOS channel construction becomes questionable and will be further discussed in the sequel by means of our results. Nevertheless, it was shown by measurements that in many indoor scenarios with low mobility the choice of larger antenna spacings supports higher and more reliable channel capacities [7]. A further aspect that must be taken into account in the case of current WLAN applications, which again contradicts the Rayleigh flat fading model, is the distinctly larger transmission bandwidth in comparison to earlier indoor and outdoor applications leading to a frequency selective fading channel instead of the narrowband flat fading one. Here the channel capacity for single channel realizations gets also frequency selective 2. Hence, we consider the following, widely deterministic description of the MIMO transmission channel. It is motivated and supported by the measurements in [7]. For a single inputsingle output frequency selective, deterministic transmission channel the equivalent baseband channel transfer function for the time invariant case is given by H(f) = K 1 k=0 a k e j2πfτ k = K 1 k=0 a k e j2π L k λ, (1) where the sum is evaluated over the K different transmission paths consisting of K 1 reflected signal parts and where the LOS signal component is incorporated by the index k = 0. The complex amplitude factor a k includes the phase information φ k, the path loss as well as the power-loss due to the complex reflection factor ξ k, i.e. a k = ξ kλ 4πL k e jφ k. The time τ k denotes the signal part s delay for the particular transmission path which is clearly linked to its path length L k by the speed of light c 0, i.e. τ k = L k /c 0. The right part of the equation additionally introduces H(f) depending on the wavelength λ. For a non-mobile scenario a k and τ k are mainly constituted by the location s geometry, i.e. primarily by the Tx s and Rx s positioning within the room as well as their arrangement in relation to the main reflection layers like walls, bottom, ceiling or large scale objects. With increasing mobility it becomes more and more appropriate for the channel to be modelled by some 2 In scenarios with high mobility, which is not the case for the measurements presented here, this frequency selectivity of course does not hold for the ergodic channel capacity as it is defined as the capacity s expectation. statistical process. In the observed scenarios described in this paper the amount of mobility is kept low and therefore no statistical modelling over time is performed. Applying equation (1) to the MIMO case, for a time invariant frequency selective M N-MIMO system consisting of N transmit and M receive antennas the vector of receive signals y(t) C M 1 is calculated by an inverse Fourier transform of the spectrum of the transmit signal vector X(f) C N 1 multiplied by the frequency selective CTM H(f) C M N, i.e. Y (f) =H(f) X(f)+Υ(f), (2) where Υ(f) C M 1 denotes the spectral vector of the additive noise η(t). The noise is assumed to be zero-mean complex Gaussian with covariance matrix R η =E[ηη H ]=σ 2 ηi M, where I M C M M denotes the identity matrix and σ 2 η is the noise power at each receive antenna. Here each entry H mn (f) in H(f) has the structure of equation (1) whereas the values K and hence, a k and τ k differ. B. Capacity calculation According to [1] and [2] if uncorrelated transmit signals and equal power at each Tx antenna are presumed the time invariant channel capacity normalized by the transmission bandwidth (unit [Bit/sec/Hz]), which in the sequel is denoted C, fora frequency selective MIMO-channel in the absence of channel knowledge at the Tx is calculated from Z ( C = 1 log B 2 [det I M + σ2 x H(f)H H (f) df, (3) 2 B where B denotes the transmission bandwidth, I M C M M is the identity matrix, σx 2 denotes the mean transmit power that is allocated to each transmit antenna. Furthermore (.) H abbreviates the complex conjugate transpose. When the overall bandwidth is partitioned into sufficiently narrow, say S, segments each segment can be treated frequency flat and the integral of equation (3) reduces to a sum over the segments capacity contributions, i.e. ( C = 1 S SX s=1 log 2 [det I M + σ2 x H[f s]h H [f s] 2. (4) Hence, the capacity C[f s ] of each frequency bin f s can be redescribed [1] in a way that is sometimes more convenient for analysis as it offers insight into the matrix constitution to a greater extent, C[f s]= 1 S UX u=1 log 2 (1 + σ2 x 2 λ u[f s]), (5) where λ u [f s ] denotes the u-th eigenvalue out of U possible eigenvalues of H[f s ]H H [f s ]. The optimal channel with respect to the capacity is characterized by identical eigenvalues [1]. In the contrary, the worst transmission channel has a CTM of rank = 1, i.e. only one eigenvalue is different from zero. A resulting possibility for the channel characterization is the channel matrix condition number which denotes the square root of the ratio of the largest to the smallest eigenvalue. In

3 the sequel we use a slightly different definition of the channel s frequency selective condition number Γ(f) as we define it as the ratio of the largest eigenvalue to the smallest eigenvalue of H(f)H H (f) that is different from zero, i.e. Γ(f) = λ max(f) λ min (f), ( ) λ min(f) =min λ 1 (f),..., λ U (f). (6) λ 0 Especially for 2 2 MIMO systems Γ(f) gives a good criterion on the channel s constitution with respect to MIMO capacity gain. With increasing system order the value of Γ(f) of course gets less meaningful because in every case only the largest and the smallest eigenvalue are taken into account while ignoring all the further eigenvalues. For such systems the same Γ(f) could be obtained although the MIMO capacity may show huge differences. This problem gets the more eminent the more nonzero eigenvalues with values the maximum and the minimum one exist. Therefore in our discussion Γ(f) is only used if we limit ourselves to low system orders. III. EVALUATION OF THE MEASUREMENTS A. Measurement equipment For the channel measurements we used a 5 5 MIMO channelsounder with a bandwidth of 80 MHz at a carrier frequency of 2.45 GHz endowed with λ/2 dipole antenna arrays positioned in form of equidistantly-spaced uniform linear arrays (ULAs). The maximum bandwidth ranges some factor above the typical bandwidths proposed for current and future indoor applications and hence, achieves a higher time resolution. The channel information was collected by sequentially deriving the entries of the CTM, i.e. by collecting the complex baseband channel impulse responses (CIRs) and channel transfer functions (CTFs). We used 196 pilot symbols and due to our sampling rate of 100 MHz we achieved a frequency resolution of 510 khz ( khz = 100 MHz), but for the capacity evaluation we reduced the considered bandwidth off line to the particular bandwidth under investigation by digital filtering. The measurement of one complete channel matrix consisting of 25 entries took about 160 µs. This measurement time is short enough by far to consider the channel invariant during that time, especially in the chosen scenarios with low mobility and it is much faster than even a single SISO network-analyser measurement which typically takes around 100 ms. A further advantage of the system is the fact that after measuring the 25 CIRs we are able to arbitrarily combine them in off-line mode and virtually built up different MIMO systems by antenna selection from the measured data, in fact ( 5 5 M)( N) combinations are possible for an M N MIMO system. Furthermore the setup enables us to estimate the DOA s for every transmit antenna separately at the Rx off line in 2 dimensions. Here we applied the Capon beamforming method [8]. B. SNR for the capacity calculation from measured data A crucial aspect when evaluating measured CTFs in terms of the channel capacity is the signal to noise ratio (SNR) in equation (3). Generally, we suggest a method that best represents the nature, which transmission systems have to cope with, keeping the user s degrees of freedom in terms of the system design. Those are mainly the transmit power per transmit antenna σ 2 x and the noise power per receive antenna σ 2 η as the latter includes the Rx noise figure which depends on the chosen hardware. Of course the ratio σ 2 x/σ 2 η does not yet meet the typical SNR which describes the receive signal to noise ratio ρ at the Rx input, but ρ clearly emanates from σ 2 x/σ 2 η if it is multiplied by the channel s path loss ζ 2 (f), i.e. ρ = σ 2 x/σ 2 η ζ 2 (f) (7) In our evaluation of the channel ζ 2 (f) is completely incorporated in the CTM which causes ρ at the Rx input being dependent upon the distance between Tx and Rx as well as the number and power of impinging waves. What is more important, the LOS signal part as well as all the reflections are included with their true power contributions. If comparing scenarios with LOS present to NLOS scenarios the LOS case is never simply penalized due to the sole consideration of the correlated channels as the SNR increase compared to the NLOS case is also correctly regarded. This approach does best represent the nature as it focusses on the true capacity available in practice including all the relevant effects. To make this statement more vivid again: if the commonly investigated NLOS Rayleigh flat fading channel for a given value of ρ is considered, the ergodic channel capacity will be comparatively high for that ρ. If now a strong LOS signal component with a CTM of rank = 1 is brought into the scene the rank of the overall CTM will decrease what initially leads to the conclusion of LOS being harmful to the capacity. Contrarily, as the value of ρ will be distinctly higher due to the LOS signal s power contribution this initial conclusion might be a severe mistake as it ignores the SNR gain. The risk of such misinterpretation especially rises if a comparison of the two channels is performed by simply substituting ρ for σx/σ 2 η 2 and in addition power-normalizing the CTM by an appropriate matrix norm 3 to unity for compensation, i.e. eliminating the path loss from the channel. If after such measure the two power-normalized channels were compared for the same value of ρ, the NLOS channel would clearly outperform the LOS channel, a result which is far from the truth in many cases. Hence, we propose to avoid any normalization of the SNR or the CTM to evaluate different channels with respect to their effective channel capacity. Only when investigating certain effects on the channel capacity without being interested in the absolute capacity value itself a normalization can sometimes be helpful. The proposed method is applied in the sequel in order to evaluate the measured data without any additional calculations and it is generally valid. Although not explicitly needed in this paper, for the reader it might be interesting to have an idea on the path loss of the measured scenarios and the resulting value of ρ without having the measured data on hand. As it is well known from theory, e.g. [3], for the LOS path of channel matrix entry L mn the free space path loss is calculated according to ζ 2(mn) LOS (λ) =(λ/4π L mn ) 2. (8) 3 for example the Frobenius matrix norm H 2 F =1

4 This LOS path loss is further increased or reduced by reflected signal parts which constructively or destructively interfere. Hence, in order to calculate the true path loss ζ 2(mn) (λ) for each matrix entry in a simplified modelling approach an additional factor acting as a gain or loss has to be introduced. We denote this factor ζ 2(mn) NLOS (λ) indicating the change of the LOS path loss due to the reflected waves, i.e. ζ 2(mn) (λ) = ζ 2(mn) LOS (λ) ζ 2(mn) (λ). (9) NLOS Clearly, the relation ζ 2(mn) NLOS (λ) > 1 holds if the receive power is increased by the reflections and the relation ζ 2(mn) NLOS (λ) < 1 is valid in the opposite case. By means of this an estimate of the SNR ρ at the Rx input can be obtained according to equation (7), substituting ζ 2 (λ) for example by ζlos 2 (mn) E{ζ2 LOS (λ c )} using the expectance over the matrix entry indices (mn) and setting a fixed value λ c for the wavelength, e.g. the carrierwavelength 4. The latter measure includes the knowledge on the only slightly varying path loss over the frequencies within the transmission bandwidth. An even simpler method for such estimation could be, to use the center of the Tx and Rx arrays as reference points and calculate the LOS path loss from the distance among them. Using this method for example for the setups 1 or 2 in figure 1 the path loss is calculated different MIMO systems but again we hold the opinion that it is more appropriate in practice. The reason is found from the fact that for each Tx-Rx antenna combination a certain channel dependent path loss has to be bridged resulting in a link budget which usually is the basis for calculating the needed Tx power. If now the Tx antenna number is increased and different signals are transmitted from each Tx antenna the power per antenna can not simply be reduced in order to keep the overall transmit power constant. If this was the case all the Tx-Rx link budgets would no longer be satisfied and the SNR at the Rx would break down for every single link. Hence, if a further Tx antenna is added to the system the overall transmit power also has to be increased in practice, meaning the power per Tx antenna has to be kept constant instead, although this aspect of implementation sometimes may complicate a fair comparison of different systems. C. Scenario description In the sequel we will discuss the results collected from two sets of measurements, both depicted in figure 1, where the larger picture for set 1 is depicted in true scale. The first set includes 10 lg ( ζ 2 LOS) 20 lg(4π 5.7 [m]/0.122 [m]) = 55.3 [db] for a carrier frequency of 2.45 GHz what meets the observation from the exemplarily depicted magnitude in figure 2. The resulting ρ can now be calculated dependent upon σx/σ 2 η. 2 For the capacity calculation presented in this paper the value of σx/σ 2 η 2 was chosen 80 db resulting in ρ =24.7dB. Choosing σx/σ 2 η 2 a particular value simply assumes a certain hardware configuration which does not necessarily have to be further specified. The only information we obtain from the measurements is the truly scaled, frequency selective and as possible noise-free CTM for the capacity calculation. If the presumed hardware would be changed also σx/σ 2 η 2 will change but those channel matrices still stay valid. Finally, for the following discussion it is useful to calculate the LOS capacity s bounds for different antenna numbers provided that any reflections are neglected. Using all the above assumptions on the SNR as well as the simplified method of calculating ζlos 2 presented in the preceding paragraph, those bounds for an M N MIMO system without channel knowledge at the Tx are calculated according to Cmax LOS = min{m,n} log 2 (1 + σ2 x ζlos 2 2 max{m,n}) Cmin LOS = log 2 (1 + σx/σ 2 η 2 ζlos 2 M N) (10) As can be seen from the two equations, if the Tx antenna number is increased, the overall transmit power is also increased as every Tx antenna emittes the power σ 2 x. Of course it can be argued that such definition is unfair in terms of comparing 4 In fact, each receiver branch m could have a different ρ m in practice. For simplicity here we assume the ρ m being equal. Figure 1: measurement scenario and antenna arrangements results from measured channel snapshots with fixed positioning of the antenna arrays centers and only varying the arrays orientations as well as the antenna spacings. For the discussion in the first part of subsection D. we limited ourselves mainly to the 2 2 MIMO case for the sake of comparability and expressiveness of the later used channel condition numbers Γ(f), which probably would be questionable for distinctly higherorder systems according to our notes for equation (6). Generally, we only exemplarily selected the typical results. The chosen arrangements cover variously appropriate setups for exploring the MIMO capacity gain in terms of LOS to examine its impact on the overall channel capacity. The measurement set 2 was a grid measurement in order to get more representative results than possible from single snapshots. During one complete grid measurement the relative positions of the Tx and Rx antenna arrays were kept unchanged, i.e. the distance of the antenna arrays centers was fixed at 3.6 m in order to keep the LOS path loss unchanged and only investigate the reflections variance in spatial domain. Here we assumed a spatial statistic

5 treating the Rx and Tx positions as a random variable. Furthermore, the arrays were oriented either broadside or perpendicular, approximately adequate to the array orientations of setup 4 and 2, respectively, with antenna spacings 22 cm at both, Tx and Rx. The mobility was always kept low. The grid results were intended to further substantiate those from the snapshots in set 1 and are mainly discussed in the second part of the following subsection. All measurements were taken within the same room which can be considered typical for a large office or conference room with dimensions m, where many people are connected to a WLAN access point that is also located within the room. It consisted of about ten tiers with twelve metallic one-person working desks and chairs. The measurement grid was chosen among those tiers. The backside of the room was shaped by large panorama windows and the front side was covered by metal whiteboards. Furthermore two vertically positioned metal pillars of 80 cm in diameter equidistantly pervaded each side wall. Except for setup 4, where the antennas were spaced λ/2 all the arrays in the setups 1-3 were spaced 22 cm (and therefore distinctly larger than λ/2). Besides, the Tx antenna positioning in setup 3, was even larger spaced as can be seen from the illustration. As we were able to arbitrarily combine any Rx-Tx antennas we had the chance to evaluate results for different antenna spacings at both, Tx and Rx, dependent upon the selected antennas. D. Discussion of the results Figure 2 exemplarily depicts the available information on the channel in form of the CTF and CIR 5 and a polar diagram of the DOAs for a typical antenna combination of setup 1. In the figure the angle 0 is equivalent to the normal on the Rx antenna array towards the Tx. It clearly can be observed that the channel is frequency selective for one channel realization H mn in the range of the measurement bandwidth with huge variations of its magnitude corresponding to reflections of varying number and power. Although not discovered for the case of the figure, beyond multiple reflections with high reflection factors ξ a short reflection at the bottom or the ceiling is probable in principle. This short reflection may fade the LOS signal part over the whole bandwidth and therefore enhance the NLOS component s impact on the sum signal. The DOA diagram shows the mean of the impinging signal energy derived over the measured frequency span and gives an impression on the main DOAs. It indicates only few but dominant reflections from discrete directions of arrival that from their angular distribution mainly find their origin in wall reflections 6. Here it should be kept in mind that one DOA not necessarily corresponds to exactly one reflection but rather incorporates all the reflections impinging from one particular direction. Nevertheless, according to the measurements the reflections seem to play an important role. This impression proves widely true if looking at the capaci- 5 here only absolute values are plotted, whereas from the measured data of course also the phase information is available 6 The Rx linear array used as a beamformer only estimates DOAs θ in the range of [ π θ < π], the DOAs outside this angular range, particularly the directions from behind the Rx array, are mapped into that range by pointreflection at the origin of the beamformer s reference coordinate system. Figure 2: information on the channel and the DOAs ties derived for different cases. To enable the reader off some principle insight into the fundamental results, figure 3 displays the capacity results for different, measured 2 2 scenarios. The chosen antenna combinations can be seen from the legend. Figure 3: capacity evaluation of 2 2 MIMO-systems In that notation the term Tx: 1 5 for example denotes the Tx antenna pair 1 and 5 corresponding to a spacing of 4 22 cm. Furthermore the abbreviation s. x describes the chosen antenna setup with index number x (figure 1). The next information given in the legend is the value C norm that denotes the measured overall channel capacity for the considered bandwidth normalized by min{m,n} what states the maximum linear capacity increase possible for an M N MIMO system. This normalization appears to be convenient to compare the MIMO gain of configurations which differ in their antenna number as it gives an idea how close the particular configuration approaches its maximum linear MIMO gain. Two further measures are used to characterize our results: at first, the optimum LOS capacity value, i.e. the capacity which was obtained provided that all reflected signals were neglected, for the different antenna numbers and the particular Tx-Rx distances 7 was 7 the distance between the arrays center points is used

6 calculated and in the sequel it is denoted C opt. LOS. Slightly dependent upon M and N its optimum normalized value applying the identical normalization as for C norm, i.e. dividing by min{m,n}, lies close to 9 [Bit/s/Hz] and it is calculated according to equation (10). Consequently C opt. LOS is equivalent to the value Cmax LOS in the equation. Secondly, we calculated the capacity values that with every particular antenna arrangements in figure 1 theoretically would have been achieved if no reflections had been existent. This value is denoted C LOS and our method of keeping σx/σ 2 η 2 fixed and leaving the path loss in the CTM (described in part B. of this section) makes it easy to exactly calculate it from the assembly s available geometric information. In the legend we provide the ratio of the calculated LOS capacity (C LOS ) of the observed antenna positioning to the theoretic maximum C opt. LOS. This ratio provides a measure how appropriate the current configuration in theory would be to achieve the maximum MIMO capacity in the absence of reflections. The figure s ordinate displays the ratio of the capacity C, which in reality was measured, and C LOS to give some insight into the procedure, how the theoretic C LOS is altered by the reflected waves. The more the measured capacity deviates from the theoretic LOS capacity the higher the reflections impact on the accessible MIMO capacity in practice must be considered. Together with the value C opt. LOS the reader gets an idea, whether the reflections impact on C is dependent upon a LOS-optimized antenna assembly or not. Last, for the curves three times the standard deviation of C/C LOS is quoted, giving some hints on the capacity s variations over frequency. The setups 4 and 2 only provide low-rank LOS channels and therefore their capacity strongly depends on the reflections. This fact is observed from the curves as they are located at high ordinate values. It is especially noticeable that setup 4 with the λ/2 spaced antenna arrays only provides a small C LOS whereas the overall capacity due to the reflections is still acceptable with strong variations over frequency. The second curve in the figure shows a result for the identical setup with the two Tx and Rx antennas spaced 2λ and obviously C LOS increases but C norm remains nearly unchanged. Only the variations over frequency decrease. Scenario 2 shows the case of nearly perpendicular oriented antenna arrays again causing a bad condition for C LOS. Nevertheless C norm also in this case only slightly decreases. Contrarily, the scenarios 1 and 3 (see figure 4) deliver higher-rank LOS constitutions due to which the overall capacity is much more dominated by the LOS part with C norm being in the range of the other configurations. Altogether it can positively be observed from figure 3 that the overall capacity per bandwidth unit C norm for the large bandwidth of 80 MHz stays fairly identical for the different setups within the location. Observing the C s variations, one can imagine that if only a narrow bandwidth was used the variations would lead to high capacity variances. In a widely static scenario the risk of staying close to the capacity s minimum is probable 8. From a practical point of view the high variations must be considered problematic as the accessible channel capacity for a single chan- 8 it must be remembered that in a 2 2 MIMO system the capacity s maximum and minimum just differ from a factor around 2 nel use and a moving Tx or Rx is hard to predict. In order to cope with this problem, our results mainly suggest two strategies: Firstly, the particular curve in figure 5 indicates a higher number of antennas to be appropriate for reducing the capacity s variations over frequency, even in low-rank LOS channels. This effect can be ascribed to the fact that a higher number of Rx antennas increases the overall receive signal energy what in the consequence raises the MIMO capacity. Furthermore the use of a higher number of antennas reduces the risk of simultaneous fading or signal loss at every antenna. This could be a promising approach in the case of narrowband transmission systems where the following, second approach can not be applied. This second approach suggests to generally use larger bandwidths in order to reduce the overall capacity s variations. More exact, the capacity per bandwidth unit can be signifi- Figure 4: condition numbers for some measured setups cantly stabilized. This gets evident already from figure 3 as one can imagine that for a bandwidth which includes as many overshoots as undershoots of the capacity will vary less even for varying antenna positions. The discussion of measurement set 2 at the end of this subsection quantifies this statement more accurate. At first, to recapitulate and better understand the fact of the generally satisfactory high channel capacity even in the case of low LOS capacity contributions figure 4 depicts the condition number Γ(f) according to section II. for selected 2 2 cases as well as for a 5 3 system. At this point it interestingly can be stated that all of the measured scenarios always provided rank{h[f s ]H H [f s ]} eigenvalues different from zero within the same order of magnitude 9. The curves illustrate the theoretically known connection among the capacity and the condition number. At the peaks of the condition number the capacity is around its minimum whereas it gets maximal for low condition numbers. As the condition number appears to generally lower and the curve varies less for the 5 3 case again the enhancing effect of using a higher number of antennas seems to be stressed. It is noticeable that the condition number almost never falls to its theoretic optimum value of 1 marking full MIMO capacity gain. Nevertheless, the capacity is accept- 9 Of course in a noisy measurement the channel always has full rank but we assured by our method of evaluating the measured data that errors coming from the noise could not significantly change the results.

7 ably high already for significantly higher condition numbers than one. This leads to the assumption that in many cases not perfectly conditioned CTMs often requiring laborious geometric optimizations are sufficient. For larger numbers of antennas the needed value for the condition number increases what can be ascribed to the increasing receive signal energy due to the larger Rx antenna number. So for higher antenna numbers the condition number gets more flat across frequency and at the same time the system tolerates even higher condition numbers for identical MIMO capacity gain. These two enhancing effects can ultimately be observed by C norm in all the figures. The results from the grid measurement set 2 are depicted in figure 5 in form of cumulative distribution functions (CDFs). Starting with the curves for the B = 20 MHz bandwidth (in- Figure 5: capacity CDFs for different MIMO systems dicating a typical WLAN channel bandwidth) the capacity s dependence on the geometric assembly and its capability to access the maximum LOS capacity can be observed. The curves for the perpendicular array orientation, belonging to a pure LOS channel of rank 1 (neglecting the reflections) are below those of the broadside array orientation, belonging to a LOS channel with higher rank. For the latter case we chose antenna spacings of 44 cm indicating a good but not the optimum antenna arrangement in terms of a high-rank LOS channel. Despite an observable dependency, only a small profit in capacity can be gained by means of geometric optimization in the presence of reflections. Comparing the curves to the theoretic LOScapacity bounds given in the table the obtained capacities are only slightly below the theoretic LOS-maximum. Especially for the low-rank LOS case the SNR as well as the channel matrix rank were distinctly increased by the reflected waves leading to higher capacities. For the higher-rank pure LOS case (broadside arrays) the SNR indeed is increased but with reflections the CTM will almost never have full rank, moreover for a full-rank pure LOS channel the reflections will even decrease the matrix rank. Hence, the capacity is not shifted above the theoretic LOS maximum but settles in the range of its 90 % value. Last, taking a look at the steepness of the curves indicating the capacity s variance over the spatial grid, the curves for B = 20 MHz are significantly steeper than the curves for B = 510 khz. This result again supports the idea to use larger bandwidths in order to include overshoots and undershoots of the capacity from its mean and stabilize the capacity per bandwidth unit. Finally, from the last curve in the figure which represents the narrowband case with increased Rx antenna number also an increasing steepness compared to the identical bandwidth with only 2 Rx antennas is observed. This proofs the statement that higher antenna numbers also help to reduce the capacity s variations, although the effect is less than for the increased bandwidth. Besides, this curve most distinctly outperforms the theoretic LOS maximum coming from the increased receive signal energy and the reduced risk of simultaneous signal fades at every antenna. Here the idea of such asymmetric systems with increased Rx antenna number being very appropriate in practice is supported 10. IV. CONCLUSION The results from in-room broadband measurements in a typical, quasi-deterministic, non-mobile office scenario with a strong LOS signal component generally showed the LOS signal part being beneficial to the channel capacity. The antenna assembly s geometric optimization for the LOS signal part [6] was shown to have a measurable but only negligible impact on the channel capacity in the presence of reflections. Instead, two main effects could be discovered: firstly, the LOS signal part distinctly increases the SNR, an effect which is irreplaceably lost in the NLOS case, and secondly, the reflections help to increase the channel matrix rank even in the case of badly constructed LOS channels. In the consequence the overall channel capacity is always satisfactory high. Furthermore, high variations of the capacity over bandwidth were observed to be an important drawback in practice. These variations can be equalized by increasing the number of antennas, but more efficient, larger bandwidths were shown to be appropriate to stabilize the overall channel capacity per bandwidth unit. REFERENCES [1] E. Telatar, Capacity of Multi-Antenna Gaussian Channels, AT&T-Bell Technical Memorandum, [2] G.J. Foschini and M. Gans, On the Limits of Wireless Comunications in a Fading Environment when Using Multiple Antennas, Wireless Personal Communication, vol. 6, 1998, pp [3] M. Paetzold, Mobile Fading Channels, John Wiley & Sons, [4] W.C. Jakes, Microwave Mobile Communications, John Wiley & Sons, [5] A.A.M. Saleh, R.A. Valenzuela, A Statistical Model for Indoor Multipath Propagation, IEEE J. Select. Areas Commun., vol.5, 1987, pp [6] F. Bohagen, P. Orten, G.E. Oien, Construction and Analysis of High-Rank Line-of-Sight MIMO Channels, Wireless Communications and Networking Conference, March 2005, pp [7] J.-S. Jiang, M.A. Ingram, Spherical-Wave Model for Short-Range MIMO, IEEE Transactions on Communications, vol. 53, No.9, September 2005 [8] J. Capon, High-Resolution Frequency-Wavenumber Spectrum Analysis, Proceedings of the IEEE 57(8), 1969, pp despite, this system does of course not provide the highest absolute capacities as the normalized curves for the 4 2 system have to be multiplied only by 2 but for the 4 4 case the multiplier is 4