CAPACITY OF MULTI-ANTENNA ARRAY SYSTEMS IN INDOOR WIRELESS ENVIRONMENT

Size: px

Start display at page:

Download "CAPACITY OF MULTI-ANTENNA ARRAY SYSTEMS IN INDOOR WIRELESS ENVIRONMENT"

Joel Reed
5 years ago
Views:

1 CAPACITY OF MULTI-ANTENNA ARRAY SYSTEMS IN INDOOR WIRELESS ENVIRONMENT Chen-Nee Chuah, Jseph M. Kahn and David Tse -7 Cry Hall, University f Califrnia Berkeley, CA {chuah, jmk, dtse}@eecs.berkeley.edu Abstract Previus studies have shwn that a wireless system using n transmitting and n receiving antennas can achieve a capacity n times higher than a singleantenna system in an independent Rayleigh fading envirnment. In this paper, we explre the capacities f multiple-element antenna arrays (MEAs) in a mre realistic prpagatin envirnment simulated via the WiSE ray-tracing tl. We impse an average pwer cnstraint and cllect statistics f the capacity with ptimal pwer allcatin,, and the mutual infrmatin with an equal pwer allcatin, I eq. In additin, we present expressins fr the asympttic grwth rates /n and I eq /n as n fr tw cases: (a) independent fadings and (b) crrelated fadings at different antennas. We find that /n and I eq /n cnverge t cnstants *andi eq *, respectively in case (a), and t C wf and I eq in case (b). We bserve that C wf and I eq predict very clsely the slpes bserved fr simulated channels, even fr mderate n (i.e., 6). I. Intrductin Signals prpagating thrugh wireless channels experience path lss, distrtin due t multipath fading, additive nise, and cchannel interference. These impairments, alng with the cnstraints n pwer and bandwidth, limit the system capacity. In the past, multiple antennas have been used at the receiver t cmbat multipath fading f the desired signal, e.g., using maximal rati cmbining [], r t suppress interfering signals, e.g., using ptimal cmbining []. Recent studies reprt that using MEAs at bth transmitter and receiver increases system capacity cnsiderably ver singleantenna systems ([3], [4]). In [4], Fschini and Gans cnsider n transmitting and n receiving antennas, with i. i. d. narrwband Rayleigh fading between antenna pairs. Assuming that a fixed pwer is allcated equally ver all transmitting elements, the MEA mutual infrmatin (I eq ) is reprted t grw linearly with n. An MEA system achieves almst n mre bps/hz fr every 3 db increase in signal-t-nise rati (SNR), cmpared t a singleantenna system, which nly achieves ne additinal bps/hz. In practice, crrelatin exists between the signals transmitted by r received at different antennas. Crrelatin may arise if the antenna elements are nt spaced far apart enugh, e.g., Lee pinted ut in [5] that the required antenna spacing t btain a crrelatin cefficient between signals t be less than.7 is apprximately 7 wavelengths fr the bradside case and 5- wavelengths fr the inline case. The presence f a dminant line-f-sight cmpnent can als affect the MEA capacities. Here, we explre the MEA capacities in a mre realistic prpagatin envirnment, where the fadings are nt necessarily Rayleigh, nr independent. We determine the capacity when the transmitter knws the channel and ptimum pwer allcatin (water-filling) isused.als, we cmpute the mutual infrmatin I eq with equal pwer allcatin f the MEA system, and we investigate the perfrmance degradatin as cmpared t. We study the behavir f MEA capacities thrugh simulatin and analysis. We emply the Wireless System Engineering (WiSE) [6] sftware tl t simulate explicitly the channel respnse between a transmitter and a receiver placed inside an ffice building. We mdel the multiple-input-multiple-utput (MIMO) Rayleigh-fading channel as a matrix H, and study hw and I eq behave as n grws large. We shw almst sure cnvergence f the asympttic grwth rates /n and I eq /n cnsidering tw cases: (a) when fadings between different antenna pairs are independent and (b) when these fadings are crrelated. The remainder f this paper is rganized as fllws. In Sectin II, we mdel the channel as a MIMO system with flat frequency respnse. Using this mathematical mdel in Sectin III, we present infrmatin-theretic results fr the capacity f MEA systems and analyze its asympttic grwth rate as n grws large. In Sectin IV, we present capacity estimates fr the simulated channels and discuss the discrepancies between these results and the asympttic capacities predicted by thery. We briefly describe hw WiSE is used t represent the indr prpagatin envirnment that ur study is based n. CnclusinsarepresentedinSectinV. II. Channel Mdel The fllwing ntatin will be used thrughut the paper: ' fr vectr transpse, fr transpse cnjugate, I fr the identity matrix, E[. n n n ] fr expectatin, and underline fr vectrs. A. Basic Channel Mdel We cnsider a single-user, pint-t-pint cmmunicatin channel with n transmitting and n receiving antennas, with n c-channel interference. We assume. WiSE is a ray-tracing tl that predicts RF prpagatin in a specific building, based n ff-line experimental measurements.

2 that the channel respnse is flat ver frequency. This apprximatin is reasnable if the cmmunicatin bandwidth, W, is much less than the cherent bandwidth. In ur simulated channels, the maximum delay spread is 4 ns. Since the cherence bandwidth is apprximately the reciprcal f the delay spread, the frequency respnse can be cnsidered flat as lng as W is much less than 4 MHz. We assume that the channel is linear time-invariant and use the fllwing discrete-time equivalent mdel: Y HX + Z. () Here, X [ x, x,, x T ]' is an n vectr whse jth cmpnent represents the signal transmitted by the jth antenna. Similarly, the received signal and received nise are represented by n vectrs, Y and Z, respectively, where y i and z i represent the signal and nise received at the ith antenna. The cmplex path gain between transmitter j and receiver i is represented by H ij, fr i,,..., n and j,,...,n. We further assume that: The ttal radiated pwer is P tt, regardless f n. The nise Z is an additive white cmplex Gaussian randm vectr. Its cmpnents, Z i, i,,..., n, are i. i. d. circularly symmetric cmplex Gaussian randm variables with variance E[ Z i ] N W. We cnsider the fllwing tw cases:. H is knwn nly t the receiver but nt the transmitter. Pwer is distributed equally ver all transmitting antennas in this case.. H is knwn at the transmitter and receiver. Therefre, pwer allcatin can be ptimized t maximize the achievable rate ver the channel. In this wrk, we treat H as quasi-static. H is cnsidered fixed fr the whle duratin f cmmunicatin, thus capacity is cmputed fr each realizatin f H withut time averaging. On the ther hand, H changes if the receiver is mved frm ne place t the ther, which happens ver a much larger time scale. The capacity and mutual infrmatin I eq assciated with H can be viewed as randm variables. III. Analysis f MEA Capacities Channel capacity is defined as the highest rate at which infrmatin can be sent with arbitrarily lw prbability f errr [8]. Since H is quasi-static, it is reasnable t assciate t a specific realizatin f H, frafixed P tt and N W. Thrughut ur analysis, we assume H ij fr i, j,,..., n, are identically distributed with the same variance υ E[ H ij EH [ ij ] ]. We assume that υ is the same fr all fading gain H ij fr all psitins f the transmitting and receiving MEAs within their respective wrk spaces.. Delay spread here refers t the difference between the arrival times f the earliest- and latest-arriving rays having appreciable amplitude. When n antennas are used, we dente the MEA capacity and mutual infrmatin as (n) andi eq (n), respectively. Fr the case with n, the capacity is: P tt ( ) I eq ( ) lg H bps/hz. () N W In the high-snr regime, each 3-dB increase f P tt /N W yields a capacity increase f bps/hz. A. Capacity With Water-filling Pwer Allcatin In this sectin, we assume the transmitter has perfect knwledge abut the channel. Thus, P tt can be allcated mst efficiently ver the different transmitters t achieve the highest pssible bit rate, which is given by: ( n) max Q bps/hz, (3) where Q is the n n cvariance matrix f X ( Q E[ XX ]), and must satisfy the average pwer cnstraint: The achievable capacity ([9]) is: where µ satisfies lg det. (4), (5), and the Λ i s are the eigenvalues f HH. The ptimal slutin that gives the capacity in (5) is analgus t the water-filling slutins fr parallel Gaussian channels [8]. B. Mutual Infrmatin With Equal Pwer Allcatin Here, we assume that equal pwer is radiated frm each transmitting antenna, which is a natural thing t d when the transmitter des nt knw the channel. The MEA mutual infrmatin is: I eq ( n) bps/hz. (6) C. Asympttic Behavir f Capacity We investigate the grwth f I eq and as n grws large fr tw cases: (a) when path gains, H ij, are independent, and (b) when H ij s are crrelated. In bth cases, we assume that H ij s are identically distributed cmplex Gaussian with variance υ. We define the average received SNR as ρ υ P tt N W.. Assuming Independence f Path Gains Fr a given H, the capacity f n-antenna MEA is given by (5). The Λ i s are randm variables that depend I n tr( Q) E X i ( n) i lg det n i n i Λ i + [ ] HQH N W P tt lg ( Λ i µ + ) µ I n P tt P tt HH nn W

3 n H. Fr each n, letf n be the fractin f Λ i less than r equal t Λ with n antennas: F n ( Λ) -- { i: ( Λ. (7) n i Λ) } Nte that I eq and depend n H nly thrugh the empirical distributin f Λ i, F n (Λ).Theasymptticprperties f (n) depends n hw the distributin F n behaves as n appraches infinity. Khrunzhy et al, and Yin studied cnvergence f F n in []-[]. The fllwing almst sure cnvergence therem is due t the wrk by Silverstein et al in []. Therem. Define G n (Λ): F n (nλ). Then, almst surely, G n cnverges t a nnrandm distributin G*, which has a density given by: g ( Λ) Λ 4 π Λ 4 therwise. The scaling by n in the definitin f F n means that the Λ i are grwing as rder n. After rescaling, the distributin cnverges t a deterministic limiting distributin, i.e. fr large n, F n (nλ) lks similar fr almst all realizatins f H. Using this therem, we derive the asymptticgrwthratefc wf (n) as n while keeping the average received SNR ρ cnstant. Prpsitin. With almst sure cnvergence, ( n) C, where n wf 4 ( lg ( µλ) ) + g ( Λ) dλ (9) 4 and µ satisfies µ ---. Λ + g ( Λ ) dλ ρ If we assume the transmitter always allcates an equal pwer P tt /n t each transmitting antenna, the mutual infrmatin is given by (6). Using Therem, we can prve the fllwing prpsitin. Prpsitin. With almst sure cnvergence, I eq ( n) I,where n eq I eq 4 (8) ( lg ( + αλ) ) + g ( Λ) dλ. () With the abve tw prpsitins, we find that (n) and I eq (n) scale like n *andni eq *, respectively. Using L Hpital s rule, it can be shwn that at lw SNR, lim , ρ I eq while at high SNR, lim I eq. ρ. Cnsidering Crrelatin between Path Gains Let Ψ T be an n n matrix whse entry Ψ T jk is the crrelatin cefficient between signals transmitted by jth antenna and kth antenna, Ψ T jk EH [ pj H pk ] E[ H pj ]E[ H pk ]. () In ur mdel, we assume that Ψ T jk des nt depend n the index f the receiving antenna, i.e. p can be arbitrary as lng as p {,,, n}. Similarly, let Ψ R be an n n matrix whse entry Ψ R pq is the crrelatin between signals at receiver p and receiver q, EH [ pj H qj ] E[ H pj ]E[ H qj ], () Ψ R pq and it is als assumed t be independent f the index f the transmitting antenna, j. T simplify ur analysis, we assume that crrelatin fr H ij s when bth transmitting and receiving antennas are different is the prduct f the tw ne-dimensinal crrelatin functins mentined abve: EH [ pj H qk ] E[ H pj ]E[ H qk ] Ψ R pq Ψ T jk. (3) We verify the validity f this assumptin thrugh WiSE simulatin. We estimate crrelatin f H ij s empirically frm realizatins f H fr n. Cmparing the prduct f Ψ T and ΨR with the actual estimate f EH [ H ], clse agreement is fund cnsistently between the tw ver the range f antenna spacings that we cnsider. The asympttic results in previus sectin can be extended t the case when the H ij s are crrelated, under certain assumptins n the cvariance matrices Ψ R and Ψ T. In particular, we assume that the empirical distributins f the eigenvalues f Ψ R and Ψ T cnverge t sme limiting distributins F R and F T, respectively. This will be true if: The crrelatin between the fading at tw antennas depends nly n the relative and nt abslute psitins f the antennas; and The antennas are arranged n a regular lattice, such as in square grids r linear arrays, and as we scale up the number f antennas, the relative psitins f adjacent antennas are fixed. Under the abve cnditins, it can be shwn that almst surely, as n, ( n) C (4a) n wf ( F R, F T, ρ) I eq ( n) and I, (4b) n eq ( F R, F T, ρ) where C wf and I eq are cnstants that depend nly n the SNR and the limiting eigenvalues distributins f Ψ R and Ψ T. While these limits can be cmputed fr arbitrary SNR [3], we shall fcus here nly n the case when the

4 SNR is high. In this regime, particularly simple expressins can be btained. It can be shwn that at high SNR, ( F R, F T, ρ) I eq ( F R, F T, ρ) lg ρ + + lg η R ( x) dx (5) where fr each x, η R (x) is the unique slutin t: F R ( y) (6) η R ( x) + xf R ( y) d y The apprximatin in (5) is in the sense that the differencegestzeras ρ.itisshwnin[3]that lg η R ( x ) dx, (7) with equality if and nly if fadings are independent at the receiver. Hence this term quantifies the capacity penalty due t crrelatin at the receiver. It can als be shwn that lg F ( T x ) dx, (8) with equality if and nly if fadings are independent at the transmitter. This term thus quantifies the capacity penalty due t crrelatin at the transmitter. IV. Ray-Tracing Channel Simulatin A. WiSE System Mdel We use the experimentally based WiSE ray-tracing simulatr [6] t generate the channel matrix H fr the indr wireless envirnment f a tw-flr ffice building in New Jersey (see Fig. ). We place the transmitting MEA n the first flr ceiling near the middle f the ffice building thrughut ur study. Receiving MEAs are placed with randm rtatins at randmly chsen psitins in Rm A, which is at intermediate distance frm the transmitter. We cnsider a carrier frequency f 5. GHz (wavelength, λ.58cm).the MEAs cnsist f multiple mnidirectinal antennas, arranged either in square grids r linear arrays within hrizntal planes. The separatin between antenna elements d is the same fr bth the transmitting and receiving MEAs. Since H varies fr different receiver lcatins, we estimate the channel variance υ, by averaging ver realizatins f H, and ver all pssible antenna pairs, j t i. We assume that the average received SNR ρ,asdefined in Sectin III-C, shuld be high enugh fr lw-errrrate cmmunicatin. If the SNR is t lw, we need excessively lng cdes t achieve a lw errr prbability. Practical cnstraints n current A/D cnverters limit the maximum SNR that can be explited effectively. Thus, lg F T ( x) dx we cnsider SNRs in the 8- db range. Fr all ur simulatins, we assume W t be MHz, and N t be - 7 dbm/hz, giving a ttal nise variance N W f -.8 dbm. The capacity and mutual infrmatin, (n) and I eq (n), are cmputed fr different n. B. Simulatin Results and Discussin. Capacity and Mutual Infrmatin f MEAs In this sectin, we cnsider square arrays fr cmpactness. The receivers are placed in rm A. We cnsider n, 4, 9, 6, 5 and 36, d.5λ, andρ 8dB. The CCDFs fr (n) are pltted in Fig. (slid lines). The rightward shift f the curves shws that (n) increases with n, because spatial diversity prvides additinal degrees f freedm fr transmissin. One perfrmance indicatr f interest is the capacity that can be supprted 95% f the time, i.e., the 5 % channel utage. Using a single antenna yields C.5 wf () 5.9 bps/hz while MEAs with fur antennas achieve C.5 wf (4) bps/hz, which is almst three and a half times larger. Fr n 36, we can get as high as 6 bps/hz. The CDDFs f I eq (n) are als pltted in Fig. (dashed lines). The advantage f having channel knwledge at the transmitter fr water-filling t be emplyed is illustrated by the hrizntal gap between the CCDFs f (n) andi eq (n). Fr small n such as n 4, the difference between C.5 wf (4) and I.5 eq (4) is nly abut bps/hz (abut 5% difference). This gap increases with n, e.g. fr n 36, C.5.5 wf is.3 % larger than I eq. The relative capacity gain f (n) ver I eq (n) is sensitive t ρ and n. C.5 wf (n)/ I.5 eq (n) are pltted in Fig. 3. The gain decreases as ρ increases, and it decreases at a slwer rate fr larger n. Whenρ is small, knwing the channel allws us t allcate pwer mre efficiently t strnger subchannels and therefre achieve higher capacity as cmpared t equal pwer distributin ver all subchannels. When ρ is large, there is sufficient pwer t be distributed ver all sub-channels, therefre the relative strength f the subchannels becme less imprtant. Fr n 4, the rati decreases frm 3 at ρ - db t at ρ 5dBfrC.5 wf (n)/i.5 eq (n).. Asympttic Behavir f MEA Capacities We study hw MEA capacity behaves as n grws large in simulated channels. We nly fcus n the high- SNR regime, ρ db.since (n)/i eq (n) isclset fr high SNR, we nly cnsider water-filling capacity. Fr simplicity, we cnsider linear arrays where the antenna-elements f MEA are equally spaced with tw antenna spacings: d.5λ and 5 λ. The transmitting MEA is placed rthgnal t the lng dimensin f the hallway ( bradside arrangement as in [5]). We esti-. Typical tw sided pwer spectral density f thermal nise at 3 K (rm temperature) fr a receiver that is mdeled as a 5 Ω resistance is -7.8dBm/Hz.

5 υ mate the variance and eigenvalues f the cvariance matrix t cmpute * and C wf using (9) and (5). The average capacity (n) frdifferentn is cmputed using realizatins f H, and is pltted fr d.5λ and 5 λ as the slid lines in Fig. 4. The dashed lines represent the capacities apprximated using the asympttic grwth rates fr the crrelated case; these are straight lines with slpe C wf.thegapbetweensimulatin results and the asympttic results grws smaller fr increasing n. Frd 5λ, (n)/n cnverges t 98 % f C wf when n 6. The dtted line represents the asympttic capacity derived assuming independent fading, which is a straight line f slpe *. We bserve that even fr, n * is significantly larger than the value f (n) fund fr simulated channels. That is, the asympttic results f Sectin III-C-, which d nt include the effects f crrelatin, verestimate MEA system capacity. If the assumptins in Sectin III-C- hld, and the crrelatin is crrectly captured by ur mdel, (n)/n shuld cnverge almst surely t C wf in the limit f large n. In Fig. 5, we illustrate this asympttic behavir f (n)/n at large SNR by pltting the empirical prbability density functins (PDFs) f (n)/n fr n 4,9 and6withd.5λ (strng crrelatin between H ij s) and d 5 λ (less crrelatin between H ij s). As n increases, the PDF becmes narrwer and has a higher peak value, i.e. (n)/n becmes less randm. In the limit f large n, we expect the PDF f (n)/n t cnverge t an impulse functin centered at the value C wf. The narrwing PDF s in Fig. 5 illustrate the almst surely cnvergence f (n)/n t C wf. Nte that when, the PDF s are narrwer and taller than when d.5 λ. This indicates that the rate f cnvergence is higher when d is larger, which is the case when the crrelatin between H ij is lwer. Further analysis is needed t understand hw crrelatin affects the validity f the asympttic results in Sectin III-C when ρ is nt large. V. Cnclusins MEA systems ffers ptentially huge capacity gains ver single-antenna systems. With perfect channel knwledge at the transmitter, water-filling slutins can be emplyed t achieve capacity. Equal pwer allcatin is easier t implement, but yields a mutual infrmatin I eq that can be significantly smaller than.the water filling gain /I eq is mst significant when the average received SNR ρ is small. C.5 wf /I.5 eq 3.5 when ρ - db, but at ρ 5 db, water filling gain is negligible, C.5.5 wf /I eq. Assuming i. i. d. path gains between different antenna pairs, theretical analysis shws that the capacity grws linearly with the number f antennas n in the limit f large n.hwever,inamrerealisticprpagatin envirnment, crrelatin des exist between antenna pairs and causes a smaller rate f grwth in capacity. Our simulatin results shw that fr.5 λ antenna spacing, the simulated average capacity is nly 79% f the predicted value n fr a bradside system with n 6 at ρ db. When the antenna spacing is increased, we see mre agreement between and n. Indeed with, (n)/n 98% when n 6. VI. Acknwledgments The authrs are grateful t Reinald Valenzuela, Jerry Fschini, Jnathan Ling and Dmitry Chizhik fr allwing us t use their WiSE simulatin tls, and fr their valuable advice & suggestins. Discussins with Jack Salz and Da-shan Shiu have been enlightening and are much appreciated. VII. References [] W. Jakes Jr., Micrwave Mbile Cmmunicatins, New Wiley, 974. [] J. Winters, Optimum Cmbining fr Indr Radi Systems with Multiple Users, IEEE Trans. Cmmun., vl. cm-35, n., pp. -3, Nv [3] G. J. Fschini and M. J. Gans, On Limits f Wireless Cmmunicatin in a Fading Envirnment When Using Multiple Antennas, accepted fr publicatin in Wireless Persnal Cmmunicatins. [4] G. J. Fschini and M. J. Gans, Capacity When Using Diversity At Transmit And Receive Sites and The Rayleigh- Faded Matrix Channel Is Unknwn At The Transmitter, WIN- LAB Wrkshp n Wireless Infrmatin Netwrk, New Brunswick, NJ, March -, 996. [5] W. C. -Y. Lee, Effects n Crrelatin Between Tw Mbile Radi Base-Statin Antennas, IEEE Trans. n Cmmunicatins, vl. cm-, N., pp. 4-4, Nvember, 974. [6]S.J.Frtune,D.H.Gay,B.W.Kernighan,O.Landrn,R. A. Valenzuela and M. H. Wright, WiSE design f Indr Wireless Systems: Practical Cmputatin and Optimizatin, IEEE Cmputatinal Science and Engineering, vl.,n., pp , March, 995. [7] G. J. Fschini and R. A. Valenzuela, Initial Estimatin f Cmmunicatin Efficiency f Indr Wireless Channels, Wireless Netwrks, vl. 3, n., pp. 4-54, 997. [8] T. M. Cver and J. A. Thmas, Elementary f Infrmatin Thery, Jhn Wiley & Sns, New Yrk, 99. [9] I. E. Telatar, Capacity f Multi-antenna Gaussian Channels, submitted t IEEE Transactins n Infrmatin Thery. [] A. M. Khrunzhy, B. A. Khruzhenk and L. A. Pastur, Asympttic prperties f large randm matrices with independent entries, Jurnal f Mathematical Physics, vl. 37, n., pp , Oct [] Y. Q. Yin, Limiting Spectral Distributin fr A Class f Randm Matrices, Jurnal f Multivariate Analysis, vl., pp. 5-68, 986. [] Jack Silverstein, Strng Cnvergence f the Empirical Distributin f Eigenvalues f large Dimensinal Randm Matrices, Jurnal f Multivariate Analysis, vl. 55, n., pp , 995. [3] David Tse, Capacity Scaling in Multi-antenna Systems, in preparatin.

6 VIII. Figures 4 meters y x Transmitting MEA 8 meters Rm A d Receiving MEA Prb ( r I eq > abscissa) n I eq & I eq (bps/hz) Fig.. Flr Plan f the ffice building mdelled in WiSE. The transmitting MEA is placed with its adjacent sides parallel t x- and y-axes, respectively. The receiving MEA is placed with a randm rientatin at each f the sample lcatins in rm A. Fig.. The CCDFs f (achieved via water -filling) and I eq (with equal pwer allcatin) fr n,4,9,6,5& 36 at received ρ 8 db. MEA antennas are arranged in square grids with d.5λ..5 /Ieq n 6 n 4 n 9.5 / I eq.5 d.5λ Average received SNR, ρ (db) (n), n *, ncwf (bps/hz) (n) n * n Bradside ρ db d.5λ Number f Antennas, n Fig. 3. Water-filling gain.5 / I eq.5 (slid lines) ver varying average received SNR, ρ, in rm L47 fr n 4, 9 and 6. Antennas at bth the transmitter and the receiver are arranged in square grids in this case. Fig. 4. Average capacity (n) versusn. Alsshwnare n * and n, which are asympttic results fr independent and crrelated H ij, respectively (see Sectin III-C). We cnsider linear arrays with the transmitting MEA placed parallel t the y-axis (bradside case). Empirical PDF fr (n)/n.8 C wf d.5λ. n (n)/n (bps/hz/antenna) C wf 6.3 d.5λ n (n)/n (bps/hz/antenna) C wf 6.3 d.5λ n (n)/n (bps/hz/antenna) Fig. 5. Empirical prbability density functin f the nrmalized capacity (n)/n fr n 4, 9 and 6. We cnsider linear arrays with antenna elements separated by.5 λ and 5 λ. The reference value is, as predicted by the asympttic thery cnsidering crrelated H ij.

ENSC Discrete Time Systems. Project Outline. Semester

ENSC Discrete Time Systems. Project Outline. Semester ENSC 49 - iscrete Time Systems Prject Outline Semester 006-1. Objectives The gal f the prject is t design a channel fading simulatr. Upn successful cmpletin f the prject, yu will reinfrce yur understanding