Switched and Sectred Beamfrming c Raviraj Adve, 2005. rsadve@cmm.utrnt.ca Intrductin Having investigated the use f antenna arrays fr directin f arrival estimatin we nw turn t the use f arrays fr data prcessing. Remember the gal f this curse was t investigate the use f arrays t supprt mre users, reduce the rate f drpped calls and increase the efficiency f the wireless infrastructure. Here we will lk at hw fixed beamfrming, i.e., nn-adaptive signal prcessing at a base statin antenna array can significantly imprve n all these figures f merit. This nn-adaptive prcessing is the traditinal apprach t array prcessing. Our analysis will be based, as well, n sme traditinal figures f merit. w 0 * w * w * N- Σ Output Signal Figure : Generalized beamfrming n receive Any beamfrmer adds the signals at the N antenna elements weighted by scalars. Cnsider the array structure shwn in Fig.. The signal at each element f the array is multiplied with a cmplex weight (the cnjugate f the weights). The utput signal is given by y = N n=0 w nx n = w H x, () where w is the length-n vectr f weights. In a fixed beamfrmer, the weights used are fixed, i.e., the weight vectr w is fixed. The structure shwn there is the general frmulatin f hw we will prcess received signals - this is beamfrming n receive. Nte that due t reciprcity, if n transmit the excitatin n Mst f the material in this review is frm Janaswamy []
4 Pattern f 4 beams using the Butler Matrix 3.5 3 2.5 Beampattern 2.5 0.5 3 2 0 2 3 ψ Figure 2: The fur utput beams f the Butler matrix the n th antenna were the weights w n, the transmit beampattern wuld be the same as the receive beampattern. S, mst f what we will say abut receive is valid n transmit t. 2 The Butler Matrix Each weight vectrs crrespnds t a particular beam pattern (beam). Frm ur earlier discussin n array thery, we knw that chsing w n = e jnψ = e jnkdcs φ has a mainlbe in the directin φ. A simple chice f the set f weights, then, is t chse N sets f w such that each f the resulting beams is equally spaced in the ψ dimensin. There is n theretical limit n hw many beams can be frmed, hwever, chsing N beams makes the frming f the beams equivalent t a FFT f the received signals. In practice, at the radi frequencies invlved, this FFT can be implemented using a Butler matrix as shwn in Fig. 3. The fur utputs in the figure crrespnd t the fur beampatterns shwn in Fig. 2). The beams are pltted in a linear, nt the usual db scale. Nte that the fur beams are equally spaced in ψ space. Furthermre, the lcatins f each mainbeam is at the nulls f the ther three beams. This is als a characteristic f the FFT peratin. This is imprtant fr cmmunicatin applicatins, because if the desired signal arrives frm a directin at the peak f ne beam, it des nt leak int any ther beam. We will see that this is the ideal situatin fr a switched beam system. Of curse, in a mre practical situatin, the signal wuld nt be perfectly at the mainbeam lcatin. There wuld be an assciated signal lss and leaking int multiple beams. 2
-90-90 -90-90 45 45-90 -90-90 -90 Figure 3: The Butler Matrix fr a fur element array 3
The signal lss is knwn as the scallping lss [2]. 2. Switched and Sectred Appraches In a cmmunicatin setting, fixed beamfrming has been explited in tw ways: switched r sectred beamfrming. Figure 4 presents the general scheme f a switched beam system. The signals frm the N antenna elements are the input t a beamfrming netwrk with P beams. In general, P N. The Butler matrix is a beamfrming netwrk that frms P = N beams. Fr each user, the signal-tnise rati (SNR) f each beam is measured and the maximum is determined. The beam assciated with the largest SNR is chsen fr further prcessing. At any given time, all channels assigned t the cell that this array is serving are available t all users. Therefre, a particular beam, at certain times, may serve several users. Beamfrming Netwrk Repeat fr each user RF Switch Output Signal SNR Measurement Cntrl Signal (max SNR) Figure 4: The general scheme f a switched beam system In a sectred apprach, n the ther hand, the channels available in the cell are equally divided between the P beams. A mbile that transmits within that beam is assigned a channel (if available) frm that beam nly. If the mbile mves int the area cvered by a different beam, a handff must ccur. A sectred apprach, therefre, inherently suffers frm greater numbers f handffs. Clearly a switched apprach is mre efficient. All channels are available t all beams. A particular beam wuld be able t cver a ht spt, an area with many users. A switched beamfrmer culd serve up t N k users (where N k is the number f channels available t the base statin). Hwever, the trade ff is that a switched beam apprach is significantly mre cmplex t implement - implementing such a system requires the ability t measure the SNR, determine the maximum and a RF switch that chse the apprpriate beam. This prcess must be repeated fr each user. 4
The sectred apprach, n the ther hand, is equivalent t P new sub-cells within the cell served by the base statin. A sectred apprach culd nly serve N k /P users within any beam. 3 Figures f Merit 4 5 D 3 2 7 R 3 4 6 4 2 5 3 2 Figure 5: Channel distributin in a cellular system T illustrate the imprvements in perfrmance due t beamfrming, we use three figures f merit: spectral efficiency, number f users that may be serviced and prbably f utage. The analysis will be perfrmed in cntext f the cellular system shwn in Fig. 5. The gegraphical area t be cvered is divided int clusters, each f N c cells. The available channels are divided amngst the cells such that n adjacent cells use the same set f channels. In the figure, 7 cells (marked thrugh 7) frm the cluster. The central cell acts as the desired cell. It is assumed that the situatin is hmgeneus, i.e., each cell is like every ther cell. If the ttal bandwidth available is B T Hz and each channel ccupies B c Hz, the ttal channels 5
available and channels per cell are simply Ttal Channels = N T = B T B c, (2) 3. Spectral Efficiency The spectral efficiency f a system is given by η s = Channels Per Cell = N k = N T N c. (3) # f channels per Hz Area cvered = B T/B c B T Area = B T/B c B T N c A, (4) where A is the area f each cell. Nte that N c, the cluster size is in the denminatr, implying that we culd imprve spectral efficiency by reducing the cluster size (r the effective cluster size). 3.2 Trunking Efficiency...... r equivalently, the number f paying users per cell. This is nt the number f active users, but the ttal number f users that can be supprted. Befre determining the number f users per cell in a switched r sectred beamfrming system, we must define sme terms frm cmmunicatin thery. The verall and per user traffic intensity are defined t be E = E u = (avg. # calls in time T) (avg. duratin f call), (5) T (avg. # calls made by an individual user in time T) (avg. duratin f call), (6) T respectively. Bth are measured in terms f Erlangs. A traffic lad f ne Erlang wuld use ne channel all the time. If a system can ffer E Erlangs f traffic intensity, the number f users that can be supprted is E/E u. In a cell with N k channels, with prbability f blcked call set at 0.0, ne culd supprt K users, where K = N k E u [ 0.855 tanh(0.07nk ).4 0 3 N 2 k exp( 0.07N k) ]. (7) Nte that the number f users per cell N k is determined by the cluster size as in Eqn. (3), i.e., it is inversely prprtinal t the cluster size. Beamfrming allws us t reduce the effective cluster size and s increase the number f channels. Cnsider the cellular system in Fig. 5. The central area, marked, is the primary cell under analysis. The ther areas marked use the same sets f frequencies - signals frm these areas 6
wuld interfere with each ther. Each cell has radius R. Assuming all base statins transmit at the same pwer level, the wrst case carrier pwer is where n is the path lss expnent. The interference pwer is given by C Rn, (8) Interference per interfering cell D n, Ttal Interference = I N I Dn, (9) Carrier t Interference Rati = C/I = ( D n, (0) N I R) where N I is the number f interfering cells (c-channel cells) using the same set f frequencies as the cell under cnsideratin (N I = 6 in the example scenari f Fig. 5). In general, it is can be safely assumed that the interference arrives frm nly the first layer f interfering cells surrunding the cell under cnsideratin. Fr the gemetry shwn, D = R 2 + (4R) 2 2(R)(4R)cs(20 ) = R 2. In general, D = R 3N c, () C/I = (3N c ) n/2. (2) N I Reversing Eqn. (2), if a certain (C/I) is required t maintain a gd cmmunicatin link, the minimum required cluster size is N creqd = 3 [N I(C/I) reqd ] 2/n. (3) 3.3 Outage Prbability The utage prbability is the prbability that an n-ging call is drpped due the carrier pwer falling belw a threshld (usually defined in terms f interference pwer). We shall cnsider a call t be drpped if the carrier pwer (C) is belw a margin (q) ver the interference pwer (I), i.e., C/I < q. Nt that we wuld usually mdel the envelpe f the signal (e.g. lg-nrmal, Rayleigh, etc.) and nt the pwer level. Interference ccurs due a user in a c-channel cell is using the same channel as the user in the cell under cnsideratin. Since we can safely assume the signals frm interfering cells are independent, if there are i c-channel cells interfering, then the interference pwer is given by i I = I k, k= 7
where I k is the interference pwer f the k th interfering cell. The prbability that a particular channel is in use in a given c-channel cell (p c ) is the traffic intensity divided by the ttal number f channels, p c = E k N k. In pint f fact, this prbability is reduced by a factr ( p b ) where p b is the prbability that a call is blcked. The prbability that i cells (f a ttal pssible N I c-channels) use the same channel is therefre Outage ccurs if P i = ( NI i ) p i c( p c ) (N I i). P ut = P(C/I < q) = P(C/I < q/i)p i = N I N I P(C/I < q/i) i= i= ( NI i ) p i c( p c ) (N I i), where P(C/I < q/i) is the prbability f utage, P(C/I < q), given there are i interfering cchannels. Nte that P(C/I < q/i) is highly dependent n the fading assciated with the channel. In general, as we are lking at relatively large distances, the mdel used t determine P(C/I < q/i) wuld assume large scale fading (such as lg-nrmal fading). 4 Perfrmance Imprvements Having determined ur figures f merit, what perfrmance gains des a sectred antenna r switched beamfrming achieve? The simplest (and ideal) beamfrmer has a flat respnse ver an angular sectin. Given an N element array, such a beam wuld cver /N f the ttal angular regin, such as shwn in Fig. 6. Assuming that the interference is distributed unifrmly in angle, n average, such a beamfrmer wuld receive nly (/N) th f the interference and (C/I) wuld increase by a factr f N. Mre rigrusly, the interference pwer is given by I mni I s.b. 2π 0 2π 0 P 0 dφ, AF 2 P 0 d φ, where P 0 is the average angular interference pwer density. The subscripts mni and s.b. refer t an mnidirectinal antenna (single element/n directivity) and switched beamfrming respectively. In terms f interference pwer, the gain in using a directive array is therefre I mni I s.b. = 2π 2π 0 AF 2 P 0 dφ = D max (4) 8
0.8 Beampattern 0.6 0.4 0.2 0 3 2 0 2 3 ψ Figure 6: An ideal sectred beam where D max is the directivity f the array (mre rigrusly, the maximum directivity f the array). If the mainbeam f the array is pinted in directin f the user, this is als the gain in terms f (C/I). Hwever, if there is a mismatch between mainbeam directin and the mbile lcatin, this gain wuld be reduced by the scallping lss. Nte: This gain is the same fr a sectred antenna system using the same beam pattern as ur switched beam system. Due t the gain in the carrier t interference rati, using Eqn. (3), the required cluster size reduces (with attendant increase in number f channels per cell), i.e. N creqd (s.b.) = 3 [ ] 2/n NI (C/I) reqd D max = N creqd (mni) D ( 2/n) max (5) N k(s.b.) = N T N creqd (s.b.) = N k(mni) D (2/n) max. (6) The gain in terms f cluster size and number f channels per cell is therefre D (2/n) max. Unfrtunately, n, the path lss expnent is usually greater than 2 and the gain is less than the maximum directivity. 4. Spectral Efficiency Equatin (4) shws that the spectral efficiency is inversely prprtinal t the cluster size, implying an increased spectral efficiency fr switched beam systems. The spectral efficiency f a switched 9
3000 2800 Number f users per cell fr sectred and switched beam antennas Omni Antenna Switched Beam Sectred Apprach 2600 2400 Number f users/cell 2200 2000 800 600 400 200 000 800 2 3 4 5 6 7 8 9 0 2 Directivity f Antenna beam system is Figure 7: Number f users in sectred and switched beam systems η s(s.b.) = # f channels per Hz Area cvered = B T/B c B T Area = B T /B c B T N creqd (s.b.) A (7) = η s(mni) D (2/n) max. (8) Since the imprvement in cluster size is the same fr switched beam and sectred systems, s is the imprvement in spectral efficiency. 4.2 Trunking Efficiency...... r the number f paying users per cell. T determine the number f users per cell that a switched beamfrming system can supprt, we use the number f channels frm Eqn. (6) in Eqn. (7). Due t the increased number f available channels, the number f pssible users increases substantially. K s.b. = N k(s.b.) E u [ ) 0.855 tanh (0.07N ( ) ] 2 k(s.b.).4 0 3 0.07N k N k(s.b.) e (s.b.). (9) Fr a sectred system, this increase in number f channels is spread ver the number f beams 0
(P). The number f channels per beam and ttal number f users that can be supprted are, N k(sect) = N k(s.b.) P, (20) N k(s.b.) ( K sect = P 0.855 tanh P E u 0.07 N k(s.b.) P ) ( ) 2 Nk(s.b.).4 0 3 e 0.07Nk (s.b.) /P. P (2) It is the nn-linear relatinship between number f users supprted and the number f channels available that gives switched beam systems the advantage ver sectred appraches. Figure 7 plts the number f users per cell in three pssible appraches - the mnidirectinal antenna system f Eqn. (7), switched beam and sectred beam systems. As is clear, the sectred beam system results in a very large increase in the number f users that can be supprted (ver the mnidirectinal antenna withut any directivity). Switched beamfrming is a much mre cmplicated (and cstly) system - unlike in a sectred system, a SNR measurement, decisin mechanism and RF switch is required fr each user in service. Hwever, as is shwn in Fig. 7, switched beam systems can supprt many mre users than a sectred beam system. This is the primary justificatin fr the added cst f a switched beam system. 4.3 Outage Prbability In a sectred r switched beam system, the individual cmpnents f the interference envelpe are weighted by the directivity, i.e., i I = I k d k, k= where d k is the directivity f the array in directin f the interference surce. As the directivity is implementatin dependent, we can nly lk at specific examples. Here we pick the easiest example - ne in which the beam pattern is the ideal beam pattern f Fig. 6, i.e. D max = N. In this case, the prbability that any c-channel cell is using the same channel as in the primary cell is weighted by the prbability that that individual cell is in within the beam selected, i.e. P ut = N I p c(s.b.) = P(C/I < q/i) i= p c(mni) N ( NI i = p c N ) (22) p i c (s.b.) ( p c(s.b.) ) (N I i), (23) = N I P(C/I < q/i) i= ( NI i ) (pc N ) i ( p c N )(N I i). (24)
Figure 8: Outage prbability in a switched beam system Figure 8 plts the utage prbability f a switched beam system in cmparisn t a mnidirectinal antenna. The plt is Fig. 8. f Janaswamy []. The fading is mdelled as lg-nrmal with tw different standard deviatins. The number m is the number f elements (ur N) while ζ is the prbability a particular c-channel cell is interfering with the primary cell (ur p c ). Z d is the SNR. As is seen, switched beamfrming results in a significantly lwer prbability f utage at a chsen SNR. 5 Summary This sectin has analyzed the impact f nn-adaptive prcessing n traditinal measures f perfrmance f a wireless cmmunicatin system. All measures f perfrmance imprve and the imprvement is dependent n the directivity f the antenna array. The array prcessing used here is fixed, i.e., is nn-adaptive. The impact f using an array was analyzed fr switched and sectred beam systems. Switched beamfrming antennas prvide fr a significant increase in spectral efficiency and large increases in number f users that can be supprted. This increase is ver and abve increases due t directivity (as explited by a sectred beam system). The primary mtivatin fr switched 2
beamfrming is the large increase in pssible number f users. A nte t remember: We have analyzed switched and sectred beamfrming in extremely simple scenaris. The results presented in the figures here are dependent n system implementatins and nly prvide a guideline t the mtivatins. In a real system, ne wuld have t include the parameters, fading and the mdulatin seen by the individual system. Fr example, a CDMA system may have a different interference pattern than a FDMA system. References [] R. Janaswamy, Radiwave Prpagatin and Smart Antennas fr Wireless Cmmunicatins. Kluwer Academic Publishers, 2000. [2] F. Harris, On the use f windws fr harmnic analysis with the Discrete Furier Transfrm, Prceedings f the IEEE, vl. 66, N., pp. 5 83, Jan 978. 3