ECE 559: Wreless Commucato Project Report Dversty Multplexg Tradeoff MIMO Chaels wth partal CSIT Hoa Pham. Summary I ths project, I have studed the performace ga of MIMO systems. There are two types of ga provded by the MIMO chaels: oe measures the crease of the capacty by the umber of degrees of freedom of the chael (spatal multplexg ga), ad the other mprovemet of the relablty by the dversty ga. Most of prevous works focus o ether oe of these two types of ga by desgg schemes to extract ether maxmal dversty ga or maxmal spatal multplexg ga. However, maxmzg oe type of gas may ot ecessarly maxmze the other. Ths make t dffcult to compare dversty-based schemes ad multplexg-based schemes. Recetly, Zheg ad Tse [] proposed a ew pot of vew that ufes these two types of gas: gve a MIMO chael, both gas ca be smultaeously obtaed, but there s a tradeoff betwee how much of each type of ga ay codg scheme ca extract. The optmal dverstymultplexg tradeoff s a fudametal performace lmt of a MIMO chael. By studyg ths fudametal tradeoff, we ca ga mportat sghts o the resource costrat of MIMO systems, as well as o specfc schemes to commucate over such systems. The tradeoff curve ca be used as a ufed framework to compare the performace of varous space-tme codg schemes for MIMO chaels ad stmulates the desg of ew schemes. For each scheme, we ca compute the achevable dverstymultplexg tradeoff curve ad compare t agast the optmal curve. By dog ths, we ca take to accout both the capablty of a scheme to combat agast fadg (through dversty ga) ad ts acheved spatal multplexg ga. Therefore, the tradeoff curve provdes a ufed framework to make far comparsos. The aalyss focus o slow fadg chael (the chael s costat durg a block) ad hgh SNR rego. The approach used for both cases of o CSIT ad partal CSIT based o evaluatg the age probablty at hgh SNR. For the case of o CSI at the trasmtter, the optmal tradeoff curve s a pecewse lear curve coectg the pots (r, (m-r)(-r)), where r s the multplexg ga ad (m-r)(r) s the dversty ga. Itutvely, we ca thk of a pot o the optmal curve as followed: If we wat to use r trasmt ateas ad r receve ateas for multplexg
ga to crease the rate, the remag (m-r) trasmt ateas ad (-r) receve ateas ca be used to gve the dversty ga of (m-r) (-r). Cosder a example of the x system. The maxmal dversty ga for ths chael s d max = 4, ad the total umber of degrees of freedom the chael s r max =. To get the maxmal dversty ga, each formato bt eeds to pass through all the four paths from the trasmtter to the recevers. The smplest way s to use repetto code, ths wll guaratee the maxmal dversty, however, t oly acheved the rate / sce oly oe symbol s trasmtted over two symbol perods. As class we see that, the Alam scheme ca also acheve the full dversty. However terms of tradeoff acheved, the Alam scheme s strctly better tha the repto scheme, sce t acheves a hgher dversty ga for ay posttve spatal multplexg ga ad ts maxmal multplexg ga s. However, the tradeoff curve acheved by the Alam scheme s stll below the optmal curve for postve values of r. Fgure shows a example of dfferet codg schemes for the x MIMO chael. Fgure. Example of tradeoff curves for dfferet codg schemes
For the case of partal CSIT, the recever, after estmatg the chael wll feedback a dex to the trasmtter. Ths dex dcates the rego to whch the chael state belogs. Based o ths formato, the trasmtter wll cotrol the power to mmze the age probablty whle rema sastfyg the log term power costrat. The tradeoff curve dervato for ths case s smlar to the case whe there s o CSI at trasmtter whch s based o the age probablty. The mportat pot ths case s fdg the optmal dex mappg for dfferet states of the chael. Aother mportat pot s the cocept of exteded approxmately uversal codto (stated the slde) whch guaratees the upper boud ad lower boud to be matched. The ma results are gve the attached slde. There are two terestg results: - The maxmal dversty ga of MIMO chael ca be creased expoetally wth the umber of quatzed regos ( d (0) = ( m), where K s the umber of quatzed regos). Note that whe K = (o feedback), the result s the same as before. Ths result suggests that, we ca use oly a few bt of feedback to get the dversty ga ad save most of atea resource for multplexg ga. - Whe the adaptve rate trasmsso scheme s used (dfferet multplexg gas r are used for dfferet regos of the chael state, we ca acheve o-zero dversty ga at maxmal multplexg ga. Ths ca ot be acheved the case of o CSIT. For K =, the full dversty ga of the o-csit case (m) ca be acheved at maxmal multplexg ga. The ext secto provdes some detaled proofs stated the presetato to derve the optmal tradeoff curve of the MIMO chael whe the trasmtter does ot kow the CSI. The approach to dervg the tradeoff curve for the partal CSIT case s the same except the troducto of exteded approxmately uversal codto ad the optmal dex mappg scheme whch results the much hgher dversty ga. The detaled of the proofs ca be foud [], whch I wll ot cover here. K k = k. Dversty-Multplexg tradeoff wth o CSI at trasmtter [] Ths secto gves some detaled dervato of the optmal dversty-multplexg tradeoff curve. As we kow from the presetato, the approach s as followed: - Formulate the age probablty - Lower boud the error probablty by age probablty - Fd the upper boud o the error probablty - The upper boud turs to match the lower boud, therefore we get the SNR expoet of the error probablty Deftos: A scheme s sad to acheve spatal multplexg ga r ad the dversty ga d f the data rate:
ad the average error probablty RSNR ( ) lm = r, deoted by R rlog SNR log SNR SNR log Pe ( SNR) lm = d, deoted by Pe SNR log SNR SNR d. Outage probablty As the presetato we have, o the scale of terest (hgh SNR): P R P I SNR R ( ) [log det( + HH ) < ] To compute the asymptotcal age probablty, we eed the followg lemma: Lemma : let R be a m x radom matrx wth..d CN(0,) etres. Suppose m, μ μ... μ be the ordered o-zero egevalues of RR, the the jot pdf of μ s s: m = m, j = < j p( μ,..., μ ) K μ ( μ μ ) e μ Where K m, s the ormalzg costat. Defe: we ca fd the jot pdf of α s: α log μ / log SNR, for all. The ( ) m + α α α j α m, = < j = p( α) = K (log SNR) SNR. ( SNR SNR ) exp[ SNR ] Now cosder the age probablty wth R = rlog SNR, let λ λ... λ be the ozero egevalues of HH, ad λ = SNR α, we have: r ( ) [log det( + HH ) < ] = [ ( + λ ) < ] = P R P I SNR R P SNR SNR At hgh SNR, we have + ( ) ( + SNRλ ) SNR α, where (x) + deotes max{x,0}. The, + ( α ) + ( ) [ < ] = [ ( α ) < ] P R P SNR R P r Sce, we are oly terested the SNR expoet of P, we ca gore the term K, (log ) m SNR. Furthermore, for ay α < 0, the term ( SNR α ) decays expoetally
wth SNR. Thus, we ca gore the tegral over the rage wth α < 0, ad defe the set A m{ m, } + + ' = { α R α α... αm{ m, } 0, ad ( α) < r}, the: ( m + ) α α α P ( log ). ( j r SNR SNR SNR SNR ) d A' = < j The, we have the followg theorem: Theorem : Let the data rate R = rlog SNR, wth r m{ m, }. The age probablty sastfes: d ( r) P ( rlog SNR) SNR α A' m{ m, } where d ( r) = f + m α Proof. See [] for detaled proof. = The mmzg α ca be explctly computed. I the case that r takes a terger value k, we have: α =, for =,..., k ad α = 0, for = k+,...,. Itutvely, a geeral mx system, a age occurs whe the chael matrx H s ear sgular. The key step s to explctly quatfy how sgular H eeds to be for age to occur. Sce the smaller sgular values have a much hgher probablty to be close to zero tha the larger oes, the typcal age evet has -k smallest sgular values λ SNR ad k largest sgular values are of order. For the case that r s ot a terger, say r ( k, k + ), we have: α α α α =, for =,..., k, = 0, for = k+,..., ad k = k+ r. Wth these optmzed value of α, we get. Error Probablty d () r = ( m r)( r) Lower Boud: Frst, we wll prove the error probablty s lower boud by the age probablty. Fx a codebook C of sze Rl mxl, ad let X C be the put of the chael, whch s uformly draw from the codebook C. Codtoed o a specfc chael realzato H = H, by Fao s equalty, we have:
Rl + P(error H = H) Rl+ I( X; Y H= H), hece I( XY ; H= H) P(error H = H) lr log SNR lr log SNR The last term goes to zero as SNR goes to fty. Averagg over H gves the average error probablty P ( ) [ (error e SNR = EH P H = H)] Defe a set Dδ { H : I( XY ; H= H) < ( r δ ) llog SNR}, for ay H D δ, the r δ probablty of error s lower bouded by + o( ), hece r ( ) r δ P () ( e SNR + o P Dδ ) r The choose the put X to mmze PD ( δ ) ad apply the theorem, we have:. r Pe ( SNR) + o() SNR r SNR δ d ( r δ ) d ( r δ ) d ( r) Take δ 0, by the cotuty of d (r), we have the lower boud Pe ( SNR) SNR Ths result suggests that codtoed o the chael age evet, t s very lkely that a detecto error occurs.. Upper Boud: Choose the put to be the radom code from the..d Gaussa esemble. We have, Pe ( SNR) = P ( R) P(error age) + P(error, o age) P ( R) + P(error, o age) The secod term ca be upper-bouded va a uo boud. Assume X(0), X() are two possble trasmtted codeword ad Δ X = X(0) X(). Suppose X(0) s trasmtted, the probablty that a ML recever wll make a detecto error favor of X(), codtoed o a certa realzato of the chael, s SNR PX ( (0) X() H = H) = P H( ΔX) w m F
Where w s the addtve ose o the drecto of H ( Δ X ) wth varace /. Usg the upper boud for the Q-fucto, we have SNR PX ( (0) X() H = H) exp H( ΔX) 4m Averagg over the esemble of radom codes, we have the average parwse error probablty (PEP) codtoed o the chael realzato [3] SNR PX ( (0) X() H = H) det I+ HH m lr Apply the uo boud for SNR codewords, we have lr SNR PX ( (0) X() H = H) SNR exp H( ΔX) 4m = l m{ m, } lr SNR = SNR ( + ) m λ l Averagg wth respect to the dstrbuto of the SNR expoet α of sgular values ad wth a smlar argumet as Theorem, we ca approxmate ths as. dg ( r, α ) P(error, o age) SNR d c ( A') m{ m, } m{ m, } + wth dg( r, α) = + m α + l ( α) r = = The probablty s domated by the term correspodg to α that mmzes dg ( r, α ). For l (m{ m, }) + m = m+, the mmum always occurs wth ( α ) = r, hece α m{ m, } d () r m ( + m ) α = d () r r G α = m{ m, } r = Therefore P( SNR) P ( R) + P(error, o age) e. SNR + SNR SNR d ( r) dg ( r) d ( r)
We see that the upper boud ad the lower boud are matched, thus d ( r) Pe ( SNR) SNR, or we have the dversty ga d () r = d () r = ( m r)( r) whch gves the optmal tradeoff curve. 3. Q&A Q: How ca the tradeoff curve be utlzed? (What s the applcato of the tradeoff curve?) A: The optmal tradeoff curve shows the optmal tradeoff betwee dversty ga ad multplexg ga provded by the MIMO chael. It s used as a ufed framework to compare dfferet space-tme codes. By computg the tradeoff curve of each specfc scheme ad compare t to the optmal curve, we ca see how optmal that scheme s. Sce some schemes are desged to maxmze the capablty to combat fadg ad the others are desged to maxmze the trasmsso rate, t s dffcult to compare these schemes. However, by computg the tradeoff curves of these schemes, we take to accout both of these capabltes that each schemes ca acheve. That make the far comparso. Besde, the optmal tradeoff curve may stmulate the desg of ew schemes to utlze the resources gve by the MIMO chaels. However, t may ot show how to acheve the optmal curve. Q: It s ot clear to me f ths result provdes some boud o geeral space-tme code? A: Yes. Because, the optmal curve s the upper boud o the dversty ga ad multplexg ga that a MIMO chael ca support. Ay space-tme code s desged to utlzed these gas ad ca ot get above the optmal curve. Q: What s the tuto behd the pecewse lear tradeoff curve? Is t some kd of tme multplexg? Ad, Q: How ca we acheve pots the curve by tme-sharg? Partcularly, how to acheve the pot (0,) Alam scheme for x system? A: A specfc pot o the curve s ot acheved the sese of tme sharg or tme multplexg. The optmal curve s derved by fxg the multplexg ga r, ad fd the maxmal dversty ga that ca be acheved at that rate. Therefore, a specfc pot o the curve llustrates that f we wat to operate the system at the rate R rlog SNR, the maxmal dversty we ca get s the dversty ga at pot (r, d(r)). Itutvely, for a mx MIMO system, f we use r trasmt ateas ad r receve ateas for multplexg ga to crease the rate, the remag (m-r) trasmt ateas ad (-r) receve ateas ca be used to gve the dversty ga of (m-r) (-r). For the Alam scheme, the pot (0,) ca be acheved by scalg the rate R as R = log SNR ad compute the error probablty whe SNR goes to fty, we ca see
that the SNR expoet of the error probablty s the same as the sgle atea case, whch gve zero dversty ga. Q: How ca a specfc pot o the tradeoff curve ca be acheved practce? A: The optmal curve does ot tell how to acheve a specfc pot practce, sce t oly gves the boud o the gas that a chael may provde. Q: Practcally speakg, whe should we choose betwee dversty ad multplexg gas? Whch should be prefered? Is there a relato betwee the relablty ad the scheme that we choose? A: I my opo, ths deped o the chael codtos ad the applcato we wat to provde. The dversty oreted schemes may be chose the case of bad fadg chael where we wat to provde depedet fadg paths to combat fadg. The multplexgoreted schemes may be chose whe we wat hgh data-rate applcatos but do ot have tght requremets o error probablty. Q: Ca radom codg be used to arrve at the lower boud o the probablty of error? A: I the age probablty formulato, the put s take as the Gaussa dstrbuto ad the the age probablty s take as the fmum over all put dstrbuto. Sce the ose s Gausa, the Gaussa put maxmze the mutual formato ad the age occurs whe the mutual formato drops below the requred rate R, ths gve the lower boud o age probablty for every put dstrbuto. Furthermore, as we showed that the error s very lkely to occur whe the age evet occurs, therefore we ca use age probablty as a lower boud for the error probablty. Q: How s the tradeoff curve affected by feedback from the recever? A: Whe the recever feedbacks the quatzed value of CSI whch dcate the rego whch the chael appears, to the trasmtter, the trasmtter ca use that formato to cotrol the power. The dversty ga acheved s surprsgly creased expoetally wth the umber of quatzed regos, the maxmal dversty ga whch ca be acheved K k s ( NN t r), where K s the umber of quatzed regos []. Aother terestg result k = s that, whe adaptve rate trasmsso s used (dfferet multplexg gas for dfferet chael state regos), we ca have o-zero dversty ga at maxmal multplexg ga, whch ca ot be acheved the case of o CSIT.
Refereces [] L. Zheg ad D. Tse, Dversty ad Multplexg: A Fudametal Tradeoff Multple Atea Chaels, IEEE Trasactos o Iformato Theory, vol. 49, May 003, pp. 073-96. [] Tug T. Km ad Mkael Slogkud, Dversty-Multplexg Tradeoff MIMO Chaels wth Partal CSIT, IEEE Tras. Iform. Theory, vol. 53, NO. 8, August 007. [3] V.Tarokh, N. Seshadr, ad Calderbak, Space-tme codes for hgh data rate wreless commucatos: Performace crtero ad code costructo, IEEE Tras. Iform. Theory, vol. 44, pp. 774-765, Mar. 998