Multiple Parameter Estimation With Quantized Channel Output

Transcription

1 010 Internatona ITG Workshop on Smart Antennas (WSA 010 Mutpe Parameter Estmaton Wth Quantzed Channe Output Amne Mezghan 1, Fex Antrech and Josef A. Nossek 1 1 Insttute for Crcut Theory and Sgna Processng, Technsche Unverstät München, 8090 Munch, Germany German Aerospace Center (DLR, Insttute for Communcatons and Navgaton, 834 Wessng, Germany Ema: 1 {Mezghan, Nossek}@nws.e.tum.de, fex.antrech@dr.de Abstract We present a genera probem formuaton for optma parameter estmaton based on quantzed observatons, wth appcaton to antenna array communcaton and processng (channe estmaton, tme-of-arrva (TOA and drecton-ofarrva (DOA estmaton. The work s of nterest n the case when ow resouton A/D-converters (ADCs have to be used to enabe hgher sampng rate and to smpfy the hardware. An Expectaton-Maxmzaton (EM based agorthm s proposed for sovng ths probem n a genera settng. Besdes, we derve the Cramér-Rao Bound (CRB and dscuss the effects of quantzaton and the optma choce of the ADC characterstc. Numerca and anaytca anayss reveas that reabe estmaton may st be possbe even when the quantzaton s very coarse. Index Terms: Quantzaton, MIMO channe estmaton, TOA/DOA estmaton, EM agorthm, Cramér-Rao Bound, stochastc resonance. I. INTRODUCTION In mutpe-nput mutpe-output (MIMO communcaton systems, where ow power and ow cost are key requrements, t s desrabe to reduce the ADC resouton n order to save power and chp area 1. In fact, n hgh speed systems the sampng/converson power may reach vaues n the order of the processng power. Therefore, coarse anaog-to-dgta converters (ADCs may be a cost-effectve souton for such appcatons, especay when the array sze becomes very arge or when the sampng rate becomes very hgh (n the GHz range. Naturay, ths generates a need for deveopng new detecton and estmaton agorthms operatng on quantzed data. An eary work on the subject of estmatng unknown parameters based on quantzed can be found n 3. In 4, 5, the authors studed channe estmaton based on snge bt quantzer (comparator. In ths work, a more genera settng for parameter estmaton based on quantzed observatons w be studed, whch covers many processng tasks, e.g. channe estmaton, synchronzaton, deay estmaton, Drecton Of Arrva (DOA estmaton, etc. An Expectaton Maxmzaton (EM based agorthm s proposed to sove the Maxmum a Posteror Probabty (MAP estmaton probem. Besdes, the Cramér-Rao Bound (CRB has been derved to anayze the estmaton performance and ts behavor wth respect to the sgna-to-nose rato (SNR. The presented resuts treat both cases: pot aded and non-pot aded estmaton. We extensvey dea wth the extreme case of snge bt quantzed (comparator whch smpfes the sampng hardware consderaby. We aso focus on MIMO channe estmaton and deay estmaton as appcaton area of the presented approach. Among others, a channe estmaton usng 1-bt ADC s consdered, whch shows that reabe estmaton may st be possbe even when the quantzaton s very coarse. In order to ease the theoretca dervatons, we restrct ourseves to reavaued systems. However, the resuts can be easy extended and apped to compex vaued-channes as we w do n Secton VI. Our paper s organzed as foows. Secton II descrbes the genera system mode. In Secton III, the EM-agorthm operatng on quantzed data s derve and the estmaton performance mt based on the Cramér-Rao Bound (CRB s anayzed. In Secton IV, we dea wth the snge-nput snge-output (SISO channe estmaton probem as a frst appcaton, then we generaze the anayss to the mutpe-antennas (MIMO case n Secton V. Fnay we hande the probem of sgna quantzaton n the context of Goba Navgaton Satete Systems (GNSS n Secton VI. Notaton: Vectors and matrces are denoted by ower and upper case tac bod etters. The operators ( T, ( H, tr(, (,Re( and Im( stand for transpose, Hermtan transpose, trace of a matrx, compex conjugate, rea and magnary parts of a compex number, respectvey. I M denote the (M M dentty matrx. x s the -th coumn of a matrx X and x,j denotes the (th, jth eement of t. The operator E s q stands for expectaton wth respect to the random varabe s gven q. The functons p(s, q and p(s q symboze the jont dstrbuton and the condtona dstrbuton of s and q, respectvey. Uness otherwse noted, a ntegras are taken from to +. Fnay, 1 bt symbozes that the equaty hods for the snge bt case. II. SYSTEM MODEL As mentoned before, we start from a genera sgna mode, descrbed by: r Q(y, wth (1 y f(x, θ+, ( where y s the unquantzed receve vector of dmenson N, f(, s a genera mutdmensona system functon of the unknown parameter vector θ, to be estmated, and the known or unknown data vector x, whe s an..d. Gaussan nose wth varance σ n each dmenson. We assume that the nose /10/$ IEEE 143

2 varance σ s known, athough ths part of the work can be easy extended to the case where σ s part of θ. The operator Q( represents the quantzaton process, where each component y s mapped to a quantzed vaue from a fnte set of code words as foows r Q(y, f y r o,r up. (3 Thereby r o and r up are the ower vaue and the upper mts assocated to quantzed vaue r. Addtonay, we denote the pror dstrbuton of the parameter vector by p θ (θ when avaabe. Smary the pror p x (x s aso known, and can for nstance be obtaned from the extrnsc nformaton of the decoder output. The jont probabty densty functon nvovng a random system varabes reads consequenty as 1 p(r, y, x, θ I D(r (y (π N σ N e y f(θ,x σ where I denotes the ndcator functon takng one f y D(r { y R N r o p x (xp θ (θ, (4 y r up ; {1,...,N} }, (5 and 0 otherwse. Note that ths speca factorzaton of the jont densty functon s cruca for sovng and anayzng the estmaton probem. A factor graph representaton of the jont probabty densty s gven n Fg. 1 to ustrate ths property. Each random varabe s represented by a crce, referred to as varabe node, and each factor of the goba functon (4 corresponds to a square, caed functona node or factor node. Naturay, the MAP souton ˆθ has to satsfy θ L(θ 0. (8 Ths condton can be wrtten as: θ L(θ θp(r, θ p(r, θ θ p(r, y, x, θ dxdy p(r, θ θ p(r, y, x, θ p(x, y r, θ p(r, θ p(x, y r, θ dxdy θ p(r, y, x, θ p(x, y r, θdxdy p(r, y, x, θ E p(x,y r,θ θ n p(r, y, x, θ 0.! (9 There s aso another way to wrte the optmaty condton, by frst ntegratng out the varabe y to obtan the condtona probabty of the quantzed receved vector p(r x, θ r up r o 1 (π N σ N (Φ( rup e y f(θ,x σ dy f (x, ˆθ Φ( ro f (x, ˆθ, (10 where Φ(x represents the cumuatve Gaussan dstrbuton readng as p(x N (0,σ x f(x, θ y I D(r (y r θ p(θ Φ(x 1 π x exp( t /dt. (11 Therefore, we aso obtan an aternatve condton as θ p(r, y, x, θdy θ L(θ dx p(r, θ θ p(r, x, θ p(x r, θ p(r, θ p(x r, θ dx θ p(r, x, θ p(x r, θdx p(r, x, θ E p(x r,θ θ n p(r, x, θ 0,! (1 Fg. 1. Factor graph representaton. III. CONSTRUCTION OF THE ESTIMATION ALGORITHM AND PERFORMANCE BOUND Gven the quantzed observaton r, and the og-kehood functon L(θ n p(r, y, x, θdxdy np(r, θ, (6 our goa s to fnd the MAP estmate ˆθ gven by ˆθ argmax L(θ. (7 θ whch can be expcty wrtten as, E p(x r,ˆθ f (x,ˆθ (e σ e (r o f (x,ˆθ σ θ f (x, ˆθ πσ (Φ( rup f (x,ˆθ Φ( ro f (x,ˆθ f (x,ˆθ θp θ (ˆθ p θ (ˆθ 0 1 bt e σ θ f (x, E x r,ˆθ r ˆθ θp θ (ˆθ 0, πσ Φ( rf(x,ˆθ p θ (ˆθ (13 where the ast step hods for the snge bt case,.e. r {±1}. 144

3 A. EM-Based MAP Souton In genera, sovng (13 s ntractabe, thus we resort to the popuar Expectaton Maxmzaton (EM agorthm as teratve procedure for sovng the condton (9 n the foowng J (e recursve way,r E x,y r,θ θ n p(r, y, x, θ (14 1 bt Thus, at each teraton the foowng two steps are performed: e E-step: Compute the expectaton g(r, θ, ˆθ E x,y r,ˆθn p(r, y, x, θ + const E x,y r,ˆθ y T f(θ, x+ f(θ, x /(σ+np θ (θ E x r,ˆθ (f (ˆθ, x+e x, r, ˆθ T f(θ, x f(θ, x /(σ+np θ (θ, (15 where E x, r, ˆθ π e 1 bt r π M-step: Sove the maxmzaton f (x,ˆθ Φ( rup f (x,ˆθ σ e (r e f (x,ˆθ σ Φ( rf(x,ˆθ o f (x,ˆθ σ Φ( ro f (x,ˆθ ˆθ +1 argmax g(r, θ, ˆθ. (16 θ In many cases, ths maxmzaton s much easer than (7, as we w see n the exampes consdered ater. B. Standard Cramér-Rao Bound (CRB The standard CRB 1 s the ower bound on the estmaton error for any unbased estmator, that can be obtaned from the Fsher nformaton matrx J(θ under certan condtons E(θ ˆθ(θ ˆθ T (J(θ 1. (17 Hereby, the Fsher nformaton matrx reads as 6 J E r θ θ L(θ T θ L(θ E r θ E x r,θ θ n p(r, x, θ E x r,θ T θ n p(r, x, θ E r θ E x r,θ E x r,θ e (r Φ( rup e f (x,θ Φ( rup up f (x,θ σ e (r f(x,θ σ e (r f(x,θ. o f (x,θ σ Φ( ro f (x,θ o f (x,θ σ Φ( ro f (x,θ T θf (x, θ T 1 θ f (x, θ πσ. (18 1 The standard CRB, n contrast to the Bayesan CRB, hods for a determnstc parameter,.e. the pror p(θ s not taken nto account. In the pot-based estmaton case (x s known, t smpfes to f (x,θ σ e (r πσ(φ( rup (f (x,θ o f (x,θ σ θ f (x, θ T θ f (x, θ f (x,θ Φ( ro f (x,θ σ θ f (x, θ T θ f (x, θ πσφ( f(x,θ Φ( f(x.θ. (19 Addtonay, n the ow SNR regme ( f (x, θ, (19 can be approxmated by J ρ Q σ where the factor ρ Q 1 π θ f (x, θ θ T f (x, θ, (0 r (e σ (ro e σ Φ( rup Φ( ro 1, (1 depends ony on the quantzer characterstc (here assumed to be the same for a dmensons and represents the nformaton oss compared to the unquantzed case at ow SNR, n the potbased estmaton case. For the snge bt case,.e., (r o,r up {(, 0, (0, }, the Fsher nformaton oss ρ Q s equa to /π, whch concdes to the resut found n 7, 8 n terms of the Shannon s mutua nformaton of the channe. For the case that we use a unform symmetrc md-rser type quantzer 9, the quantzed receve aphabet s gven by r {( b 1 + kδ; k 1,, b } R, ( where Δ s the quantzer step-sze and b s the number of quantzer bts. Hereby the ower and upper quantzaton threshods are and r o r up { r Δ for r Δ (b otherwse, { r + Δ for r Δ (b + otherwse. In order to optmze the Fsher nformaton at ow SNR (0 and get cose to the fu precson estmaton performance, we need to maxmze ρ Q from (1 wth respect to the quantzer characterstc. Tabe I shows the optma (non-unform step sze Δ opt (normazed by σ of the unform quantzer descrbed above, whch maxmzes ρ Q for b {1,, 3, 4}. If we do not restrct the characterstc to be unform, then we get the optma quantzaton threshods whch maxmze ρ Q n Tabe II. We note that the obtaned unform/non-unform quantzer optmzed n terms of the estmaton performance s not equvaent to the optma quantzer, whch we woud get when mnmzng the dstorton, for a Gaussan nput 9. In addton, contrary to the quantzaton for mnmum dstorton the performance gap between the unform and non-unform quantzaton n our case s qute nsgnfcant, as we can see from both tabes. 145

4 Tabe I OPTIMAL UNIFORM QUANTIZER. b Δ opt ρ Q 1 - /π Ths resuts obtaned by numerca optmzaton of the Fsher nformaton concde wth the resuts found n 5 through observatons at asymptotcay arge N Tabe II OPTIMAL NON-UNIFORM QUANTIZER. b Optma threshods ρ Q 1 0 /π 0;± ;± ;± ;± ;± ;± ;± ;± ; ± ;± ;± Fsher Informaton h In the foowng, the theoretca fndngs w be apped to the channe estmaton probem and for a GNSS probem wth quantzed observatons. IV. EXAMPLE 1: SISO CHANNEL ESTIMATION We frst revew the smpe probem of SISO channe estmaton consdered n 5. A. Pot-based snge bt estmaton (one-tap channe The SISO one-tap channe mode s gven by r sgn(y sgn(hx +, for {1,...,N}, (3 where N s the pot ength and x { 1, 1} s the transmtted pot sequence wth normazed power. The channe coeffcent h R s here our unknow parameter,.e. θ h. It can be shown by sovng the optmaty condton (13 (wth unform pror p θ (θ that the ML-estmate of the scaar channe from the snge bt outputs r, s gven by 5 ĥ σerf 1 ( rt x. (4 N Besdes, the Fsher nformaton (19 becomes n ths case Ne h σ J(h πσφ( h Φ( h. (5 Ths expresson of the Fsher nformaton s shown n Fg. for N 00 as functon of h /σ. In Fg. 3 the CRB,.e. 1/J and the reatve exact mean square error (MSEoftheMLestmate from N 00 observatons, both normazed by h, are depcted as functon of the SNR h /σ. We nterestngy observe that above a certan SNR, the estmaton performance degrades, whch means that nose may be favorabe at a certan eve, contrary to the unquantzed channe. Ths phenomenon s known as stochastc resonance, whch occurs when deang wth such nonneartes. We can naturay seek the optma SNR that maxmzes the normazed Fsher nformaton,.e. mnmzes CRB/h : h σ opt Ne γ argmax γ πγφ( γφ( dB. γ ( h /σ (near Fg.. Fsher Informaton vs. σ for a SISO channe, b 1, N 00. (CRB, MSE/h x 10 3 MSE CRLB h /σ (near Fg. 3. Estmaton error vs. σ for a SISO channe, b 1, N 00. B. Pot Based Estmaton (two-tap channe Now et us consder a more genera settng wth a two-tap nter-symbo-nterference (ISI channe r sgn(y sgn(h 0 x +h 1 x 1 +, for {1,...,N}, (7 where h 0 and h 1 are the channe taps. Agan we utze a bnary amptude pot sequence x { 1, 1} N and we try to fnd the the ML-estmate of the parameter vector θ h 0,h 1 T n cosed form. Ignorng the frst output r 1, the ML-condton (13turnstobe(p(θ 1 r x r x 1 e (h 0 x +h 1 x 1 σ Φ( r(h0x+h1x 1 0, e (h 0 x +h 1 x 1 σ Φ( r(h0x+h1x 1 0. (8 146

5 Takng the sum and the dfference of these equatons devers respectvey r (x + x 1 e r (x x 1 e (h 0 x +h 1 x 1 σ Φ( r(h0x+h1x 1 0, (h 0 x +h 1 x 1 σ Φ( r(h0x+h1x 1 0. (9 Next, we mutpy the numerator and denomnator of each equaton by Φ( r(h0x+h1x 1 to get r (x + x 1 r (x x 1 Then, usng the fact that Φ( r x h 0+h 1 Φ( h0+h1 Φ( h0+h1 0, Φ( r x h 0+h 1 Φ( h0 h1 Φ( h0 h1 0. (30 h 0 + h 1 Φ( r x 1 r x erf( h 0 + h 1, (31 σ where erf( denotes the Gaussan error functon, we get (x + x 1 r (N 1+ x x 1 erf( h 0 + h 1, σ (x x 1 r (N 1 x x 1 erf( h 0 h 1. σ (3 Fnay, sovng the ast equatons wth respect to h 0 and h 1, we get the ML souton. σ (x +x 1 r (x x 1 r ĥ 0 erf 1 N + N +erf 1 x x 1 N + N, x x 1 σ (x +x 1 r (x 1 x r ĥ 1 erf 1 N + N +erf 1 x x 1 N + N. x x 1 The souton conssts n qute smpe computatons (apart of the fna appcaton of erf 1 snce we ony have to do wth bnary data (r,x {±1}. C. Non-Pot Aded (Bnd Estmaton Suppose now that a unknown bnary symbo sequence x {+1,.1} s transmtted over an addtve whte Gaussan nose (AWGN channe wth an unknown rea gan h. The anaog channe output s y h x +, (33 where the varance of the nose, σ, s aso unknown. Addtonay, the recever s unaware of the transmtted symbos x. Based on N quantzed observatons r Q(y,wewsh to estmate the parameter vector θ h, T. We note an nherent ambguty n the probem: the sgn of the gan h and the sgn of x cannot be determned ndvduay. We aso note that at east bts are needed n ths case, because a snge bt output does not contan any nformaton about h. Snce the ML probem s ntractabe n cosed form, we resort to the EM approach. The EM-update for h can be obtaned from the genera expressons n (15 and (16 as ĥ +1 1 N ĥ Φ( rup xĥ ˆ Φ( rup ĥ ˆ,x {+1, 1} Φ( ro xĥ ˆ Φ( ro ĥ ˆσ π 1 N e ĥ ˆσ, xˆσ π (e (r ĥ +Φ( rup ˆ +ĥ ˆ o ĥ ˆσ, up xĥ ˆσ, up +ĥ ˆσ, e (r Φ( ro +ĥ ˆ o xĥ ˆσ, o +ĥ e (r e (r +e (r ˆ,, Φ( rup ĥ Φ( ro ˆ ĥ +Φ( rup ˆ +ĥ Φ( ro ˆ +ĥ ˆ (34 whe the update for the nose varance foows from the expectaton ˆσ +1, 1 N Φ( rup ĥ ˆ Φ( rup ĥ ˆ ĥ e Φ( ro + ĥ e Φ( ro E x, r,h,ˆσ, ˆσ, ĥ ˆ, (r o ĥ e (r o ĥ +Φ( rup ˆ +ĥ ˆ ˆσ N π ĥ ˆσ, Φ( ro +ĥ ˆ, (r o + ĥ e (r o ĥ +Φ( rup ˆ +ĥ ˆ where we used the condtona dstrbuton ˆ, +ĥ ˆ +ĥ ˆσ, Φ( ro +ĥ ˆ +, (35 ˆ, I D(r( +ĥ e +I p( r, ˆ, ĥ πˆσ D(r(, ĥ e πˆ, Φ( rup ĥ Φ( ro ˆ ĥ +Φ( rup ˆ +ĥ Φ( ro ˆ +ĥ ˆ (36 The Cramér-Rao Bound that can be easy obtaned from the kehood functon L(h, n (Φ( rup h Φ( ro h +Φ( rup + h Φ( ro + h, (37 as we as the MSE of the estmates ĥ and found by Monte Caro smuaton, both normazed by h, s depcted n Fg. 4 as functon of h /σ, where the pot ength s N 100 and the quantzer resouton s b 3. Ceary the MSE of the estmate ˆ aso exhbts the non-monotonc behavor mentoned before wth respect to the SNR.. 147

6 (MSE, CRB/h 10 3 MSE h MSE CRB h CRB 10 4 As exampe, we took the specfc channe matrx 1.5 H. ( Fg. 5 shows aso the MSE when usng the anaog (unquantzed output, whch s exacty MSE b σ tr ( (x1, x T x 1, x 1. (4 Ceary, the estmaton error under quantzaton does not ncrease monotoncay wth hgher σ, as we know aready from the SISO case h /σ (near Fg. 4. MSE and CRB of the bnd estmates of h and vs. h /σ for a SISO channe, b 3, N Unquantzed 1 bt CRB 1 bt smu. MSE V. EXAMPLE : PILOT-BASED MIMO CHANNEL ESTIMATION Now, we consder the MIMO case. We begn frst wth the probem of estmatng a MIMO channe from snge bt outputs, snce t can be aso soved n a cosed form, as shown ater on. MSE A. Snge bt estmaton of a MIMO Channe As exampe, et us consder a pot-based estmaton of a rea vaued channe matrx assumng a snge-bt quantzer r sgn(h 1 x 1 + h x +, (38 where x 1, x { 1, 1} N are the pot vectors transmtted at each Tx antenna, whe r 1, r { 1, 1} N are the receved vectors at each Rx antenna. The maxmum kehood (ML channe estmate ˆθ T ĥ11, ĥ1, ĥ1, ĥ11 can be found by sovng (13 n cosed form, smary to the -tap SISO channe (see Subsecton IV-B, as ( σ ĥ j erf 1 (x1 + x T r N + x T 1 x +erf 1 (xj x j T r N x T 1 x, (39 wth, j {1, }, j 3 j and erf 1 the nverse functon of the error functon. We can see that the HW mpementaton of the estmaton task s st consderaby smpe, snce ony shft regsters, counters and a ook-up tabe for erf 1 woud be necessary. Fg. 5 shows the Monte Caro smuaton and the CRB of the estmaton error j E(h j ĥj for a gven channe as functon of the nose varance σ. Thereby, the Fsher nformaton matrx can be obtaned from (19 as J (N + xt 1 x e (h 1 +h σ 4πσ Φ( h1+h Φ( h1+h (N x T 1 x e (h 1 h σ 4πσΦ( h1 h Φ( h1 h ( Fg. 5. Estmaton error vs. σ for a rea vaued channe, b 1, N 00. B. Pot-based MIMO channe estmaton of arbtrary sze Let us consder now a more genera settng of a quantzed near MIMO system and y vechx +, (43 r Q(y, (44 wth a channe matrx H R L M and X R M N contans N pot-vectors of dmenson M. Hereby, we stack the unquantzed, quantzed and the nose sgnas nto the vectors y, r and, respectvey. Our unknown parameter vector s therefore θ h vech and we have the system functon f(h, X Xh, (45 where the new matrx X R M N M L contans agan the pot-vectors n a proper way. Furthermore we assume, contrary to the prevous cases, that a pror dstrbuton p(h s gven accordng to h N(0, R h. Wth ths defnton the EM-teraton (15 and (16 reads n ths case as 148

7 E-step: Compute for 1,...,N M-step: b π e Xĥ Φ( rup σ e (r o Xĥ σ Xĥ Φ( ro Xĥ (46 ĥ +1 (X T X + σr 1 h 1 X T (Xĥ + b. (47 Let us at ths pont vadate the convergence of the EMagorthm to a unque optmum souton. For ths, we wrte the og-kehood functon expcty (Φ( rup x T h Φ( ro x T h h T R 1 h. L(θ n (48 Ths og-kehood functon s a smooth convex functon wth respect to θ. Ths foows from the og-concavty of Φ(b z Φ(a z, (49 b>a, wth respect to z, snce t s obtaned from the convouton of the Gaussan densty and a normazed boxcar functon ocazed between a an b, whch are both og-concave 10. Therefore, the statonary pont of the EM-teraton fufng the condton (13 s the unque optma souton s aso shown for comparson. Obvousy, at medum and ow SNR, the coarse quantzed does not affect the estmaton performance consderaby. VI. EXAMPLE 3: QUANTIZATION OF GNSS SIGNALS The quaty of the data provded by a GNSS recever depends argey on the synchronzaton error wth the sgna transmtted by the GNSS satete (navgaton sgna, that s, on the accuracy n the propagaton tme-deay estmaton of the drect sgna (ne-of-sght sgna, LOSS. In the foowng we w study the effect of quantzaton n terms of smuatons and the CRB as derved n Secton III-B. We assumed an optma unform quantzer as gven n Tabe I. We w frst assess the accuracy of a standard one-antenna GNSS recever n case no mutpath s present. Secondy, we w assess the behavor of array synchronzaton sgna processng n a mutpath scenaro appyng the nnovatve dervaton of the EM agorthm as shown n Secton III-A. Ths assessment s based on the work presented n 11. In the foowng we assume a GPS C/A code sgna wth a chp duraton T c ns, a code ength of 1 ms and a bandwdth of B 1.03 MHz. The receved sgna s samped wth the sampng frequency f s B. We ony use one code perod as an observaton tme where the channe s assumed constant durng ths observaton tme. The synchronzaton of a navgaton sgna s usuay performed by a Deay Lock Loop (DLL, whch n case no mutpath sgnas are present, effcenty mpements a maxmum kehood estmator (MLE for the tme-deay of the LOSS τ 1. In Fg. 7, the ower bound of the RMSE of τ 1, for dfferent the b 1 b b 3 b 4 MSE b b4 b3 b b og 10 (σ CRB(ˆτ1 c0 meter 10 1 Fg. 6. Estmaton error vs. σ for a rea vaued 4 4 MIMO channe, b 1, N 1000, R h I 16, x,j { 1, +1}. Fg. 6 ustrate the average MSE defned by h MSE E ĥ, (50 under dfferent bt resouton for a 4 4 MIMO channe wth..d. unt varance entres. Hereby, we chose an orthogona pot sequence,.e. X T X R h I 16 wth x,j { 1, +1}. The estmaton error n the unquantzed case, whch s gven by ( MSE b σtr (X T X + σr 1 h 1 ( Fg. 7. RMSE of ˆτ c 0 vs. bt resouton and SNR wth one antenna f s.046mhz. One code perod s used for estmaton. number of bts, s gven n terms of the CRB(ˆτ 1 n meters where c 0 denotes the speed of ght. A nomna SNR for a GPS C/A sgna s approxmatey 0 db. In Fg. 7 one can observe that the CRB(ˆτ 1 does not sgnfcanty decrease further for more than 3 bts, thus a rather smpe hardware mpementaton s suffcent for such a GNSS recever. Now, we assess the EM agorthm as derved n Secton III- A wth p θ (θ beng a unform dstrbuton, hence consderng a ML estmator. We consder a two path scenaro where the 149

8 LOSS and one refectve mutpath sgna are receved by an unform near antenna array (ULA wth M 8sotropc sensor eements. We defne CRLB, LOSS RMSE( ˆφ 1 CRB( ˆφ RMSE( ˆφ θ Re{γ} T, Im{γ} T, τ T, ν T, φ T T, (5 wth the vector of compex amptudes γ γ 1,γ T,the vector of azmuth anges φ φ 1,φ T, the vector of tmedeays τ τ 1,τ T, and the vector of Dopper frequences ν ν 1,ν T. The parameters wth the subscrpt 1 refer to the LOSS and parameters wth the subscrpt refer to the refecton. The refected mutpath and the LOSS are consdered to be n-phase, whch means arg(γ 1 arg(γ, and the sgna-to-mutpath rato (SMR s 5dB. Sgna-tonose rato (SNR denotes the LOSS-to-nose rato and we assume SNR.8dB. The DOAs for the LOSS and the mutpath are φ 1 30 and φ 6 respectvey. Further, we defne the reatve tme-deay between the LOSS and the mutpath as Δτ τ 1 τ 0.3T c and reatve Dopper Δν ν 1 ν 0Hz. In Fg. 8 the RMSE of ˆτ 1 and ˆτ vs. the bt resouton s depcted. In Fg. 9 the RMSE of ˆφ 1 RMSE c0, CRB c0 meter CRLB, LOSS RMSE(ˆτ 1 CRLB, MULT. RMSE(ˆτ Fg. 8. RMSE of ˆτ 1 c 0 and ˆτ c 0 (n meter vs. bt resouton for M 8, φ 1 30, φ 6, Δτ 0.3T c, SNR.8dB, SMR 5dB, Δν 0Hz. One code perod s used for estmaton. and ˆφ vs. the bt resouton s shown. Based on the resuts presented n Fg. 8 and Fg. 9 one can derve the mportant statement that 4 bts seem to be suffcent for hgh-resouton estmates wth respect to the consdered channe condtons. VII. CONCLUSION A genera EM-based approach for optma parameter estmaton based on quantzed channe outputs has been presented. It has been apped n channe estmaton and for GNSS. Besdes, the performance mt gven by the Cramér-Rao Bound (CRB has been dscussed as we as the effects of quantzaton and the optma choce of the ADC characterstc. It turns out that the gap to the dea (nfnte precson case n terms of estmaton performance s reatvey sma especay at ow SNR. Ths hods ndependenty of whether the quantzer s unform or not. Addtonay, we observed that the addtve b RMSE, CRB Fg. 9. RMSE of ˆφ 1 and ˆφ (n degree vs. bt resouton for M 8, φ 30, φ 6, Δτ 0.3T c, SNR.8dB, SMR 5dB, Δν 0Hz. One code perod s used for estmaton. nose mght, at certan eve, be favorabe when operatng on quantzed data, snce the MSE curves that we obtaned were not monotonc wth the SNR. Ths s an nterestng phenomenon that coud be nvestgated n future works. REFERENCES 1 R. Schreer and G. C. Temes, Understandng Deta-Sgma Data Converters, IEEE Computer Socety Press, 004. D. D. Wentzoff, R. Bázquez, F. S. Lee, B. P. Gnsburg, J. Powe, and A. P. Chandrakasan, System desgn consderatons for utra-wdeband communcaton, IEEE Commun. Mag., vo. 43, no. 8, pp , Aug R. Curry, Estmaton and Contro wth Quantzed Measurements, M.I.T Press, T. M. Lok and V. K. W. We, Channe Estmaton wth Quantzed Observatons, n IEEE Internatona Symposum on Informaton Theory, Cambrdge, MA, U.S.A., August 1998, p M. T. Ivrač and J. A. Nossek, On Channe Estmaton n Quantzed MIMO Systems, In Proc. ITG/IEEE WSA, Venna, Austra, Feb A. Papous, Probabty, random varabes, and stochastc processes, McGraw-H, fourth edton, A. Mezghan and J. A. Nossek, On Utra-Wdeband MIMO Systems wth 1-bt Quantzed Outputs: Performance Anayss and Input Optmzaton, IEEE Internatona Symposum on Informaton Theory (ISIT, Nce, France, June A. Mezghan and J. A. Nossek, Anayss of 1-bt Output Noncoherent Fadng Channes n the Low SNR Regme, IEEE Internatona Symposum on Informaton Theory (ISIT, Seou, Korea, June J. G. Proaks, Dgta Communcatons, McGraw H, New York, thrd edton, S. Boyd and L. Vandenberghe, Convex Optmzaton, Cambrdge Unversty Press, 004, frst edton, F. Antrech, J. A. Nossek, G. Seco, and A. L. Swndehurst, Tme deay estmaton appyng the Extended Invarance Prncpe wth a poynoma rootng approach, n Proceedngs of the Internatona ITG/IEEE Workshop on Smart Antennas WSA, Bern, Germany, February 009. b 150