Super-Resolution Blind Channel Modeling

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1 MITSUBISHI ELECTRIC RESEARCH LABORATORIES hp:// Super-Resoluion Blind Channel Modeling Pun, M-O.; Molisch, A.F.; Orlik, P.; Okazaki, A. TR2-28 June 2 Absrac We consider he problem of exracing a wide-band channel model when only measuremens in pars of his band are available, specifically in disjoin frequency sub-bands. Convenional channel modeling echniques canno model a all hose pars of he band where no sounding signals are available; or, if hey use convenional inerpolaion, suffer from poor performance. To circumven his obsacle, we develop in his paper a hree-sep super-resoluion blind algorihm. Firs, he pah delays are esimaed by exploiing super-resoluion algorihms such as MUSIC or ESPRIT based on he ransfer funcion of each sub-band, separaely. Exploiing such a se of delay esimaes, he proposed algorihm performs blind (i.e., wihou raining signal) channel esimaion over he unmeasured sub-bands, and subsequenly derives he frequency response over he whole wide-band channel. Finally, esimaes derived from differen sub-bands are combined via a sof combining echnique. Compuer simulaions show ha he proposed super-resoluion blind algorihm can achieve a significan performance gain over convenional mehods. IEEE Inernaional Conference on Communicaions (ICC) This work may no be copied or reproduced in whole or in par for any commercial purpose. Permission o copy in whole or in par wihou paymen of fee is graned for nonprofi educaional and research purposes provided ha all such whole or parial copies include he following: a noice ha such copying is by permission of Misubishi Elecric Research Laboraories, Inc.; an acknowledgmen of he auhors and individual conribuions o he work; and all applicable porions of he copyrigh noice. Copying, reproducion, or republishing for any oher purpose shall require a license wih paymen of fee o Misubishi Elecric Research Laboraories, Inc. All righs reserved. Copyrigh c Misubishi Elecric Research Laboraories, Inc., 2 2 Broadway, Cambridge, Massachuses 239

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3 Super-Resoluion Blind Channel Modeling Man-On Pun, Andreas F. Molisch, Philip Orlik and Akihiro Okazaki Misubishi Elecric Research Laboraories (MERL), Cambridge, MA 239, USA Ming Hsieh Depar. of Elecrical Engineering, Universiy of Souhern California, Los Angeles, CA 889, USA Misubishi Elecric Corporaion, 5-- Ofuna Kamakura, Kanagawa 24785, Japan. Absrac We consider he problem of exracing a wideband channel model when only measuremens in pars of his band are available, specifically in disjoin frequency subbands. Convenional channel modeling echniques canno model a all hose pars of he band where no sounding signals are available; or, if hey use convenional inerpolaion, suffer from poor performance. To circumven his obsacle, we develop in his paper a hree-sep super-resoluion blind algorihm. Firs, he pah delays are esimaed by exploiing super-resoluion algorihms such as MUSIC or ESPRIT based on he ransfer funcion of each subband, separaely. Exploiing such a se of delay esimaes, he proposed algorihm performs blind (i.e., wihou raining signal) channel esimaion over he unmeasured subbands, and subsequenly derives he frequency response over he whole wideband channel. Finally, esimaes derived from differen subbands are combined via a sof combining echnique. Compuer simulaions show ha he proposed super-resoluion blind algorihm can achieve a significan performance gain over convenional mehods. Index Terms Super-resoluion channel modeling, blind channel esimaion, sof combining. I. INTRODUCTION Accurae characerizaion of wireless propagaion channels plays a criical role in designing high-performance wireless sysems. As demonsraed in Shannon s seminal work, he fundamenal performance limis of wireless ransmission are dicaed by he wireless channel characerisics. Hence, an indeph undersanding of he underlying channel can faciliae sysem archiecs o design, opimize and subsequenly analyze pracical wireless sysems [], [2]. For he purpose of sysem developmen, channel models based on measuremens are essenial. Convenionally, channel measuremens are conduced by sending and measuring sounding signals over he whole frequency band of ineres. However, here are ofen challenging siuaions in which sounding signals can be ransmied only over some pars of he frequency band of ineres, raher han he whole band. Such challenges arise in a number of pracical siuaions including regulaory resricions, measuremens wih inerference and reuse of narrowband measuremens. Firs of all, as exising (legacy) wireless services such as analog TV broadcasing are eliminaed or relocaed from paricular frequency bands, he freed-up bands may be re-grouped o provide various broadband services. Thus, channel models for hese wideband channels are required o develop fuure applicaions even before he legacy services are erminaed. However, measuremens of he channel characerisics can only be performed in he whiespace beween he exising channels while he legacy services are sill operaing. Secondly, for many measuremens, i is impossible o guaranee absence of inerference over he whole desired bandwidh, which is paricularly rue for ISM (Indusrial, Scienific, and Medical) bands due o heir license-free operaion. Tradiionally, all measuremens conaminaed by inerference have o be discarded, despie he fac ha he bandwidh of he inerference is ofen smaller han he measuremen bandwidh. Given he high cos incurred during channel measuremens, i is hus highly desirable if channel models can be direcly derived from he inerference-free measuremens over some pars of he desired frequency band. Thirdly, each generaion of wireless daa sysem occupies more bandwidh han he previous one, and needs herefore more broadband channel models. While such broadband channel models can be derived hrough new measuremen campaigns, he enormous effors incurred make i worhwhile o invesigae wheher or no exising narrowband measuremens in adjacen frequency bands can be reused. Thus moivaed, a naural quesion o ask is wheher i is feasible o measure a wideband wireless channel by sending only narrowband sounding signals. In his work, we demonsrae ha his ask can be successfully achieved. Taking advanage of he proposed super-resoluion blind algorihm, we show ha a wideband wireless channel can be accuraely measured and modeled using narrowband sounding signals ransmied in disjoin subbands. More specifically, by exploiing a parameric channel model o exrac channel informaion such as mulipah componen (MPC) delays and ampliudes, we propose o esimae he channel ransfer funcion ha is valid over a larger bandwidh han he original measuremens. Furhermore, since his esimaion can be performed in each separae subband, esimaions derived from differen subbands are hen weighed and combined for channel model exracion. The main conribuions of his paper are as follows: we presen he (o he bes of our knowledge) firs algorihm ha obains consisen channel measuremens from measuremens in disjoin subbands (separaed by more han a coherence bandwidh of he channel). Furhermore, his work explicily incorporaes he effecs of pracical pulse shaping filers (raised-cosine filers) ino he high-resoluion algorihms such as muliple signal classificaion (MUSIC) algorihm or frequency-domain Esimaion of Signal Parameers via Roaional Invariance Technique (ESPRIT), which disinguishes his work from oher applicaions of hese high-resoluion algorihms for channel esimaion such as [3], [4]. Finally,

4 Subband # PN sequences Subband #K Upsampling ht() TX Pulse shaping ht() j2 fk e x() j 2 f w () e x () K s () h () Channel r () e e j2 f j 2 f K y () () h () R Mach filering h () R y ( K ) () RX Super-resoluion Blind Channel Modeling Fig.. Block diagram of he measuremen sysem under consideraion. unlike [3], [4] ha solely concenrae on pah delay exracion, our work furher characerizes he MPC ampliudes and subsequenly he channel ransfer funcion over a larger bandwidh. Noaion: Vecors and marices are denoed by boldface leers. ( ), ( ) T and ( ) H sand for he Moore-Penrose pseudoinverse, ranspose operaion and Hermiian ransposiion, respecively. denoes he ampliude of he enclosed complexvalued quaniy while x is he maximum ineger less han x. Furhermore, [A] i,j denoes he i-h row and j-h column enry of marix A whereas A(q, :) he q-h column of marix A. Finally, I N is he N N ideniy marix while F N is he N-poin discree Fourier ( ransform ) (DFT) marix wih enries [F ] n,k = N exp j2πnk for n, k N. N II. SIGNAL MODEL We consider a channel measuremen sysem shown in Fig.. The sysem consiss of K disjoin narrowband subbands separaed by guard bands (also referred o as he blind regions). Fig. 2 illusraes a paricular case wih K = 2 subbands. Noe ha he bandwidh of each subband or guard band can differ. SIG Channel response? Blind Region SIG2 f f 2 Fig. 2. Illusraion of measuremen channel for K = 2. Freq. (Hz) As shown in Fig., a sounding signal comprised of G repeaed pseudo-noise (PN) sequences is firs up-sampled before being fed ino a pulse shaping filer h T () such as a square-roo raised cosine filer. Afer ha, he pulsed-shaped signal is up-convered o f k and ransmied hrough he k-h subband. As a resul, he ransmi signal s() is a superposiion of muliple narrowband sounding signals residing in differen subbands. We consider a frequency-selecive channel comprised of L discree MPCs (see Sec. V for a discussion of his assumpion). Thus, he channel impulse response can be expressed as h() = L α l δ( τ l ), () l= where δ( ) is he dela funcion while α l and τ l are he pah gain and delay of he l-h MPC, respecively. Noe ha we have implicily assumed ha he channel remains approximaely saic over he G PN sequences. The receiver obains r(), he convoluion of he ransmi signal s() wih his complex channel impulse response h(), plus addiive whie Gaussian noise w() ha is modeled as a circularly symmeric complex Gaussian variable wih sandard deviaion σ. Upon receiving r(), he receiver firs down-convers he received signal ino each subband followed by mached filering he down-convered signals wih h R (). The resuling k-h subband oupu is y (k) (), for k =, 2,, K. Denoe by H(f) he frequency response of h(). Clearly, a sraighforward leas-squared (LS) esimae of H(f) can be derived as follows. Ĥ(f) = R(f) S(f), (2) where R(f) and S(f) are he Fourier ransforms of r() and s(), respecively. However, as shown in he laer simulaion, since S(f) over he blind region, he esimae Ĥ(f) derived from (2) will incur subsanial esimaion errors over he blind region. Noe ha his convenional mehod can be slighly improved by linear (or oher) inerpolaion-based echniques beween he measured subchannels. However, he improvemen is minor if he widh of he blind region is larger han he coherence bandwidh of he channel. In he sequel, he mehod shown in (2) is referred o as he convenional mehod. In he nex secion, we will propose a super-resoluion blind algorihm o derive he channel frequency response H(f) by exploiing sounding signals in disjoin subbands. For presenaional clariy, we will concenrae on he case of K = 2 as illusraed in Fig. 2 in he sequel. However, i should be emphasized ha he following discussion can be exended o K > 2 in a sraighforward manner. 2

5 y () () y (2) () Super-Resoluion Delay Esimaion Super-Resoluion Delay Esimaion III. PROPOSED ALGORITHM Sep Sep 2 Sep 3 Blind Channel Esimaion Blind Channel Esimaion Sof Combining Channel model exracion ouside [, +] as compared o he ideal auocorrelaion funcion. In oher words, a smaller rolloff facor resuls in beer band-limiing performance a he cos of more inerference for super-resoluion delay esimaion Ideal β= β=.5 β=. Fig. 3. Block diagram of he proposed super-resoluion blind channel modeling algorihm Fig. 3 illusraes he block diagram of he proposed superresoluion blind channel modeling algorihm. v(τ) A. Sep One: Super-resoluion delay esimaion ( k y ) () x ( ) Fig. 4. f s f s ( k y ) [ n] x [ n] U n ( k z ) ( ) Delay MUSIC or Freq. ESPRIT Super-resoluion delay esimaion by exploiing y (k) (). ˆq In he firs sep, super-resoluion delay esimaion is performed by exploiing eiher delay-domain MUSIC [3] or frequency-domain ESPRIT [4]. Denoe by T c and U he PN sequence chip duraion and he number of chips per PN sequence, respecively. In conras o he convenional PN correlaion mehod in which he resoluion of pah delay esimaion is limied by T c, he super-resoluion delay esimaion proposed in [3], [4] can provide esimaes of resoluion of a fracion of T c. In paricular, he ESPRIT algorihm is more compuaionally advanageous han MUSIC since i does no require exhausive search [4]. Nex, we propose an improved ESPRIT algorihm based on [4]. Two key differences disinguish he improved algorihm from [4]: () we need o ake he pulse shaping ino accoun; (2) raher han direcly applying ESPRIT o he received signal as proposed in [4], we consider applying he ESPRIT algorihm afer correlaing he received signal wih he ransmied PN sequence. Le y (k) () and τ-delayed x() denoe firs D- ime oversampled a f s = /(DT c ) as shown in Fig. 4. Then, y (k) [n] is correlaed wih x τ [n] and summed over one PN sequence. The resuling z (k) (τ) akes he following form z (k) (τ) = L α l e j2πτ lfk v(τ) + ψ (k) (τ), (3) l= where v(τ) is he auocorrelaion funcion of he pulse-shaped PN sequence and ψ (k) (τ) is he addiive noise afer correlaion. Fig. 5 illusraes he funcion v(τ) for raised cosine pulseshaped PN sequences wih differen values of rolloff facor β. I is ineresing o observe from Fig. 5 ha he auocorrelaion funcion associaed wih a smaller β enails larger ripples τ Fig. 5. Auocorrelaion funcions of raised cosine pulse-shaped PN sequence wih differen values of rolloff facor β. Then, we conver z (k) (τ) ino he frequency domain before performing he frequency ESPRIT as follows. Afer deconvoluion, we have J (k) (f) = Z(k) (f) V (f) = L α l e j2πτ lf k + Ξ (k) (f), (4) l= where Ξ (k) (f) = Ψ(k) (f) V (f) wih Z (k) (f), V (f) and Ψ (k) (f) being he Fourier ransforms of z (k) (τ), v(τ) and ψ (k) (τ), respecively. N samples of J (k) (f) are aken from is main lobe a f =,, 2,, (N ). I can be shown ha he noise correlaion marix is given by [R Ξ (k)] p,q = σ2 F N (p, :) R FN H (q, :) V (p ) 2, (5) where p, q N and R is he pulse-shaped noise covariance marix wih [R ] p,q = v(τ p τ q ). Subsiuing (5) ino he frequency-domain ESPRIT algorihm in [4], we can exrac { super-resoluion } esimaes of pah delays denoed by by ˆτ (k) q, where q =, 2,, Q wih Q L. B. Sep Two: Blind channel esimaion { } Upon aaining ˆτ q (k), wo approaches can be uilized o derive he MPC ampliudes and hus he channel impulse response, namely delay-domain and frequency-domain approaches. In he delay-domain approach, we firs collec I samples a he correlaor oupu before forming a vecor z (k) = [ z (k) (T ) z (k) (T 2 )... z (k) (T I ) ] T. From (3), i is sraighforward o show ha z (k) can be rewrien in he following marix form: z (k) = B (τ ) α + Ψ (k), (6) [ ] T [ where α = α α 2... α L, B (τ ) = v(τ ) v(τ 2 )... v(τ L ) ] and v(τ l ) = [ v(t τ l ) v(t 2 τ l )... v(t I τ l ) ] T. 3

6 As a resul, he LS esimae of α can be derived as ˆα = [B (ˆτ )] z (k). (7) However, one possible drawback associaed wih he delaydomain approach is ha he channel frequency response derived from (7) may exhibi large deviaion from ha esimaed in (2) over he k-h subband. Thus moivaed, we nex discuss he frequency-domain approach ha exracs he channel ampliudes by exploiing he esimaes derived from (2). This approach is shown o provide more accurae frequency response esimaion over each measuremen subband. We firs define he channel impulse response vecor as h N = [h, h,, h N ] T where only L elemens are non-zero. In he following, we exploi he fac ha H(f) = F N h N = F N T h L, (8) where h L conains only he L non-zero elemens of h N and T is an N L marix whose l-h column is he f s τ l -h column of I N. Thus, F N T is a sub-marix of F N wih only he corresponding columns. Since {τ l } is no available, we { replace {τ l } wih ˆτ (k) q } and (8) becomes Ĥ (k) (f) = F N T (k) h (k) Q + η, (9) where η is he addiive noise and T (k) is an N Q marix whose q-h column is he f sˆτ q (k) -h column of I N. Thus, we have ĥ (k) Q [F = N T (k)] Ĥ(k) (f). () However, recall ha esimaes of Ĥ(k) (f) derived from (2) are reliable only over he k-h subband. Thus, in (), we should ake M (k) > Q samples of Ĥ (k) (f) only over he k- h subband derived from (2). Finally, subsiuion of ĥ (k) Q ino (9) resuls in Ĥ(k) (f) over he whole channel bandwidh. C. Sep Three: Sof combining The las sep is o combine Ĥ(k), k =, 2, o provide an accurae channel esimae over he whole wideband channel. Clearly, he resuling esimae has o saisfy a leas he following wo requiremens. Firs of all, he combined esimae should render a coninuous frequency response over he whole channel. Second, he combined esimae should provide good esimaes over he blind regions as well as he measuremen subbands. A sof-combining approach can be esablished as follows: K Ĥ(f) = ρ k (f) Ĥ (k) (f), () k= where ρ k (f) are he weighing coefficiens a frequency f wih ρ 2 k (f) =. I is easy o see ha {ρ k(f)} should be designed o accuraely reflec he reliabiliy of Ĥ(k) (f). Noe ha Ĥ (k) (f) becomes less reliable as f falls far from he k-h subband. Inspired by his observaion, a simple bu effecive design example of {ρ k (f)} is shown in Fig. 6 where ρ k (f) remains uniy over he k-h subband and linearly decreases o zero over he blind region. Alernaive combinaion mehods ha aim o minimize he mean-square error of he esimae a each frequency will be described in our fuure works. Fig. 6. SIG Blind Region SIG2 f f 2 Freq. (Hz) Examples of weighing coefficiens employed for sof combining. IV. SIMULATION RESULTS In his secion, simulaion resuls are shown o demonsrae he performance of he proposed super-resoluion blind algorihm. As shown in Fig., we simulae a sysem of K = 2 equal-bandwidh subbands wih f = Hz and f 2 = 2 Hz, i.e. boh subbands and heir guard band have bandwidh of Hz. The received signal is 8-ime oversampled. Thus, he subband #, subband #2 and he blind region reside in he following normalized [ frequency band (wih respec o he sampling frequency), 6, + ] [ 6, + [ 6, + 6] 3 and + 3 6, + 6] 5, respecively. Unless oherwise specified, raised cosine pulse shaping wih β =.5 is employed in he following simulaion. Furhermore, for illusraion purposes, a frequency-selecive complex fading channel wih L = 5 pahs is simulaed. The pah delays normalized wih respec o T c are se o τ = {.,.6, 2.4, 3., 4.} whereas he pah gains are randomly generaed in each simulaion run wih pah power of {,,, 3, 3} db. I is worh noing ha he convenional correlaion-based delay esimaion approach will fail under his simulaion seup since here are MPCs separaed by less han one T c. Fig. 7 depics he esimaed channel frequency response obained wih he convenional mehod as shown in (2). Inspecion of Fig. 7 suggess ha he convenional mehod incurs subsanial esimaion errors over he blind region as well as he edges of he signal subbands due o raised cosine pulse shaping. Indeed, he peak shown in he blind region is always presen, regardless of he underlying channel model. This is because ha he peak is induced by S(f) in (2) over he blind region. Fig. 7. Power (db) Esimaed (convenional) True..2.3 Normalized frequency Performance of he convenional channel esimaion mehod. 4

7 Fig. 8 shows he esimaed Ĥ(2) using he sounding signal in subband #2. While Fig. 8 shows improved esimaes of Ĥ (2) over he blind region as compared o Fig. 7, he esimaion performance degrades rapidly beyond subband #2. Similar observaion can be also obained from he esimaed Ĥ () derived from subband #. acual channel. However, i is very likely ha null channel responses exis over he blind region. As a resul, he RMSE of he convenional mehod is dominaed by large errors due o error magnificaions over he null responses by dividing zero. Furhermore, we can observe ha he performance of he proposed algorihm is insensiive o SNRs. 2 4 Proposed (SIG2) True 3 Power (db) 2 3 Relaive MSE (%) 2 Convenional Proposed Normalized frequency SNR (db) Fig. 8. Performance of he proposed super-resoluion blind algorihm by exploiing received signal from subband #2. Fig.. RMSE as a funcion of SNR over ime-invarian mulipah fading channels. Nex, Fig. 9 shows he esimaed Ĥ by combining Ĥ() and Ĥ(2) using he weighing coefficiens shown in Fig. 6. Compared o Fig. 8, Fig. 9 indicaes ha he proposed superresoluion blind algorihm can provide balanced good performance over boh measuremen subbands and he blind region. Power (db) Proposed (SIG+SIG2) True..2.3 Normalized frequency Fig. 9. Performance of he proposed super-resoluion blind algorihm afer combining boh wo subbands. Finally, Fig. shows he performance of he convenional mehod and he proposed super-resoluion blind channel modeling algorihm in erms of relaive mean squared error (RMSE) over he blind region as a funcion of signal-onoise raio (SNR) defined { as /σ } 2. More specifically, we Ĥ H 2 define RM SE = E. The resuls are averages H 2 over runs. Fig. shows ha he proposed algorihm subsanially ouperforms he convenional mehod. This can be explained by he fac ha he convenional mehod always esimaes a peak over he blind region, regardless of he V. CONCLUDING REMARKS In his paper, we have presened a mehod for reconsrucing propagaion channels from measuremens in disjoin subbands of he band of ineres. By using high-resoluion esimaion of he mulipah parameers, and suiable combining of he resuls, we arrived a a model ha accuraely inerpolaes beween he measured subbands. The mehodology was verified by means of a synheic channel model, where he correc descripion was exacly known. There are several exensions of his sudy ha can be furher explored. One fundamenal assumpion in his work is he validiy of he apped-delay-line represenaion shown in () wih a finie number of aps. Therefore, for channels of very large bandwidh and/or diffuse MPCs, i is imporan o develop effecive means o confirm he validiy of his assumpion before applying he proposed algorihm [5]. Furhermore, he performance of he proposed algorihm hinges on he accuracy of super-resoluion delay esimaion. Thus, i deserves furher invesigaion on improving he delay esimaion accuracy. Finally, racking he esimaed channel model over ime may help idenify and remove channel modeling arifacs due o esimaion errors. REFERENCES [] M. Ibnkahla (ed.), Digial Signal Processing for Wireless Communicaions Handbook. CRC Press, 24. [2] A. Molisch, M. Shafi, and L. J. Greensein, Propagaion issues for cogniive radio, Proceedings of he IEEE, vol. 97, pp , March 29. [3] T. Manabe and H. Takai, Superresoluion of mulipah delay profiles measured by PN correlaion mehod, IEEE Trans. on Anennas and Propagaion, vol. 4, pp. 5 59, May 992. [4] H. Saarnisaari, TLS-ESPRIT in a ime delay esimaion, in Proc. IEEE 47h Vehicular Technology Conference, Phoenix, AZ, May 997. [5] P. A. Bello, Characerisaion of randomly ime-varian linear channels, IEEE Trans. Commun. Sysems, vol. CS-, pp , Dec