Equivalent Tapped Delay Line Channel Responses with Reduced Taps

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1 Equivalent Tapped Delay Line Channel Responses with Reduced Taps Shweta Sagari, Wade Trappe, Larry Greenstein {shsagari, trappe, WINLAB, Rutgers University, North Brunswick, NJ 892, USA. Abstract Typically, a ultipath channel response can be characterized as a su of Rayleigh-fading rays, each defined by a tie delay and a ean-square aplitude. Therefore, the channel response can be largely described by a power delay profile (PDP), which is the set of ean-square ray aplitudes and relative delays. Here, we address the following question: Given an actual (or true ) PDP, PDP(τ), which ay have any rays, is there a 3-ray (i.e. 3-tap) equivalent response, derivable fro PDP(τ), that can be used to accurately estiate the average bit error rate,, vs. receiver input signal-to-noise ratio, SNR? The results reported here give an affirative answer, e.g., for values down to =, the required SNR using a 3-tap equivalent channel response is less than. db larger than that required for the true channel. This agreeent can be iproved upon, suggesting further work on deriving and evaluating equivalent 3-tap channels. We discuss the benefits of such siplifications for hardware eulators as well as for siulation and analysis. Index Ters channel odels, eulators, power delay profile, tapped delay line I. INTRODUCTION A. Background and Motivation Multipath channel responses coe in any varieties, depending on physical environent, path geoetry, frequency band and signal bandwidth. In ost wireless scenarios, they are typically represented by a ulti-tapped delay line, with Rayleigh-fading tap gains, and tap spacings of /W, where W is the signal bandwidth. If the RMS delay spread of the channel is τ rs, the nuber of significant tap gains is soe ultiple of, which can be quite large for even oderately wideband channels. Here, we investigate whether, for any given channel, a 3-tap odel can be found which accurately predicts the average link perforance over that channel. Specifically, we assue a channel can be characterized by a power delay profile (PDP) coposed of a set of tap delays, {T }, and a set of corresponding ean-square tap gains, {P }, where =,, 2... The tap delays need not be evenly spaced, though we will assue they are in the scenario we study here. Also, the fading statistics of the tap gains can be arbitrary, though here they will be assued to be coplex-gaussian (Rayleigh fading). We will assue a true PDP that follows a widely used channel odel; and we will consider two approaches to atching it with 3-tap equivalent PDPs. The latter are characterized by the sets {T,T,T 2 } and {P,P,P 2 }, where, with no loss in generality, we set T = and P = P P 2. TABLE I SOME NOTATIONS AND DEFINITIONS Bandwidth Power Delay Profile as a function of delay τ RMS delay spread of the channel longest delay aong significant echoes in the channel (axiu tie dispersion in the ultipath channel) Gain for the th-delayed channel echo Power, < g 2 >,forthe th -delayed channel echo Ipulse response of the channel Ricean K-Factor Noise Peaking Factor Multipath averaged bit-error-rate Signal-to-noise ratio W PDP(τ) τ rs τ ax g P h(t) K r NPF, y SNR We believe the proposed solution would be of potential value to eulator design, coding of siulation platfors, and coding of analysis progras. For eulators, it siplifies the needed firware, which leads to lower cost. For coding of either siulation or analysis progras, it both siplifies the coding and allows the code to be universal, i.e., the sae lines of code for all channel conditions (power-delay profiles); the only thing that changes fro one channel to another is the set of input paraeters {P,P 2,T,T 2 }. The advantage of reduced-tap responses is even ore proinent in eulator design, especially those built for large systes with any nodes. For exaple, given a syste with N nodes, there will be not only N direct paths in a given scenario, but N(N 2) potentially interfering paths as well. In such cases, the cost of the taps can doinate the cost of the eulator syste. Assuing the nuber of taps needed for the true channel is 5 and the channel is only oderately wideband, e.g., =2, the reduction in taps can produce significant cost savings. To suarize, we present two new ethods for atching a 3-tap PDP to a true one with any taps. One atches the first 4 oents of the true PDP and produces excellent perforance agreeent up to about =2; the other is an even sipler ethod that produces excellent agreeent for arbitrarily large values of. The 3-tap approach reduces the cost of large-scale eulators. Also, it siplifies coputer coding for analysis and siulation progras: The sae coding applies to all channel conditions, requiring only channel-specific changes in the input paraeters /3/$3. 23 IEEE

2 B. Organization Section II describes related work. Section III introduces a widely used power delay profile for ultipath fading channels, which we use here to represent the true channel and fro which we derive a ulti-tap channel ipulse response. In Section IV, we introduce a 3-tap channel that atches the true channel in the first four oents of the PDP and ight therefore serve as an equivalent. We also introduce an alternative 3-tap channel response, which we consider along with the oent-atching channel response. In Section V, we describe a ethod to copute and copare receiver perforance over the true and 3-tap channels (specifically, their bit-error-rates averaged over fading) to assess the accuracy of 3-tap equivalents as a function of bandwidth. Section VI gives nuerical results, and Section VII discusses further work. Notations and definitions used throughout the paper are partly suarized in Table I. II. RELATED WORK Analysis/siulation of tapped-delay-line channel odels is a well studied proble and several approaches can be found in the literature. Hoeher [] studies the discrete-tie ultipath channel to copute tap-gain values using a delay-weight-andsu ethod. Several studies [2] [4] investigate challenges of coputational coplexity and siulation accuracy while ipleenting tapped-delay-line channel odels on hardware. Borries et al. [3] proposes a scalar resolution technique in siulator odules to achieve accuracy, dyanic range and speed for FPGA-based hardware siulators. Kahrs and Zier [4] discusses ultipath signal generation for ipleentation on a digital signal processor based on replacing coplexvalued fading functions with aggregate noise functions which can be generated by using Doppler filtering. Mehlfuhrer and Rupp [5] propose a fraework for path reduction based on an ad hoc path cobining ethod and takes an optiization approach to atch total power, ean delay, and the RMS delay spread specified by the tapped-delay-line odel. Tripathi, et al. [6] eploy tap reduction schees in the channel estiation algorith to reduce coplexity of the receiver, based on axiizing the correlation between the band-liited channel response functions of the estiated and reduced taps. They also propose an ad-hoc approach to find a subset of arbitrarily spaced taps which ay receive significant energy. III. BACKGROUND Here, we describe the generic channel ipulse response and power delay profile of a true channel, which we will use as a baseline to develop 3-tap channel approxiations in later sections. A. Generic Channel Ipulse Response Fro Nyquist sapling theory, we know that any linear channel of bandwidth W can be represented by a tapped delay line, with the taps spaced by /W or less. Assuing a tap spacing of exactly /W, the ipulse response for such a channel can be written as h(t) = g δ(t /W ); =,...,M(all channels) () where is a suation over fro to M; M is chosen so that M/W represents the longest delay, τ ax, aong the significant echoes in the channel; and g is the coplex gain (which can be slowly tie-varying) for the th-delayed channel echo. By slowly, we ean that the gains hardly change over the duration, τ ax, of the ipulse response. The value of M should be the sallest integer equal to or greater than Wτ ax ; if this nuber is very large, then hardware ipleentation becoes a ajor challenge. Proceeding further with the general case, we now odel the teporal variation, g (t), of the gain of the th tap. We assue that the tap gain is a Ricean process with a Ricean K-factor, K r.ifk r is infinite, then the tap gain is constant (nonfading channel); at the other extree, if K r is zero, then the tap gain is a zero-ean coplex Gaussian process (Rayleigh fading channel). We will assue the latter case for purposes of this study, but the ore general Ricean odel for g (t) could just as well be assued. Fro the above, we can write the th tap gain as g (t) =(P ) /2 u (t); =,...,M (2) where P is the ean-square value of g (t) over tie (i.e., it is the average power gain of the channel echo at delay /W ); u (t) is a coplex Gaussian process of zero ean and unit variance; and each u (t) is low-pass-filtered to have the desired Doppler spectru. The su of the P s is the average path gain, whose negative db value is the path loss. B. The Power Delay Profile A siple way to characterize a ultipath channel is through the set of power gains {P }, i.e., if this set is known, then cobining it with Eq. 2 totally deterines the ipulse response, Eq.. The only inforation needed, in that case, is how P varies with. A popular assuption is that P decays exponentially with increasing delay, /W in Eq., [7]. We can then say that the P s are uniforly-spaced saples of a decaying exponential function of delay. Following tradition, we assue the PDP has unit area and that, for the exponential case, it has the for PDP(τ) =/τ rs exp( τ/τ rs ); τ (True Channel) (3) where τ rs is the RMS delay spread of the channel. Use of this exponential shape is supported by several reported easureents, e.g., [8], [9], and is widely popular for that and other reasons, such as siplicity of use. We will therefore odel the actual channel as a tapped delay line filter, with taps spaced of /W, where the th tap gain is zero-ean and coplex Gaussian with ean-square value proportional to PDP(/W ), Eq. 3. We call this the true channel and, in the next section, will describe a 3-tap filter that attepts

3 to approxiate it for purposes of eulation, siulation or analysis. Finally, we note that the particular PDP we are using has infinite extent. The value used for M therefore depends on where the designer/analyst chooses to truncate the PDP: If the choice is to go out, say, to 5 ties the RMS delay spread, then M will be the nearest integer for which M/W =5τ rs, i.e., M 5. Including the tap at delay, this eans that, for =, the filter would consist of 6 taps. IV. CHANNEL WITH REDUCED TAPS A. A Moent-atching 3-tap Channel Approxiation An ipleentation of the tapped-delay line filter above can get large as increases significantly beyond the order of, corresponding to significantly greater than 6 taps. What we do here is propose a 3-tap filter that atches the true channel in soe iportant ways. Specifically, the first, second, third and fourth oents of its PDP are the sae as those for the true channel. It is easy to show that these four oents for the exponential profile of Eq. 3 are τ rs, 2(τ rs ) 2, 6(τ rs ) 3 and 24(τ rs ) 4, respectively. For a 3-tap channel, the PDP is PDP(τ) = P δ(τ T ); =,, 2 (3-tap Channel). (4) The design issue thus coes down to finding {P,T } for =,, 2 such that the first 4 oents of this PDP atch those of the true channel s PDP. The solution ust satisfy the conditions P + P + P 2 =and T =. Without going through the algebra, the results are as follows: (P,T )=(.3333, ), (P,T )=(.622,.2679τ rs ), (P 2,T 2 )=(.447, 4.738τ rs ). The tie-varying tap gains for the filter eulation can be obtained by foring a set of i.i.d. coplex Gaussian processes, {u (t)}, =,, 2, that have the desired Doppler spectru, and then applying Eq. 2 and Eq. 5. B. An Ad-hoc 3-tap Channel Approxiation We considered another plausible 3-tap channel, which is independent of the PDP oents and therefore ore general. We will see, oreover, that for the PDP of Eq. 3 at least, it actually produces soewhat better perforance predictions. Again subject to the conditions T =and P +P +P 2 =, the set of values for {P,T } for =,, 2 is (P,T )=(/3, ), (P,T )=(/3,τ rs ), (P 2,T 2 )=(/3, 2τ rs ). The PDP for this channel atches that for the true channel in the first oent; alost atches it in the second oent (.67τ 2 rs instead of 2τ 2 rs); and falls well short in atching the third and fourth oents. The choice of this second 3-tap channel was basically ad hoc and based on intuition. (5) (6) V. EVALUATION METHOD A. Evaluation Metrics It is easy to copute H(f) for each of the above channels. It is H(f) = (P ) /2 u exp( j2πf/w); =,...M for the true channel, and siilarly for the 3-tap channel. Note that there is not one frequency response for a channel, but an infinite nuber, depending on the coplex values chosen for the coplex Gaussian, noralized tap gains, i.e, {u,...,u M } for the true channel, and {u,u 2,u 3 } for the 3-tap channel. So the siulation consists of generating these sets, say, 2 ties; coputing a perforance etric for each set; and then getting a CDF of the values coputed. What should the coputed etric be, and how should we copare it for the two kinds of channels? One etric that is eaningful, easy to copute, and also independent of the particular odulation forat, received power and noise level, is the noise peaking factor (NPF) that arises when the channel response is fully equalized, i.e., when the receiver filter is adapted to be /H(f) over the signal bandwidth (this is the zero-forcing equalizer). This kind of filter totally neutralizes the ultipath response, but it causes an increase in the receiver output noise power. That increase is just the average of [/ H(f) 2 ] over the signal bandwidth and, indeed, it requires no assuptions about the signal odulation, received power, or other link details. By coputing the NPF, which we will denote here by the sybol y, for a few thousand realizations of the sets {u,u 2,...}, we get a population of y-values that can be described by a CDF. Doing this for both kinds of channels, we can see how closely the 3-tap filter predicts the NPF statistics. We will show in the next section, for a single-carrier link using M-ary quadrature aplitude odulation (M-QAM), how to relate the NPF to the ultipath-averaged bit-error-rate, as a function of the link signal-to-noise ratio, SNR. This is the ultiate etric we will use to copare true channels with their 3-tap approxiations. Finally, we note that the CDF of y and the resulting curves of vs. SNR depend solely on. The accuracy of the 3-tap odel in predicting perforance is likely to decline with increasing. We find that in fact it does, but only up to a point and then it levels off. B. The Siulations We begin with a recipe for how to copute the CDF of y: ) Start with the true channel and choose a low value (say,.5) for. 2) Generate the coplex Gaussian set u ; copute H(f); and copute y, as above. 3) Repeat until there are 2 saples of y. (Each of these 2 trials should generate a new, independent saple

4 Wtrs =.5 Wtrs = 2 = 2 = Wtrs = Wtrs = 2 2 = 4 2 = True Channel 3 tap Channel Tap Channel True Channel 5 2 Fig.. CDFs of the Noise Peaking Factor (NPF) coparing with 3-tap oent-atching channel Fig. 2. BER vs. SNR for 4-QAM Signaling coparing with 3-tap oentatching channel of u at each, so the Doppler spectru is irrelevant for this siulation.) 4) For a CDF of y, and plot it. 5) Now repeat the above for the 3-tap channel, and copare the two channels. 6) Repeat the above several ties, with increasing in each new round. At soe value of, the CDFs for the true and 3-tap channel will start to separate. This ay denote the liitation of the 3-tap approxiation, i.e., the approxiation ay only be sufficient for up to the value where the CDF separation begins. Soe results are shown in Fig., which copares CDFs for the true and 3-tap channels, for each of four values of. We see that separation between the two CDFs of y start to becoe significant at =2. However, there is a further step needed to truly test the 3-tap approxiation: Using the set of y-values for a given to generate curves of vs. SNR, where is the bit error rate averaged over the channel realizations (e.g., the 2 realizations generated in the above siulations); and SNR is the signal-to-noise ratio at the receiver input. If we include interference, then SNR represents the signal-to-(interferenceplus-noise) ratio. In either case (with or without interference), we can copute vs. SNR curves as follows: ) Choose a channel type (true or 3-tap) and a low value of such as.5. - Start with a low value of SNR, such as ( db). - Copute using the set of saples for y and the siple forula shown below. - Increent SNR in 2-dB steps, up to SNR= (2 db), coputing for each case. -Plot vs. SNR for the given channel type and value of. 2) Repeat for the other channel type. 3) Repeat and 2 for successive doublings of. A close approxiation to the instantaneous bit-error-rate, i.e., the value of BER conditioned on the value of the noise peaking factor, y, is given for the case of M-QAM odulation by [].5 BER y =.2exp( SNR/y); (7) (M ) and is coputed as the average of this quantity over the nuber of y-values coputed in the earlier siulation. Note that SNR/y is the receiver output signal-to-noise ratio, which is diinished fro its value at the receiver input (SNR) by a factor y, i.e., the noise increase caused by the equalizing filter. VI. RESULTS AND DISCUSSION Soe results of the bit-error-rate coputations are shown in Fig. 2. The separation between the solid and dashed curves for a given value of show how well the approxiation works for that value. The cases shown are =, 2, 4 and 8, forsnr as high as 2 db and as low as 8. We see that, for up to 2, thesnr values required to achieve = 8, as predicted for the true channel and its 3-tap approxiation, differ by less than db. Next, we coputed results for = 6 and 32, and found that these discrepancies reain below 2.7 db. At the ore practical bit-error-rate level =, the discrepancies are even saller, lying below.5 db. Assuing these are acceptable discrepancies in the prediction of link

5 TABLE II DISCREPANCIES (db) IN SNR AT = 8 Moent-atching Ad-hoc 3-tap 3-tap channel channel perforance, we can say that, even for signal bandwidths as high as 25 MHz, the 3-tap channel approxiation can be used for RMS delay spreads up to.3 icroseconds. In ost cases, systes with 25-MHz bandwidths will be liited to ranges so short that RMS delay spreads hardly, if ever, exceed this value. Pursuing the subject further, we considered values for of 64 and 28 and found the discrepancies to be no higher than for = 32. In fact, the discrepancy sees to level off asyptotically with increasing.the reason is that the variable y in Eq. 7 becoes ore and ore narrowly distributed about a single value as increases without bound. (The reason, in turn, for this narrowing is that the nuber of channel correlation bandwidths within the signal bandwidth increases without bound, so that the rando variable y converges toward its ean.) We repeated the above exercise for the ad-hoc 3-tap channel. As given in Table II, beyond the value =, we find the discrepancies in required SNR to be even saller than for the oent-atching 3-tap channel, though not by uch: As increases to very large values, the SNR discrepancies for = 8 level off at about 2. db; at = they level off at about. db. We are currently developing a ore rigorous solution for the optial 3-tap channel. Meantie, the botto line of this overall investigation sees clear: For a single-carrier transission with receiver equalization, a suitable 3-tap approxiation to the true channel can be found for any cobination of bandwidth and channel delay spread of likely practical interest. Whether this finding applies for other odern-day signaling forats, such as OFDM, reains to be studied. signal forats besides the single-carrier forat assued here, notably, OFDM forats; and (3) devising a ore rigorous approach to optiizing reduced-tap equivalent channels. ACKNOWLEDGMENT We would like to acknowledge valuable discussions with Mike Cagney, db Corporation. REFERENCES [] P. Hoeher. A statistical discrete-tie odel for the WSSUS ultipath channel. Vehicular Technology, IEEE Transactions on, 4(4):46 468, nov 992. [2] Fei Ren and Y.R. Zheng. A low-coplexity hardware ipleentation of discrete-tie frequency-selective rayleigh fading channels. In Circuits and Systes, 29. ISCAS 29. IEEE International Syposiu on, pages , ay 29. [3] K.C. Borries, G. Judd, D.D. Stancil, and P. Steenkiste. FPGA-based Channel Siulator for a Wireless Network Eulator. In Vehicular Technology Conference, 29. VTC Spring 29. IEEE 69th, pages 5, april 29. [4] M. Kahrs and C. Zier. Digital signal processing in a realtie propagation siulator. Instruentation and Measureent, IEEE Transactions on, 55():97 25, feb. 26. [5] C. Mehlfuhrer and M. Rupp. Approxiation and resapling of tapped delay line channel odels with guaranteed channel properties. In Acoustics, Speech and Signal Processing, 28. ICASSP 28. IEEE International Conference on, pages , 3 28-april [6] V. Tripathi, A. Mantravadi, and V.V. Veeravalli. Channel acquisition for wideband CDMA signals. Selected Areas in Counications, IEEE Journal on, 8(8): , aug. 2. [7] A. F. Molisch. Wireless Counications. Wiley and Sons, 2nd ed., 2; Scn [8] D.C. Cox. 9 MHz urban obile radio propagation: Multipath characteristics in New York city. Vehicular Technology, IEEE Transactions on, 22(4):4, nov 973. [9] D. Cassioli, M.Z. Win, and A.F. Molisch. The ultra-wide bandwidth indoor channel: fro statistical odel to siulations. Selected Areas in Counications, IEEE Journal on, 2(6): , aug 22. [] A.J. Goldsith and Soon-Ghee Chua. Variable-rate variable-power MQAM for fading channels. Counications, IEEE Transactions on, 45():28 23, oct 997. VII. CONCLUSION The use of 3-tap channel approxiations has been shown to be quite proising. Part of the reason is that, in a wellequalized radio link, the specifics of the ultipath channel response have a reduced ipact on the details of the perforance. That is, the receiver adaptation renders the details of the channel s ultipath structure and fading statistics less iportant to the perforance outcoe. Even so, further work needed to confir our current findings would include () trying other true channel characteristics, notably, channels that have sparse ultipath, as in any ultra-wideband (UWB) scenarios; (2) testing the perforance of the 3-tap approxiation on other