Characterization of WSSUS Channels: Normalized Mean Square Covariance

Transcription

1 IEEE TANSACTIONS ON WIELESS COMMUNICATIONS, VOL. X, NO. XXX, XXX Characterization of WSSUS Channels: Normalized Mean Square Covariance Do-Sik Yoo, Member, IEEE, Wayne E Stark, Fellow, IEEE Abstract A set of second order statistics collectively called normalized mean square covariance (NMSV) is defined to characterize the frequency /or the time selectivity of wide-sense stationary uncorrelated scattering (WSSUS) channels. Normalized frequency mean square covariance (NFMSV) quantifies the frequency selectivity, while normalized time mean square covariance (NTMSV) characterizes the time selectivity. Normalized frequency-time mean square covariance is defined to characterize the combined effect of frequency time selectivities. The NMSV s of a WSSUS channel can easily be computed from the scattering function. We show that there is a very close relationship between the NMSV of a WSSUS channel the performance of various diversity combing scheme. Also we discuss, with practical system design problems, how useful the parameters are for efficient system design. Index Terms normalized mean square covariance (NMSV), frequency selectivity, time selectivity, root mean square (rms) delay spread, coherence bwidth, correlation time, wide-sense stationary uncorrelated scattering (WSSUS) channel I. INTODUCTION Increasing dem for high speed mobile communications has motivated rapid development of wideb wireless networks. One of the most important performance degrading factors in such a network is signal fading, which is usually divided into shadowing multipath fading. Shadowing is a phenomenon due to obstacles immediate to the receiver, while multipath fading is a cumulative effect of scattering, diffraction reflection due to various objects surrounding transmitter receiver. In typical packet-based mobile communications, the effect of shadowing usually lasts longer than the duration of a packet so that its effect can be treated separately from that of multipath fading as an attenuation factor. The effect of shadowing can usually be described by a single romly varying parameter with a suitable choice of probabilistic distribution such as the log-normal distribution. In contrast, multipath fading requires significantly more complex descriptions due to its frequency time dispersive nature. In this paper, we propose a set of parameters to characterize effectively the qualities of multipath fading channels. Since a radio link acts as a linear medium for electromagnetic waves, a multipath fading channel can be described by a time-variant linear system hence characterized effectively Manuscript received January xx, 003. xxx D. Yoo with WESCOMM LLC W. E. Stark is with the Dept. Electrical Engineering Computer Science, The University of Michigan, Ann Arbor, MI, dyoo@umich.edu stark@eecs.umich.edu. This research was supported in part by Ericsson Inc., in part by DAPA under grant SA , in part by National Science Foundation under grant ECS by its system functions. Because of the practical unpredictability the effectiveness in system performance analysis, the system functions are often regarded to be romly timevariant described mathematically by rom processes. For intuitive physical interpretations, we often divide system functions into specular diffuse components. The specular component is defined as the mean of the system function hence is a deterministic function. The diffuse component describes the romly varying fluctuations of the system function from the specular component. In general, the characterization of a multipath fading channel dems the specification of the specular component the statistical properties of the diffuse component. A non-zero specular component usually implies the existence of reliable signal paths in a fading environment. Consequently, the case of a zero specular component generally implies the worst case scenario for system performance. In fact, such adverse situations occur frequently in practical mobile communication environments. In this paper, we shall confine ourselves to the characterization of multipath fading channels with zero specular components. 1 We will assume that the diffuse components satisfy the wide-sense stationary uncorrelated scattering (WS- SUS) conditions have Gaussian probability distribution. Consequently, we will consider only (complex) Gaussian wide-sense stationary uncorrelated scattering (WSSUS) channels []. Either the WSSUS assumption or the Gaussian approximation may not be valid in practical channels. For example, as the signal bwidth or the time duration becomes larger, the WSSUS assumption becomes less valid. Also the Gaussian approximation can be inaccurate for various reasons. To resolve such discrepancy with practical situations, more realistic channel models can be developed. However, the performance analysis under such models are usually more difficult provide only isolated results rather than drawing complete picture of system performance in terms of channel quality. Consequently, to be able to focus on the relation between channel characteristics the system performance, we confine ourselves to the class of complex Gaussian WSSUS channels rather than delving into more realistic but isolated channel models. Complex Gaussian WSSUS channels can be specified completely by their scattering functions. However, with infinite variety of scattering functions, it is impractical to investigate the direct relation between system performance scattering functions. Consequently, derived quantities such as rms (root 1 Characterization of multipath fading channels with non-zero specular components is discussed in [1].

2 IEEE TANSACTIONS ON WIELESS COMMUNICATIONS, VOL. X, NO. XXX, XXX 004 mean square) delay spread or correlation time are often used to represent the channel quality [3]. The rms delay spread of a channel is defined to be the variance of the time delay with the (normalized) delay power profile of the channel as the probability density function. Because frequency selectivity is a result of time dispersion, it makes some sense to consider the rms delay spread as its measure. However, as shown in Section IV-A, there is not a close relationship between rms delay spread system performance when the bwidth of the channel is wide. The coherence bwidth is a more direct measure of frequency selectivity. There are several ways to define the coherence bwidth of a channel. We can define the coherence bwidth of level-κ by the largest frequency separation up to which the correlation of the channel frequency response is greater than κ. While the number κ can be chosen to be any number between 0 1, it is usually taken to be close to 1 so that the coherence bwidth actually means the bwidth over which the channel response is highly correlated. In many cases, we often use the following definition of coherence bwidth: B c = k where k is some constant is the rms delay spread [4], [5]. Though mathematically less compelling, this definition gives a convenient measure of the coherence bwidth as a function of the rms delay spread. However, regardless of the definition, there is no close relationship between the coherence bwidth the system performance, when the channel bwidth is wide relative to the coherence bwidth. Such poor relations between existing parameters system performance stem from the fact that they fail to characterize the correlation property of a channel over the whole region of interest. For example, the coherence bwidth is a local measure of the correlation function hence cannot make an overall measure of the frequency selectivity of a channel when the coherence bwidth is small compared to the whole bwidth. Observing this, we propose, as a measure of frequency selectivity, a parameter called normalized frequency mean square covariance (NFMSV), which is the absolutesquared auto-covariance of the frequency response averaged over all frequency regions in consideration so that it can represent the overall covariance of the channel. We propose similar parameters called normalized time mean square covariance (NTMSV) normalized frequency-time mean square covariance (NFTMSV) for the characterization of the time selectivity the combined frequency time selectivity of a given channel, respectively. The three parameters, namely, NFMSV, NTMSV NFTMSV, will collectively be called the normalized mean square covariances (NMSV s) of the channel. The rest of this paper is organized as follows. In Section II, we define NFMSV, NTMSV, NFTMSV for general multipath fading channels. In Section III, we provide simple formulae that express the NMSV s in terms of the correlation functions of WSSUS channels. In Section IV, we show how closely the NMSV is related to the performance of practical diversity combining schemes through simulations. In Section (1) V, we provide a few examples to illustrate the practical utility of the parameters in efficient system design. Finally, we draw conclusions in Section VI. II. NOMALIZED MEAN SQUAE COVAIANCE In this section, we consider a multipath fading channel described as a romly time-variant linear system []. We briefly describe various system functions to establish notational conventions then define the normalized mean square covariances. A. omly Time-Variant Linear System Model We denote by h(τ, t) the complex representation of the channel response at time t due to a unit impulse at time t τ. Then, the output y(t) of the channel due to an input signal x(t) is given by y(t) = h(τ, t)x(t τ) dτ. () It is in many ways useful to analyze the spectral content of the impulse response h(τ, t) in both t τ variables. As usual, the time variant frequency response or the transfer function H(f, t) is defined by H(f, t) = h(τ, t) e jπfτ dτ. (3) By manipulating () (3), we can obtain y(t) = H(f, t)x(f)e jπft df. (4) Similarly, we define the delay Doppler spread function k(τ, ν) the Doppler spread function K(f, t), respectively, by k(τ, ν) = h(τ, t)e jπtν dt (5) K(f, ν) = k(τ, ν)e jπfτ dτ. (6) To model the romly time-varying behavior of the channel, we regard the system functions as stochastic processes. We assume that the time variant frequency response H(f, t) has well defined first second order statistics denote by V H (f, t; f, t ) the covariance function of H(f, t), i.e., V H (f, t; f, t ) = E[H(f, t)h (f, t )] E[H(f, t)]e[h (f, t )]. Before proceding, we note the following Cauchy-Schwarz inequality: V H (f, t; f, t ) V H (f, t; f, t) V H (f, t ; f, t ). (8) B. Definition In this subsection, we define parameters collectively called normalized mean square covariances for the channel described above. Note that other system functions such as k(τ, ν) are often assumed to have impulsive second order statistics. However, in most applications, we assume that H(f, t) has well-behaved first second order statistics. (7)

3 IEEE TANSACTIONS ON WIELESS COMMUNICATIONS, VOL. X, NO. XXX, XXX ) Normalized Frequency Mean Square Covariance (NFMSV): To characterize the frequency selectivity of the channel over a particular frequency region of interest at a particular time, we define the normalized frequency mean square covariance (NFMSV). Let represent the frequency region of interest such that 0 < V H (f, t; f, t) df < (9) at time t. Then, we define by V H (f, t; f, t) df df V f (; t) = [ ] (10) V H (f, t; f, t) df the normalized frequency mean square covariance of the channel over the frequency region at time t. If the relevant communication signal is transmitted over the frequency b between frequencies A B, then we usually choose = [A, B]. For a system such as a frequency hopping spread spectrum system, the region of interest can be chosen to be a union of intervals to designate the frequency hopping region. ) Normalized Time Mean Square Covariance (NTMSV): Similarly, we introduce the normalized time mean square covariance (NTMSV) to describe the time selectivity of the channel over a certain time region at a certain frequency. Let T be the time region of interest such that 0 < V H (f, t; f, t) dt < (11) T at frequency f. Then, the normalized time mean-square covariance (NTMSV) of the channel over the time region T at frequency f is defined by V H (f, t; f, t ) dt dt V t (f; T ) = T T [ T ]. (1) V H (f, t; f, t) dt Usually, the time region T of interest is chosen to be the region over which the relevant packet is transmitted. In many cases, it is just an interval of time. However, it can represent a union of time intervals in systems such as time-division multiplex access (TDMA) systems. 3) Normalized Frequency-Time Mean Square Covariance (NFTMSV): NFMSV NTMSV are defined to characterize frequency time selectivity separately. To quantify the combined effect of frequency time selectivity, we introduce the normalized frequency-time mean square covariance (NFMSV). Let be the frequency-time region of interest such that 0 < V H (f, t; f, t) df dt <. (13) Then, we define the normalized frequency-time mean square covariance (NFTMSV) V ft () of the channel over the frequency-time region by V H (f, t; f, t ) df dt df dt V ft () = [ ]. (14) V H (f, t; f, t) df dt As in the case of the time or the frequency region of interest, the frequency-time region often denotes a rectangle or a union of rectangles. It is very important to note that the NFMSV, NTMSV, NFTMSV depend not only on the covariance function V H but also on the region of interest, which implies that two different systems using different frequency/time regions can experience, under the same physical channel, different frequency/time selectivity. This property can be exploited judiciously in channel resource allocation. Note that the amount of channel resource used by a given system is described by the area of the region of interest. Before proceeding, consider the case when = T. If V H (f, t; f, t ) does not change appreciably over time in T or frequency in, then V ft () is approximately the same as V f (; t) (t T ) or V t (f; T ) (f ), respectively. In particular, we have lim V ft { (t δ, t + δ)} = V f (, t) (15) δ 0 lim V ft {(f δ, f + δ) T } = V t (f; T ). (16) δ 0 III. NMSV FO WSSUS CHANNELS The normalized mean square covariances are defined for a general romly time-variant linear channel. However, the parameters are the most useful for channels with zero specular components. 3 Consequently, we shall only consider channels with zero specular components from now on. We further assume that the diffuse components satisfy the WSSUS channel model. Hence, in the following discussion, we consider only WSSUS multipath fading channels [6], [7], [8]. 4 In this section, we show how the NMSV s are represented by various correlation functions of WSSUS channels. We start with brief descriptions of WSSUS channel models for notations definitions before studying the relations between the NMSV s the correlation functions. A. WSSCUS Channel Consider a WSSUS channel with impulse response h(τ, t). Then, E[h(τ, t)] = 0 (17) E[h(τ, t)h (τ, t )] = p(τ, t t )δ(τ τ ) (18) where p(τ, t) is the delay cross-power spectral density of the channel []. Note that p(τ, 0) is the delay power profile that describes how the average received power is distributed with respect to the time delay. 3 The specular components of a fading channel is defined by the mean functions of the system function. For example, the specular component of the impulse response h(τ, t) is defined by E[h(τ, t)], i.e., by the mean function of h(τ, t). The diffuse component is then defined by h(τ, t) E[h(τ, t)]. 4 Wide-sense stationarity (in time t) uncorrelatedness (between delay τ) assumptions become less valid as the observed time duration become longer the channel bwidth become wider, respectively. However, the idealized assumptions of WSSUS models provide mathematical simplicity enable insightful system performance analysis. WSSUS assumptions are particularly meaningful for complex Gaussian channels for which the first the second order statistics provide complex statistical specifications.

4 IEEE TANSACTIONS ON WIELESS COMMUNICATIONS, VOL. X, NO. XXX, XXX It is easy to see that the time-variant transfer function H(f, t) satisfies that E[H(f, t)h (f, t )] = P (f f, t t ) (19) where P (f, t) is the time-frequency correlation function defined by P (f, t) = p(τ, t)e jπfτ dτ. (0) Note that the uncorrelatedness in the delay τ implies the widesense stationarity in the frequency f. The converse is also true, since the Fourier transform is invertible. Note that the transfer function H(f, t) is wide-sense stationary in both time frequency for a WSSUS channel. Similarly, the Doppler delay spread function k(τ, ν) satisfies E[k(τ, ν)k (τ, ν )] = q(τ, ν)δ(ν ν )δ(τ τ ) (1) where q(ν, τ) is the scattering function defined by q(τ, ν) = p(τ, t)e jπνt dt. () We see the wide-sense stationarity of h(τ, t) in time t implies the uncorrelatedness of k(τ, ν) in frequency ν. Finally, we see that the Doppler-spread function K(f, ν) satisfies E[K(f, ν)k (f, ν )] = Q(f f, ν)δ(ν ν ) (3) where Q(f, ν) is the Doppler cross-power spectral density defined by Q(f, ν) = q(τ, ν)e jπfτ dτ. (4) Note that P (f, t) = = B. NMSV s Correlation Functions q(τ, ν)e jπ(fτ νt) dτ dν (5) Q(f, ν)e jπνt dν. (6) In this subsection, we show how we can represent V f (; t), V t (f; T ), V ft () in terms of various correlation functions. First of all, since V H (f, t; f, t ) = P (f f, t t ), we have V ft () = V f (; t) = V t (f; T ) = T T P (f f, 0) df df [ P (0, 0) P (0, t t ) dt dt [ P (0, 0) T ] (7) df ] (8) dt P (f f, t t ) df dt df dt [ ]. (9) P (0, 0) df dt Since the channel is wide-sense stationary in both time frequency, NFMSV NTMSV are independent of the time t the frequency f, respectively. Consequently, we shall write, from now on, V f () V t (T ) for V f (; t) V t (f; T ), respectively. It is not difficult to rewrite (7) - (9) in terms of p(τ, t), q(τ, ν) or Q(f, ν) using (0), (5) (6). Of these variations, the formulae with the scattering function q(τ, ν) are the most useful because of the ease of physical interpretation evident duality. To manifest the symmetry the role of the scattering function, we define a set of kernel functions K f (; τ), K t (ν; T ) K ft (; ν, τ) by 5 e jπfτ df K f (; τ) =, (30) df e jπνt dt T K t (ν; T ) =, (31) dt K ft (; ν, τ) = T e jπ(fτ νt) df dt, (3) df dt respectively. Then, as shown in Appendix I, V f (), V t (T ), V ft () can be rewritten as in (33), (34), (35) (see next page). Note that q(τ, ν) dν = p(τ, 0) is the delay power profile of the channel. Consequently, NFMSV can easily be obtained from the delay power profile. Similarly, NTMSV can be obtained from the Doppler power sepctrum Q(0, ν) = q(τ, ν) dν of the channel. From (33) - (35), we see that the NMSV s are dependent not only on the scattering function but also on the regions of interest through the kernel functions. Although the kernel functions can be calculated in principle for any well-shaped regions, we are mostly interested in some rectangular shaped regions. As an example, consider a frequency region r defined by r = [A + k(v + w), A + k(v + w) + w]. (36) k=0 where A, v w are non-negative real numbers. So r is a union of K intervals of length w. Consequently, the sum of the lengths of the intervals is Kw regardless of the choice of v. In other words, the total frequency resource occupied by this frequency region is Kw regardless of the value of v. In Section V-A, we consider the problem of determining the suitable value for v. It is not difficult to show (see Appendix II) [ ] sin{πk(w + v)τ} K f ( r ; τ) = sinc (πwτ). (37) K sin{π(w + v)τ} 5 Note that if = T, then K ft (; ν, τ) = K f (; τ)k t (ν; T ).

5 IEEE TANSACTIONS ON WIELESS COMMUNICATIONS, VOL. X, NO. XXX, XXX K f (, τ τ )p(τ, 0)p (τ, 0) dτ dτ V f () = [ ] (33) p(τ, 0) dτ K t (ν ν ; T )Q(0, ν)q (0, ν ) dν dν V t (T ) = [ ] (34) Q(0, ν) dν K ft (; ν ν, τ τ )q(τ, ν)q (τ, ν ) dτ dν dτ dν V ft () = 4 [ ] (35) q(τ, ν) dτ dν In particular, K f ([A, B]; τ) = sinc {π(b A)τ}. (38) By definition, NFMSV NTMSV are properties at a specific time at a specific frequency, respectively, while NFTMSV is an overall channel property. Normally there is not a simple relation between these three parameters. However, one simple case is worth mentioning in which we can factorize the scattering function q(τ, ν) into delay power profile T (τ) Doppler power spectrum N(ν) as 6 In this case, it follows that q(τ, ν) = T (τ)n(ν). (39) K f (; τ τ )T (τ)t (τ ) dτ dτ V f () = [ ], (40) T (τ) dτ K t (ν ν ; T )N(ν)N (ν ) dν dν V t (T ) = [ ], (41) N(ν) dν V ft ( T ) = V f ()V t (T ). (4) Although this factorization is usually not applicable to practical channels, this shows the product relationship between NFMSV, NTMSV NFTMSV implying that a channel has significantly higher frequency-time selectivity when it exhibits both frequency time selectivities. IV. NMSV AND SYSTEM PEFOMANCE The NMSV s are defined as average covariances over regions of interest (with proper normalization) to obtain useful insight on the quality of multipath fading channels. Since the quality of a channel means the reliability of communications 6 Delay power profile is a function of the distribution of the signal travel distance, while Doppler power spectrum is related to the angular distribution of the received signal power. Consequently, such a factorization is possible only if the angular distribution of the scatters is independent of the signal travel distance. it permits to a system, it is essential to investigate the relationship between NMSV s system performance. In this section, we verify that there are meaningful relations between NMSV s system performance of popular diversity combining schemes. Since purely analytical performance evaluations of practical systems under practical channel environments are usually very difficult, we first investigate such relationship through simulations. In the upcoming paper [9], conceptual foundations idealized mathematical analysis are provided for more eidetic grounds for such relations. For simulations, we consider frequency hopping direct sequence spread spectrum systems. Throughout this section, we consider only discrete-time circularly symmetric complex Gaussian WSSUS channels with finite number of resolvable paths. As shown in Section III, such channels can be described by the scattering functions. The fading levels at different times for a given path are realizations of a zero mean circularly symmetric complex Gaussian rom variable are correlated according to the Doppler power spectrum corresponding to the path. All the systems considered employ binary phase shift keying (BPSK) modulation. We assume there exist additive white Gaussian noise at the receiver that the channel fading levels corresponding to different packets are independent. We assume that the receivers know the exact channel impulse responses for coherent demodulation. A. Frequency Hopping Spread Spectrum Systems In this subsection, we consider time non-selective but frequency selective channels study the relations between NFMSV performance of frequency hopping spread spectrum (FHSS) systems. For numerical analysis, we consider 71 channels described by discrete-time delay power profiles. The delay power profiles were obtained through actual measurements 7 performed in four airfield-type nine urban/suburban-type environments. Various combinations of antenna heights between m carrier frequencies among 40.7, 77.5, 37.5, MHz with 0MHz 7 These measurements were performed by the SPAWA Systems Comm in San Diego, CA in USA.

6 IEEE TANSACTIONS ON WIELESS COMMUNICATIONS, VOL. X, NO. XXX, XXX = 10dB NFMSV BE = 1dB = 14dB rms delay spread [µ sec] Fig. 1. elation between NFMSV rms delay spread. Each point corresponds to the (rms delay spread, NFMSV) pair for one of 71 frequency selective fading channels. There is not a close relation between rms delay spread NFMSV NFMSV Fig.. BE vs NFMSV for a FHSS system with a (31, 15) S code. Each point corresponds to the (NFMSV, BE) pair for one of 71 frequency selective fading channel at = 10, 1 or 14 db. We observe a roughly monotonic relation between the NFMSV the BE. bwidth were used to measure the characteristics in the thirteen different physical locations resulting in total of 71 different delay power profiles. Among the 71 delay power profiles, 41 have 40 tabs, 3 have 300 tabs remaining 7 have 400 tabs. For all the 71 profiles, the tab-spacing is 31.5nsec. We assume that a contiguous MHz is allocated to the systems it is sliced into 33 equal bwidth frequency slots. We assume that during a packet duration the carrier hops many times over the 33 slots using pseudo-romly generated hopping patterns. We define the frequency region of interest by the whole frequency b of MHz. Figure 1 shows the relation between NFMSV rms delay spread for these 71 channels. We find that there is a very weak relation between the two parameters implying that at least one of the two parameters is bound to exhibit a weak relation with the system performance. Since frequency hopping alone cannot exploit the frequency selectivity, we need to employ channel coding for meaningful results. A (31, 15) eed-solomon (S) code a rate 1/ convolutional code are used as examples of block trellis codes, respectively. Since both codes as well as the channel are linear, we assume that all zero information bits are transmitted. The simulations generate at least 0, 000 packets, that is, at least 0, 000 realizations of channel fading levels then proceed to generate more until we count 1, 000 bit errors before computing the bit error rate (BE). 1) eed-solomon Code: First, we consider a (31, 15) S code [10] as an example of block codes. In this coding scheme, a set of 15 3-ary symbols are encoded to a block of 31 3-ary symbols. Consequently, each codeword consists of 155 coded bits for 75 information bits. We assume that each packet consists of 0 codewords so that there are 3100 coded bits in a packet, which is S-symbol interleaved by a block interleaver of size 6 by 10. The coded symbols are written column-wise are read row-wise. Each row of coded symbols is transmitted over the same frequency slot BE = 5dB = 7dB = 9dB NFMSV Fig. 3. BE vs NFMSV for a FHSS system with a (r=1/, K=5) convolutional code. Each point corresponds to the (BE, NFMSV) pair for one of 71 frequency selective fading channel at = 5, 7 or 9 db.we observe a roughly monotonic relation between the NFMSV the BE. but different rows are transmitted over romly independently chosen frequency slots. This implies the hopping rate is 6 hops/packet that the hopping pattern is rom. We assume that bounded distance decoding algorithm is used with hard decisions made on the encoded bits. Figure depicts the relationship between NFMSV BE of the system. Each solid dot corresponds to the (NFMSV, BE) pair for one of the 71 channels at a particular signal to noise ratio (SN). We clearly see that the BE is strongly related to the value of NFMSV. ) Convolutional Code: As an example of trellis codes, we consider the rate 1/, constraint length 5 convolutional code with 3 35 as the generators in octal from [6]. Each packet is again assumed to have 3100 coded bits which correspond

7 IEEE TANSACTIONS ON WIELESS COMMUNICATIONS, VOL. X, NO. XXX, XXX BE = 3dB = 8dB = 13dB NFTMSV Fig. 4. BE vs NFTMSV for a DSSS system with -stage SIC. Each point corresponds to the (NFTMSV, BE) pair for one of 00 channels at = 3, 8 or 13 db. Circles correspond to the channels with time selectivity only, while x-mark to the channels with frequency selectivity only. The solid dots are results for the channels with both frequency time selectivity. We observe a roughly monotonic relation between the NFTMSV the BE to 1550 information bits, the output of the convolutional encoder is bit interleaved by a block interleaver of size 6 by 50. Coded bits are written column-wise are read rowwise. Again each row of coded bits is transmitted over the same frequency slot different rows will be transmitted over romly independently chosen frequency slots. The receiver uses soft-decision Viterbi algorithm for decoding. Figure 3 shows the simulation results. Again we find there is also very a strong relation between NFMSV BE. B. Direct Sequence Spread Spectrum System In the previous subsection, we studied the relationship between NFMSV the performance of FHSS systems under frequency selective but time non-selective channels. 8 In this subsection, we consider doubly selective (namely frequency time selective) channels as well as frequency-only selective time-only selective channels investigate the relationship between NFTMSV performance of a direct sequence spread spectrum (DSSS) system with successive interference cancellation (SIC). By spreading, DSSS systems obtain more resolvable paths for rake reception. This is the way how DSSS benefits from the frequency selectivity. Time selectivity can also be exploited by interleaving channel coding. Consequently, a practical DSSS system can exploit both frequency time selectivity. We show that there exists a nice relationship between NFTMSV the performance of DSSS systems considered regardless whether channels exhibit frequency (or time) selectivity. In the DSSS system considered, a multi-stage successive interference cancellation (SIC) is employed to mitigate the detri- 8 It is not unreasonable to expect similar relations between NFTMSV the performance of FHSS systems if channel state varies during a packet duration (but not during a hop duration.) For DSSS systems, the combined effect of frequency time selectivity is less obvious. mental effect (namely intersymbol interference) of frequency selectivity. Each packet consists of, 000 binary symbols which are spread by a pseudo-rom spreading code with spreading gain 5. Each set of 50, 000 chips is interleaved by a pseudo rom interleaver before transmission. 9 Since chip interleaving is employed, the spreading/despreading process, that can be regarded as a process of repetition coding/decoding, benefits from the time selectivity of channels. The receiver is assumed to employ rake reception to exploit the frequency selectivity of channels. The signalling rate is assumed to be 5 mega-chips per second. The multistage SIC is executed as follows. After detecting the first symbol using the usual threshold test, its (estimated) contribution is cancelled from the received signal (assuming the decision on the first symbol is correct) then the second symbol is detected. After detecting the second symbol, its contribution is cancelled similarly then detection is made for the third symbol so on. This is the first stage of the SIC employed in the system. After detecting the last (i.e. 000 th ) symbol subtracting its contribution, noise estimates are obtained that equals to the actual additive noise if all symbols were correctly detected. The second stage begins with the addition to the noise estimates of the contribution due to the first symbol assuming that the detection in the first stage is correct. At this point, we have noise estimates plus the signals corresponding to the first symbol regardless whether the decision for the first symbol in the first stage is correct or not. 10 Now, assuming this signal as the received signal, the rake receiver detects the first symbol subtracts its contribution again. Now, new noise estimates are obtained to which the contribution is added due to the second symbol. The second stage continues until this process reaches the last symbol. Similarly, higher stages can be processed. In this paper, we stop at the end of the second stage compute the BE. Instead of the measured delay power profiles considered in the previous subsection, we use 00 romly generated scattering functions to model doubly selective channels. To generate the 00 scattering functions, we first generate 500 delay power profiles romly then choose 150 delay power profiles with wide range of NFMSV values. Then, 50 scattering functions are directly defined by 50 of the 150 delay power profiles assuming time non-selectivity. With the remaining 100 delay power profiles, we generate 100 frequency time selective scattering functions by multiplying the relative gain for each path by a romly generated Doppler spectrum. emaining 50 are defined to be timeonly selective scattering functions by romly generating 50 Doppler spectra. Consequently, 50 channels are time nonselective, 50 channels frequency non-selective, remaining 100 channels doubly selective. The process of rom generation of the delay power profiles the Doppler spectra is described as follows. First, 9 For this system, no error correction coding is employed. 10 For this reason, multi-stage SIC performs decently in many cases [11], [1]. This is also consistent with the simulation results shown in Figure 4 since no manifest evidence of detrimental effect of frequency selectivity is displayed.

8 IEEE TANSACTIONS ON WIELESS COMMUNICATIONS, VOL. X, NO. XXX, XXX the number of resolvable paths are chosen uniformly among 1 30 the delay spreads are chosen uniformly between 0 0 microseconds. The relative amplitude gains for the paths (or collectively delay power profiles) are chosen by r n X b where n is the index indicating the order of the path (i.e. n th path), r one of 0.8, , X a stard normal rom variable, b one of 1, 3. Finally, the resultant delay power profiles are properly normalized. To generate the Doppler spectrum for each path, we first consider the Jakes spectrum [13] of uniformly distributed scatterers with Doppler frequency f m = 00 Hz romly choose a scale factor s between 0 1. Then, we truncate from the Jakes spectrum the region outside the frequencies between sf m sf m, which is then properly normalized. The resultant Doppler Spectrum S(f) is S(f) = arcsin s f m (f f c ) for f f c < sf m 0 otherwise, (43) where f c is the carrier frequency of the signal. When the carrier frequency is GHz, the Doppler frequency 00Hz corresponds to the vehicular speed 30m/s. For these 00 channels, the NFTMSV is computed from the formula (35) assuming the system occupies an interval of 5MHz frequency b. The BE of the system is computed by Monte Carlo simulations under each of the 00 channels for three different SN s. We first generate 1000 packets enumerate the number of total symbol errors. Then, the process of packet generation continues until the number of bit errors exceed 1000 before computing the BE. The results of the simulations are depicted in Figure 4. In the figure, sts for the energy per bit N 0 for the one-sided noise power spectral density (PSD) of the additive white Gaussian noise at the receiver. Each point corresponds to the (NFTMSV, BE) pair for each of the 00 channel at = 3, 8 or 13 db. We observe that the BE does not depends heavily on the detailed shape of the scattering functions if the NFTMSV s are the same. In particular, the performance does not depend much on whether the channel has time or frequency selectivity if the NFTMSV is the same. We can also see there exists a roughly monotonically increasing relationship between the NFTMSV the BE. For this reason, we propose the NFTMSV as a measure of channel quality. 11 V. NMSV AND SYSTEM DESIGN Differences in service requirements channel characteristics have led to a variety of communication systems. The optimality of system performance is especially crucial in mobile communications due to various stringent requirements such as high energy spectral efficiency. However, the diversity in the characteristics of mobile communication channels makes it difficult to investigate the optimality of communication systems. In the past, it was very difficult to evaluate the 11 As a minor point, we note that the BE is slightly higher with the channel with time selectivity only. As mentioned earlier the system exploits the time selectivity by the interleaver the repetition. Hence, the results imply the rom interleaver considered is not effective. In contrast, we can conclude that the -stage SIC is quite effective in treating the intersymbol interference. NFMSV v=0 v=w v=w v=5w Fig. 5. The usage of NFMSV in the frequency allocation. We see that we can lower the overall NFMSV of the channel for a FHSS system by separating each hop slots by inserting some amount of guard b. However, the NFMSV does not decrease dramatically as the separation bwidth become greater than the bwidth of each hop slot. system optimality based on performance analysis under a small number of communication channels. However, it is not very difficult to see that the close relationship between NMSV system performance can greatly simplify the system design process in many cases. Most of all, such close relationship directly enables us to investigate the optimality of various systems based on the performance analysis under a finite number of channels with diverse NMSV values. Moreover such relationship provides us with useful insight for system design optimization. In this section, we consider two simple examples to illustrate the utility of NMSV. We first consider the problem of frequency allocation for an FHSS system. Next we consider the effect of frequency hopping rate in an FHSS system. A. Frequency Allocation for a FHSS System In this subsection, we consider the problem of frequency b allocation for an FHSS system. In Section IV-A, we assumed that the 33 frequency slots are allocated to form an interval of frequency b of size MHz. In other words, frequency hopping is made within the interval in this system. However, by introducing a space between adjacent slots, we can lower the NFMSV of the resultant frequency region keeping the total bwidth the same. Assume there are K frequency slots of bwidth w with a space of bwidth v between adjacent slots. This frequency region is represented by the r given by (36) the NFMSV V( r ) can be calculated from (33) with (37). By calculating V( r ) for the 71 channels in introduced Section IV-A, we obtain Figure 5. Here, we assume that K = 33 w = 31.5kHz. From the figure, we can tell that we have smaller NFMSV with larger v, namely, with larger spacing. Since the BE performance is closely related to NFMSV, we can also tell, from the BE v.s. NFMSV curves (in Section IV-A), how much gain is obtained by introducing a space between slots. The space between slots

9 IEEE TANSACTIONS ON WIELESS COMMUNICATIONS, VOL. X, NO. XXX, XXX BE (NFMSV) BE for Channel NFMSV for Channel Hops / Packet Fig. 6. Bit error rate (BE) 1/NFMSV versus number of hops for Channel 1. The NFMSV of the total frequency b for Channel 1 is 0.. We note that the two curves exhibit similar saturation tendency. BE (NFMSV) BE for Channel NFMSV for Channel Hops / Packet Fig. 7. Bit error rate (BE) 1/NFMSV versus number of hops for Channel. The NFMSV of the total frequency b for Channel 1 is We note that the two curves exhibit similar saturation tendency. can be allocated to other systems. For example, if we choose v = w, then we can allocate slots to uplink downlink systems alternately. We can allow more space by considering neighboring cells together in the allocation of frequency slots without wasting frequency resources. However, increasing v larger than 5w does not decrease V( r ) appreciably. Consequently, we expect saturation of the performance enhancement after v becomes larger than 5w. B. Choice of Frequency Hopping ate Choosing optimal frequency hopping rate is an important problem in FHSS system design. In this subsection, we show how we can use NMSV to solve this problem. To illustrate the basic idea, we consider a frequency hopping system with (55, 17) eed-solomon code [10] under two different frequency selective time non-selective WSSUS channels called Channel 1 Channel. We assume that each packet consists of one codeword, namely, 040 binary digits. The total bwidth of the channel is assumed to be 10.4MHz which is divided into N = 55 slots of bwidth 40kHz. We assume that the fading is flat over each of the 55 slots. For definiteness, we assume the system is allocated the frequency region between A = 000 B = 010.4[MHz]. The two channels are so chosen that V f ([A, B]) = 0.0 V f ([A, B]) = 0.06 for Channel 1 Channel, respectively. The number K of frequency hopping per packet is chosen between The hopping pattern is chosen to be roughly uniformly spaced. For example, if K = 7, the 1 st, the 37 th, the 73 th, the 110 th, the 146 th, the 183 th, the 19 th slots are used for transmission. We denote by K the frequency region consisting of the K slots on which the system hops. Similarly, the 55 symbols are divided roughly equally so that the 1 st the 36 th symbols are transmitted over the 1 st slot, the 37 th the 7 th symbols over the 37 th slot so on. As a performance measure of the system, we consider the bit error rate (BE). For performance evaluation, we use a systematic (55, 17) eed Solomon code assume that the all zero information bits are transmitted. The receiver first makes a hard-decision on each coded symbol. Then, it counts the number of coded-symbol errors assume the errors are not corrected by the decoder if the number of symbol errors exceeds the error correcting capability, in which case the number of information bit errors in the systematic parts are counted toward the total errors. By simulations, we obtain the BE vs. hopping rate. Similarly, we plot V( K ) as a function of K. Figures 6 7 show the results. For the two channels, different SN s are chosen to make the saturated BE roughly the same. For Channel 1, V( K ) is virtually the same for K > 0 we observe that the BE of the system is saturated around K = 0. Similar coincidence between the tendencies of NFMSV BE can also be observed for Channel, although it is very difficult to find out the point of obvious saturation. Generally, it is much more time consuming to evaluate the system performance compared to the computation of the NFMSV. However, as shown in the figures, we can predict the performance saturation point with the NFMSV saturation point. VI. SUMMAY AND CONCLUSION In this paper, we defined three parameters called normalized frequency mean square covariance (NFMSV), normalized time mean square covariance (NTMSV), normalized frequencytime mean square covariance (NFTMSV) to characterize WS- SUS channels. From the definitions, it is easy to underst how these parameters characterize the overall frequency /or time selectivities of WSSUS channels. We showed that there exists very close relationship between NMSV s performance of practical systems such as FHSS or DSSS systems. Due to the close relationship with the system performance, the NMSV s can be regarded as useful measures of channel quality are very useful for efficient system design. In the upcoming paper [9], conceptual foundations idealized mathematical analysis are provided for more eidetic grounds.

10 IEEE TANSACTIONS ON WIELESS COMMUNICATIONS, VOL. X, NO. XXX, XXX V ft () = 4 e jπf(τ τ )+jπ(ν ν )(t t ) df dt q(τ, ν)q (τ, ν ) dτ dν dτ dν [ ]. (44) q(τ, ν) dτ dν df dt APPENDIX I NMSV IN TEMS OF SCATTEING FUNCTION In this section, we derive (33) - (35). From the definition of P (f, t) in (0), we have P (f f, t t )P (f f, t t ) df dt df dt = )(τ τ )+jπ(ν ν )(t t ) = 4 4 e jπ(f f q(τ, ν)q (τ, ν ) dτ dν dτ dν df dt df dt (45) e jπf(τ τ )+jπ(ν ν )(t t ) df dt q(τ, ν)q (τ, ν ) dτ dν dτ dν (46) P (0, 0) = q(τ, ν) dτ dν. (47) Consequently, by plugging (46) (47) into (9), we obtain (44) for V ft () (see above). For V f (; t) V t (f; T ), choose = (t δ, t + δ) = (f δ, f + δ) T take the limit δ 0 in (44). APPENDIX II CALCULATION OF A KENEL Let I k = [A + k(v + w), A + k(v + w) + w]. Then, e jπfτ df (48) = e jπfτ df e jπf τ df (49) k=1 k =1 I k I k = w sinc (πwτ)e jπτ(w+v)(k k ) (50) k=1 k =1 = w sinc (πwτ) k=0 e jπτ(w+v)k (51) [ ] sin{kπ(w + v)τ} = w sinc (πwτ). (5) sin{π(w + v)τ} Since df = Kw, we have (37). EFEENCES [1] D.-S. Yoo W. E. Stark, Characterization of multipath fading channels: Channels with specular components, IEEE Transactions on Wireless Communications, submitted. [] P. A. Bello, Characterization of romly time-variant linear channels, IEEE Transactions on Communication Systems, vol. CS-11, pp , Dec [3] J. G. Proakis, Digital Communications, 4 th ed. McGraw-Hill, 000. [4] W. C. Y. Lee, Mobile Communications Engineering, nd ed. McGraw- Hill Book Company, [5] M. J. Gans, A power spectral theory of propagation in the mobile raido environment, IEEE Transactions on Vehicular Technology, vol. 1, pp. 7 38, Feb [6] J. G. Proakis, Digital Communications, 3 rd ed. McGraw-Hill, [7]. S. Kennedy, Fading Dispersive Communication Channels. John Wiley & Sons, [8]. L. Peterson,. E. Ziemer, D. E. Borth, Introduction to Spread Spectrum Communications. Prentice Hall, [9] D.-S. Yoo W. E. Stark, Normalized mean square covariance diversity combining, IEEE Transactions on Wireless Communications, submitted. [10] F. J. MacWilliams N. J. A. Sloane, The Thoery of Error-Correcting Codes. North-Holl, [11] D.-S. Yoo, A. Hafeez, W. E. Stark, Trellis-based multiuser detection for ds-cdma systems with frequency-selective fading, IEEE Wireless Communications Networking Conference, vol., pp , [1] D.-S. Yoo W. E. Stark, Interference cancellation for multirate multiuser systems, IEEE Vehicular Technology Conference, Spring 001. [13] W. C. Jakes, Microwave Mobile Communications. IEEE Press, Do-Sik Yoo received B.S. degree in electrical engineering M.S. degree in physical from Seoul National University, Seoul, Korea in , respectively. He received M.S. Ph.D. degrees in electrical engineering from the University of Michigan, Ann Arbor, in , respectively. His research interests consist of diverse aspects of communications networking including adaptive signal processing, coding modulation, information theory, multiple access resource allocation, wireless networking. In the last few years, he has developed a turbo coded FH-OFDM receiver with interpolated joint channel estimation decoding power control algorithms that tracks timing errors Doppler shifts. Wayne E. Stark received the B.S. (with highest honors), M. S., Ph.D. degrees in electrical engineering from the University of Illinois, Urbana- Champaign, in 1978, 1979, 198, respectively. Since September 198, he has been a Faculty Member in the Department of Electrical Engineering Computer Science at the University of Michigan, Ann Arbor, where he is currently a Professor. His research interests include the areas of coding communication theory, especially for spreadspectrum wireless communication networks. Dr. Stark is a member of Eta Kappa Nu, Phi Kappa Phi, Tau Beta Phi. He was involved in the planning organization of the 1986 International Symposium on Information Theory of the IEEE TANSACTIONS ON COMMUNICATIONS in the area of spread-spectrum communications. He was selected by the National Scicence Foundation as a 1985 Presidential Young Investigator. He was the Principal Investigator of an Army esearch Office Multidisciplinary University esearch Initiative (MUI) Project on low-energy mobile communications.