Curtailed-Gaussian and Cosine Functions for Multihop Doppler Spectrum Modeling

Wireless channels are characterized among others by their Doppler spectrum. In the cooperative diversity, one of diversity branch may consist of several mobile relays forming multihop link which each hop introduced Doppler shift. With employing amplify-and-forward (AF) relays, the Doppler shift keeps accumulating to the end of the link. Doppler shift value affects the time varying channel rate, which is a challenge in broadband mobile communication system. Hence, the Doppler parameter is very important and must be considered in broadband mobile communication system design and analysis. Unfortunately, it is hard to derive the expressions of this Doppler spectrum in a closed form since a special function under integration such as complete elliptic integral exists. To solve this problem, curve-fitting method base on least-square is used. In this process, curtailed-Gaussian and cosine functions are proposed as an approximation function. Then, from Kullback-Leiber divergence test, it is showed that both proposed functions, i.e., curtailed-Gaussian and cosine functions have a good approximation as Doppler spectrum modeling of Multihop mobile channel with all gain relays assumed as 1 and all mobile terminals are assumed move with almost same velocity.


Introduction
Multihop transmission with moving relays has attracted many researchers recently [1]- [8], due to its useful applications to achieve cooperative diversity for mobile cellular communication network.Such mobile-to-mobile communication systems differ from the conventional cellular radio systems, where one terminal, i.e., the base station (BS), is stationary, and only the mobile station (MS) is moving.Although the received signal envelope is Rayleigh faded under non-line-of-sight (N-LOS), but the mobility of both the transmitter (Tx) and the receiver (Rx) result in different statistical properties of the wireless channels, as described first by Akki and Haber [9], [10].Their analysis was extended by Patel et al in [11] by proposing a "double-ring" model, which was developed based on sum-of-sinusoids (SoS) models for simulating such channels.This channel model was used by Patel and Stuber in [12] to analyze statistical properties of amplify-and-forward relay fading channels.They noticed that the closed form expression for the Doppler spectrum for dual-hop; i.e. a mobile-to-mobile link in cascade with a mobile-to-fixed link, remains an open problem.
In cooperative communications, there are different schemes of relay processing namely non-regenerative amplify-and-forward (AF) and regenerative decode-and-forward (DF) scheme.The AF scheme is widely used for cooperative diversity, because it is easy to implement and yields satisfactory performance [1].It also has been shown in [1], [13] that the AF scheme has about the same quality as the DF scheme at high SNR in terms of its diversity.
In the cooperative diversity, the cooperation link, which is the link that involves the use of relays, is often modeled as dual-hop, i.e. using a single relay.Whereas in cellular networks, a lot of mobile terminals are available to perform the relaying function and assist other users in delivering the message to remote destination.The existing dual-hop model can be extended to Multihop with all the relays assumed to be mobile.To the best of our knowledge, very little is known about the Doppler spectrum of Multihop mobile-to-mobile link, based upon which a cooperative system can be evaluated.
Our contribution is two-fold: firstly, a new Doppler spectrum models for mobile-to-mobile Multihop channels with AF relays and secondly, a mathematical approximation of the Doppler spectrum model.More specifically, as motivated by the fact that the overall channel consists of mobile-to-mobile channels in cascade and can be modeled as the product of N fading amplitudes, the Doppler spectrum is also propagated and accumulated from one hop to the next up to the destination.Closed-form expressions of the desired Doppler spectrum seem to be intractable and therefore, we use curve-fitting method to the Doppler spectrum curve which was obtained numerically.We propose functions as simple and close as possible to the Doppler spectrum as models.
The remainder of the paper is organized as follows.Section II introduces the Multihop mobile-to-mobile channels based on double-ring scatterers channel model and presents numerical results of Multihop Doppler spectrum models.The curve-fitting formulation of the model is presented in section III.Results and analysis are presented in section IV.Finally, section V provides some conclusions.

Proposed Method
In this section, Multihop mobile to mobile fading channel and doppler spectrum are explored.

Multihop Mobile to-Mobile Fading Channel
The Multihop uplink transmission is depicted in Figure 1 in which a mobile station (MS) source (i.e.node 0) is sending data via MS relays (i.e.node 1 to node ) to the base station (BS) destination (i.e.node n ).The Multihop transmission might constitute a diversity branch in cooperative communication systems that consist of relaying chain process.
For the scenario of n-hop transmissions, S, R 1 ,..,R n-1 , and D, denote a source, relays, and a destination, respectively.The transmitted signal, ‫ݏ‬ሺ‫ݐ‬ሻ, after travelling through ݊-hop channel and amplify (݊ − 1) times by (݊ − 1) will received as ‫ݎ‬ ሺ‫ݐ‬ሻ.Analogy with received signal ‫ݎ‬ ଶ ሺ‫ݐ‬ሻ, of two-hop as derived in [12], the received signal, ‫ݎ‬ ሺ‫ݐ‬ሻ, of ݊-hops can be derived as follow: Where ‫ܣ‬ሺ‫ݐ‬ሻ denotes the relay gain which is assumed to be identical for all relays; ℎ ሺ‫ݐ‬ሻ, the fading factor of the ݇-th channel that ends at the ݇-th relay or the BS for ݇ = ݊, ‫ݖ‬ ሺ‫ݐ‬ሻ the zeromean additive white Gaussian noise (AWGN) with power spectrum density ܰ at node ݇.All noises introduced by relays and destination have identical statistic properties.For further discussions and simplicity, the time index is dropped.All amplification factors are, ‫ܣ‬ሺ‫ݐ‬ሻ, considered fixed and normalized as 1.
For the mobile-to-mobile fading channel, the channel models derived by Patel in [11] is used and is given as: ݂ ଵ = ‫ݒ‬ ଵ /ߣ ଵ and ݂ ଶ = ‫ݒ‬ ଶ /ߣ ଶ are the maximum Doppler shifts induced by the motion of the source MS with speed ‫ݒ‬ ଵ and carrier wavelength of ߣ ଵ and the relay MS with speed ‫ݒ‬ ଶ and carrier wavelength of ߣ ଶ .The index ‫‬ refers to the ‫-‬th propagation path from the Tx to the ‫-‬th of the ܲ scatterers located on the Tx end ring, while the index ‫ݍ‬ refers to the ‫-ݍ‬th paths from the ‫-ݍ‬th of the ܳ scatterers on the Rx end ring to the Rx.The phases ߠ , ~ܷሺ−ߨ, ߨሿ are independent for all ‫,‬ ‫ݍ‬ pairs.ߙ and ߚ are angles of departure and arrival to/from the scatterers at Tx and Rx, respectively.For the Multihop link, each mobile-to-mobile hop is modeled by this channel model which is essentially the same model for all hops except the last one.The last hop is a mobile-tofixed channel, for which a single-ring scatterers channel model derived in [14] is used.This channel model is given as follow: The dynamic behavior of the relay channels can be described by the time autocorrelation function.For dual-hop transmission with fixed gain relay, the overall impulse response is ℎ = ℎ ଵ ℎ ଶ , where ℎ ଵ is a mobile-to-mobile channel and ℎ ଶ is a mobile-to-fixed channel.The auto-correlation function can be derived as follow, Where Eሾ. ሿ is the statistical expectation operator, * denotes the complex conjugate operator, ‫ܬ‬ ሺ. ሻ is the zeroth order Bessel function of the first kind.In the second line we exploit the independence of ℎ ଵ and ℎ ଶ and substitute both autocorrelation value of ℎ ଵ and ℎ ଶ with autocorrelation function of mobile-to-mobile channel derived in [11] and autocorrelation function of mobile-to-fixed derived in [14], respectively.Hence the autocorrelation function consists of the product of three Bessel functions with different maximum Doppler shift i.e., ݂ ଵ as source maximum Doppler shift, ݂ ଶ and ݂ ଷ as relay maximum Doppler shift at received and transmitted state, respectively.
Analogous to the dual-hop, the auto-correlation function can be derived as the product of ݊ Bessel functions with possibly different maximum Doppler shifts and a scale factor of 2 ିଵ .
Following the expression for rms Doppler spread, ߚ ௗ , as defined in [15], it can be easily shown that for ݊-hop,

Multihop Mobile to-Mobile Doppler Spectrum
Motion in the wireless environment causes transmitted signals to experience Doppler shifts as a function of the angle between the signal path and the direction of motion.Signals arrive at a receiver from moving mobile terminals through reflections from scatterers at a variety of angles.The superposition of these multipath components having various Doppler shifts and received powers appears as the Doppler spread at the receiver.The exact form of the Doppler spectrum depends on the wireless environment and is the Fourier transform of auto-correlation function of the channel response.The well-known U-shaped Doppler spectrum of cellular channels [9] with normalized channel power gain is given by, The Doppler spectrum for mobile-to-mobile channels is obtained as function of the complete elliptic integral of the first kind ‫ܭ‬ሺ.ሻ given by [11], where ݀ = ݂ ଶ /݂ ଵ is defined as the Doppler ratio for single-hop.Hence for ݊-hop that consists of ሺ݊ − 1ሻ-hop mobile-to-mobile fading channels and 1-hop of mobile-to-fixed fading channel, the closed-form of Doppler spectrum may be obtained by convolution operation of ሺ݊ − 1ሻ Doppler Spectrum in (8) and a Doppler spectrum in (7) as mentioned above or by taking the Fourier Transform of (5), i.e. products of ሺ2݊ − 1ሻ Bessel functions.Unfortunately, finding a closed form expression for the Doppler spectrum using either of these two operations is rigorous since they both have a special function under integration such as complete elliptic integral.To overcome this difficulty, we use numerical method to find the shape of Doppler spectrum.For simplicity, the Doppler spectrum is expressed as functions of the source' maximum Doppler shift ݂ ଵ and the Doppler ratio ݀ defined as, Where ݂ ே is the maximum Doppler shift of the last hop.Therefore, for fixed ݂ ଵ , different spectrum is obtained by varying the Doppler ratio ݀ .For dual-hop channels, the Doppler spectrum has several shapes with one or two discontinuity points and is more concentrated near zeros when the two mobile stations move with almost identical velocity vectors [12].For ݊-hop link, the Doppler spectrum is obtained by taking the Fourier Transform of the time's autocorrelation function that consists of the product of ሺ2݊ − 1ሻ Bessel functions.If ݀ has a value near one (such as, ݀ = 1 ± 0.2), it means all mobile stations move with almost identical velocity.The Doppler spectrum has a shape as seen in Figure 2.This figure shows that the Doppler spectrum shape does not change much and is still concentrated near zero as in dual-hop (in this figure, all Doppler ratio is set to 1 and 1 f = 70 Hz) and the maximum Doppler shift, , max f is defined as: Figure 2. Doppler spectrum for Multihop mobile-to-mobile channel with double-ring scatterers

Research Method
Curve-fitting method is applied to the Doppler spectrum of Multihop mobile-to-mobile channel for getting the approximation models.Several functions that best fit the normalized Doppler spectrum in the least-square sense are used.Based on figure 2, the Doppler spectrum shape can be approximated with a curtailed-Gaussian and a cosine functions as the Multihop number increased.So with this reason, we choose these functions to express the models.
Another reason that supports this idea (choosing the curtailed-Gaussian function for the approximation) is described as follow: Given the product of M Bessel functions: Then its Fourier Transform can be expressed as: ܵ ெ ሺ݂ሻ is as defined in ( 7) and ( 8) and * denotes convolution operator.This form resembles that of pdf (probability density function) of a random variable, which is the sum of ‫ܯ‬ independent random variables of the same shape but with different parameters.Consequently, by invoking the Central Limit Theorem [16], the Fourier transform of the product (12) approaches a curtailed-Gaussian function.This is true when ‫ܯ‬ is sufficiently large, i.e. the number of hops with moving relays is large.Cosine function also proposed as alternative models because the Doppler spectrum curve is nonlinear and the shape also close to half cycle cosine function, cosሺ݂ሻ , that has maximum value at ݂ = 0 in the interval −ߨ/2 ≤ ݂ ≤ ߨ/2.
Curve fitting with curtailed-Gaussian and cosine functions is done by choosing several coefficients associate with Doppler parameter such as maximum Doppler shift and number of hops.Curve-fitting procedures can describe as follow, 1.First, the basic equation is set up for each approximation function, Curtailed-Gaussian function: Cosine function: Each function has several independent variables, consists of ‫,ܩ‬ ߙ, ܽ, ݂ and ݂ ௫ for curtailed-Gaussian and ‫,ܩ‬ ߚ, ܾ, ݂ and ݂ ௫ for cosine.݂ ௫ is related to ݂ ଵ and ݀ through equation (10).݂ ଵ and ݀ are defined first with several value of ݂ ଵ from 10 to 600 Hz (commonly, Doppler shift in cellular system is in this range) and ݀ set as random with ݀ ~ܰሺߤ = 1.ߪ ଶ = 0.1ሻ.2. Doppler spectrum is computed and plotted based on numerical computation, for each ݊. 3.Each of curtailed-Gaussian and cosine function in step 1 is fitted to the Doppler spectrum in step 2 based on least squares methods.Matlab toolbox program nlinfit is used in fitting process.4. From step 3, coefficients in ( 13) and ( 14) are obtained for several ݊ and ݂ ଵ values.5.Each coefficient in step 4 (i.e., ‫,ܩ‬ ߙ, ߚ) is plotted as function of ݂ ଵ for different ݊. 6. Approximation functions with independent variable ݂ ଵ are created for each coefficient curve in step 5.Each coefficient is related to several coefficients (i.e., ܽ ଵ , ܽ ଶ , ܽ, ܾ ଵ , ܾ ଶ , ܾ, ݃ ଵ and ݃ ଶ ) through following equations, 7.

Results and Discussion
In Figure 3, the Doppler spectrum curve obtained from numerical method and from the proposed models (curtailed Guassian and cosine) for multihop mobile channel seem have a good approximation.For this reason, the Kullback-Leiber divergence (denoted by ‫ܦ‬ ) is used to validate the models.Here, the ‫ܦ‬ is used as a measure of the difference between reference (numerical) Doppler spectrum curve and Doppler spectrum curve obtained from proposed models.From [17] is defined:

79
ܸሺ݅ሻ and ܹሺ݅ሻ are Doppler spectrum from the reference (numerical) and the proposed models, respectively.The Kulback-Leiber divergence is always non-negative and equal to zero if and only if ܸ = ܹ.So, the closer ‫ܦ‬ is to 0, the better is the approximation of the model with the theory.The results of this comparison, presented in Table II, show that the Doppler spectrum of our models provides a good approximation to the numerical Doppler spectrum.
Further investigation on n -Multihop shows that if ݉ out of ݊ − 1 mobile relays are not moving (stopping during traveling), than the number of Bessel factors is decreased by 2݉ and the Doppler spectrum width will reduce to ሺ݊ − ݉ሻ-hop Doppler spectrum, but the power remains high, which is close to ݊-Multihop power.If Doppler ratio is assumed as random with normal distribution, ݀ ~ܰሺߤ = 1, ߪ ଶ = 0.1ሻ, the models still provide a good approximation.
The Doppler spectrum models can be used to analyze system performance with adaptive modulation applications [18], because M-ary modulation with high M has narrow bandwidth and more sensitive to spectral spread due to Doppler.

Conclusions
From the discussion, the Doppler spectrum models of Multihop communication with mobile amplify-and-forward relays can be approximated with two simple functions, i.e., curtailed-Gaussian and cosine functions.Both models are valid if all mobile terminals are moving with almost identical velocity.It is shown that the approximation models can be used to generate mobile Multihop channels for evaluation of Doppler effects and to determine the delay spread in the study of cooperative diversity systems.Our future work is evaluating ICI in cooperative-OFDM due to high Doppler frequency introduced by Multihop mobile channel and how to mitigate the ICI in such environment.

Figure 3 .
Figure 3. Validation result: The comparison of approximation models to the numerical value of Doppler spectrum (f 1 =70 Hz)

Table 1 .
Doppler spectrum coefficients of curtailed-Gaussian & cosine model