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Doppler Spread Estimation in Frequency Selective Rayleigh Channels
Doppler Spread Estimation in Frequency Selective Rayleigh Channels for OFDM Systems Athanasios Doukas, Grigorios Kalivas {adoukas, kalivas}@ee.upatras.gr Department of Electrical and Computer Engineer, University of Patras Campus of Rion, Achaia, 26500, Greece Abstract- In this paper, we present a method for In addition, in OFDM systems, if the channel varies estimating the Doppler spread (DS) in Wireless Local Area considerably within one OFDM symbol because of high Networks (WLAN) using Orthogonal Frequency Division Multiplexing (OFDM). DS gives a measure of the fading rate MU mobility, orthogonality between subcarriers is lost, of the wireless channel, which can be used to adjust the leading to inter-carrier interference (ICI) [4], degrading channel estimation rate and create specifically designed system performance. Doppler information can help in channels estimators to combat Inter-Carrier Interference selection of appropriate transmission characteristics to (ICI) induced due to loss of orthogonality that DS imposes on combat ICI including proper channel estimators to OFDM systems. The estimation is based on the autocorrelation function of time domain channel estimates enhance reception. Specifically designed channel over two OFDM symbols and since that most of the receiver estimators can be applied and the rate of appliance can be algorithms require knowledge if the receiver moves or not we chosen to improve throughput. When mobility is low the divide the operation region into two modes: still mode(S- rate of estimation can be lowered and increase throughput mode) and moving mode (M-mode). The estimation accuracy, this way. In the other way, an increase in the estimation examined in environments with different PDPs, including channel sparsity, using several constellation schemes is quite rate can help to lower BER. accurate from low SNR values of 5 dB. Doppler spread estimation is widely examined in the literature. Level Crossing Rate (LCR) of the received Keywords: Doppler estimation, fading channels. signal envelope is proportional to the Doppler frequency [5] and thus used in Doppler estimation. However, the I. INTRODUCTION fading nature of the wireless channel decreases the Orthogonal Frequency Division Multiplexing (OFDM) estimator’s accuracy in low Doppler values. Other [1] has been widely applied in the last years for various methods, highly associated to LCR, are the Zero Crossing wireless communication systems such as Digital Video Rate (ZCR), which uses the in-phase or quadrature phase Broadcasting (DVB) and wireless local area networks (I/Q) signal part [6], and some other higher order (WLANs) ensuing great success. These systems however, crossings of the signal envelope [7]. Switching rate of should be capable of working efficiently in wide range of diversity branches is used for the velocity estimation in [8], operating conditions, such as large range of mobile unit but in [9] its sensitivity to the fading channel is shown. In (MU) speeds, different carrier frequencies in licensed and [10] and [11], velocity estimation algorithms that use un-licensed bands, various delay spreads, and wide pattern recognition and wavelets respectively are proposed. dynamic signal to-noise ratio (SNR) ranges. This way Another type of estimators are the covariance-based assessing the channel quality and its rate of change is of estimators, which estimate the Doppler frequency from great importance in adapting the system parameters to the autocovariances of powers of the received signal continuously changing channel conditions [2], [3]. envelope [12], or from sums of the I/Q components [13]. The previously mentioned reasons motivated the use of The estimators presented in [14], based on the second adaptive algorithms in wireless communication systems. spectral moment of the I/Q components, and the Scope of the adaptive algorithms is to optimize wireless correlation-based estimators in [15] and [16], share the systems performance, fully exploit channel capacity and same basis. utilize available resources into the most efficient manner In all the previous works, crossing-rate estimators are to maximize their throughput for a given quality of service less efficient than their covariance-based counterparts (QoS), which in most of times is measured in terms of Bit over short estimation windows, due to the fact that the Error Rate (BER). However, adaptation necessitates the observed signal does not experience many level crossings accurate knowledge of some wireless channel’s in a small time window [14]. This observation is parameters. important for practical applications, as a wireless channel One crucial parameter in adaptation of wireless can only be assumed to be wide-sense stationary (WSS) systems is the maximum Doppler spread. It provides over short intervals.Furthermore most of the estimators information about the fading rate of the channel. suffer from insufficient performance in low Doppler Knowledge of Doppler spread can improve detection and values and for low SNR values and are computationally aid into transmission optimization in both physical layer high, needing a significant load of measurements. and higher levels of the protocol stack [2].In particular, a In the previous works OFDM is used in DVB systems, power control update approach can be applied, adjust of where the Doppler frequency is high. Yet in WLAN interleaving length to reduce reception delays and so on. systems the mobility is lower, ranging from 0 to 40Hz. Thus the most critical for a WLAN system is to know ∞ whether is standing still (S-mode) or moving (M-mode) H ( t , ƒ ) ≡ ∫ h ( t , τ)e − j2 π ƒ τ dτ = ∑ γ l ( t )e − j2 π ƒ τl . (3) and when moves if retains its velocity. Such an estimator −∞ l is presented only in [17], where the operation mode is divided into three different states, of low, medium and fast Hence, the correlation function of the frequency mobility. response for different times and frequencies is In this work we present a Doppler estimator that uses the time correlation of only two OFDM symbols, { rH (∆t, ∆ ƒ ) ≡ Ε Η (t + ∆t , ƒ + ∆ ƒ )Η * ( t ,ƒ ) } satisfying the WSS channel assumption and achieving low = ∑ rγ l (∆t )e − j2 π∆ ƒ τl complexity, which manages to clearly distinguish the two l modes of mobility. A Doppler estimator that achieves (4) ⎛ ⎞ these two goals, to the best of authors’ knowledge, is the = rt (∆t )⎜ ∑ σ l e − j2 π∆ ƒ τl ⎟ 2 first time that is presented. We examine its estimation ⎝ l ⎠ performance using various constellation schemes with = σΗ rt (∆t )rƒ (∆ ƒ ) 2 different bit rates in environments with a variety of Power Delay Profiles (PDP), including channel sparsity. The where σ H = 1 is the total average power of the channel 2 estimator in most of the cases works satisfactory from SNR values as low as 5 dB. impulse response defined as The rest of the paper is organized as follows. In Section II the channel model along with the OFDM system used σΗ ≡ ∑ σl 2 2 (5) are described. The estimator and its characteristics are l derived in Section III. Estimation results are presented in σ l − j2 π∆ ƒ τl 2 Section IV. Finally concluding remarks are given in rƒ (∆ ƒ ) = ∑ e (6) l σΗ 2 Section V. 2 II. FADING CHANNEL AND OFDM SYSTEM Without loss of generality, we also assume that σ Η = 1 , which, therefore, can be omitted from (4). Prior to examining Doppler estimation for OFDM From (4), the correlation function of H(t,ƒ) can be systems in WLAN radio channels, we briefly describe the separated into the multiplication of a time domain channel statistics, emphasizing the separation property of correlation rt(∆t) and a frequency domain correlation wireless channels, which is crucial employing our rƒ(∆ƒ). rt(∆t) is dependent on the vehicle speed or, estimator. In this section we also describe the OFDM equivalently, the Doppler frequency, while rƒ(∆ƒ) depends system used for our estimator. on the multipath delay spread. With the separation A. Wireless Channel Model property, we are able to propose our Doppler estimator The complex baseband representation [18] of a wireless described in the next section. channel impulse response can be described by For an OFDM system with block length Tƒ and tone spacing (subchannel spacing) ∆ƒ, the correlation function for different blocks and tones can be written as h ( t , τ) ≡ ∑ γ l ( t )δ(τ − τl ) (1) l rΗ [i, k ] = rt [i]rƒ [k ] (7) where τℓ is the delay of the ℓ-th path and γℓ(t) is the corresponding complex amplitude and δ is the delta where function. Due to the motion of the vehicle, γℓ(t)’s are wide-sense stationary (WSS) narrowband complex rt [i] ≡ rt (iTƒ ) (8) Gaussian processes, which are independent for different rƒ [k ] ≡ rƒ (k∆ ƒ ) (9) paths. We assume that γℓ(t) has the same normalized correlation function rτ(∆t) for all ℓ and, therefore, the and rt and rƒ are the time and frequency correlation respectively. Equation (7) is valid for an exponentially same normalized power spectrum pt(Ω). Hence decaying multipath PDP, which is used in the simulations. { l } rγ l (∆t ) ≡ Ε γl (t + ∆t )γ * ( t ) = σl rt (∆t ) 2 (2) Exponential PDP is the most commonly accepted and accurate model for indoor channels [19]. From Jakes’ model where σl is the average power of the k-th path. 2 Using (1), the frequency response of the time-varying rt [n ] = J 0 (nωd ) ≡ rJ [n ] (10) radio channel at time t is where J0(x) is the zeroth-order Bessel function of the first kind, and its Fourier transform (FT) is ⎧2 1 ⎪ω , if ω < ωd p J (ω) = ⎨ d 1 − (ω ωd )2 (11) ⎪ ⎩0, otherwise In the above expression ωd=2πΤƒƒd, and ƒd is the Doppler frequency, which is related to the vehicle speed υ and the carrier frequency ƒc by υƒ c ƒd = (12) c where c is the speed of light. B. OFDM System Figure 1. OFDM Baseband Model The baseband model of the OFDM system employed N r −1− ρ into this paper is shown in Fig. 1. Time domain samples of 1 an OFDM symbol can be obtained from data symbols as rt [ρ] = Nr − ρ ∑Υ i =0 i ,k ⋅ Υi* ρ ,k + (16) x i ,n = IFFT{X i ,k } where Nr is the number of used samples and ρ is the time N −1 (12) difference of the used samples. = ∑ X i ,k e j2 πnk / N n = 0, K , N − 1 Firstly we set Nr=2 and ρ=0 in (10) and (16). Thus we k =0 get where Xi,k is data symbol of the k-th subcarrier of the i-th rt [0] = J 0 (0 ⋅ ω d ) = J 0 (0 ) = 1 (17) OFDM symbol, and N is the number of subcarriers. After 2−1− 0 1 the guard interval addition, the samples xi,n are transmitted 1 1 over the linear time dependent channel described in eq. 1 , rt [0] = 2− 0 ∑Υ n =0 i ,k ⋅ Υi* 0 ,k ⇒ rt [0] = + ∑ Υi,k ⋅ Υi*,k 2 i =0 (18) with additive white Gaussian noise (AWGN) zi,n, with zero mean and variance of σ 2 . In this paper, we assume z It is clear that this step is actually a normalization the channel to be constant over an OFDM symbol, but purpose step. In the next step we set Nr=2 and ρ=1 in (10) time-varying across OFDM symbols, which is a and (16). Thus we get reasonable assumption for low mobility. At the receiver, assuming perfect synchronization, the rt [1] = J 0 (1* ωd ) = J 0 (2πTƒ ƒ d ) (19) received samples can be expressed as 2−1− 1 1 y i , n = x i , n ∗ h ( t , τ) + z i ,n (13) rt [1] = 2−1 ∑Υ i =0 i ,k ⋅ Υi* 1 ,k ⇒ rt [1] = ∑ Υ0,k ⋅ Υ1*,k (20) + where * stands for convolution. Combining (17) to (20) we get Doppler frequency by After removing the guard interval, the receiver de- multiplexes the received samples by using the FFT as ⎛ ⎞ ⎜ ⎟ 1 −1 ⎜ ∑ Υ0,k ⋅ Υ1,k * Yi ,k = FFT{y i ,n } ƒd = J0 ⎟ (21) 2πTf ⎜ 1 1 ⎟ ⎜ ∑ Υi ,k ⋅ Υi ,k * 1 N−1 (14) ⎟ = ∑ yi,n e− j2πnk / N N n =0 k = 0, K, N − 1 ⎝ 2 i =0 ⎠ IV. SIMULATION RESULTS and the demodulated signal Yi,k can be expressed as In this section we are going to describe the simulation Yi ,k = X i ,k ⋅ H( t ,ƒ ) + Zi ,k . (15) results of the Doppler Estimator. We have tested the estimator in BRAN Channels A and B [20], which are typical for indoor WLANs. These channels are quite harsh III. DOPPLER SPREAD ESTIMATOR and include sparsity. In Fig. 2 the estimation results for BRAN Channel A In this section we describe the functions used to derive with 5 dB SNR using BPSK modulated data for different the Doppler estimator estimator. We are going to use the Doppler Spread values, ranging from 0 to 40 Hz, are time correlation function expressed in (10) and the time depicted. From this figure we can see that the scope of the correlation function using the received samples expressed estimator, to clearly distinguish the two operational modes, by the following equation is succeeded. Especially from subcarriers 5 to 35, the distinction is clearer. -3 Doppler estimation with SNR=10dB in BRAN Channel A for QPSK modulated data -3 Doppler estimation with SNR=5dB in BRAN Channel A for BPSK modulated data x 10 x 10 7.5 8 7 7 6 5 6.5 E s t im a t io n Estimation 4 Doppler estimation 0 Hz Doppler estimation 10 Hz Doppler estimation 20 Hz 6 Doppler estimation 0 Hz 3 Doppler estimation 30 Hz Doppler estimation 10 Hz Doppler estimation 40 Hz Doppler estimation 20 Hz 2 Doppler estimation 30 Hz 5.5 Doppler estimation 40 Hz 1 0 5 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 Subcarrier Index Subcarrier Index Figure 2. Doppler estimation results in BRAN Channel A with BPSK Figure 4. Doppler estimation results in BRAN Channel A with QPSK modulation with 5dB SNR. modulation with 5dB SNR -3 Doppler estimation with SNR=10dB in BRAN Channel B for BPSK modulated data -3 Doppler estimation with SNR=5dB in BRAN Channel A for BPSK modulated data x 10 x 10 8 8 7 7 6 6 Doppler estimation 0 Hz 5 Doppler estimation 10 Hz 5 Doppler estimation 20 Hz E s tim a tio n Doppler estimation 30 Hz Estimation 4 Doppler estimation 40 Hz 4 Doppler estimation 0 Hz Doppler estimation 10 Hz Doppler estimation 20 Hz 3 3 Doppler estimation 30 Hz Doppler estimation 40 Hz 2 2 1 1 0 0 0 5 10 15 20 25 30 35 40 45 50 0 5 10 15 20 25 30 35 40 45 50 Subcarrier Index Subcarrier Index Figure 5. Doppler estimation results in BRAN Channel B with BPSK Figure 3. Doppler estimation results in BRAN Channel A with BPSK modulation . modulation with 10 dB SNR. Analogous results are taken in the case of 10 dB SNR, are presented. Likewise the case of QPSK data in Fig. 4, depicted in Fig. 3. In this figure, the distinction between the estimator can not distinguish the two modes of the two modes is clearer because of the higher SNR. From operation. This result is due to the harsher transmission these two figures and taking under consideration that in conditions that are imposed into this case. BRAN Channel most of the WLAN standards such as 802.11a, B is a harsher transmission environment than BRAN HIPERLAN/2, there is a part of two BPSK modulated Channel A. Thus the noise added in the received signal useful symbols in the preamble part, we can say that this results to insufficient estimation results. Again the method can be applied on the preamble data and insertion of a channel estimator would probably improve distinguish the two mobility states. This way the system the estimation results. will be able to adapt its transmission scheme before the V. CONCLUSIONS useful packet of data is transmitted and thus increase its performance. We have presented a Doppler estimator for low In Fig. 4 the estimation results for BRAN Channel A mobility OFDM systems in Frequency Selective Rayleigh with 5 dB SNR using QPSK modulated data for different Fading Channels, using only two OFDM symbols for the Doppler Spread values, ranging from 0 to 40 Hz, are time correlation in wireless OFDM systems. The estimator depicted. In this case even that the estimator manages to instead of trying to estimate the accurate value of the slightly distinguish the different Doppler spreads, the Doppler frequency divides the mobility into a still and a difference is negligible. This result can be explained from moving mode, which for the case of WLANs is the most the constellation type used. QPSK is more sensitive to important. We have examined its performance in wireless noise than BPSK. 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