Low Complexity Residual Phase Tracking Algorithm for OFDM-based WLAN Systems
Suvra S. Das†,Ratnam V.Rajakumar§, Muhammad I.Rahman†, Arpan Pal‡, Frank H.P.Fitzek†, Ole Olsen†,Ramjee Prasad† † Department of Communications Technology, Aalborg University,e-mail: [email protected] ‡ Tata Consultancy Services,§ IIT Kharagpur,India
Abstract— This paper presents the design of an efficient low complexity Low Pass ADC LNA Time Synch Frequency Synch residual phase tracking algorithm for Orthogonal Frequency Division X Filter converter Multiplexing (OFDM) based Wireless Local Area Network receivers. In Low Noise Amplifier this paper the focus is on mitigation of the residual carrier frequency Local Channel Phase Tracking FFT synchronization offset and sampling frequency offset. We propose a novel Oscillator Equalization algorithm that uses piecewise linear approximation to estimate the com- Front End(Inner Receiver) Fast Fourier Transform plex exponential of the phase angle at pilot locations instead of estimating De-Mapper Channel Decoder Bit(s) output the actual phase angle by highly complex (costly) angle computation and Back End (OuterReceiver) search functions. This helps in reducing the implementation cost of an OFDM receiver. By means of analysis and simulation we show that our Fig. 1. OFDM receiver front end architecture design combines both high performance and low complexity.
I.INTRODUCTION blocks placed at the receiver front end are not able to estimate the Wireless Local Area Networks (WLAN) are becoming part exact carrier frequency offset due to circuitry noise and fixed word of omnipresent communication infrastructures. WLANs are being length effects. Moreover due to sampling frequency offset there is applied in hotels, airports, cafes and various other locations. Different a slowly increasing timing offset. The receiver has to continuously types of terminals are already (or planed to be) equipped with WLAN track and compensate for these effects in order to improve the such as laptops, PDAs, and mobile phones. After the first phase efficiency of the system. Literatures describing correction algorithms where WLANs were penetrating the consumer market, now we are in use the search function argmax [2], [3], [6], [7] after complex– the next phase supporting enhanced quality of services (QoS). First conjugate–multipy–add operations. They also need to compute the systems were based on direct sequence or frequency hopper spread inverse tangent [3], [5], [8], [9], [10] to find the phase angles. spectrum technology. Currently Orthogonal Frequency Division Mul- The implementation complexity is very high for all these necessary tiplexing (OFDM) is the chosen technology for enhanced and future function blocks [3]. high data rate WLAN systems. In this paper we propose a novel algorithm for residual phase OFDM is a key technology to mitigate the multi-path effect of the tracking without using either of the afore mentioned complex (costly) wireless channel. It has been already used in DVB–T, IEEE 802.11a arithmetic functions. It computes the complex exponential of the and 16a. It is being considered for IEEE 802.20 and IEEE 802.11n as phase angle at the pilot-tone locations instead of the phase angles well. OFDM will remain as the key enabling technology for achieving to minimize the implementation complexity. The design of our higher data rates in wireless packet based communication in next few algorithm is such that it can be very easily applied to any coherent years to come [1]. Targeting the mass market of wireless modules low OFDM demodulation scheme for WLAN type of networks. cost solution has to be found, having in mind the tradeoff between The remainder of the article is organized as follows. In Section II efficiency and price. Cost of an OFDM receiver largely depends on we describe the system under investigation. Then we briefly present the implementation complexity of the synchronization and channel the well known mathematical analysis describing residual synchro- estimation algorithms. High accuracy is needed in synchronization nization errors in OFDM systems. Our proposed novel algorithm is since coherent demodulation of OFDM is extremely sensitive to such described in Section IV. A detailed discussion on the performance errors. Carrier and sampling frequency acquisition and maintenance evaluation of our scheme is given in section V. with high accuracy is vital for successful transmission of long packets. Residual Carrier Frequency Offset (CFO) and Sampling II.SYSTEM UNDER INVESTIGATION Frequency Offset (SFO) tracking (phase tracking Figure 1) are thus very critical part of OFDM receivers. But, the tracking module is A. Frame Format highly complex [2], [3], and thus has significant potential for cost OFDM systems vary greatly in their implementation. It is thus optimization of OFDM receivers. important that we describe the frame format referred in this article. The synchronization impairments that an OFDM receiver has We consider the IEEE802.11a [11] frame format as described in to mitigate are Frame Timing Offset (FTO) or Symbol Timing Figure 2. A transmitted packet has an all pilot (known data at Offset (STO), taken care of in the Time Synch, CFO compensated Transmitter (Tx) and Receiver (Rx)) training sequence known as by the Frequency Synch block of Figure 1. Such algorithms are very PREAMBLE at its beginning. It is used for packet start identification, well discussed in several literatures [4], [5] and many more. In this automatic gain control system, symbol timing synchronization, initial article, we deal with the residual CFO and SFO errors, jointly termed carrier frequency synchronization and channel estimation. A Guard as Residual Phase Errors. Residual Phase Error is the combined Interval (GI) follows the PREAMBLE which in turn is followed by error due to non-exact carrier frequency offset correction and existing the SIGNAL field. The SIGNAL field has information about the packet sampling frequency offset. In a real environment the synchronization length and modulation format used in the frame. It is a Binary Phase G G G G S I G N A L D A T A D A T A where n(t) is additive white gaussian noise. Here, Hk is the channel P R E A M B L E I I I I transfer function (CTF) for kth subcarrier. N-1 S Time U III.RESIDUAL PHASE ERRORINTHE RECEIVER B Pilots (Amplitde +/-1) Residual Phase Error has already been defined in Section I. FTO C Zero (Amplitude 0 ) A and CFO corrected signal after FFT can be expressed from [2], [5] R N −1 Data Sub-carrier j 2π (N +lN )φ j(π d φ +θ) sin(πφk) R R = H X e N CP sym k e N k I l,k k l,k “ ” πφk E sin Nd R S 0 + Nl,k (4)
Fig. 2. OFDM frame format Where φk ≈ kζ + ξTd; θ the carrier phase offset; ξ = δ(Frx − Ftx); Ftx and Frx are the local oscillator frequencies at the transmitter and the receiver respectively and δ implies residual error after initial car- rier frequency offset correction. ζ is the receiver sampling frequency Shift Keying (BKSP) modulated OFDM symbol. Then follows a 0 0 sequence of DATA fields separated by GIs, i.e the OFDM symbols offset defined through Ts = Ts(1 + ζ); where Ts is the receiver carrying information. There are 64 subcarriers used in the Fast Fourier sampling period and Ts the transmitter sampling period. Residual Transform (FFT) block of the OFDM system under consideration. carrier frequency offset in the signal even after CFO correction Not all subcarriers are used to carry information. Some subcarriers together with the sampling frequency offset cause phase rotation in such as the zero frequency component (to avoid carrier transmission) each subcarrier to increase with OFDM symbol index l as can be and the higher frequency subcarriers are made zero in order to avoid seen from Equation 4. Cumulative phase increment severely limits the filtering effect on the subcarriers due to analogue components. the number of OFDM symbols that can be transmitted in one packet. Hence only 52 subcarriers carry non-zero power, among which, pilots The receiver thus has to continuously track and compensate for the tones (values ±1) are present at four distinct locations (subcarriers effect (Phase Tracking block of Figure 1). -21,-7,7 and 21). Thus only 48 subcarriers carry data information. Since the time invariant terms are inseparable from the channel transfer function, we can write 0 B. OFDM Transmission Signal Model jlφkC Rl,k = Xl,kHke + Nl,k (5) An OFDM symbol consists of a sum of subcarriers that are 0 2π Nd−1 sin(πφ ) j N NCP φk j(π N φk+θ) k where Hk = Hke e “ πφ ” modulated by using any linear modulation method, such as Binary sin k Nd Phase Shift Keying (BPSK) or Quadrature Amplitude Modulation Nsym th and C = 2π N . (QAM). The transmitted baseband signal for l OFDM symbol, sl(t) 0 2 So, we can call Hk as the equivalent channel transfer function. σN = can be expressed as [2]: 2 E{|Nl,k| } is the additive noise power [2], where E{...} denotes N/2−1 2π expected value. There is an ICI term present in the received signal 1 X j T k[t−lTsym−TCP ] sl(t) = √ Xl,ke d (1) which can be represented as additional noise term. This leads to T d k=−N/2 degradation in the available SNR. The power of the ICI term of the th th th k sub-carrier of the l OFDM symbol is given by [6] Where Xl,k are constellation points to IDFT input at k subcarrier of th π2 l OFDM symbol; Tsym, Td, TCP and Ts are duration of complete σ2 ≈ (kζ)2 (6) OFDM symbol, data part, Cyclic Prefix (CP or GI) and sampling l,k−ICI 3 period respectively. Similarly, Nsym, Nd, NCP defines samples for In OFDM-WLAN environment of 64 sub-carriers OFDM symbol, −5 2 complete OFDM symbol, data part and CP respectively. Tsym = this effect is very small at values of ζ ∼ 10 and thus σl,k−ICI Td + TCP , Nsym = Nd + NCP and N = Nd. Further on we shall can be omitted now for the algorithm under discussion. We need to omit the scaling factor for simplification of representation. The signal mention here that if the slowly increasing sampling timing drift due in Equation 1 is transmitted over frequency–selective fading channel, to SFO reaches one sampling period, then we either miss a sample which is characterized by its low–pass–equivalent impulse response or oversample it. This leads to irreducible ISI. Rob-stuff [12] method h(τ, t) plus AWGN n(t). The channel is considered to be quasi–static is used to combat such an effect. This particular situation is not during the transmission of a complete packet, thus h(τ, t) simplifies addressed by our algorithm. to h(τ) [5]. It is further assumed that the effect of channel response IV. OFFSET CORRECTION ALGORITHM h(τ) is restricted to the interval t ∈ [0,TCP ], in another words, the length of CP is chosen to be longer than the maximum possible In this section we elaborate on the proposed residual phase delay spread, τmax. In this way the guard interval (the cyclic prefix) correction algorithm. First, after channel equalization the estimate jlφ C is able to completely absorb the tail of the pulse of the previous of the exponential part (e k ) of Equation 5 is computed at the symbol. Received baseband signal at the receiver antenna can be pilot locations, instead of estimating the phase (lφkC). Then a written from [5] running time-averaging is done to increase the SNR of the estimates. Finally using the estimates at pilot locations we piece-wise-linearly- τmax Z interpolate (Figure 3) the compensating complex exponential at all r(t) = sl(t − τ)h(τ)dτ (2) the data subcarriers. These are then multiplied with Rl,k after channel 0 equalization. N/2−1 2π The effect of noise in the channel estimate can be reduced to as X j T k[t−lTsym−TCP ] = HkXl,ke d + n(t) (3) low as 0.41dB by using a channel estimator Gain of 10dB [2]. Thus k=−N/2 for now we assume ideal channel compensation. If the estimated Piece–wise–linear–interpolation is done to find the complex multi- plication factor for each subcarrier for compensating residual phase error as states below.
yl,k = βbl,p − ml(p − k) (13) Where m indicates the nearest pilot index to kth subcarrier. The for 00 0 ∗ compensation we use Rl,k = Rl,k · yl,k. The maximum errors that may occur will be at the farthest subcarriers where the difference is largest because of larger sub-carrier index (see Equation 4 & 5).
V. PERFORMANCEOF ALGORITHM In [13] received signal after compensation by the proposed algo- rithm for small angles is given by Fig. 3. Piece wise linear interpolation 00 2 Rl,k = Xl,k + Xl,k{lB(p − k)}
≈ Xl,k (14) 0 −5 channel transfer function is Hbk we assume Hbk ≈ Hk. It is to be where B = ζC. Since ζ is in the order of 10 the degradation noted for IEEE 802.11a type wireless networks a basic assumption is due to the approximation used in the proposed algorithm described that the channel quasi–static in the whole packet duration. The Long in section IV is almost negligible. The noise power, for small angles, Training Sequence [11] present in the PREAMBLE is used to estimate has been shown in [13] to be almost equal to the noise power present the channel once every packet. This estimate is valid for equalizing in the system before applying the algorithm. If Eb is the energy per all OFDM symbols in entire packet since channel variability is much bit and No/2 the noise power density and since for BPSK, symbol slower than the packet duration. We can write the received subcarriers energy and bit energy are the same we can say that the probability of q after FTO, initial CFO correction and channel compensation as 2Eb error for BPSK modulation is given by Pe = Q{ } [14] does No 0 0 l 00 Rl,k = Rl,kHk/Hbk = Xl,k(αl,k) + Nl,k (7) not degrade significantly for low residual phase errors. This is, can
0 be understood in the light of Equation 14. where Rl,k is the received subcarrier after timing, frequency and 00 jφkC A. Simulation Parameters channel compensation and Nl,k as the new noise term; αl,k = e jlφkC and we define βl,k = e . If Pl,p (values ±1) is the pilot tone Simulations were performed with parameters from IEEE 802.11a th th at p subcarrier index of the l OFDM symbol. For all further WLAN standard: N = Nd = 64, NCP = 16 , BPSK modulation computation we have p at pilot indexes only. The algorithm is stated with 1/2 code rate convolution code, but constraint length was kept as as 3; Carrier frequency = 5.4 GHz; Sampling Frequency Offset 0 0 = 50 ppm. We have taken the largest SFO as per the standard’s α = να + (1 − ν)R · R ∗ (8) bl,p bl−1,p l,p l−1,p requirement since, very small SFO does not have significant effect αbl,p = αbl,p/|αbl,p| (9) on the performance of the receiver. Residual carrier frequency error 0 ∗ after CFO correction was varied from ∼ 30Hz to ∼ 4kHz to test the βbl,p = νβbl−1,pαbl,p + (1 − ν)Rl,p · Pl,p (10) algorithm’s performance for small ( 800bits) and larger (4000bits) β = β /|β | (11) bl,p bl,p bl,p packet lengths. with initial conditions α = β = 1 where x indicates estimate b0,p b0,p b B. Performance Comparison of x; x∗ denotes complex conjugate of x; ν is the memory factor used for averaging. Equation 8 estimates the increment from (l−1)th For comparison in correspondence to our objective, an algorithm OFDM symbol to lth OFDM symbol. It updates previous estimates. that estimates phases angles lφkC at the pilot locations using inverse Equation 10 estimates the compensating complex exponential for the tangent function is taken with reference to [3], [8]. A linear interpola- lth OFDM symbol. Equation 8 and 10 uses averaging to increase tion of the phase is done for all data sub-carriers. The compensation is −jlφ C the SNR of the estimate. A detailed derivation and analysis of the done by multiplying the received signal of Equation 5 by e \k , algorithm is given in [13]. It has to be noted that we are not computing where lφ[kC is the estimated phase angle. In Figure 4 and 5, we the phase angle, rather the complex exponential of the phase angle. denote our algorithm as alg-1 (dotted line) and the algorithm used Then we interpolate the real and imaginary parts separately. The as reference for comparison as alg-2 (solid line). A reference Bit straight solid line in Figure 3 is the actual phase that needs to be Error Rate (BER) level of 10−5 was chosen from IEEE802.11a [11] estimate, and the curve is the sinusoid of the phase. We are estimating standard for performance comparison . this sinusoid at the pilot locations. There will be a sine and a cosine 1) Low Residual Phase Error: Figure 4 shows the BER vs SNR term, only one component is shown to reduce the complexity of curve for both algorithms for low residual phase error. It can be the figure. We approximate them to be piece–wise–linear for small observed that both algorithms perform very closely for residual carrier angles. Then the mean slope of each of them is estimated as given frequency error in the range of ∼ 30Hz (0.005ppm) to ∼ 300Hz below in Equation 12. Note that we are estimating the slope as a (0.05ppm) with sampling frequency offset of 50 ppm for both small complex entity in one single equation since the real and imaginary and medium sized packet of length 800 and 4000 bits. BER of parts do not interact in the equation stated below. ∼ 10−5 is achieved by both algorithms at almost same SNR of about (7dB). This confirms our analysis for small angles as stated in 1 βbl,p21 − βbl,p−7 βbl,p7 − βbl,p−21 ml = ( + ) (12) the beginning of section V. It may be noted that the estimation of the 2 21 − (−7) 7 − (−21) phase angle by angle functions is not perfect because of corruption BER Vs SNR for low Cfo for OFDM based WLAN receivers to replace the complex (costly) 0.1 inverse tangent and argmax function implementation. It has been observed that for low residual phase errors the proposed 0.01 algorithm performs almost identical to the algorithm using phase
0.001 angle estimation by inverse tangent functions. For higher residual phase errors the proposed algorithm’s performance is still stable when 0.0001 the other scheme almost fails. By being robust to larger residual BER errors it reduces stringent performance requirement of the Freq Synch 1e-05 block. This creates provision for lower performance requirement of the initial freq synch block followed by our tracking algorithm as 1e-06 given in Figure 1. Higher accuracy performance requirements of the 1e-07 Freq Synch block increases complexity [3] and implementation cost 1 2 3 4 5 6 7 of the receiver. Further, our algorithm does not use the argmax or SNR inverse tangent functions which are very costly in terms of hardware alg 1 800bits 0.005Cfo alg 1 4000bits 0.005Cfo alg 1 800bits 0.05Cfo alg 1 4000bits 0.05Cfo implementation. It may be mentioned here that low complexity alg 2 800bits 0.005Cfo alg 2 4000bits 0.005Cfo alg 2 800bits 0.05Cfo alg 2 4000bits 0.05Cfo implementation of inverse tangent function may be done using table lookup or cordic structure. With increased required resolution, table Fig. 4. BER Vs SNR for low residual phase look up uses more space and cordic suffers from higher latency. In contrast, our algorithm does not have such limitations. We have thus BER Vs SNR for high Cfo seen for WLAN type of packet based wireless network using OFDM 1 scheme the proposed algorithm for residual phase tracking can prove 0.1 highly effective in reducing cost of receivers without compromising
0.01 on performance.
0.001 REFERENCES
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