Low Cost Residual Phase Tracking Algorithm for OFDM-based WLAN Systems Suvra Sekhar Das† ,R.V. Rajakumar§ , Muhammad Imadur 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 orthogonal frequency division multiplexing receiver targeting wireless local area networks. The focus is on mitigation of the residual frequency synchronization error and sampling frequency offset. We propose a novel algorithm that uses an approximation instead of implementing the complex and vulnerable angle functions. By means of analytical description and simulation results we show that our design combines both low complexity and sufficiently high and robust performance.
LNA
X
Low Pass Filter
Local Oscillator
ADC converter
Phase Tracking
Time Synch
Channel Equalization
Frequency Synch
FFT
Front End(Inner Receiver) Back End (OuterReceiver) De-Mapper
Fig. 1.
Channel Decoder
Bit(s) output
OFDM receiver front end architecture
I. I NTRODUCTION Wireless Local Area Networks (WLAN) are becoming part of omnipresent communication infrastructures. WLANs are being applied in hotels, airports, cafes. Different types of terminals are already or planed to be equipped with WLAN such as laptops, PDAs, and mobile phones. Currently Orthogonal Frequency Division Multiplexing (OFDM) is the chosen technology for enhanced and future high data rate WLAN systems. OFDM is a key technology to mitigate the multi-path effect of the wireless channel. It has been already used in DVB–T, IEEE 802.11a [15] and 11g. Furthermore it is being considered for IEEE 802.20 and IEEE 802.11n. It will also remain the key enabling technology for achieving higher data rates in wireless packet based communication in next few years to come [1]. Targeting the mass market of wireless modules low cost solution has to be found, having in mind the tradeoff between efficiency and price. Cost of an OFDM receiver largely depends on the implementation complexity of the synchronization and channel estimation algorithms. Coherent demodulation of OFDM is extremely sensitive to synchronization errors. Carrier and sampling frequency acquisition and maintenance with high accuracy is vital for successful transmission of long packets. Residual carrier frequency offset and sampling frequency offset tracking(phase tracking Figure 1) is thus very critical part of OFDM receivers. But, the tracking module is highly complex [2][8], and thus has significant potential for cost optimization of OFDM receivers. The synchronization impairments that an OFDM receiver has to mitigate are Frame Timing Offset (FTO) or Symbol Timing Offset (STO), taken care of in the Time Synch (Figure 1), Carrier Frequency Offset (CFO) is compensated by the frequency synch of Figure 1. In this article, we deal with the residual CFO and SFO errors, jointly termed as Residual Phase Errors (phase tracking
block in Figure 1. Residual Phase Error is the combined error due to non-exact carrier frequency offset correction and existing sampling frequency offset. In a real environment the synchronization blocks placed at the receiver front end are not able to estimate the exact carrier frequency offset due to circuitry noise fixed word length effects. More over due to sampling frequency offset there is a slowly increasing timing offset. The receiver thus has to continuously track and compensate for these effects. Literatures describing correction algorithms use the search function argmax [2], [8], [10], [13], [14] after complex–conjugate–multipy–add operations. They also need to compute the inverse tangent [3], [5], [8], [16], [17] to find the phase angles. The implementation complexity is very high for all these necessary function blocks [8]. In this paper we propose a novel algorithm for residual phase tracking without using either of the costly arithmetic functions mentioned above. It computes the complex exponential (Section III) itself at the pilot-tone locations (Section II) instead of its phase angles to minimize the implementation cost. The design of our algorithm is such that it can be very easily applied to any coherent OFDM scheme. The remainder of the article is organized as follows. In Section II we describe the system under investigation. Then we briefly present the well known mathematical analysis describing residual synchronization errors in OFDM systems. Our proposed novel algorithm is described in Section IV. A detailed discussion on the performance evaluation of our scheme through simulation in section. II. S YSTEM U NDER I NVESTIGATION In this section we provide a brief overview of the system under investigation. OFDM systems vary greatly in their implementation. It is thus important that we describe the frame format referred in this article. We consider the
G I
PREAMBLE
S U B C A R R I E R S
S I G N A L
G I
D A T A
N-1
G I
G I
DATA
Time Pilots (Amplitde +/-1) Zero (Amplitude 0 ) Data Sub-carrier
B. Frequency Selective Channel
0
Fig. 2.
OFDM receiver front end architecture
IEEE802.11a [15] frame format as described in Figure 2. A packet transmitted has an all pilot {known data at Transmitter (Tx) and Receiver (Rx)} training sequence often known as the PREAMBLE at its beginning. The PREAMBLE is used for packet start identification, automatic gain control system, symbol timing synchronization, initial carrier frequency synchronization and channel estimation. A Guard Interval (GI) follows the PREAMBLE which in turn is followed by the SIGNAL field. The SIGNAL field has information about the packet length and modulation format used in the frame. It is a Binary Phase Shift Keying (BKSP) modulated OFDM symbol. After this follows the sequence of DATA fields separated by GIs, i.e the OFDM symbols carrying information. There are 64 subcarriers used in the Fast Fourier Transform (FFT) block of the OFDM system under consideration. Not all subcarriers are used to carry information. Some subcarriers such as the zero frequency component (to avoid carrier transmission) and the higher frequency subcarriers are made zero in order to avoid the filtering effect on the subcarriers due to analogue components. Hence only 52 subcarriers carry non-zero power. Among these, pilots tones (values ±1) at four distinct locations (subcarriers -21,-7,7 & 21) are used to enable the receiver track cumulative residual phase errors. Thus only 48 subcarriers carry information. In the next subsection we describe the analytical model of the OFDM system to provide the reader with the fundamentals related to our proposed algorithm. A. OFDM Transmission Signal Model An OFDM symbol consists of a sum of subcarriers that are modulated by using any linear modulation method, such as Binary Phase Shift Keying (BPSK) or Quadrature Amplitude Modulation (QAM) [5]. The transmitted baseband signal for lth OFDM symbol, sl (t) can be expressed as [2]: 1 sl (t) = √ Nd
2π k[t−lTsym −TCP ] jT
Xl,k e
d
The signal in Equation 1 is transmitted over frequency– selective fading channel, which is characterized by its low– pass–equivalent impulse response h(τ, t) plus AWGN n(t). The channel is considered to be quasi–static during the transmission of a complete packet, thus h(τ, t) simplifies to h(τ ) [3, Section 4.2.3.3]. It is further assumed that the effect of channel response 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 delay spread, τmax . In this way the guard interval (the cyclic prefix) is able to completely absorb the tail of the pulse of the previous symbol. This way the GI preserves the orthogonality of the OFDM symbols in a packet, hence reducing ISI. Here the GI is the last 16 of the 64 samples, generated by the IFFT, placed at the beginning of the OFDM symbol. C. OFDM Receiver Model Baseband signal received at the receiver antenna can be written from [3] τZ max r(t) = sl (t − τ )h(τ )dτ (2) 0 N/2−1
=
X
(1)
k=−N/2
The symbols used in Equation 1 and all the subsequent equations are defined as: Xl,k as constellation points to IDFT input at k th subcarrier of lth OFDM symbol; Tsym , Td , TCP and Ts are duration of complete OFDM symbol, data part, Cyclic Prefix (CP) and sampling period respectively. Similarly, Nsym , Nd , NCP defines samples for complete OFDM symbol, data part and CP restively. Hence, Tsym = Td + TCP and Nsym = Nd + NCP . N denotes total number of subcarriers.
2π k[t−lTsym −TCP ] jT
Hk Xl,k e
d
+ n(t)
(3)
k=−N/2
Where n(t) is additive white gaussian noise added by device circuitry. Here, Hk is the channel transfer function (CTF) for k th subcarrier and τmax is the maximum delay spread of the channel. Next we look at the well known synchronization errors present in the OFDM symbol. III. R ESIDUAL P HASE E RROR IN THE R ECEIVER Residual Phase Error has already been defined in Section I. FTO and CFO corrected signal after FFT can be expressed from [2], [3], [7] 2π
Rl,k = Hk Xl,k ej N (NCP +lNsym )φk ej(π + Nl,k
N/2−1
X
k and l are used for subcarriers index and OFDM symbol index respectively. Also note that N and Nd can be used interchangeably. Further on we shall omit the scaling factor for simplification of representation.
Nd −1 φk +θ) N
sin(πφk ) k sin πφ Nd (4)
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 carrier frequency offset correction. ζ is the receiver sampling frequency offset defined through 0 0 Ts = Ts (1 + ζ); where Ts is the receiver sampling period and Ts the transmitter sampling period. Residual carrier frequency offset in the signal even after CFO correction together with the sampling frequency offset causes phase rotation in each subcarrier to increase with OFDM symbol index as can be
seen from Equation 4. Cumulative phase increment severely limits the number of OFDM symbols that can be transmitted in one packet. The receiver thus has to continuously track and compensate for the effect (phase tracking block in Figure 1). Since the time invariant terms are inseparable from the channel transfer function, we can write 0
Rl,k = Xl,k Hk ejlφk C + Nl,k 0
2π
where Hk = Hk ej N NCP φk ej(π
Nd −1 φk +θ) N
(5)
sin(πφ k ) πφ sin N k d
N
and C = 2π sym N .0 So, we can call Hk as the equivalent channel transfer function as seen by the receiver channel Equalization block. There is an ICI term present in the received signal which can be represented as additional noise term. This leads to degradation in the available SNR. The power of the ICI term of the k th sub-carrier of the lth OFDM symbol is given by [10]
pilot indexes only. The algorithm is stated as 0
π (kζ)2 3
(8)
αl,m = αl /|αl,m |
(9) 0
∗ βl,m = νβl−1 αl,m + (1 − ν)Rl,m · Pl,m
(10)
βl,m = βl,m /|βl,m |
(11)
where ν is the memory factor used for averaging. Equation 8 estimates the increment from l − 1th ofdm symbol to lth OFDM symbol. It updates previous estimates. The normalization is done to reduce the effect of the noise term which would otherwise go on adding to the power of α and β. Equation 10 estimate the compensating complex exponential for the lth OFDM symbol. It uses averaging to increase the SNR of the estimate. It is to be noted that we are not computing the phase angle, rather the complex exponential itself. Then we interpolate the real and imaginary parts separately. The straight Piece wise linear interpolation Pilot
2
2 σl,k−ICI ≈
(6)
KzlC
-21
In OFDM-WLAN environment of 64 sub-carriers OFDM symbol, this effect is very small at values of ζ ∼ 10−5 2 and thus σl,k−ICI can be omitted now for the algorithm under discussion. We need to mention here that if the slowly increasing sampling timing drift due to SFO reaches one sampling period, then we either miss a sample or oversample it. This leads to irreducible ISI. IV. O FFSET C ORRECTION A LGORITHM Here we elaborate on the proposed residual phase correction algorithm. First the estimate of the exponential part (ejlφk C ) of Equation (5) is computed at the pilot locations, instead of estimating the phase (lφk C). Then a running time-averaging is done to increase the SNR of the estimates. Finally using the estimates at pilot locations we piece-wise-linearly-interpolate the compensating complex exponential at all the data subcarriers. These are then multiplied with Rl,k after channel equalization as explained below. The effect of noise in the channel estimate can be reduced to as low as 0.41 dB by using a channel estimator Gain of 10dB [2]. Thus for now we assume ideal channel compenb k we sation. If the estimated channel transfer function is H b k ≈ H 0 . We can write the received subcarriers after assume H k FTO, initial CFO correction and channel compensation as 0
0
b k = Xl,k (αl,k )l + N ” Rl,k = Rl,k Hk /H l,k 0
0
∗ αl,m = ναl−1 + (1 − ν)Rl,m · Rl−1,m
(7)
where Rl,k is the received subcarrier after timing, frequency 00 and channel compensation and Nl,k as the new noise term; αl,k = ejφk C and we define βl,k = ejlφk C . If Pl,m (values ±1) is the pilot tone at mth subcarrier index of the lth OFDM symbol. For all the computation now onwards we have m at
-7 7
KzlC
K
Actual Phase
21 (KzlC+lCT x) d
Sine(KzlC+lCT x) d
LCTdx
Mean slope
Fig. 3.
Piece wise linear interpolation
line in Figure 3 is the phase that needs to be estimate, and the curve is the sinusoid of the phase. We are estimating this sinusoid at the pilot locations. We approximate it to be piecewise-linear. Then the mean slope is estimated as γl =
βl,m7 − βl,m−21 1 βl,m21 − βl,m−7 ( + ) 2 21 − (−7) 7 − (−21)
(12)
Where the suffix of m denote the pilot location. Piece wise linear interpolation is done to find the complex multiplication factor for each subcarrier for compensating residual phase error as states below. yl,k = βl,m − γl (m − k)
(13)
Where m indicates the nearest pilot index to k th subcarrier. The for compensation 0
” Rl,k = Rl,k · yk∗
(14)
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. S IMULATION R ESULTS AND D ISCUSSION A. Simulation Parameters Simulations were performed with parameters from IEEE 802.11a WLAN standard: number of OFDM subcarriers, N = 64, length of cyclic prefix (CP), NCP = 16 samples, BPSK symbol mapping with half-rate convolutional coding(for
It is found from the simulations that an error 10 ppm is not correctable for any of the schemes, as the noise floor is reached at around 10−2 . It should be noted that 10 ppm residual CFO means an error of 54kHz, which is very high and implies failure of the freq synch block.
B. Performance Comparison
BER Vs SNR for High CFO
0
10
−1
10
−2
10
BER
same packet length in bits, OFDM packet duration gets larger for BPSK as compared to higer order QAM modulation. Thus, the cumulative residual phase rotation will be the larger for BPSK case and hence provides a worse case analysis situation); 5.4 GHz carrier frequency; maximum SFO = 50 ppm was taken for simulations. The standard specifies a tolerance range for carrier frequency error as 50 ppm (±25 ppm on each side), that equals to 5.4GHz±135kHz. residual carrier frequency error after CFO was varied from 0.1 to 10 ppm to test the algorithm’s performance.
−3
10
−4
For comparison in correspondence to our objective, an algorithm that estimates phases angles at the pilot locations using inverse tangent function and then linearly inter/extrapolates the phases to all data sub-carriers was taken as reference. The compensation is done by multiplying the received signal of b Equation 5 by e−j φk , where φbk is the estimated phase angle. In Figure 4 and 5, we denote our algorithm as alg-1 (solid line) and the algorithm used as reference for comparison as alg-2( dotted line). We have compared the algorithms for different ranges of residual CFO. The values for low CFO is taken as 0.1-0.5 ppm (540Hz to 2700Hz), and for high CFO as 2-10 ppm (10.8kHz54kHz). 1) Low residual CFO Conditions: Figure 4 shows us the BER Vs SNR curve for both algorithms when very low amount of CFO is left for correction in phase tracking block. When residual CFO = 0.1 or 0.3 ppm, then Algo-2 clearly outperforms our algorithm for packet length of 1000 bits. When the error is increased to 0.5 ppm, the performance gap between the two algorithms is ∼ 2 dB at BER rates ∼ 10−5 ; However at relatively larger residual error of 0.7 ppm, Algo-2 almost fails. Compared to this, we can see that our algorithm still performs in a steady manner. Figure 4 also shows that Algo-2 performs very badly when packet length is increased to 10kbits. It is shown in (4) that the residual phase increases along with the packet length. BER Vs SNR for Low CFO
0
10
−1
10
−2
BER
10
−3
10
−4
10
cfo0.1 cfo0.5 cfo0.1 cfo0.5 cfo0.3 cfo0.5 cfo0.3 cfo0.5
−5
10
−6
10
0
5
Fig. 4.
10
15 SNR
20
1kb pkt alg1 1kb pkt alg1 1kb pkt alg2 1kb pkt alg2 10kb pkt alg1 10kb pkt alg1 10kb pkt alg2 10kb pkt alg2
25
30
BER Vs SNR for low residual phase
2) High CFO Conditions: Figure 5 shows the simulation results when comparatively higher CFO is left after Frequency Synch is performed. Algo-2 performs very unsatisfactorily when CFO is 0.7 ppm up to 5 ppm. At this level, our algorithm still reaches a BER 10−5 at an SNR of 23dB for 5 ppm error.
10
cfo0.7 cfo2.0 cfo5.0 cfo10 cfo0.7 cfo2.0 cfo5.0 cfo10
−5
10
−6
10
0
alg1 alg1 alg1 alg1 alg2 alg2 alg2 alg2 5
Fig. 5.
10
15 SNR
20
25
30
BER Vs SNR for high residual phase
VI. D ISCUSSION AND C ONCLUSION It has been observed that for low residual phase errors the proposed algorithm using very low complexity performs very close to algorithms using accurate phase estimation by inverse tangent functions. For higher residual phase errors (larger packet lengths and higher residual CFO) the proposed algorithm’s performance is still stable and performs satisfactorily when the other scheme almost fails. Thus it is seen that this algorithm is very robust and can tolerate larger residual CFO errors. Thus the proposed algorithm gives double fold benefit. One it relaxes otherwise stringent performance requirement of the freq synch block of Figure 1. For higher accuracy of freq synch block (as required by the other scheme) the complexity and the cost of implementation of the receiver increases [8]. Thus low complexity freq synch block can be used with the proposed algorithm. Secondly, as described our algorithm does not use the arg max or inverse tangent functions which are very costly in terms of hardware implementation. For inverse tangent using a table lookup requires more space, while using cordic algorithm increases the latency. Both the parameters, space and latency requirement increases with increased required resolution. Fixed word length plays a significant role in the performance of the nonlinear inverse tangent function. Phase angle estimates can be very noisy at angles close to ± pi 2 . Thus inverse tangent function is vulnerable at large angle estimates. Larger angles occur for larger packet duration and higher residual phase errors. It is seen in the simulation as well how the alg-2 suffers at higher residual phase errors and larger packet lengths, where as alg-1 does not. Thus we have seen for WLAN type of packet based wireless using OFDM scheme the proposed algorithm for residual phase compensation can prove highly effective in reducing cost of receivers. R EFERENCES [1] [2] [3]
Richard Van Nee & Ramjee Prasad, OFDM for Wireless Multimedia Communications, Artech House Publishers, 2000. M. Speth, S.A. Fechtel, G. Fock & H. Meyr, Optimum Receiver Design for Wireless Broad-Band Systems Using OFDM - Part I, IEEE Transactions on Communications, vol. 47, no. 11, November 1999. Klaus Witrisal, OFDM Air Interface Design for Multimedia Communications, PhD thesis, Delft University of Technology, The Netherlands, April 2002.
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