Time Domain Synchronization using Newman Chirp Training Sequences in AWGN Channels Sandrine Boumard and Aarne M¨ammel¨a VTT Electronics Kaitov¨ayl¨a 1, P.O. Box 1100, FIN-90571 Oulu, Finland Tel: +358 8 551 2111, Fax: +358 8 551 2320 Email:
[email protected],
[email protected] Abstract— We propose a new time-domain simultaneous fractional frequency and sampling offsets estimation algorithm using the characteristics of down-sampled chirp training signals. The training sequence is composed of one up and one down chirp symbols. We also propose a new estimation algorithm to estimate the integer timing and frequency offsets in the time domain. The algorithms use the outputs of matched filters and autocorrelators. We present Monte Carlo simulations results in AWGN channels and show the good performance of our algorithm, especially for the integer timing and frequency estimates compared to known methods. Indeed, we obtain perfect results for SNR of 5 dB and higher and normalized frequency offset smaller than 10, when no fractional offsets are present. The fractional timing and frequency estimator does not perform as well as known methods, but is much simpler to implement. Keywords— Synchronization, Chirp modulation, time domain measurements, frequency estimation, timing measurements, matched filters, correlation.
I. I NTRODUCTION In this paper, we focus on synchronization in AWGN channels. We present new algorithms for timing and frequency estimation using training signals having two or more symbols, each based on a chirp signal. We now review the characteristics of the chirp signals and thus give the reasons for using them as training signals. Chirp signals have been used in radar [1], data transmission [2] and channel sounding [3] due to their good autocorrelation and frequency domain characteristics. Chirp, or linear frequency modulation (FM), signals have a knife-edge or ridge ambiguity function [1]. The chirp waveform is frequency invariant, thus a frequency-shifted signal will still be compressed properly by a filter matched to the original chirp, although the peak will be diminished in amplitude and will appear at the output shifted in time. The timing-frequency coupling can be resolved by sweeping the linear FM up and down on alternate pulses [1]. Thus sending up and down training signals allows the decoupling of the timing and frequency synchronization problems. Chirp signals also have good peak-to-average power ratio (PAPR) characteristics. We first notice that the power of the chirp signal is a constant, which means that it eases the stress on the power amplifier at the transmitter and the automatic gain controller at the receiver. It can be shown, using the results from [1], [3], [4], that the fast Fourier transform (FFT) of an up chirp signal is approximately equal to its
down chirp fixed phase shifted equivalent, and vice-versa. This means that the spectrum of the chirp signal is flat. It also means that the signal can be transmitted either in the time or frequency domain. In the discrete frequency domain and the orthogonal frequency division multiplexing (OFDM) literature, this modulation is referred to as the Newman phases [4] - [6] and has been widely studied for PAPR reduction. In those studies, the Newman phases were proven suboptimal for PAPR reduction purposes, but the number N of subcarriers was small. In our case, we focus on larger N. Chirp signals have been studied and used for synchronization [7]. Their extension to multiple-input multiple-output (MIMO) systems via the Frank codes and other similar sequences [1], [8] is an interesting topic for synchronization of OFDM systems in MIMO environments [9]. Finally, the chirp signals are optimal for channel estimation [9]. Channel estimation is not covered herein but it proves that the choice of chirp-like training signals will also provide good results for channel estimation acquisition. In this paper, we propose a new simultaneous fractional frequency offset and sampling offsets estimation algorithm using the characteristics of the chirp signals. It utilizes the great potentials of chirp signals to accurately estimate the timing and frequency error at the receiver side in a AWGN channel. We also propose a new estimation algorithm to estimate the integer frequency and timing offsets. In Section II, we present the system model. In Section III, we express the outputs of the matched filters and autocorrelators, as well as the estimation algorithms. We present Monte Carlo simulations results and compare those performance with those of known methods in Section IV. We finally draw some conclusions and discuss the extension of the results in Section V. II. S YSTEM M ODEL The block diagram of the system and the synchronization blocks is shown in Figure 1. We focus here on a single-input single-output (SISO) system. A. Transmitter The transmitted training sequence is composed of two chirp signals conjugate of each others. The multiplexer takes N samples from each input. The discrete time signal at the output
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d1(m) r1(m)
Newman Vector (lenght N)
Newman Vector (lenght N)
Delay N/4
channel s(m)
N (.)*
e
−
τ0
moving average length N/4 (.)*
1
j 2πf c m N
Process phases
δˆτ
c
c
moving average length N/4
r2(m)
c0
δˆ f
Delay N/4
d2(m)
(.)*
di(m)
n(m) MF X*(k)
r(m) (.)*
Fig. 1.
MF X(k)
jπm2 N
s(m) = e−
jπ(m−N )2 N
for m = 0, . . . N − 1,
(1)
for m = N, . . . 2N − 1,
where N = 4p with p an integer. N is called the dispersion factor, or time-bandwidth product, of the chirp signal. The transmitter’s low-pass filter, not shown in Figure 1, is supposed ideal, thus the continuous time signal is given by jπt2
sc (t) = e NTs2 2
− jπt 2
sc (t) = e
NTs
for t ∈ [0, N Ts [,
(2)
for t ∈ [N Ts , 2N Ts [,
where Ts is the sampling period. It comes from the fact that the frequency domain of a chirp signal is a frequency domain chirp signal [4] and the output of the ideal low-pass filter of a discrete exponential is a continuous time exponential. B. Channel Model We consider an AWGN channel model as shown in Figure 1. The output of the channel r(m) is expressed as: r(m) = e−
j2πfc m N
· c0 sc ((m − τ0 )Ts ) + n(m),
(3)
where c0 is the complex gain of the channel and τ0 is the delay and models the timing offset, fc is the frequency offset and n(m) is the complex AWGN of variance 2σ2 . fc is normalized by the inverse of the sampling rate 1/Ts and τ0 is normalized by the sampling rate Ts . The timing and frequency offsets are decomposed into their integer and fractional parts as follows fc = mfc + δfc τ0 = mτ0 + δτ0
with −0.5 < δfc ≤ 0.5, with −0.5 < δτ0 ≤ 0.5.
u2(m)
Process peaks
mˆ f c
m ui(m)
mˆ τ c
m
Block diagram of the system.
of the transmitter s(m) has a length of 2N and is expressed as: s(m) = e
u1(m)
(4)
The only assumption is a negligible sampling rate error. For HIPERLAN/2 system parameters, the carrier frequency is 5.8 GHz and the sampling frequency is 20 MHz. When the same oscillator provides both the frequencies, even with a 5 MHz carrier frequency offset, the duration of one training signal of 64 samples is 3.203 ms instead of 3.2 ms. This error equals 1/20th of a sample duration, which is negligible.
C. Receiver At the receiver, we focus on the synchronization blocks, see Figure 1. We suppose a time of arrival (TOA) block gives us a rough estimate of the start of the frame, using some power estimation algorithm. The de-multiplexer outputs alternately one sample to each output. Those down-sampled versions of r(m), r1 (m) and r2 (m), are fed to autocorrelators with delay of N/4 and moving averaging over N/4 samples. The phases of the outputs of the autocorrelators at their peak is used to estimate the fractional timing and frequency offsets. The received signal is also fed to matched filters (MF). The peak positions at the outputs of the MFs give an estimate of the integer frequency and timing offsets. The processing is done in the time domain, allowing low complexity and low delays for the synchronization. Details follow in the next section. III. S YNCHRONIZATION By synchronization, we mean timing and frequency offset estimation and correction for the acquisition phase at the receiver side. We focus on synchronization based on autocorrelators and matched filters. A. Fractional Frequency and Timing Offset Estimation As shown in Figure 1, we first down-sample the received signal by 2 and feed those samples to 2 autocorrelators with delay N/4 and moving averaging length of N/4. Indeed, thanks to the properties of the chirp signal, when downsampling by 2 the symbol of length N we obtain two consecutive chirp signals with a dispersion factor of N/4. To simplify the equations, we do not consider the noise. We obtain, at the output of the autocorrelators, at each peak named here mij , with i = 1, 2 the branch at the output of the down-sampler and j = 1, 2 the peak position, d1 (m11 ) d1 (m12 ) d2 (m21 ) d2 (m21 )
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= = = =
N
(−1) 4 |c0 |2 ejπ(fc +τ0 ) , N (−1) 4 |c0 |2 ejπ(fc −τ0 ) , N −(−1) 4 |c0 |2 ejπ(fc +τ0 ) , N −(−1) 4 |c0 |2 ejπ(fc −τ0 ) .
(5)
No
u1 (m )
2
Legends, with i=1,2: - N: sequence length. - OW: observation window for the detection of the highest peak. - PeakOW: observation window around the highest peak. - PeakT: threshold to detect additional peaks in PeakOW. - PeakMinT: minimum threshold to detect additional peaks in PeakOW when the number of additonal peaks for u1 and u2 differ. - posi: highest peak position for ui (updated at the end of the algorithm). - HPi: value of the power at the highest peak for ui. - AddPeaksi: number of additional peaks detected for ui. - posPeaksi: table of the positions of the peaks detected for ui (the highest peak is in the middle and the table is of size PeaksOW).
No
Highest Peak during OW?
u2 (m )
2
Highest Peak during OW?
Yes
Yes
highest peak position pos1 (value HP1)
highest peak position pos2 (value HP2)
look for peaks with |u1(m)|2 > PeakT*HP1 in window of PeakOW centered on pos1 -> AddPeaks1 -> posPeaks1[PeakOW]
look for peaks with |u2(m)|2 > PeakT*HP2 in window of PeakOW centered on pos2 -> AddPeaks2 -> posPeaks2[PeakOW]
filling up gaps in peak position vector update: AddPeaks1 posPeaks1[PeakOW]
filling up gaps in peak position vector update: AddPeaks2 posPeaks2[PeakOW]
running count
PeakOW HPi starting value for posi
add highest non selected peak in posPeaks2 if |u2(m)|2 > PeakMinT*HP2 update AddPeaks2 anyway
PeakT*HPi Yes
AddPeaks1 > AddPeaks2?
PeakMinT*HPi m
No OW add highest non selected peak in posPeaks1 if |u1(m)|2 > PeakMinT*HP1 update AddPeaks1 anyway
Yes
AddPeaks1 < AddPeaks2?
Fig. 2.
No
update pos1 and pos2 according to the middle of the detected peaks
N − pos 2 + pos1 mˆ fc = floor + 0.5 2
The matched filters peak estimation algorithm.
We can easily see that we can extract an estimate of the fractional timing and frequency offset 2 1 X (arg(di (mi1 )) + arg(di (mi2 ))) , δˆfc = 4π i=1 2 1 X δˆτ0 = (arg(di (mi2 )) − arg(di (mi1 ))) . 4π i=1
(6)
B. Integer Frequency and Timing Synchronization For the integer frequency and timing synchronization we use matched filters. We here use a matched filter matched to the first chirp symbol, but we can equivalently use a MF matched to the second one. To simplify the equations, we do not consider the noise. The squared amplitude of the outputs has roughly the shape of a sinc2 function with sharp peaks with the following positions m1 = N + round (τ0 + fc ) for |u1 (m)|2 m2 = 2N + round (τ0 − fc ) for |u2 (m)|2 ,
(7)
where the function round(x) gives the integer closest to x. When correcting the received signal r(m) using the estimates for the fractional timing and frequency offsets, we can extract the integer frequency offset by measuring the number of sampling periods m1 − m2 + N (8) m ˆ fc = 2 between the two peaks at the output of the matched filters. The timing synchronization signal is created by waiting for ˆ fc is m ˆ fc samples period after the peak from u1 (m) if m ˆ fc is negative. positive, or u2 (m) if m
To avoid delays, we do not correct the received signal, thus τ0 and fc are not integer numbers. We use an estimation algorithm that detects a group of peaks and extract their middle position. We then can use (8) where m1 and m2 , the highest peaks, are replaced by the middle positions defined by the estimation algorithm. This algorithm is depicted in Figure 2. Having detected the highest peak in OW using the algorithm presented in [10], we then search for secondary peaks in a window of PeakOW inputs around the highest peak. The selected secondary peaks are those higher than a threshold, equal to PeakT times the power at the highest peak. We keep track of the positions of those secondary peaks and we compare the number of detected peaks for each MF. If not equal, we select additional peaks from the outputs of the MF that has less peaks. But we do not select peaks that are too low, meaning that their power is smaller than a threshold, equal to PeakMinT times the power at the highest peak. This reduces the probability to choose peaks mainly due to noise. We can finally correct the position of the highest peaks by calculating the middle position from the positions of all the selected peaks. C. Alternative Fractional Frequency and Timing Offset Estimation When the fractional frequency offset estimate has to be very accurate, we can add a third training symbols. In this case the first two training symbols are identical and we use an autocorrelator to estimate the fractional frequency offset. This autocorrelator makes full use of the symbols sent, with a delay of N and a moving averaging window of length N . This algorithm has been widely used for e.g. in OFDM
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0.5 0.12
0.45 0.4
0.2 0.15
0.06
20
0.1 Standard deviation
0.25
SNR = 0 dB SNR = 5 dB SNR = 10 dB SNR = 15 dB SNR = 30 dB
0.08
Mean
Standard deviation
0.3 Mean
25
0.1
0.35
0.12
SNR = 0 dB SNR = 5 dB SNR = 10 dB SNR = 15 dB SNR = 30 dB
15
10
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0.1
5
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0.04
0.02
0.05 0
0
0
0.1 0.2 0.3 0.4 Fractional frequency offset
0
0.1 0.2 0.3 0.4 Fractional frequency offset
Fig. 3. Mean and standard deviation of the fractional frequency offset estimate in AWGN channel.
SNR = 0 dB SNR = 5 dB SNR = 10 dB SNR = 15 dB SNR = 30 dB
25
15
10
x 10
0.9
0.6 0.5 0.4
0
10 20 Integer frequency offset
0
10 20 Integer frequency offset
Values for integer offsets 66 3 0.4 0.1 -
Values for fractional offsets [10] 7 1
0.3
A. AWGN Channel
0.1 0
0
Fig. 5. Mean and standard deviation of the integer frequency offset estimate in AWGN channel with a fractional timing offset of 0.5.
OW PeakOW PeakT PeakMinT backward shift time
0.7
0.2
5
10 20 Integer frequency offset
Parameters
0.8 Standard deviation
Mean
20
0
TABLE I VALUES OF THE PARAMETERS OF THE ESTIMATION ALGORITHMS .
−6
1
0
0
0
10 20 Integer frequency offset
Fig. 4. Mean and standard deviation of the integer frequency offset estimate in AWGN channel when the fractional frequency and timing offset are null.
systems [11]. We correct the last training symbol received knowing the frequency offset estimate and use the small autocorrelators, as presented in Section III-A, to estimate the fractional timing offset. IV. S IMULATIONS R ESULTS We now present the Monte Carlo simulation results in an AWGN channel. The results are given in terms of mean and standard deviation, which is the square root of the variance of the estimated value. We compare the results with those for algorithms designed for OFDM systems. We use N = 64 and a sampling rate of 20 MHz. Thus the fractional frequency offset is in the range ] − 156.25 kHz, 156.25 kHz] and the fractional timing offset in the range ] − 25 ns, 25 ns]. The results are averaged over 1000 estimates. The values for the algorithms’ parameters are shown in Table I. The backward shift time determines the position, relative to the detected peak, of the output we use to measure the fractional offsets [10].
The results for the fractional timing offset are the same as those for the fractional frequency offset, see Figure 3, due to the time-frequency duality in the estimation algorithm. For an SNR of 5 dB, the maximum standard deviation of the fractional frequency offset estimate is about 0.04. Although not shown in Figure 3, with the alternative estimator, the standard deviation is 0.013 for the same SNR. The alternative solution is more accurate but it needs more training symbols. For the integer frequency offset, when both fractional frequency and timing offsets are null, see Figure 4, the standard deviation is 0 for SNRs of 0 dB and higher and frequency offset smaller than 16, which gives a very good acquisition range. In the presence of a timing offset, see Figure 5, we obtain the same results but for SNRs of 5 dB and higher. The integer timing offset estimate algorithm shows good accuracy, with a standard deviation below 1 sample, see Figures 6 and 7. The true integer timing offset is 0 and the x axis shows the dependence towards the integer frequency offset. B. Comparison with Existing Algorithms Due to the attractive low PAPR characteristics of the chirp signal, we focus on algorithms for OFDM systems. The timing offset estimation algorithm presented in [12] shows a variance of 20 sample2 in an AWGN channel with a SNR of 5 dB. This is clearly much worse than our results. In [13], the integer frequency offset estimate rms in a AWGN channel for an SNR
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2
−0.05
0.035
1.8
−0.1
0.03
−0.2 −0.25 SNR = 0 dB SNR = 5 dB SNR = 10 dB SNR = 15 dB SNR = 30 dB
−0.3 −0.35 −0.4
0
10 20 Integer frequency offset
1.6 1.4
0.025
1.2 Mean
Mean
−0.15
SNR = 0 dB SNR = 5 dB SNR = 10 dB SNR = 15 dB SNR = 30 dB
0.02
1 0.8
0.015
0.6
0.01
0.4
1 Standard deviation
0.04
Standard deviation
0
0.8
0.6
0.4
0.2
0.005
0.2 0
0
10 20 Integer frequency offset
Fig. 6. Mean and standard deviation of the error for the integer timing offset estimate in AWGN channel when the fractional frequency and timing offset are null.
of 5 dB is higher than 0.01. This shows that our algorithms is again performing very well. The fractional timing offset estimate is usually extracted from the phase difference between equalized outputs of the FFT [14]. This algorithm performs much better than our algorithm, since we average over a smaller number of values. But our algorithm is much simpler and does not require the complex signal processing blocks that are FFT and equalization, which also delay the synchronization process. Finally, we can clearly see that our fractional frequency offset estimate performs roughly four-fold worse than the alternative estimator. This is due to the ratio between the peak value and the noise variance, our averaging length being four-fold smaller than the one of the alternative estimator. V. C ONCLUSIONS In this paper, we have presented a new simultaneous time domain fractional frequency offset and sampling offset estimation algorithm using the characteristics of down-sampled chirp training signals. The training sequence is composed of one up and one down chirp symbols. We also propose a new estimation algorithm to estimate the integer timing and frequency offset in the time domain. The algorithms use the outputs of matched filters and autocorrelators. The integer timing and frequency offset estimator shows good performance in the AWGN channel, providing perfect results for SNR of 5 dB and higher and frequency offset smaller than 16 subcarrier spacings, when fractional offsets are null. The algorithm still shows a good performance in the presence of fractional offsets, especially for the integer frequency offset estimate. The fractional estimates do not show the same range of performance, due to the smaller averaging length compared to known methods. However, the algorithms are less complex and work in the time domain, which reduce the processing time and delay. The next step is a performance study in a multipath fading channel. In that case, the fractional timing offset estimate
0
0
10 20 Integer frequency offset
0
0
10 20 Integer frequency offset
Fig. 7. Mean and standard deviation of the error for the integer timing offset estimate in AWGN channel with a fractional timing offset of 0.5 and a fractional frequency offset of 0.2.
might need some major changes since we receive a sum of delayed and phase shifted versions of the transmitted symbols. ACKNOWLEDGMENT The authors would like to thank their colleague Pertti J¨arvensivu for his help. R EFERENCES [1] G. W. Deley, “Waveform Design,” in Radar Handbook, ed. M. I. Skolnik, 1st ed., New-York: McGraw-Hill, 1970, pp. 3-1–3-47. [2] M. Alles and S. Pasupathy, “Suboptimum detection for the two-wave Rayleigh-fading channel,” IEEE Trans. on Commun., vol. 42, no. 11, pp. 2947–2958, November 1994. [3] R. G. Vaughan and N. L. Scott, “Super-resolution of pulsed multipath channels for delay spread characterization,” IEEE Trans. on Commun., vol. 47, no. 3, pp. 343–347, March 1999. [4] A. S. Master, “Nonstationary sinusoidal model frequency parameter estimation via Fresnel integrals analysis,” Stanford University EE 391 Report (August 2002). [5] S. Boyd, “Multitone signals with low crest factor,” IEEE Trans. on Circuits and Systems, vol. 33, no. 10, pp. 1018–1022, October 1986. [6] C. Tellambura et al., “Optimal sequences for channel estimation using discrete Fourier transform techniques,” IEEE Trans. on Commun., vol. 47, no. 2, pp. 230–238, February 1999. [7] L. H`azy and M. El-Tanany, “Synchronization of OFDM systems over frequency selective fading channels”, in Proc. VTC’97, vol. 3, pp. 2094– 2098, May 1997. [8] N. Suehiro and M. Hatori, “Modulatable orthogonal sequences and their applications to SSMA systems”, IEEE Trans. on Information Theory, vol. 34, no. 1, pp. 93–100, January 1988. [9] J. Ha et al., “LDPC coded OFDM with Alamouti/SVD diversity technique”, in Proc. WPMC’01, vol. 3, pp. 1345–1350, September 2001. [10] T. Onizawa et al., “Fast synchronization scheme of OFDM signals for high-rate wireless ATM”, IEICE Trans. on Commun., vol. 82, no. 2, pp. 455–463, February 1999. [11] T. M. Schmidl and D. C. Cox, “Robust Frequency and Timing Synchronization for OFDM,” IEEE Trans. on Commun., vol. 45, no. 12, pp. 1613–1621, December 1997. [12] H. Minn et al., “A robust timing and frequency synchronization for OFDM systems”, IEEE Trans. on Wireless Commun., vol. 2, no. 4, pp. 822–839, July 2003. [13] H. Kobayashi, “A novel symbol frame and carrier frequency synchronization for burst mode OFDM signal”, in Proc. VTC’00 Fall, vol. 3, pp. 1392–1396, September 2000. [14] M. Speth et al., “Optimum receiver design for OFDM-based broadband transmission - Part II: a case study”, IEEE Trans. on Commun., vol. 49, no. 4, pp. 571–578, April 2001.
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