Estimation of Symbol Rate from the Autocorrelation Function Y.T. Chan1, B.H. Lee1, R. Inkol2, F. Chan1 Department of Electrical and Computer Engineering Royal Military College of Canada, Kingston, Ontario, Canada email: {chan-yt, lee-h, chan-f}@rmc.ca 2 Defence Research and Development Canada, Ottawa, Ontario, Canada email:
[email protected] 1
d = 0.5Ts. This is because for d ≠ 0.5Ts, the circuit destroys the Ts periodicity of y(t). Hence at d = 0.5Ts, the spectral peaks are more distinct relative to the other frequency components. The BPF is narrowband with a centre frequency at 1/Ts. For optimal settings of the delay and the BPF, it is necessary to have some a priori knowledge about Ts. An improvement to [5] is in [6], which replaces the delay and multiply operation by a nonlinear envelope detector. The output has a periodic component at 1/Ts. This component is a result of squaring a sequence of pulses (for example raised cosine) of duration Ts. Any nonlinear operation, however, produces non-zero mean noise, and introduces biases in the estimates. Since s(t) is a cyclostationary process, [7] uses this property and develops a second order statistics scheme to obtain spectral lines. But the cyclostationarity degrades when the symbol pulse has a sharp edge, i.e., a small roll-off factor. The spectral lines then become unstable. As a remedy, [8] proposes passing s(t) to a bank of LPFs, of different bandwidths covering all possible symbol rates. The output of each LPF goes to a fourth order nonlinear unit which then produces a spectral line at 1/Ts. More recently, [9] gives a near maximum likelihood (ML) estimator. The output of a bank of matched filters initializes a coarse search followed by a fine search of Ts to maximize the ML function. However, the computational cost can be high. For (ii), [10] compares a reference clock against the baseband symbol sequence and adjusts the clock frequency to obtain a timing match. For carrier modulated symbol sequences, [11, 12, 13] locate phase discontinuities by the Haar Wavelet Transform (HWT), which is a digital differentiator. Applying HWT at different scales to s(t), [11, 12, 13] then select the highest magnitude of HWT coefficient c(a, τ). The coefficients, of magnitude c, scale a, and translation τ contain information on Ts. An autocorrelation (AC) or Fourier transform of c(a,τ) then gives Ts or 1/Ts. This paper presents a new Ts estimator. Noting that the AC of a baseband symbol sequence always has a first discontinuity at Ts, the estimator first takes the AC of the sequence, and then locates its discontinuity by least squares. Since the estimator operates on a baseband signal, a
ABSTRACT The estimation of the symbol rate (or symbol duration Ts) of a symbol sequence in noise has important applications in symbol timing recovery and radio surveillance. Since a symbol sequence has spectral peaks at intervals 1/Ts, Ts estimation centers mostly on the establishment of spectral rate lines from the sequence. Examples include the delay and multiply circuit, and the cyclostationarity based estimators. This paper presents an alternative. It first computes the autocorrelation function of a sequence, which has a first discontinuity at lag Ts. Locating this discontinuity then gives an estimate of Ts. Simulation results show that, for the range of signal-to-noise-ratios under consideration, errors of less than 3% of Ts are achievable. Index Terms— Parameter estimation, correlation, least squares methods 1. INTRODUCTION An important problem in electronic warfare [1] and civilian spectrum monitoring concerns the estimation of signal parameters such as carrier frequency, modulation type, symbol rate (or duration Ts), and signaling schemes. This paper is on the blind estimation of Ts, which has other applications in network synchronization [2], software defined radio [3], variable data rate satellite communications [4], and in the detection of low probability of intercept signals [5, 6] Estimators for Ts divide broadly into two classes: (i) Those that obtain, via signal processing, an output spectrum that contains spectral lines at integer multiples of 1/Ts [5-9]; (ii) Those that detect the transitions between symbols to determine Ts [10-13]. The basis for (i) is that the baseband spectrum of a symbol sequence, s(t), has peaks at integer multiples of 1/Ts. The delay and multiply circuit in Figure 1 sharpens these peaks for easy detection. The output of the lowpass filter (LPF) is s(t) and the input to the bandpass filter (BPF) is y(t). Now y(t) = s(t)s(t + d) produces a sharp spectral peak at
978-1-4244-3508-1/09/$25.00 ©2009 IEEE
547
or a periodic sequence of ones and zeroes, of symbol duration 0.5Ts. Thus inequality (6) holds. Figure 3 shows a typical ACF. Since discontinuities exist at τ = Ts, 2Ts, … , locating the first discontinuity gives an estimate for Ts.
prerequisite is that appropriate demodulation processing has taken place, or that the signal envelope is available from the magnitude of its Hilbert transform. Sections 2 and 3 present the development of the estimator and the simulation results, respectively. Section 4 gives concluding remarks. 2. ESTIMATION OF TS BY AUTOCORRELATION
2.2. Finding Ts by least squares
2.1. Autocorrelation function of a symbol sequence
The ACF in Figure 3 suggests that checking for slope changes can locate the first discontinuity in R(τ). However, simulation experiments indicate that this is unreliable. The problem is that if R(τ) is noisy, computing the slope by differentiation tends to accentuate the effects of noise. The following describes an alternative that is less sensitive to noise. From Figure 3, it is apparent that the discontinuity at Ts is also the intersection of two straight lines. A least squares (LS) fit of the noisy R(τ) points to the left of Ts gives one line, and similarly another line to the right of Ts. Their intersection will give a point close to Ts.
Let s(t) be a baseband random sequence of unipolar, nonreturn-to-zero symbols. The autocorrelation function (ACF) is R(τ ) =
1 NTs
NTs
∫ s(t )s(t −τ )dt
(1)
0
where N, Ts, and τ are the number of symbols, the symbol duration, and the correlation time shift, respectively. Now
MA2 (2) N where M is the number of ones in s(t) and A is the pulse amplitude. For ease of illustration, the pulses are rectangular. However, Section 3 has a simulation example where the pulses are trapezoidal, demonstrating the applicability of the estimator to other pulse shapes. Further, Figure 2 shows that M⎛ L τ ⎞ (3) ⎟ R(τ ) = A2 ⎜1 − R (0) =
The procedure is: (i) Fit a LS straight line to the R(τ) points between τ = 0 and τ1 ≈ 2Ts. The exact value of τ1 is not critical as long as τ1>Ts. This assumes that some a priori knowledge of possible values for Ts is available. Let this line be A in Figure 4. (ii) Compute the residuals between line A and R(τ), for 0≤ τ ≤ τ1. Find the location of the maximum residual. Let this point be τ′ in Figure 4. (iii) Fit two LS straight lines, line B to the left of τ′, i.e., 0 ≤ τ ≤ τ′ and line C to the right of τ′, i.e., τ′ ≤ τ ≤ 2τ′. (iv) Find the intersection of B and C. Let it be τ″. (v) Repeat (iii) with τ″ replacing τ′, and then (iv). Iterate until successive differences are less than two sample periods. In the simulation, this usually takes three or four iterations.
M Ts ⎟⎠
N ⎜⎝
where L is the number of 0 to 1 transitions and L ≤ M. At τ = Ts M⎛ L ⎞ ⎧ 0, if L = M (4) R(Ts ) = A 2 ⎜1 − ⎟ = ⎨ N⎝ M ⎠ ⎩> 0, if L < M The slope of the ACF is Δ dR (τ ) L , 0 ≤ τ ≤ Ts (5) R (τ ) = − A2 NTs dτ For shifts of 2Ts ≥ τ ≥ Ts, there is no general expression for R(τ ) or R (τ ) , since the symbols are random. However, it is possible to show that (6) R (0 ≤ τ ≤ Ts ) ≠ R (Ts < τ ≤ 2Ts )
Figure 5 illustrates how the successive intersections of lines B and C can converge towards Ts. Suppose τ′ is to the left of Ts. The R(τ) samples to the right of τ′ will cause the magnitude of the slope of line C to be smaller than R (τ ) , for τ ≥ Ts. Thus C will intersect B at τ″, which is closer to Ts than τ′. A similar argument holds if τ′ is to the right of Ts. Finally, when τ″≈Ts, the intersection of B and C is close to Ts. For LS fitting of a straight line, the standard equation is
Let τ = Δ (0 ≤ τ ≤ Ts) and 1 R (τ ) = NTs
NTs
∫ 0
s(t )
ds(t − Δ) dt , 0 ≤ τ ≤ Ts dΔ
(7)
Similarly, let 1 R (τ ) = NTs
NTs
∫ 0
s(t )
ds (t − Ts − Δ) dt , Ts < τ ≤ 2Ts dΔ
(8)
⎡ m⎤ G⎢ ⎥ = g ⎣α ⎦
But (7) and (8) are equal only if
ds (t − Δ ) ds (t − Ts − Δ ) = dΔ dΔ
(9)
where m is the slope and α is the intersection, and
This implies that s(t − Δ) = s(t − Ts −Δ) + constant, i.e., s(t) is periodic in Ts. But this requires that s(t) is a constant,
548
(10)
REFERENCES
1⎤ ⎡ R(0) ⎤ ⎢ R(1) ⎥ 1⎥⎥ , (11) ⎥ g=⎢ ⎢ # ⎥ #⎥ ⎥ ⎢ ⎥ 1⎦ ⎣ R( k )⎦ where R(i), i=0,…,k are samples of R(τ). The LS estimate is ⎡mˆ ⎤ −1 T T (12) ⎢αˆ ⎥ = (G G ) G g ⎣ ⎦ ⎡0 ⎢1 G=⎢ ⎢# ⎢ ⎣k
[1] D. C. Schleher, Introduction to Electronic Warfare, Artech House, 1986. [2] S. Bregni, “A historical perspective on telecommunications network synchronization,” IEEE Commun. Mag. 36(6), pp.158-166, 1998. [3] F. K. Jondral, “Cognitive radio – A communications engineering view,” Wireless Commun., pp.28-33, August 2007. [4] G. Karam, C. Paxal, and H. Sari, “A variable-rate QPSK demodulator for digital satellite TV reception,” CP397, Amsterdam, Netherlands, pp.646-650, 1994. [5] D. E. Reed and M. A. Wickert, “Minimization of detection of symbol-rate spectral lines by delay and multiply receivers”, IEEE Trans. Commun. 36(1), pp.118-120, 1988. [6] D. E. Reed and M. A. Wickert, “Symbol-rate detection by a power-series-nonlinear envelope detector receiver,” Phoenix Conf. on Computers and Communications, 1988. [7] P. Ciblat, P. Loubaton, E. Serpedin, and G. B. Giannakis, “Asymptotic analysis of blind cyclic correlationbased symbol-rate estimators,” IEEE Trans. Inf. Theory 38(7), pp.1922-1934, 2002. [8] Z. Yu, Y. Q. Shi, and W. Su, “Symbol-rate estimation based on filter bank,” IEEE International Symposium on Circuits and Systems, pp.1437-1440, 2005 [9] C. Mosquera, S. Scalise, and R. Lopez-Valcarce, “Nondata-aided symbol rate estimation of linearly modulated signals,” IEEE Trans. Signal Process. 56(2), pp.664-674, 2008. [10] R. D. Rattlingourd, Digital Timing Recovery System, US Patent 4,280,099, July 21, 1979. [11] K. C. Ho, W. Prokopiw, and Y. T. Chan, “Modulation identification of digital signals by the wavelet transform,” IEE Proceedings - Radar, Sonar and Navigation 147(4), pp.169-176, 2000. [12] Y. T. Chan, J. W. Plews, and K. C. Ho, “Symbol rate estimation by the wavelet transform”, IEEE International Symposium on Circuits and Systems, pp.177-180, 1997. [13] X. Jun, Fu-ping, W. Wang, and W. Zan-ji, “The improvement of symbol rate estimation by the wavelet transform,” International Conf. on Communications, Circuits and Systems, 2005.
ˆ bτ + αˆ b and C: mˆ cτ + αˆ c their Given two lines B: m intersection is at αˆ − αˆ b (13) τ~ = c mˆ b − mˆ c 3. SIMULATION STUDIES
This section describes three experiments aimed at assessing the performance of the proposed estimator. The signal s(t) is a random sequence of ones and zeros whose amplitudes are A and 0, respectively. Each sequence has N symbols and K samples per symbol. i.e., K=Ts. The estimator processes s(t) + n(t), where n(t) is a zero mean Gaussian random variable of variance σ2. The signal-to-noise ratio is SNR=A2/σ2. Each experiment consists of 500 trials using the same s(t) with different n(t) for each trial. The estimation root mean square error, as a fraction of K, is 500
E 1 = K K
∑ ( Kˆ (i) − K )
2
i =1
(14)
500
where Kˆ (i ) is the Ts estimate at the i-th trial. Experiment 1. Figure 6 plots E/K versus SNR, with N as a variable and K = 100 for rectangular pulses. As expected, E/K decreases as N increases, and E/K < 0.03 in all cases. Experiment 2. Figure 7 shows the effects of K on the estimates with N = 40 for rectangular pulses. As K increases, E increases and E/K< 0.03 in all cases. Experiment 3. The pulses are trapezoidal with identical rise and fall times which are less than 5% of K(=100). Figure 8 shows that E/K is quite close for rectangular or trapezoidal pulses. These results suggest that the estimator is applicable to other pulse shapes. In all the experiments, biases in the estimates were negligible. 4. CONCLUSIONS
Estimation of the symbol period Ts from a noisy sequence normally requires the establishment of spectral lines at k/Ts from the sequence. By showing that the sequence always has a slope change at Ts, Section 2 develops a computationally efficient estimator, which fits straight lines to the ACF and takes their intersection as the Ts estimate. Simulation results show that the estimation errors are less than 3% for the range of SNRs considered.
Figure 1. The delay and multiple symbol rate estimator. Normally d = 0.5Ts.
549
Figure 2. Illustrating the ACF.
Figure 6. Simulation results for Experiment 1.
Figure 3. A typical ACF of a symbol sequence.
Figure 7. Simulation results for Experiment 2.
Figure 4. Fitting for line A.
Figure 8. Simulation results for Experiment 3.
Figure 5. Intersection of lines B and C to estimate Ts.
550
