Blind Symbol Rate Estimation and Testbed Implementation for Linearly Modulated Signals Sudhan Majhi
Weidong Xiang
School of Electrical and Electronic Engineering Nanyang Technological University, Singapore Email:
[email protected]
Electrical and Computer Engineering University of Michigan-Dearborn, MI, USA Email:
[email protected]
Abstract—Blind symbol or baud rate estimation is one of the most important parts of the blind signal detection process. Most of the blind symbol rate estimation algorithms provided in the literature either estimate symbol rate from the baseband signal or require prior knowledge of transmitted signal parameters. Moreover, these algorithms had been mostly limited to theoretical studies. In this paper, a cyclic correlation based blind symbol rate estimation is considered for linearly modulated signals. The estimation is performed directly from an intermediate frequency (IF) signal and without having prior knowledge of transmitted signal parameters. The algorithm is implemented on a National Instruments (NI) hardware to measure its performance in a realistic environment. Theoretical and experimental results are provided and compared with existing methods. Index Terms—Blind symbol rate estimation, baud rate estimation, measurement, testbed implementation, cyclic correlation.
I. I NTRODUCTION Blind symbol rate estimation is an important task when performing passive signal analysis. It has many potential applications including monitoring of variable data rate transmissions, blind modulation classification, timing offset estimation, software defined radio and blind synchronization of high-speed distributed networks where the receiver has to be synchronized without pilot or training sequences. Most of the systems assume that symbol rate is known at symbol timing recovering circuite. However, for variable data rate systems [1], symbol rate estimation must be known before performing the symbol timing recovery. Therefore, symbol rate estimation is crucial to estimate symbol rate without having prior knowledge of the transmitted signals. A good survey of blind symbol rate estimation techniques is available in [2]–[4]. An algorithm based on wavelet transform is proposed in [5] which requires prior knowledge of the modulation scheme. Another method based on inverse Fourier transform (IFT) [6] consists of a two-stage process and the performance is improved through a timing recovery and closed loop process. Symbol rate estimation using filter banks is proposed in [3] over a additive white Gaussian noise (AWGN) channel which presumes prior knowledge of the signal bandwidth. The number of filter banks varies with the fluctuation of estimated signal bandwidth. Designing an optimal filter bank is a complex procedure over a fading channel and computationally very expensive. In the recent years, cyclostationary based blind symbol
rate estimation method has become more popular due to its simplicity [7]–[10]. The cyclic autocorrelation approach in [7] is proposed over a noise free environment. Cyclic correlation (CC) [8] is a well established method. However, it requires several time delays to compute the second-order cyclic cumulant and performs inverse-matrix operation at each discrete Fourier transform (DFT) frequency, resulting in a very heavy computation burden. In addition, the estimator is not guaranteed to be global maximum at low signal-to-noise ratio (SNR) and the inverse matrix becomes ill-conditioned at high SNR regime. Maximum likelihood (ML) method [10] estimates symbol rate by exploiting information on the cyclostationary of the modulated signal as well as knowledge of the received pulse shape. Overall, all the existing estimators require a prior knowledge of the carrier frequency. To the best of our knowledge, no experimental studies for blind symbol rate estimation based on cyclostationary approaches have been published in the literature. In this paper, CC based blind symbol rate estimation method is simplified and implemented on a testbed with measurements done in a realistic environment. The estimator uses an arbitrary intermediate frequency (IF) signal to estimate the symbol rate. Instead of inverse-matrix, a non-linear operator is used to improve the performance of the estimator in small rolloff factor scenario. The performance of the estimator over a short burst signal is enhanced by using an interpolation method. The proposed method is used in our recent universal receiver testbed [11]. method It does not require prior knowledge of carrier frequency, modulation schemes, pulse shape filter, timing synchronization, frequency offset and carrier phase. The proposed modified CC (MCC) has been verified for binary phase shift keying (BPSK), quadrature phaseshift keying (QPSK), π/4-QPSK, 8PSK, and 16-quadrature amplitude modulation (16-QAM) linearly modulated signals though simulation and experimental results. II. S YSTEM
MODEL AND PROBLEM STATEMENT
Cyclostationary signals occur commonly in communications due to carrier modulation, and possess a periodically timevarying, mean, and covariance [12]–[14]. In fact, most manmade signals belong to the cyclostationary process. It has been proven that cyclic cumulant based estimation is computationally more efficient than the ML and NLLS [10], [15].
In this paper, a digitized IF signal is used for the symbol rate estimation due to the practical difficulties of having digitized RF signal. In this context, we consider a discrete-time cyclostationary signal. Let us start with a received continuous-time linearly modulated signal y˜(t) = Re
L−1 X l=0
a[l]g(t − lT − τ )e
−j(2πfc t+φ)
+ v˜(t) (1)
y˜[n] = Re
l=0
a[l]g[n − lP − N1 ]e−j(2πfc n+φ)
+ v˜[n] (2)
is a cyclostationary signal for oversampling factor P > 4 [8] and N1 is the number of samples corresponding to the timing offset τ . The over sampling factor is defined as P = Fs /fs = T /Ts = N/L where Fs is the receiver sampling rate, fs is the symbol rate, N is the total number of samples and Ts is the sampling period at the receiver which is small enough for all possible transmitted symbol period, so that the oversampled signal is free of intersymbol interference. In practice, receiver uses much higher sampling rate compared to the conventional symbol rate or signal bandwidth, therefore, receiver does not need to have prior knowledge of rough signal bandwidth. The real signal in (2) can be represented in complex form as y[n] = =
RATE ESTIMATION
The CC symbol rate estimation method given in [8] is based on a complex baseband signal and it uses several time delays to compute cyclic frequency. In contrast, the proposed MCC method is based on the IF signal and no delay is introduced for cyclic frequency computation. The second-order time-varying correlation function of a complex IF signal with zero time-lag can be expressed as c[y,2,1] [n; 0] = E{y[n]y ∗ [n]}.
where L is the number of symbols, a[l] is lth data symbol and assumed zero-mean, independent and identically distributed (i.i.d) with unit variance (E{|a[l]|2 } = 1), T is the symbol period, timing offset τ and carrier phase φ are uniformly distributed in |τ | ≤ T /2 and |φ| ≤ π, respectively, fc is the carrier frequency 1 and v˜(t) is AWGN with two sided power spectral density with variance σv2˜ = N0 /2. Without loss of generality, we assume that g(t) = h˜ g (t) where g˜(t) is the root raised cosine filter and h is channel gain, and g(t) is unknown at the receiver. The roll-off factor is β, 0 ≤ β ≤ 1. The bandwidth of y˜(t) is assumed to be in the interval [−(1 + β)/2T, (1 + β)/2T ]. It follows that −1/T , 0 and 1/T are the non-zero cyclic frequency. The signal y˜(t) is over sampled at rate P/T to yield the discrete-time signal L−1 X
III. S YMBOL
y˜[n] + j yˆ ˜[n] (sr [n] − jsi [n])ej(2πfc n+φ) + v[n]
(3)
ˆ where PL−1 y˜[n] is the Hilbert transform PL−1 of y˜[n], sr [n] = l=0 ar [l]g[n−lP −N1 ], si [n] = l=0 ai [l]g[n−lP −N1 ], a[l] = ar [l] + jai [l] and v[n] is a complex AWGN. The problem statement is that N samples, {y[n]}N 1 , are given, and the symbol rate has to be estimated without having prior knowledge of the parameters, {a[l], g[n], N1 , fc , φ}. 1 We refer carrier frequency, f , as the frequency of the IF signal. In order c to avoid possible aliasing effects, fc is chosen greater than twice the symbol rate and smaller than one-eighth the sampling rate.
(4)
The term E{y[n]y ∗ [n]} can be written as n n 2 o 2 o E{y[n]y ∗ [n]} = E sr [n] + E si [n] n 2 o +E v[n]
(5)
The second-order time-varying correlation function of sr [n] with zero time-lag can be expressed as follows c[r,2,1] [n; 0] =
L−1 X L−1 X
l=0 m=0
E ar [l]ar [m] g[n − lP − N1 ]
× g[n − mP − N1 ] =σr2
L−1 X
(6)
g 2 [n − lP − N1 ].
l=0
where σr2 = E ar [l]ar [m] . Similarly, we can obtain secondorder time-varying correlation function of si [n] and v[n]. The second-order time-varying correlation function of y[n] can thus be obtained by using the additive property as
c[y,2,1] [n; 0] = σa2
L−1 X
g 2 [n − lP − N1 ] + σv2
(7)
l=0
where σa2 = σr2 + σi2 . The cyclic cumulant at cyclic frequency αk can be obtained by taking the Fourier series of the above expression as [16] C[y,2,1] [αk ; 0] =
L−1 2 σa2 X G[l/P ] δ[αk − l/P ]e−j2πN1 /P P (8) l=0
+σv2 δ[αk ]
where δ[.] is the dirac delta function G[αk ] is the discrete Fourier transform (DFT) of g 2 [n] and since g 2 [n] is a symmetric in nature, G[αk ] is a real function. Therefore, the term |C[y,2,1] [αk ; 0]| is maximized at αk = l/P . However, due to the bandwidth of the usual shaping filters, we will suppose that C[y,2,1] [αk ; 0] has only one non-zero positive cyclic frequency, i.e. at αk = 1/P . The symbol rate, fs , is estimated as C[y,2,1] [αk ; 0] . (9) fs = arg max αl′ 100 Hz and |fs − fˆs | > 600 Hz. We presumed an estimation error has occurred if difference between
5
10
15 Average received SNR (dB)
20
25
Fig. 3. Error probability of symbol rate estimation of CC and MCC methods over a flat fading channel with roll-off factor 0.5.
true and estimated value exceeds the above thresholds. It is observed that error probability of the CC method is higher than the proposed MCC for both the threshold values. As frequency resolution of CC method is around 500Hz, the event |fs − fˆs | > 100 has occurred all the time i.e. 100% failure rate. The event |fs − fˆs | > 600 has occurred due to the noise and fading of the channel, but not for a low frequency resolution. The event has failure rate about 10%. However, the proposed MCC method has only 1% failure rate for both the events at 10dB SNR and the failure rate reduces with increase in SNR. It is because of that MCC has very high resolution, 3Hz, and the failure occurs only by a deep fading of the channel. The number of deep fading reduces with increase in transmit power, thus failure rate reduces with SNR. Fig.4 provides the percentage error for both the estimation processes. We can see that MCC method has better performance than the CC for α = 0.5. The performance of CC for α = 0.2 at 10 dB is due to the ill-condition matrix [8]. The average error at 25dB SNR for MCC and CC are about 3Hz and 500Hz, respectively. At α = 0.5, estimation performance for 5MHz and 2MHz IF signals are shown by solid and dotted lines, respectively. It is seen that the percentage error for both IF signals has the same performance. It implies that the proposed estimator independents with the carrier frequency as well as with frequency offset. Similar conclusion can be obtained for carrier phase and timing offset. Thus the algorithm is a robust to frequency offset, carrier phase and timing offset. Again, as the performance of the estimator for a small roll-off factor, 0.2, is not much different than 0.5, the proposed estimator can be used for small excess bandwidth signals. Fig.5 provides simulation and experimental results in terms
2
1
10
10
BPSK QPSK 16−QAM CC for α=0.5
1
10
0
10
Measurement NLOS
Percentage error
Percentage error
0
MCC for α=0.2 −1
10
10
Simulation flat fadign
−1
10
−2
10
−2
10
BPSK MCC for α=0.5
QPSK
Measurement LOS
16−QAM −3
−3
10
5
10
15 Average received SNR (dB)
20
25
Fig. 4. Percentage error of symbol rate estimation of the CC and MCC methods over a flat fading channel.
of percentage error. Two scenarios have been considered for the experimental results- one is for LOS and other is for NLOS. For the LOS, the channel does not have a very deep fade compared to the simulated channel which is Rayleigh fading. However, for the NLOS case, channel has almost the same fading distribution as the simulated channel. From the extensive studies, we have observed that the LOS provides the same results with Rician flat fading channel, however, omitted here due to the space constraint. It is obvious that LOS should have better performance than the NLOS. VI. C ONCLUSION In this paper, we have proposed a MCC blind symbol rate estimation algorithm which is well suited for linearly modulated signals and much simpler than CC method. It is directly applied on an IF signal. The proposed method has been implemented on an NI testbed and carried out theoretical and measurement studies over Rayleigh flat fading channel and real indoor environments. It has been observed that the proposed scheme shows robust performance under the presence of frequency offset, timing offset, carrier phase and small excess bandwidth. R EFERENCES [1] C. Lee, “Variable data rate modem for low earth orbiting satellite (LEOS) communication,” in Military Communications Conference, vol. 3, 1995, pp. 1234 – 1238. [2] M. M. H. Meyr and S. A. Fechtel, Digital communication receivers: Synchronization, channel estimation and signal processing. New York: Wiley, 1998. [3] Z. Yu, Y. Shi, and W. Su, “Symbol-rate estimation based on filter bank,” in IEEE International Symposium on Circuits and Systems, May 2005, pp. 1437 – 1440. [4] J. Sills, “Maximum-likelihood modulation classification for PSK/QAM,” in IEEE Military Communications Conference Proceedings (MILCOM), vol. 1, 1999, pp. 217 –220 vol.1.
10
5
10 15 Average received SNR (dB)
20
25
Fig. 5. Percentage error of symbol rate estimation of the proposed MCC over a flat fading channel and the experimental results over an LOS and NLOS with roll-off factor 0.5.
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