Synchronization of Bursty Offset QPSK Signals in the Presence of Frequency Offset and Noise K. Vasudevan Dept. of Electrical Engg. Indian Institute of Technology, Kanpur Email:
[email protected] Abstract—This paper deals with the problem of synchronization of bursty offset QPSK signals in the presence of frequency offset and additive white Gaussian noise. The main tasks of the receiver are to detect a burst (a frame of data) with a low acquisition time, to estimate and cancel the frequency and phase offset (carrier recovery), perform timing recovery, matched filtering and finally recover the data such that the bit-error-rate (BER) is close to the theoretical limit. The key contribution of this paper is to develop synchronization techniques in discretetime that perform effectively even at signal-to-noise ratio (SNR) close to 0 dB in additive white Gaussian noise (AWGN) channels. These techniques are well suited for implementation on a digital signal processing (DSP) platform. Index Terms—Offset QPSK, synchronization, matched filtering, additive white Gaussian noise.
I. I NTRODUCTION Synchronization is an important aspect of digital communications. With the advent of turbo codes, it is now possible to achieve very low BER at SNR close to 0 dB. Hence it seems logical that the synchronization techniques must also be able to perform well at such low SNR. The general features of the synchronization techniques proposed in the literature are: 1) The existing synchronization techniques do not detect the presence of a signal, rather they assume that the signal (plus noise) is already present. 2) The carrier recovery procedures suffer from phase ambiguities, resulting in the need for differential encoding at the transmitter. This work attempts to address both above problems. In particular, since our ultimate goal is to perform turbo decoding, it becomes essential to perform coherent demodulation (absence of any phase ambiguities) even at very low SNR. The synchronization and detection techniques presented here are developed in discrete-time, which are well suited for implementation on a DSP platform. An improved phase lock loop that operates effectively at medium to low SNR is proposed in [1]. A data-aided carrier recovery loop for duobinary encoded offset QPSK is given in [2]. A tutorial on carrier and timing synchronization is given in [3]. Joint carrier recovery and equalization of digitally modulated signals is discussed in [4], [5]. A data-aided carrier recovery algorithm for estimating phase and frequency offsets is discussed in [6], [7] and also in [8], [9] for digital land mobile radio and satellite communication. Detection of bursty QPSK signals at low SNR is described in [10]. A digital
modem for offset-QPSK is dealt with in [11]. Carrier synchronization for trellis-coded signals is given in [12]. A digital PLL for QPSK signals is described in [13] and an all-digital implementation of carrier synchronization for digital radio systems is proposed in [14]. A comparison of different carrier recovery techniques is presented in [15]. A carrier recovery algorithm for M -ary QAM with a capability to track large frequency offsets is discussed in [16]. Non-data-aided carrier and timing recovery at very low SNR, with applications for turbo-coding is discussed in [17]. Our approach differs from [17] in the following ways: 1) The matched filter is used as an interpolator. 2) The frequency offset is 30% of the symbol-rate. A non-data-aided carrier recovery method for modified 128QAM is proposed in [18]. More recently, iterative timing recovery is proposed in [19]. In this paper we deal with only timing acquisition and not tracking, since the timing assumed to be constant over a burst. This paper is organized as follows. Section II discusses the system model. In Section III we discuss the various receiver algorithms. The performance results are discussed in Section IV. Finally, in Section V we give our conclusions and future work. II. S YSTEM M ODEL Preamble Lp
Data Ld Fig. 1.
Postamble Lo
The burst structure.
We assume that the data to be transmitted is organized into bursts (or frames). The burst consists of a known preamble of length Lp (QPSK) symbols, followed by data of length Ld symbols and a known postamble of length Lo symbols. Thus, the total length of the frame is L = Lp + Ld + Lo . The received signal can be written as: √ r(t) = 4B s˜(t)e j [2π(Fc +ΔF )t+θ0 ] + w1 (t)
(1)
(2)
where {·} denotes the real-part, 2B is the bandwidth of s˜(t)e j 2πΔF t , Fc is the nominal carrier frequency, ΔF is the
frequency offset (which can be positive or negative), θ0 is the carrier phase and w1 (t) is additive white Gaussian noise with two-sided power spectral density N0 /2 (watts/Hz). The term s˜(t) (we use tilde to denote complex quantities) in (2) is the complex envelope of the offset QPSK signal and is given by: s˜(t)
=
L−1
Sk, I p(t − kT − α)
k=0
+j
L−1
Sk, Q p(t − kT − T /2 − α)
(3)
k=0
where Sk, I ∈ ±1 and Sk, Q ∈ ±1 are the in-phase and quadrature components of the QPSK symbol, 1/T is the symbol-rate and p(t) is the impulse response of the transmit filter, which is assumed to have the root-raised cosine spectrum with 40% roll-off. The variable α denotes the random timing phase which is assumed to be uniformly distributed in [0, T ). The task of the receiver is to estimate the QPSK symbols such that the error-rate is close to the theoretical limit. Since we deal with discrete-time signals in this paper, the first task is to convert r(t) in (2) into a discrete-time signal. This is accomplished by passing r(t) through a bandpass filter (BPF) followed by bandpass sampling. For convenience of subsequent analysis, we assume an ideal BPF having unit energy. This ensures that the noise power at the BPF output is N0 /2. Assuming a sampling frequency of Fs = 1/Ts , the bandpass sampling requirements can be stated as follows: 2π(Fc − B) Fs 2π(Fc + B) Fs
A. Up-Sampling and Matched Filtering (a)
p(t − α) p(nTs − α)
t n=1
0
≥ kπ ≤ (k + 1)π
sampling frequency of 1/Ts . The receiver in Figure 2 operates in the following steps: 1) Demodulate r(nTs ) with ω ˆ0 = ω ˆ f = 0 in Figure 2 and detect the start of frame using the preamble. Store the samples r(nTs ) corresponding to a frame. Obtain a coarse estimate of the frequency offset ω ˆ0. 2) Demodulate the stored values of r(nTs ) using π/2 + ω ˆ 0 . Estimate the start of frame for the second time. We demonstrate via simulations that the second estimate of the start of frame is better than the first estimate obtained in step (1), for large values of ω0 . 3) Obtain the maximum likelihood estimate of the residual ˆ 0 . Denote frequency offset which is equal to ωf = ω0 − ω this estimate as ω ˆf . 4) Detect the data by canceling θ0 and ωf . Before we proceed to elaborate on the above steps, let us look into the operation of up-sampling and matched filtering.
n=2
n=3
n=4
α
Ts p(5Ts − t − α) p(5Ts − nTs − α)
(b)
(4)
n=5
where k is a positive integer. We further assume that 2πFc = kπ + π/2 = π/2 mod 2π Fs
(5)
if k is even. Thus the output of the analog-to-digital (A/D) converter after bandpass sampling can be written as r(nTs ) = s˜(nTs )e j [(π/2+ω0 )n+θ0 ] + w(nTs ) (6)
t n=1
0
n=2 Ts
n=4
n=5
p(5Ts − t)
(c)
where
n=3
p(5Ts − nTs1 )
α = 2Ts1
s˜(nTs ) = s˜(t)|t=nTs w(nTs ) = w(t)|t=nTs ω0 = 2πΔF/Fs mod 2π
(7)
where w(t) is noise at the BPF output. Note that w(nTs ) denotes samples white Gaussian noise with variance N0 /2. We further assume that T /Ts = M is an even integer so that the symbols can be delayed by T /2 ≡ M/2 samples in discrete-time. III. R ECEIVER Let ω ˆ 0 and ω ˆ f denote the “coarse” and the “fine” estimates of the frequency offset respectively, corresponding to the
t 0
Ts
Ts1
Fig. 3. (a) Samples of the received pulse. (b) Filter matched to the received pulse. (c) Filter matched to p(t) sampled at a higher frequency. Observe that one set of samples (shown by solid lines) correspond to the matched filter.
For ease of exposition, we assume p(t) to be a triangular pulse. The samples of the transmitted pulse is shown in Figure 3(a) by solid impulses (Kronecker delta function). We
ω ˆf ˆ ) θ(lT 2 cos(n(π/2 + ω ˆ 0 ))
Tim. and freq. offset est.
↑I
Est.
ˆ
ˆ f M k/2+θ(lT )) e−j (ω
θ(lT )
{·}
r(nTs )
p(N Ts − nTs1 )
x ˜(nTs1 )
Delay T /2
t = n2 Ts1 + kT /2 {·}
↑I
Sˆi, I
Sˆi, Q
−2 sin(n(π/2 + ω ˆ 0 )) Fig. 2. The discrete-time receiver which up-samples the local oscillator output by a factor of I and then performs matched filtering at a sampling frequency Fs1 .
assume that p(t − α) spans over N + 1(= 6) (0 ≤ n ≤ N ) samples and the sampling frequency 1/Ts is such that it satisfies the Nyquist criterion for no aliasing of the spectrum of p(t). The corresponding discrete-time matched filter is shown in Figure 3(b) by solid impulses. The matched filter output can be obtained using the frequency-domain approach. Let P (F ) denote the Fourier transform of p(t), assumed to be bandlimited to |F | ≤ B. Then the discrete-time Fourier transform of p(nTs − α) is obtained as follows: p(t − α)
= =
g(t) ˜ ) G(F P˜ (F )e−j 2πF α 0
−B ≤ F ≤ B otherwise
Therefore = =
g(nTs ) ˜ (F ) G P ˜ )/Ts G(F 0
0 ≤ |F | ≤ B B ≤ |F | ≤ Fs /2
g1 (nTs1 ) GP (F I).
where ˜ P, 2 (F1 ) G ˜∗ P (F1 ) −j 2πF1 N Ts = Ts1 e 0
(12)
0 ≤ |F1 | ≤ B B ≤ |F1 | ≤ Fs1 /2. (14)
(9)
˜ P (F ) denotes a periodic function of frequency, that where G is ˜ P (F ) = G ˜ P (F + kFs ). G
where g(nTs ) is defined in (9). The discrete-time Fourier transform (DTFT) of g1 (nTs1 ) is [20]: Let us define a new frequency variable F1 = F I with respect to the new sampling frequency Fs1 . Now, if p(N Ts − t) is sampled at a rate Fs1 the DTFT of the resulting sequence is: ˜ P, 2 (F1 ) p(N Ts − nTs1 ) = g2 (nTs1 ) G (13)
(8) p(nTs − α)
Let us denote Ts /Ts1 = I which is referred to as the interpolation factor. Construct the up-sampled sequence from the incoming signal p(nTs − α) as follows: g(nTs /I) for n = mI g1 (nTs1 ) = (11) 0 otherwise
(10)
Now, when p(nTs − α) is convolved with p(N Ts − α − nTs ), the peak occurs at t = N Ts , independent of α. Note that p(N Ts −α−nTs ) represents the discrete-time causal matched filter for p(nTs − α). The peak value is equal to Rpp (0)/Ts , where Rpp (t) is the continuous-time autocorrelation of p(t). In practice however since the receiver does not know α, it is not possible to obtain the exact matched filter as in Figure 3(b). The solution lies in sampling p(N Ts − t) at a higher frequency (say Fs1 = 1/Ts1 ) compared to 1/Ts , as illustrated in Figure 3(c).
The convolution of g1 (nTs1 ) with g2 (nTs1 ) can be written as [21]: g1 (nTs1 ) g2 (nTs1 ) B 2 1 ˜ = P (F1 ) Ts Ts1 Fs1 F1 =−B
× e j 2πF1 (nTs1 −α−N ITs1 ) dF1 .
(15)
Clearly if α = n0 Ts1 , where n0 is an integer, the above convolution becomes Rpp ((n − n0 − N I)Ts1 ) g1 (nTs1 ) g2 (nTs1 ) = (16) Ts with a peak value equal to Rpp (0)/Ts , occurring at (n0 + N I)Ts1 . In the subsequent analysis, we assume that the correct sampling instants are at (n0 + N I)Ts1 + kT /2.
B. Synchronization and Data Detection This section is an elaboration of the four steps listed in section III. Recall that in the first step, the start of frame and a coarse estimate of the frequency offset is obtained. Let δ = (n0 + N I)Ts1 .
(17)
We proceed by making a key observation that at the right instants, the matched filter output can be approximated as: ˜ x ˜(δ + kT ) ≈ β(kT )e j (ω0 M k+θ0 ) + v˜(δ + kT )
(18)
following rule: choose that value of nTs1 which maximizes |˜ y (nTs1 )|2 . Mathematically, this can be stated as [22], [23]: nTs1
i=0
(19)
where Sk, I is defined in (3) and γk denotes the intersymbol interference (ISI) in the quadrature arm. It is assumed that Sk = 0 for k < 0 and k > L − 1. Since the preamble and ˜ ) can be precomputed and p(t) are known, γk and hence β(kT stored at the receiver. In other words, γk is a function of the past, current and future quadrature symbols as given by: Sk+LISI −1−j, Q hj
(20)
j=0
where LISI denotes the span of Rpp (t) (in symbol durations) on the either side of Rpp (0) and hj denotes coefficients of the filter having a raised-cosine frequency response. In the simulations, LISI was taken to be 3. The term v˜(·) in (18) denotes samples of zero-mean Gaussian noise with autocorrelation [21]
ω ˆ 0 = (1/M ) arg [˜ y (n1 Ts1 )] .
(1/2)E [˜ v (kT )˜ v (kT − mT )] = N0 δK (mT )
1.0e-02
1.0e-03
1.0e-04 2
4
6
8
10
Eb /N0 (dB) Normalized (wrt T ) var. of timing error vs SNR.
Fig. 5.
= x ˜(nTs1 )β˜∗ (kT ) Lp −LISI −1
=
1.0e-01
μ ˜∗ ((n + iM I)Ts1 , i)
i=0
×μ ˜((n + (i + 1)M I)Ts1 , i + 1). (22) Based on (18), the start of frame is detected using the 120000 100000
RMS estimation error (radians)
y˜(nTs1 )
1st attempt 2nd attempt
(21)
where δK (·) is the Kronecker delta function. Define
(25)
We refer to (25) as the “differential” method of estimating the frequency offset.
0
∗
μ ˜(nTs1 , k)
therefore
Normalized variance of timing error
˜ β(kT ) = Sk, I + j γk
2L ISI −1
(23)
Note that n1 Ts1 is an estimate of δ in the first attempt. The result of applying the detection rule in (23) is depicted in Figure 4. At the correct instant, say n1 Ts1 , and in the absence of noise we have Lp −LISI −1 2 j ω0 M ˜ )β((i ˜ + 1)T ) y˜(n1 Ts1 ) = e (24) β(iT
for 0 ≤ k ≤ L − 1 where
γk =
2
|˜ y (nTs1 )| .
n1 Ts1 = max
Diff. 1st attempt Diff. 2nd attempt ML, w_s=1*10^{-4} ML, w_s=5*10^{-4}
1.0e-02
1.0e-03
1.0e-04
|˜ y (nTs1 )|2
1.0e-05 0
80000
2
4
6
8
10
Eb /N0 (dB)
60000
Fig. 6.
40000 20000 0 0
5000
10000
15000
20000
25000
Time Fig. 4.
Frame detection at 0 dB SNR.
30000
RMS frequency offset estimation error vs SNR.
However, the timing and frequency offset estimates obtained from (23) and (25) are not very accurate when ω0 is large, e.g. 0.3 radians, which is clear from the “1st attempt” plots in Figures 5 and 6. The main reason is due to the approximation in (18), which gets better as ω0 gets smaller. This is the why we need to do a “2nd attempt” on the timing and the frequency offset.
The second attempt is initiated by first demodulating r(nTs ) using the local oscillator frequency as π/2 + ω ˆ 0 . Thus the ˆ 0 ) is quite small. The resultant frequency offset (ωf = ω0 − ω effect of applying (23) and (25) is again illustrated by the curves marked “2nd attempt” in Figures 5 and 6. Whereas there is a significant improvement in the timing estimate, the accuracy of the frequency offset estimate is still inadequate. For example, in Figure 6 at an Eb /N0 = 0 dB, the second attempt yields a root mean square (RMS) error equal to 6 × 10−3 radians. With T /Ts = M = 6, the phase change over 10 symbols is 0.006 × 6 × 10 = 0.36 radians, which is too fast for a phase tracking loop. This motivates us to use the maximum likelihood (ML) method of estimating ωf . Assuming that in the second attempt the outcome of the right-hand-side of (23) is n2 Ts1 , the ML rule for estimating the frequency offset can be stated as follows: set ω ˆ f = ωi if ωi maximizes Lp −LISI ∗ −j ωi M k ˜ (26) x ˜(n2 Ts1 + kT )β (kT )e k=0 where x ˜(n2 Ts1 + kT ) is given in (18) with δ replaced by n2 Ts1 , ω0 replaced by ωf and ωmin ωmin ωmax
< ωi < ωmax = ω ˆ 0 − 0.05 = ω ˆ 0 + 0.05
x ˜(n2 Ts1 + kT /2)
˜ ≈ β(kT /2)e
0.6
(27)
j(ωf M k/2+θ0 )
+ v˜(n2 Ts1 + kT /2)
= x ˜(n2 Ts1 + kT /2)e−jˆωf M k/2 ˜ = β(kT /2)e j θ(kT /2) + v˜ (kT /2)
0.5 0.4 0.3 0.2 0.1 0 0
100
200
300
400
500
600
Time Fig. 7.
Envelope of regular QPSK.
0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4
(28)
where ωf = ω0 − ω ˆ 0 . Since ωf is of the order of 6 × 10−3 radians (see the curves labelled “1st attempt” in Figure 6), we need to first cancel it using ω ˆ f . Consider x ˜ (kT /2)
where sgn [·] denotes the signum function. 0.7
where ωs denotes the step size. In the simulations two different step sizes were considered, that is, ωs = 10−4 and ωs = 5×10−4 radians. Observe that the FFT cannot be used in (26), since the search is only over a narrow portion of the digital spectrum centered around ω ˆ 0 . From Figure 6 we find that the RMS estimation error for the ML method at an Eb /N0 = 0 dB, is only 1.8 × 10−4 for ωs = 5 × 10−4 radians and 1.1 × 10−4 for ωs = 10−4 radians. Such accuracies are sufficient for the phase tracking loop. For example, with an estimation error (residual frequency offset) of 1.8 × 10−4 radians, the phase change over 50 symbol durations is only 0.054 radians, assuming that T /Ts = M = 6. Thus we can assume the phase to be nearly constant over a large interval of time, facilitating the use of very narrowband lowpass filters to average out the effects of noise. This is discussed next. After demodulation by π/2 + ω ˆ 0 and the second estimate of δ (yielding n2 Ts1 ), the matched filter output can be written as:
0.35 0.3 0.25 0
100
200
300
400
Time Fig. 8.
(29)
(30)
ˆ ) in The parameter ρ was taken to be 0.95. Note that θ(lT (30) is computed in the range [0, 2π), hence there is no phase ambiguity. Finally, the QPSK symbols at time i = l+LISI are estimated ˜ as follows (note that β(kT ) in (30) is a function of several symbols, as given in (20), hence we assume that these symbols have already been estimated):
ˆ Sˆi, I = sgn x ˜ (iT )e−j θ(lT )
ˆ Sˆi, Q = sgn x (31) ˜ (iT + T /2)e−j θ(lT )
|˜ s(nTs )|
= ωmin + iωs
z˜avg (lT ) = ρ˜ zavg ((l − 1)T ) + (1 − ρ)˜ z (lT ) ˆ ) = arg [˜ θ(lT zavg (lT )] .
|˜ s(nTs )|
ωi
where θ(kT /2) is a slowly varying phase which can be assumed to be constant over a large number of symbol durations. The T -spaced autocorrelation of v˜ (·) is identical to that of v˜(·), as given in (21). The (data-aided) phase tracking loop operates at symbol-rate (k = 2l) as follows: 2 ˜ ) z˜(lT ) = x ˜ (lT )β˜∗ (lT )/ β(lT
Envelope of offset QPSK.
500
600
1.0e-01
Theory w_s=10^{-4} w_s=5*10^{-4}
Bit error rate
1.0e-02
1.0e-03
1.0e-04
1.0e-05
1.0e-06 0
2
4
6
Eb /N0 (dB)
Fig. 9.
8
10
BER vs SNR.
IV. P ERFORMANCE R ESULTS In the simulations, p(t) was taken to be the pulse having the root-raised cosine spectrum with a roll-off of 0.41 and truncated to N + 1 = 37 samples. The parameter T /Ts = M = 6 samples per symbol and interpolation factor I = 3. Further Lp = 200, Ld = 1500 and Lo = 12 QPSK symbols. Simulations were carried out over 105 frames. The frequency offset ω0 = 0.1π (ΔF T = 0.3). We do not assume that α is an integer multiple of Ts1 in the BER simulations. The SNR per bit is defined as [21]: (32) Eb /N0 = 10 log10 |Sk |2 /(4N0 ) . Figures 7 and 8 show the envelope variations of regular QPSK to be more than that of offset QPSK. Finally, the BER results are presented in Figure 9 for different values of ωs . We find that the performance of the proposed system is as good as the theoretical performance. V. C ONCLUSIONS AND F UTURE W ORK In this work, we have presented discrete-time algorithms for synchronization and detection of bursty offset QPSK signals. These algorithms can be readily implemented on a DSP processor. The simulation parameters chosen in this paper may not be optimum in the sense of reducing the computational complexity without compromising the BER performance. We have also shown via simulations that the acquisition time for a burst is equal to the preamble length and is much smaller than the results in [17]. The next step would be to utilize the proposed algorithms for detecting turbo-coded signals. ACKNOWLEDGEMENT This work is supported by the Defence Electronics Applications Lab (DEAL)-DRDO, Dehradun, India, under Grant DEAL/02/4043/2005-2006/02/016. R EFERENCES [1] J. D. Daley, “Improved Phase-Lock-Loop Performance at Medium to Low SNR,” Electronics Letters, vol. 10, no. 17, pp. 365–366, Aug. 1974. [2] D. P. Taylor and D. C. C. Cheung, “A Decision-Directed Carrier Recovery Loop for Duobinary Encoded Offset QPSK Signals,” IEEE Trans. on Commun., vol. 27, no. 2, pp. 461–468, Feb. 1979.
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