Joint time-frequency domain cyclostationarity-based approach to blind estimation of OFDM transmission parameters
© Sun et al.; licensee Springer. 2013
Received: 1 February 2013
Accepted: 3 April 2013
Published: 29 April 2013
Blind estimation of the transmission parameters of orthogonal frequency division multiplexing (OFDM) signal is an important issue for various civilian and military applications. This paper presents a new cyclostationarity-based approach to blind estimation of OFDM signal parameters. Analytical expression for extracting transmission parameters from second-order cyclic cumulant (CC) of OFDM signal is derived first, with the consideration of the effects of time dispersion and consistent estimator errors on CC. The approach exploits the cyclostationarity of OFDM both in time delay domain and cyclic frequency domain; a statistic hypothesis testing framework was formulated to determine the threshold for detecting the presence of cycles of OFDM signal. The simulation results reveal that the proposed approach is robust for both previous synchronization and channel condition concurrently.
KeywordsOFDM Blind parameter estimation Cyclostationarity Second-order cyclic cumulant
Blind estimation of communication signal parameters plays a key role in various civilian and military applications, such as cognitive radio (CR) and noncooperative communication [1, 2]. The cognitive radio aims at improving the spectrum utilization by allowing the secondary user to sense and use the licensed spectrum, which needs to have the intelligence of dynamically changing its parameters. Therefore, it is necessary for the blind parameter estimation for CR to efficiently demodulate the signals in hostile wireless channel. In noncooperative communication, hostile communication signal must be detected, estimated, and recognized by its transmission parameters, which are vital for decisions involving electronic warfare operations and other tactical actions. Over the recent decades, orthogonal frequency division multiplexing (OFDM) has been widely employed in many wireless communication systems, owing to the advantages of high spectral efficiency and robustness against channel fading. OFDM is also a good alternative transmission scheme for underwater communication that both remedies the problem of ISI and provides low complexity solutions. For the OFDM signal, the recognized parameters include the symbol periodic, subcarrier spacing, subcarrier number, and cyclic prefix (CP) length.
The cyclostationarity properties of OFDM signal was widely exploited for various purposes: signal detection, classification [3–6], synchronization [7–9], and parameter estimation [10–13], in which the blind estimation of OFDM parameters have been investigated only in recent years, and these literatures generally achieved the estimation parameters by finding the cycles present in the cyclic statistics of OFDM signal. The algorithm proposed in  is based on the signal empirical distribution function. In , the cyclic correlation of OFDM signal is used to estimate the sampling frequency and CP length. A blind parameter estimation algorithm exploits the cyclostationary property of OFDM while considering the effect of time-dispersive channel . The work in  pursues more accurate blindly estimated results by processing the candidate set of cyclic frequencies with average idea. However, in all of these literatures, the detection procedure is overly empirical and not rigorous for estimation accuracy, e.g., by checking the cycle presence based on a predefined threshold. In addition, the exploitation of cyclic statistics only applied in single domain (time or frequency) does not make full use of the cyclostationarity properties of OFDM signal .
This paper presents a cyclostationarity-based approach to the blind estimation of OFDM transmission parameters, which derives the estimated parameters from both time (delay) and frequency domain using the second-order cyclic cumulant (CC) in a similar way as . It is shown that detection and estimation in two domains can improve the accuracy probability of estimating evidently. To ensure the preciseness of the approach, a statistic hypothesis testing was developed to determine the threshold for detecting the presence of cycles of OFDM signal, by which the theoretical relation between the threshold value and constant false alarm rate (CFAR) was derived. Another contribution of this paper is that it derived the analytical expression of second-order CC of OFDM signal in both additive white Gaussian noise (AWGN) and multipath channel, and consequently, the condition of estimating the parameters from second-order CC are achieved with the consideration of the effects of time dispersion and consistent estimator errors.
The paper is organized as follows: we first give the OFDM signal model in AWGN and multipath channels in Section 2. In Section 3, the analytical expression of second-order CC of OFDM signal and the condition of estimating the parameters are derived. The proposed cyclostationarity-based approach to the blind estimation of OFDM transmission parameters is given in Section 4. Simulation results are presented in Section 5, and finally, conclusions are drawn in Section 6.
2. Signal model
A discrete-time baseband OFDM signal, r(u), can be obtained by oversampling r(t) at a sampling frequency fs = ρK/T u , with ρ as the oversampling factor per subcarrier.
3. Second-order cyclostationarity of OFDM signal
3.1. Derivation for second-order cyclic cumulant
The cyclostationarity of OFDM signals has been proved in previous literature , which could be introduced by CP, symbol, or pilot. In the following, we derived the analytical expressions for the second-order cyclostationarity features of OFDM signal, which is specifically used by the blind parameter estimation approach proposed in Section 3. Basically, the higher-order cumulant needs the more computation complexity; hence, we only take the second-order cyclic cumulant of OFDM signal as our reference target, on which transmission parameter extraction executes under both AWGN and multipath channels.
where ⊗ represents convolution operation and δ(t) is the Dirac delta function. cs,2,1 represents the second-order cumulant of the signal constellation, cs,2,1 = E[sk,lsk,l + τ/T]. c w (t; τ) is the second-order cumulant of Gauss noise. Due to the wide-sense stationary of the noise, the cumulant of the noise does not depend on time; for simplicity, we omit the noise term in the following analysis.
where and D = ρK(1 + TcpT u −1) represents the number of samples over an OFDM symbol.
Similarly, the result in (13) omits the term for noise cumulant.
3.2. Analytical expression of CC properties for parameter extraction
This section will elaborate on the analysis on the CC properties in both delay and CF domain in order to efficiently extract and estimate the transmission parameters.
when T0 ⇒ + ∞, the estimator is close to theoretical C r (β; τ) asymptotically. However, because of the limitation of the sample amount, the CC magnitude is seldom exactly zero even if β is not a cyclic frequency.
From (15) and (16), the property of second-order CC of OFDM signal in time domain is concluded to be equal to zero unless when the delay is m × T u . Note that the estimation error because of limitation of the sample number also results in (15) being almost not established in practice. Apart from that, the effect of time dispersion came from multipath propagation which also appeared around τ = m × T u .
The analysis above employs the CC for continuous-time OFDM signal. For discrete-time OFDM signal, the derivation above and the conclusions for which the second-order CC still holds on are valid except that CF and delay should use the discrete expression. In other words, the τ = ± ρK only when β = lD−1, and (16) establishes when τ = ± ρK.
4. Joint delay and CF domain estimation for OFDM parameters
Based on the analysis of the second cyclic statistics of OFDM signal in Section 3, the number of subcarriers K can be estimated by searching the peak of second-order CC at delay domain, while the estimation of the duration of CP can utilize the discrete behaviors of CC in the cyclic frequency domain. Hence, the approach proposed for the joint blind estimation of the OFDM signal transmission parameters is comprised of two steps. We assume that the parameter estimation is applied after the process of signal detection, so the signal is already oversampled based on an estimate of the signal bandwidth. Moreover, we assume that sampling does not yield aliasing in the spectral and cyclic frequency domains. For the case of aliasing existing, it can be referred to  for an elimination method.
4.1. Estimation of subcarrier number K and symbol duration T u
The number of subcarriers of the OFDM signal, K, is estimated by exploiting the presence of the CP-induced peak in the CC magnitude at delay ρK and zero CF.
On the condition of β = 0, the CC magnitude of the baseband-received OFDM signal is checked over the delay range [ρKmin, ρKmax]. Herein, Kmin and Kmax represent the minimum and maximum number of subcarriers to be estimated, respectively. The parameter ρKmax should be chosen large enough to cover the largest candidate peak while avoiding introducing more amount of unnecessary calculation process. The parameter ρKmin should be chosen far from τ = 0 in order to reduce the influence caused by the channel time dispersion (See Figure 2), but larger ρKmin leading to some small candidate peak cannot be covered. With respect to the consideration, we propose an approach to estimating the number of subcarrier K as follows.
Firstly, we should set a predefined threshold Tth for checking the presence of nonzero second-order CC at interest delay, and the determination of the threshold value will be derived in Section 4.3. Next, we search the candidate second-order CC for the amplitude value larger than the threshold over the [ρKmin, ρKmax] delay range. Among the set of the discrete values larger than the threshold, we choose that with the largest delay value, depicted by τ K . This delay value is exactly what we will utilize to extract the number of subcarriers K.
The duration of OFDM symbol is also calculated as T u = ρK/f s accordingly.
4.2. Estimation of CP duration and OFDM symbol period T
where b is the minimum integer between 1 and bmax for which a positive value is obtained for CP. Note that although the maximum peak is achieved for b = 1 theoretically, another peak may be actually selected under both small observation intervals and low signal-to-noise ratio (SNR) conditions.
In the above analysis, we can see that the proposed algorithm requires neither carrier, waveform, and symbol-timing recovery nor estimation of noise and signal power. It means that the proposed algorithm is robust for both channel fading and synchronization results.
4.3. Determination of threshold value Tth
Because of the existence of noise and estimation error for CC, the CC amplitude is almost not a zero value even under the condition of (15). For this cause, it needs to preparedly set up a threshold for the test problem. However, the threshold relied on the experiment, and the observation (e.g., in [11–13]) is not rigorous, for which it lacks the relationship with the correct recognizing probability. In the following, by referring to the result in [17, 18], we give a method to determine the threshold, by which the threshold value will be derived from the constant false alarm rate.
The distribution of ϵ r is unknown, and it is close to zero when T0 → ∞. In , it is stated that is an asymptotical complex Gaussian variable, and C r (β; τ) is nonrandom. Therefore, it can be proved that the independent distribution of r(t), which is the distribution of ϵ r , is asymptotical Gaussian with zero mean. Consequently, for hypothesis H0 in (23), Fst can be viewed as a chi-square distribution variable with 2N degrees of freedom, where N is the number of collected samples with different delay used by outputting the consistent estimator in (14).
In other words, the threshold η th is uniquely determined by the given CFAR probability P f , by which the theoretical relation between the threshold value and constant false alarm rate was achieved.
5. Simulation results
In the simulation, the OFDM useful symbol duration K is set to 128, 256, 512, and 1,024, and the CP is set to 1/4, 1/8, 1/16, and 1/32. The OFDM subcarrier features a quadrature phase shift keying constellation and a bandwidth of 800 KHz. At the transmitter side, a root-raised cosine pulse shape with a roll-off factor of 0.025 is employed before the A/D converting. In addition, the oversampling frequency f s is set to 3.2 MHz, Δf K to 320KHz, and bmax to 4. All results are simulating under multipath fading channels.
Accuracy rate of estimation of subcarrier number K
Accuracy rate of estimation of CP length
This paper has investigated the blind estimation of OFDM parameters based on the signal’s cyclostationarity. Firstly, the possibility of extracting the transmission parameters from second-order CC is derived theoretically, which analyzes both the continuous and discrete signal models under the AWGN channel and multipath channel, respectively. The proposed estimation approach jointly exploits the delay and cyclic frequency domain properties of second-order CC and develops a CFAR-based statistic hypothesis testing framework to detect the presence of CP-introduced cyclostationarity. The approach only assumes that the OFDM signal have the cyclic prefix and does not require any priori condition, so it can be easily extended to most cases that need blind estimation of OFDM parameters.
This work was supported by the National Natural Science Foundation of China under grant 61171109 and the BUPT Research Innovation Project under grant 2013RC0104.
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