Fine Residual Carrier Frequency and Sampling Frequency Estimation in Wireless OFDM Systems
Abstract
This paper presents a novel algorithm for residual phase estimation in wireless OFDM systems, including the carrier frequency offset (CFO) and the sampling frequency offset (SFO). The subcarriers are partitioned into several regions which exhibit pairwise correlations. The phase increment between successive OFDM blocks is exploited which can be estimated by two estimators with different computational loads. Numerical results of estimation variance are presented. Simulations indicate performance improvement of the proposed technique over several conventional schemes in a multipath channel.
I Introduction
Although the OrthogonalFrequencyDivisionMultiplexing (OFDM) technique significantly enhances the system performance under frequencyselective fading channels, it is vulnerable to synchronization nonidealities, including the symbol timing offset (STO), carrier frequency offset (CFO), and sampling frequency offset (SFO).
The previous works including [1, 2, 3] deal with the coarse STO and CFO estimation in time domain before Fast Fourier Transform (FFT). However, due to the imperfections of compensation, after FFT, the residual part of CFO remains to be corrected. Also, at this stage, SFO should be estimated and removed; otherwise, it would lead to a phase rotation not only proportional to the tone index within one OFDM block (interblock increment), but also grows linearly for successive OFDM blocks (intrablock increment) [4].
In literature, several schemes are proposed to estimate or track the residual CFO and SFO in frequency domain with the assistance from pilot subcarriers [5, 6, 7]. In [5], Speth et al. utilize the symmetric locations of pilots to estimate CFO and SFO jointly. However, its performance degrades in the multipath channels. [6] suggests three estimators with the help of the least square estimation (LSE). An improved weighted LSE variant is proposed by Tsai et al. in [7] which requires the secondorder statistics of the channel state information (CSI). In general, these schemes mainly rely on the linearly growing interblock increment.
This paper proposes a novel technique to make use of the intrablock increment spanning a number of OFDM blocks. By dividing the subcarrier index into several regions, the method is capable of exploiting the pairwise correlation which leads to accurate results after applying least square fitting. Two variants differing in computation complexity are presented with their numerical variances derived.
Ii OFDM Signal Model with CFO and SFO
We consider an OFDM system where the transmitted data is modulated by an point Inverse FFT (IFFT). Assuming a total of OFDM blocks to be delivered and each block consists of data samples (), the complex baseband signal is described by
(1) 
where are the locations of the data subcarriers; for , is either pilot or null subcarrier. is the length of the guard interval, the total length of an entire OFDM block given by , and the sampling interval. is the rectangular function defined as
(2) 
The multipath channel is
(3) 
where is the total number of taps, the independent and Rayleigh distributed complex channel gains, the timing delay of each path, and the delta function. Here, we assume that .
Upconverting to a carrier frequency , the postchannel equivalent signal takes the form
(4) 
where the notation stands for linear convolution, and the complex, identically independently distributed (i.i.d.), additive white Gaussian noise (AWGN) with zero mean and variance ; also, it is wide sense stationary (WSS), with independent real and imaginary part, and equal variance in both parts (). Now, assuming a CFO and a SFO given as
(5)  
(6) 
where is the deviated carrier frequency and the deviated sampling interval at the receiver. The received th sample in the th OFDM block is
(7) 
After discarding the samples in the guard interval, the complex data for the th block and on the th subcarrier is [8]
(8) 
where is the channel transfer function (CTF) in frequency domain; and the normalized CFO to the subcarrier spacing; is the amplitude attenuation approaching unity and can be safely neglected; the intercarrier interference (ICI) due to distorted orthogonality of subcarriers; is the WSS i.i.d. Gaussian noise in frequency domain with independent real and imaginary parts. Without loss of generality, and can be regarded as the residual part of CFO and SFO after coarse synchronization or imperfect channel estimation and equalization.
Iii Proposed Technique
Define the full set of subcarrier index as , which can be further divided into equallyspaced regions (assuming even and , and is divisible by ), denoted as where
(9) 
Here, denotes integers. Ignoring disturbances of ICI, using equation (II), for the th segment in the th OFDM block, the pairwise correlation is
(10) 
where
(11)  
(12)  
(13)  
(14)  
(15) 
and . Clearly, the extra phase rotation of the useful part in (13) is irrelevant to subcarrier index and ; it is only pertinent to the OFDM block index and segment index . The cross terms are the main disturbance in estimation. In practice, the contribution of signal and channel () should be replaced by
(16) 
where
(17) 
The notation is the full set of pilots and the full set of null subcarriers. is the estimated , obtained by the decision feedback device. is the estimated CTF. could be combined using a certain weight given as
(18) 
where . See Appendix A for the details of such selection for the weighted . Computation of weighted requires the secondorder statistics of signal, channel and noise, avoided by the simplified scheme. For constantmodulus modulation, the weighted reduces to
(19) 
where . To obtain estimation of , we coherently stack by
(20) 
where denotes the conjugation of its argument and
(21) 
is the left half of while the right one; is the absolute complement of given by ; in general, the twodimensional set . can be estimated by
(22) 
The vector can be linearized into
(23) 
where is the observation matrix expressed by
(24)  
(25)  
(26) 
The vectors take the form
(27) 
where ; is the associated estimation error vector. By least square fitting, is given by
(28) 
Note that both and give estimation of ; denotes the th entry of a vector/matrix. Here, we choose and arrange all into the vector which leads to
(29) 
where , is the error vector, and
(30) 
is the observation matrix. Another least square fitting yields
(31) 
The estimated and are and respectively. A simple sketch with is drawn in Fig. 1.
Assuming correctness in tackling the phase ambiguity in the linearization process, derived in Appendix A, estimation using either the weighted or simplified is unbiased. On the other hand, the numerical variances of and are
(32)  
(33) 
where
(34)  
(35)  
(36)  
(37)  
(38) 
Appendix A validates and thus , . The equality establishes if and only if (iff)
 A1:

the channel experiences flat fading ()
 A2:

constant modulus modulation ()
Otherwise, weighted estimation always outperforms simplified estimation. To achieve the best performance, further assuming
 A3:

equal and maximal cardinality of each set , denoted by
The variances under conditions A1A3 are
(39)  
(40) 
where ; .
Remarks:
i) Apparently, an immediate way to enhance the performance is to raise , which leads to asymptotically decreasing variances in cubic scale. Nevertheless, if the application is realtime oriented rather than quality preferred, where and should be tracked in the fastest manner, and should be replaced by their minimums as .
ii) If is used for estimation, (39) and (40) are rewritten into
(41)  
(42) 
which are significantly higher than (39) and (40) increasing squarely with . Therefore, it is reasonable to use .
iii) According to (10), statistically, the proposed technique only relies on the independency and equal variance of the real and imaginary parts, and WSS assumptions of noise; it does not require the power spectrum density (PSD) of noise to be strictly flat (white), since the expectation of the cross terms is zero.
Iv Simulation
In this section, we consider a wireless OFDM system with FFT size . The guard interval is . Thus, the length of an entire OFDM block is . The total number of OFDM blocks is , and the total number of segments is . The carrier frequency is set at GHz. The sampling period is . For brevity and to exploit the best performance of the proposed estimators as well as other conventional pilotassisted schemes, all subcarriers are regarded as pilots; otherwise, notations in (17) must be used which varies with the accuracy of decision feedback device. The signal is modulated from 16PSK constellation. The channel consists of Rayleigh taps, which are statistically independent distributed with a power delay profile decaying exponentially:
(43) 
For the proposed estimators and the scheme in [7], CTF is assumed to be known perfectly as well as ^{1}^{1}1For OFDM systems containing null subcarriers, could be estimated, which is omitted in this paper., unless otherwise mentioned. Mean squared error (MSE) results are used to benchmark the performance, defined as and where denotes the expectation of its argument.
Iva Comparison of
Fig. 3 highlights the comparison of among the proposed estimators with other schemes in [5, 6, 7]. Numerical result of in (39) is drawn by assuming A1A3. In the multipath channel, both of the weighted and simplified estimator achieve the best performances. In the flat fading scenario, the weighted estimator reduces to the simplified one. provides a tight bound in moderate .
IvB Comparison of
IvC with a Varying
Fig. 5 shows the deviation of when changes under dB in multipath channel. For the proposed estimators, is asymmetric for negative and positive due to the presence of a positive ; for , the performance degrades gradually with a higher since the ICI is increasing simultaneously. For a major part, [7] and the simplified estimator entangle with each other.
IvD with a Varying
IvE with Different Estimation Accuracy
Fig. 7 shows the performance with different channel estimation accuracy in multipath channel. Employing the same philosophy of [9], the inaccurate channel estimation takes the form
(44) 
where represents the estimation accuracy, and the additional complex noise with zero mean and variance in its real and imaginary part respectively, independent from . The weighted estimator degrades significantly when severe inaccuracy occurs, which hinders the performance improvements especially in moderate to high region. For moderate to high , the weighted estimator outperforms the simplified one under different . Similar conclusion can be drawn for .
IvF and under Mobility
Fig. 7 exhibits the and in presence of terminal mobility with dB under multipath channel. Merely the CSI pertinent to the first moment of the Rayleigh fading channel in simulation (=0) is assumed to be known a priori; for the ensuing frames, the same CSI is used which entails a loss in channel estimation accuracy. The Doppler bandwidths with respect to terminal speed of are Hz. In general, the performance deviation is insignificant if not imperceptible even the terminal velocity reaches , since the maximal value of the product between the Doppler bandwidth and the duration of an OFDM block is , a relatively small value. Thus, the CSI is sound enough to secure an excellent estimation.
V Concluding Remarks
In this paper, we propose a joint estimation technique to deal with residual CFO and SFO estimation. By dividing the subcarrier index into a number of regions and exploiting the pairwise correlation, we estimate the phase increment between adjacent OFDM blocks, which yields accurate estimation after times of least square fitting. Extensive simulations indicate better performance over several conventional pilotassisted schemes.
Appendix A Bias and Variance of Estimation
First of all, consider the case of the simplified in (III) and the statistical information of in (22). For the th element of (), subtracting the left hand side of (22) yields
(A.45) 
If is in vicinity of , using the approximation for small enough, we may write
(A.46) 
where and represent the real and imaginary part of the arguments. Expectation of in (A.46) is
(A.47) 
which holds if . In fact, each component in contains either , , or (see (10)) and therefore, , which finally leads to the unbiasedness of in (31) since only linear intermediate operations are involved.
For the numerical variance of , we could use
(A.48) 
if . Standard calculations yield
(A.49)  
(A.50) 
Note that, for visual clearance, we abbreviate the notation with . Thus,
(A.51) 
where is defined in (34). Using the linear intermediate manipulations, we derive (32) and (33). Substituting with backward produces the weighted version of in (III) after lengthy calculations.
To prove , we invoke the CauchySchwarzInequality [10]. The essential steps are listed below.

To prove
missing(A.52) ,k_missing2
is equivalent to prove
(A.53) 
Now, using and , the CauchySchwarzInequality gives
(A.54) 
where the right hand side of (A.54) is exactly the right hand side of (A.53). The ’’ holds iff
(A.55) 
and we have the conditions A1, A2 ^{2}^{2}2Actually, it is merely a sufficient condition of (A.55). Specifically, consider that there is a sage at the transmitter side who could render where C is a constant, then (A.55) also establishes. However, it is not insightful to be pursued.. Therefore, we verify that .
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