CFO Estimation with Autoregressive Process

1 Introduction

The OFDM modulation has been widely used in modern communication systems, because of its robustness against the frequency selectivity in wireless channel. With inserting Cyclic Prefix (CP), OFDM system can convert
a frequency selective channel into a parallel collection of frequency flat channels, leading to a great simplified equalizer which increases the robustness of OFDM system against inter symbol interference (ISI). OFDM has been used in important applications like Wireless LAN IEEE 802.11a, IEEE 802.11g, IEEE 802.11n, IEEE 802.11ac, and IEEE 802.11ad, Digital Audio Broadcasting (DAB), Digital television DVB-T/T2 (terrestrial), DVB-H (handheld), DMB-T/H, DVB-C2 (cable), Worldwide Interoperability for Microwave Access (WiMAX), Asymmetric Digital Subscriber Line (ADSL) and The LTE and LTE Advanced 4G mobile phone standards.

OFDM systems are very sensitive to frequency synchronization. ­This is due to Doppler shift (caused by relative motion) and a mismatch between carrier frequency by the non-synchronized local oscillators or between transmitter and receiver. The CFO can be several times larger than the subcarrier spacing. It is usually divide into an integer part and a fractional part. Fractional part of CFO introduces inter-carrier interference (ICI) between subcarriers. It corrupts the mutual orthogonality between subcarriers and results in bit error rate (BER) degradation. Integer CFO does not introduce ICI between subcarriers, but does introduce a cyclic shift of data subcarriers and a phase change proportional to OFDM symbol number. The orthogonality between subcarriers is still maintained but the received data symbols, which were mapped to the OFDM spectrum, are now in the wrong position.

Imprecise CFO estimation causes to a severe performance degradation because of the reduction of the signal
amplitude at the output of each matched filter and to the interference between adjacent subchannels [1]. In the most recent years so many works have been directed toward the advancement of CFO synchronization methods for OFDM systems operating in both data-aided and non-data-aided (or blind) contexts. Additionally, a combination of data-aided and blind methods has been recently considered, which referred to semiblind approach. For example, blind methods with using the orthogonality between null subcarriers padded in the transmitted OFDM symbol and the information-bearing subcarriers, stick out in [2,3], and in [4] redundant information held within the cyclic prefix (CP) preceding the OFDM symbols made possible estimation without additional pilots, joint symbol timing and CFO maximum likelihood (ML) estimator has been derived. In [5] frequency offsets are estimated by inserting null subcarriers into a single OFDM block and a deterministic maximum likelihood (ML) approach for CFO estimation has been derived.

In this paper we describe a method for estimating the CFO with exploiting an autoregressive (AR) process from a finite number of noisy measurements of received signal[6]. The method utilizes a modified set of Yule-Walker (YW) equations that lead to a quadratic eigenvalue problem that, when solved, gives estimates of the AR parameters and then we can estimate CFO from AR parameters.

2 SIGNAL MODEL:

In OFDM systems, data are sent block by block. A series of complex data is split into blocks and allocated to subcarriers. Let be the complex data belonging to the – OFDM block, the – block of the OFDM signal can be expressed as

,                                                                                                    (1)

where , is the normalizing factor. and are the symbol duration and subcarrier spacing of OFDM, respectively. For simplicity we omit the index . We can easily perform OFDM with DFT, which can be efficiently implemented by low complexity fast Fourier transform (FFT).

The sampling space of an OFDM signal of bandwidth B is

                                                                                                                               (2)

The discrete-time transmit signal, is

                                                                                                                        (3)

which is known as the DFT of the sequence  if . Hence, the IFFT of the data block is

     ,                                                                                            (4)

After removing the Cyclic Prefix and taking FFT, the complex envelope of the baseband received signal in an OFDM block in the absence of timing offset and channel distortion can be described as

                                                                                        (5)

where is the channel response at the k-th subcarrier frequency, is additive noise, which is assumed to be zero-mean, uncorrelated, circular complex Gaussian random vector (C-CGRV) [7], with variance

 .

We can write (5) as

                                                                                                                        (6)

If we use training symbols for transmitting data (or we can simply multiply to ) then we can assume that (or ), now we can easily write as

                                                                                                                             (7)

3 Quadratic Eigenvalue CFO Estimation

We want to estimate CFO as a frequency based estimation problem which leading to a Quadratic Eigenvalue problem.

Consider this received signal without noise component. Note that α is a constant and can be omitted.

                                                                                                                                            (8)

we can easily assume then we can rewrite the equation (8) to:

                                                                                                                                         (9)

which (9) yield that

                                                                                                                                (10)

By multiplying (10) by and taking expectation gives the Yule-Walker equations of order one as follows

                                                                                                                (11)

we can use (7) to have

.                                                                                                     (12)

substituting (12) to (11) yields

                                                                                     (13)

rearranging (13) we have first order AR process

                                                                                           (14)

now we can expand (14) from lag=0 to lag=q and

                                                                                                  (15)

Now we can write in matrix form

                                                                                                                                  (16)

where

, ,       and .                                                              (17)

where is a column vector having zeros, and so that a unique solution is guaranteed. The dimensions of  , B, and V are and respectively. By multiplying both sides of (16) by leads to the quadratic eigenvalue problem [6]

                                                                                                                       (18)

where                                                                             (19)

Each of the matrices in (19) has dimension . There are so many methods to solve the quadratic eigenvalue problem [8]. One method is to define

                                                                                                              (20)

It is clearly confirmed that solving (18) is equal to solving the 4-dimensional linear eigenvalue problem

                                                                                                                                         (21)

Let be eigenvectors solving (21) these, along with their corresponding eigenvalues, appear in complex conjugate pairs [8]. The proposed subspace method can be solved as follows [6].

1) Form estimates of the autocorrelation matrix defined in (17).

2) Form the matrices and defined in (19).

3) Form the matrices P and Q defined in (20).

4) Solve the generalized eigenvalue problem

                                                                                                                                                (22)

5) The estimation of AR parameter, of the generalized eigenvector associated with the generalized eigenvalue having minimum modulus. The estimation of is given by

where denote the th entry of .

Now we can estimate the CFO by

.

References:

[1] P.H. Moose, A technique for orthogonal frequency division multiplexing frequency offset correction, IEEE Trans.
Comm. 42 (October 1994) 2908-2914.
[2] H. Liu, U. Tureli, A high-efficiency carrier estimator for OFDM communications, IEEE Trans. Comm. Lett. 2 (April
1998) 104-106.
[3] U. Tureli, H. Liu, M.D. Zoltowski, OFDM blind carrier offset estimation: ESPRIT, IEEE Trans. Comm. 48 (9)
(September 2000) 1459-1461.
[4] J.J. van de Beek, M. Sandell, P.O. Bo ¨ rjesson, ML estimation of time and frequency offset in OFDM systems, IEEE Trans.
Signal Process. 45 (July 1997) 1800-1805.

[5] M. Ghogho, A. Swami and G.B. Giannakis, “Optimized null-subcarrier selection for CFO estimation in OFDM over frequency-selective fading channels”, vol. 1, pp. 202-206.

[6] C.E. Davila, “A subspace approach to estimation of autoregressive parameters from noisy measurements”, IEEE Trans. Signal Processing, vol. 46, no. 2, pp. 531-534, 1998.

[7] B. Picinbono, On circularity, IEEE Trans. Signal Process. 42 (December 1994) 3473-3482.

[8] F. Chatelin, Eigenvalues of Matrices. New York: Wiley, 1993.