- Research Article
- Open Access
Estimation of CFO and Channels in Phase-Shift Orthogonal Pilot-Aided OFDM Systems with Transmitter Diversity
© C. Ribeiro and A. Gameiro. 2009
- Received: 1 July 2008
- Accepted: 23 January 2009
- Published: 3 March 2009
We present a CFO estimation algorithm and an associated channel estimation method for broadband OFDM systems with transmitter diversity. The CFO estimation algorithm explores the TD structure of the transmitted symbols carrying pilots and data, relying solely on the data component present on the symbols to estimate the CFO, thus avoiding additional overhead like training symbols or null subcarriers. An intermediate output of the CFO algorithm provides an easy-to-get initial CIR estimate that will be improved with the utilization of a TD LMMSE filter. The feasibility of the investigated methods is substantiated by system simulation using indoor and outdoor broadband wireless channel models. Simulation results show that the joint algorithms provide a near optimal system's performance.
- Channel Estimation
- Carrier Frequency Offset
- Cyclic Prefix
- Channel Impulse Response
- OFDM System
Future mobile broadband applications will require reliable high data-rate wireless communication systems. In recent years, multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) transmission systems [1–4] emerged as the scheme with the potential to fulfill these conditions, with bandwidth efficiency and robustness to frequency selective channels, common in mobile personal communication systems.
Various forms of OFDM have been adopted in different standards: WIMAX, LTE, IEEE.802.11a/g , IEEE.802.16 , and DAB/DVB . However, the long symbol duration makes OFDM systems particularly sensitive to carrier frequency offsets (CFOs) that always exist between the base station (BS) and mobile terminal (MT). The presence of CFO destroys the orthogonality among subcarriers leading to intercarrier interference (ICI), that causes severe degradation of the system's bit error rate (BER) [7–9].
The estimation and removal of the CFO has been the focus of a considerable number of works published in recent years. The algorithms can be categorized as blind or data-aided. The first category explores the properties of the received symbols (commonly the cyclic prefix (CP)) [10–12]. The data-aided algorithms use dedicated training symbols [13, 14] or exploit the presence of null subcarriers [15, 16].
The accurate extraction of the channel state information is crucial to realize the full potential of the MIMO-OFDM system. The performance of the channel estimator is vital for diversity combining, coherent detection and decoding, and resource allocation operations. The cochannel interference inherent to the system, where the received signal is the superposition of the signals transmitted simultaneous from the different antennas, puts an additional challenge on the design of the channel estimation algorithm.
A decision-directed channel estimation scheme that attempted to minimize the cochannel interference was published in . The proposed algorithm exhibits a high computational load. A simplified and enhanced algorithm, introducing a data-aided scheme for the data transmission mode, is presented in . The topic attracted a significant attention and has been the focus of investigation in multiple publications [19–21] and references therein.
The design of training symbols and pilot sequences with the ability to decouple the cochannel interference and minimize the channel estimation mean square error (MSE) for MIMO-OFDM was addressed in several publications [18, 22, 23].
Most publications on the topic of training-signal or pilot-aided channel estimation use the frequency-domain (FD) least squares (LS) estimates as the starting point for the analysis of the estimation algorithm or the design of the training sequence. It was established in  that in single-input single-output (SISO) OFDM a time-domain (TD) equivalent to LS estimate could be obtained using a simple linear operation on the received signal, if the used pilot sequence fulfills certain conditions.
This paper contains a proposal for a CFO estimation algorithm and associated channel estimation method for OFDM systems with transmitter diversity that exploits a standardized transmission format, where FD pilot symbols are regularly spread in the OFDM symbols. To minimize the pilot overhead, the pilot subcarriers are shared among all transmit antennas. To mitigate the resulting cochannel interference, the system adopts phase-shifted pilot sequences per transmit antenna . By exploring the TD properties of the received symbols, the proposed algorithms are able to estimate and remove the CFO, separate each of the CIRs, and generate the final channel estimate, without requiring any additional overhead (training symbols or null subcarriers). By performing most of the operations on the TD received symbols and sharing operations, the overall computational load required to implement both algorithms is affordable for real-time implementations.
The paper is organized as follows. Section 2 gives a brief introduction to the wireless multipath channel and the OFDM baseband model. In Section 3, the investigated CFO and channel estimation algorithms are developed. The feasibility of the developed method is substantiated by simulation results presented in Section 4. Finally, conclusions are drawn in Section 5.
Before introducing the investigated method, we will briefly overview the mobile wireless multipath channel and the considered OFDM baseband model.
Throughout the text, the notation is used for TD vectors and elements, and its absence denotes frequency-domain (FD) vectors and elements. The index denotes TD elements and FD elements. Unless stated otherwise, the vectors involved in the transmission/reception process are column vectors with complex elements. The superscripts and denote transpose and Hermitian transpose, respectively.
2.1. The Wireless Multipath Channel
where is the total number of subcarriers of the OFDM system.
2.2. OFDM Baseband Model
To assist in the channel estimation process, pilot symbols are added in each transmit antenna path. The vectors hold the pilot values for each path. The pilots are transmitted in dedicated subcarries (vectors and contain nonzero values in disjoint positions). The resulting FD signal transmitted by antenna is . All transmit antennas use the common set of subcarriers to convey the overlapping pilot sequences. The pilots are regularly spread every subcarriers. The pilot separation can range from 1 (particular case where all subcarriers in the OFDM symbol are dedicated to transmit pilots–training symbol) to , fulfilling the condition .
where and is the first pilot subcarrier.
The inverse discrete Fourier transform (DFT) block present in each antenna path transforms the input vector into the TD vector , using an efficient -points inverse fast Fourier transform (FFT) algorithm.
where is the DFT matrix, and is the matrix that adds the CP, with denoting the identity matrix and denoting the last columns of . The TD vectors and collect, respectively, the components of the data symbols and pilot symbols present in . The vectors are simultaneously transmitted to the receiver's antenna.
Let be the normalized angular CFO, where is the frequency offset due to the frequency mismatch of the oscillators of the transmitter and the receiver, and is the sampling interval.
where is the circulant matrix with circulant vector and, due to the properties of the DFT, , with the elements of defined by (2).
It is clear that if , then and the CFO is completely removed. As it will be demonstrated in the next section, the CFO ambiguity remaining after this block is an integer multiple of the pilot subcarrier separation , where is the subcarrier separation. This acquisition range should be sufficient for current OFDM systems; however coarse CFO estimation techniques  can be used to tackle this limitation, if proven necessary.
where is the resulting FD noise vector. The remaining phase-rotation is naturally removed in the channel estimation process, assuming that the pilot-aided scheme calculates the LS estimates (back-rotated received signal).
The deframing block separates the signals in the subcarriers conveying pilots and data symbols. The values in the data subcarriers are collected in vector and fed to the decoding block. Together with the channels' estimate , this block is now able to decode the received symbols, according to some decision rule, and generate the estimate of the transmitted data .
The algorithms implemented in this block estimate both the channels over which the transmission occurred and the CFO that affects the received signal. The inputs to the CFO estimation algorithm are the TD symbols carrying both pilots and data, according to the model defined in the previous section. The channel estimation algorithm reuses an intermediate output of the previous operation to attain an initial CIR estimate with minimal computational load.
3.1. Analysis of the TD Symbol's Structure
where and hold the data-dependent and pilot-dependent components in , respectively.
it becomes clear that it is made up of frequency-shifted and scaled replicas of each of the CIR. Moreover, the replicas of each CIR are separated by samples and transmit antenna CIR replicas are time-shifted samples from the reference position .
3.2. CFO Estimation
The CFO estimation method introduced in the following uses the pilot structures, introduced primarily for channel estimation purposes, to estimate the CFO present in the received samples. Therefore, it is absolutely bandwidth efficient, as it does not require any additional specific overhead. The algorithm exhibits a fast acquisition, being able to output an estimate with low deviation from a single OFDM frame. It proves adequate for burst mode transmission, where the frequency offset varies from frame to frame.
that clearly shows that the pilot-dependent samples are limited to the sets of samples (with elements), where the corresponding phase-shifted CIRs have energy. The remaining samples will depend only on the transmitted data and noise.
where is the initial candidate frequency offset.
We can conclude that has minimum values spread Hz, with maximum magnitude values in between, separated by Hz.
where is the estimated CFO value. The exhaustive line search is computationally demanding, depending on the search's granularity. Hence, there is a tradeoff between complexity and estimate's variance.
The cost function has a closed form expression, and its behavior is perfectly described. In the acquisition range, there are maximum values; in the interval limited by the maximum values that surround the perfect estimate, presents a smooth shape with a single minimum. Using the knowledge we possess of the cost function, we propose a 2-step approach to find its minimum value. The initial step performs a coarse line search to locate the global minimum interval. Testing candidate CFO values evenly spaced by Hz should suffice. The candidate CFO will be the one with the lowest cost. If the number of elements in is small and SNR is very low, the probability of wrong identification may not be negligible and the number of candidate CFO values can be increased thus decreasing the wrong identification probability. In the final step, we use the gradient descent method  to find the global minimum.
3.3. Channel Estimation
Assuming that the CFO is completely eliminated, the output of the initial operation of the CFO algorithm is made up of the pilot-dependent component and noise . The data-dependent component was eliminated from this vector, opening way to easily obtain an initial CIR estimate.
In , it was demonstrated that for a single transmitting antenna OFDM system with perfect synchronization, (30) is the TD counterpart of FD LS estimate. By using phase-shifted pilot sequences that allow the separation of the different CIRs, the same result holds in the present model.
This initial estimate can be significantly improved by incorporating a TD linear minimum MSE (LMMSE) filter to reduce the estimate's error, taking advantage of the CIR energy concentration. The improvements provided by this filter are especially significant for low values of SNR.
The resulting CIR and CFR estimates are, respectively, and .
A simulation scenario was implemented using an Alamouti OFDM system with modulated subcarriers, sampling interval nanoseconds and a CP with 100 samples. The transmitted OFDM symbols carried pilots and data, with a pilot separation . The OFDM frame consists of 16 symbols. The CFO value was randomly generated in each frame with a value inside the acquisition range . The CFO estimation and removal was performed on a frame basis. Two channel models with exponentially decaying power delay profile (PDP) were used to simulate indoor (50 nanoseconds rms delay spread) and outdoor environments (250 nanoseconds rms delay spread). To validate the proposed method, several simulations were performed using Eb/N0 values in the range of 0 dB to 20 dB.
The channel estimation MSE improvement that can be observed for higher-order modulations is due to the fact that the ratio between the powers in the pilot symbols and data symbols is kept constant in all simulations. The large increase of delay spread between both channels is the origin of the ~3 dB MSE degradation when moving from the indoor channel to the outdoor channel plots. This acceptable degradation shows the ability of the estimator to deal with the increasing channel delay spread by always weighing the energy of channel taps versus noise variance. The channel estimation BER plots present a degradation of ~1,2 dB that can be largely attributed to the 12.5% pilot overhead.
The joint CFO and channel estimation MSE is an effective measure of the degradation caused by both algorithms. In these plots, the estimated channel was compared against the true channel affected by the same CFO that distorted the received signal (according to (10)). The results plotted in Figures 6 and 7 show that the performance degradation of the joint process is marginal when compared with channel estimation only, substantiating the performance of the proposed algorithms.
We have investigated a CFO estimation algorithm and an associated channel estimation block for OFDM with transmitter diversity that explores the TD structure of transmitted symbols carrying pilots and data. The CFO algorithm relies solely on the data component present on the symbols to estimate the CFO, avoiding additional overhead like training symbols or null subcarriers. Simulation results show that the residual CFO has a minimal impact in the system's performance, confirming that the CFO estimates have minimal deviation from the true value. The definition and shape of the cost function determine a very low-complexity scheme. An intermediate output of the CFO algorithm provides an easy to get initial CIR estimate minimizing the overall complexity. By incorporating a TD LMMSE filter, the initial CIR estimate is significantly improved. Simulation results of the joint algorithms confirm a reduced degradation of the system's performance when compared with the ideal scenario.
The authors wish to thank Fundação para a Ciência e a Tecnologia that partially supported this work through the project "PHOTON—Distributed and Extendible Heterogeneous Radio Architectures using Fibre Optic Networks" (PTDC/EEA-TEL/72890/2006).
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