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.
1. Introduction
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 [5], IEEE.802.16 [6], and DAB/DVB [1]. 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].
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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 [17]. 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 [18]. 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].
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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 [24] 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 [18]. 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.
2. OFDM in Mobile Wireless Channels
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
Let us consider that the system transmits over multipath Rayleigh fading wireless channels modeled by the discrete-time channel impulse response (CIR):
(1)
where is the number of channel paths, and are the complex value and delay of path , respectively. The paths are assumed to be statistically independent, with normalized average power, , where is the average power of path . The channel is time variant due to the motion of the mobile terminal (MT), but we will assume that the CIR is constant during one OFDM symbol. The time dependence of the CIR is not present in the notation for simplicity. Assuming that the insertion of a long enough cyclic prefix (CP) in the transmitter assures that the orthogonality of the subcarriers is maintained after transmission, the channel frequency response (CFR) can be expressed as
(2)
where is the total number of subcarriers of the OFDM system.
2.2. OFDM Baseband Model
Consider the OFDM baseband system with transmit antennas depicted in Figure 1. The vectors hold the -ary PSK or QAM coded data to be transmitted.
Figure 1
OFDM baseband system 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 .
The system uses distinct phase-shifted pilot sequences in each transmit antenna to allow the separation of the sequences in the receiver. The th element of the vector is defined by
(3)
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.
An samples long guard interval, in the form of CP, is prefixed to vector , resulting in the TD transmitted vector
(4)
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.
The th received signal sample of the th symbol can be expressed as
(5)
where is a sample of independent and identically distributed (iid) zero mean additive white Gaussian noise (AWGN) with variance . Collecting the samples of the symbol,
(6)
where the vector collects the noise samples that affect the th symbol, the vector represents the intersymbol interference (ISI) caused by the channel dispersion, and the matrix is the lower triangular Toeplitz channel convolution matrix with first column (column -vector with the discrete-time CIR (its elements are defined by (1)) padded with zeros). The diagonal matrix that holds the phase rotation that affects each symbol sample is
(7)
The receiver starts by removing the CP from the received symbol. Dropping the symbol index, the resulting vector is
(8)
where is the matrix that removes the CP with representing the null matrix, is the resulting TD noise column -vector, and is the common phase that affects all samples of the th symbol. The last step in (8) was possible considering the structure of the matrices involved
(9)
With the assumption that the length of the CP is larger than the duration of CIR, the ISI is completely removed, and (8) can be written as
(10)
where is the circulant matrix with circulant vector and, due to the properties of the DFT, , with the elements of defined by (2).
The CFO and channel estimation block is responsible for estimating both the CFO that affects the received samples and the channels that disturbed the transmission process. Both estimation algorithms will be introduced in the next section. Moreover, this block is also responsible for reducing the CFO, using the estimated CFO value . This operation can be described by
(11)
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 [25] can be used to tackle this limitation, if proven necessary.
The DFT block transforms the vector to FD with an efficient FFT operation. Assuming that the CFO is completely eliminated, the resulting FD column -vector can be expressed as
(12)
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 .
3. CFO and Channel Estimations by Exploring the TD Properties of Phase-Shifted Pilot Sequences
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
From (10), each element of the TD received symbols (carrying pilots and data), after CP extraction, can be expressed by
(13)
where the elements of the TD data vector are
(14)
where is the th element of (complex data symbol conveyed by the th subcarrier of the transmit antenna path), and the elements of the TD pilot vector are
(15)
Replacing (14) and (15) in (13),
(16)
where and hold the data-dependent and pilot-dependent components in , respectively.
By expanding the pilot-dependent vector
(17)
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.
The algorithm requires a search within the acquisition range to find the minimum value of the cost function. An initial candidate angular frequency offset is applied to the input signal , together with the TD equivalent of the FD multiband filter that selects the pilot subcarriers [24] (phase-shifted sum of the samples in the same relative position in all segments of samples). This operation can be described by
(18)
where the matrix , the column -vectors and hold the data-dependent and pilot-dependent components in , respectively, and is the resulting noise vector. The elements of can be expressed by
(19)
If channel maximum delay (normalized to the system's sampling interval ) is short enough so that the adjacent CIR replicas in (19) to not overlap (fulfils the sampling theorem),
(20)
(19)
can be further simplified to
(21)
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.
The elements of can be expressed by
(22)
where
(23)
where is the initial candidate frequency offset.
The elements of the noise vector can be expressed by
(24)
Figure 2 depicts an example of the constitution of vector for a system with 2 transmission antennas, subcarriers and pilot separation . The plots put in evidence that the CIRs energy is limited to 2 sets of samples and the data-dependent component spans the entire symbol duration.
Figure 2
Example of the constitution of vector.
×
A careful inspection of (23) reveals that the factor (and the data-dependent component) is zero for
(25)
independently of the considered sample. Keeping in mind that and , the solution for (25) is
(26)
where is the subcarrier separation. It should be noted that the solution in (26) presents a periodicity and includes the condition when the CFO is completely eliminated . A similar analysis reveals that the factor has maximum magnitude for
(27)
We can conclude that has minimum values spread Hz, with maximum magnitude values in between, separated by Hz.
Let us define the column -vector that collects the samples of with no CIRs energy (only data-dependent and noise; example depicted in Figure 2) and the cost function as the energy in :
(28)
The definition of the cost function guarantees that, if within the acquisition range, its minimum value will converge to the true estimate as the number of elements in increases (and the noise term tends to a floor in ). The elements in may be obtained from one OFDM symbol or a set of symbols (with data and pilots) if higher accuracy on the estimate is required. From the previous analysis of the factor , it is clear that the acquisition range of our cost function is . The CFO estimate can be found by a line search within the acquisition range to find the minimum value of the cost function:
(29)
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 [26] to find the global minimum.
Figure 3 depicts an example of the cost function for a Alamouti OFDM system with subcarriers, sampling interval nanoseconds and pilot separation . The values in the plot were acquired using an dB. In Figure 3(a), the separation of 390 kHz between consecutives minimum values is visible. Figure 3(b) shows in detail the interval around . It is clear that it has a unique global minimum that is easy to find (no problem with local minimum values). In Figure 3(c), the maximum values between consecutive minimum values are clearly visible. It also shows in detail that the separation of the maximum values around is 195 kHz.
Figure 3
The cost function.
×
Figure 4 shows the evolution of the estimated CFO standard deviation with the number of samples used in estimation algorithm (elements of ). The plots depict the standard deviation when the number of samples goes from 200 to 2000, in steps of 200 samples.
Figure 4
Estimated CFO standard deviation versus the number of samples in.
×
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.
The channel estimation algorithm starts by isolating each of the phase-shifted CIRs from and removing the modulating exponential factor. The elements of the resulting vectors can be expressed by
(30)
In [24], 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 .
4. Simulation Results
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.
Figure 5 shows the remaining CFO at the output of the CFO mitigate block when using the indoor channel model. The dashed lines represent the average remaining CFO. The solid lines represent the standard deviation of the CFO estimate. In our method, the gradient descent was stopped for a step of 10 HZ. For the method in [16], one null subcarrier was added to each OFDM symbol, and an exhaustive search was performed with a 10 HZ step. The results show that our method is unbiased for the all range of Eb/N0 values. The algorithm generates estimates with small deviation from the true value using a limited number of symbols (16). The estimate deviation is ~1% of that of [16] for low values of SNR. The method in [16] requires a large number of symbols and/or more receive antennas to generate accurate estimates. This result shows that the investigated method is quite adequate for burst systems.
Figure 5
Remaining CFO.
×
Figure 6 depicts the MSE of the channel estimation algorithm (blue plots) when the system has perfect synchronization and the MSE of the joint CFO and channel estimation process (red plots), when using QPSK, 16-QAM, and 64-QAM modulations. Figure 6(a) presents the results for the indoor channel and Figure 6(b) for the outdoor channel.
Figure 6
Joint channel and CFO estimation MSE. IndoorOutdoor
×
Figure 7 depicts the system BER for 3 scenarios: the ideal situation where the receiver has perfect channel state information (CSI) (with no pilot overhead) and the CFO is absent (green plots), the situation when the receiver must estimate the channel from the received samples (blue plots), and the more realistic scenario where the receiver needs to estimate both the CFO and channel (red plots). Simulations results were obtained also for QPSK, 16-QAM, and 64-QAM modulations as identified in the figures. Figure 7(a) presents the results for the indoor channel and Figure 7(b) for the outdoor channel.
Figure 7
System BER performance. IndoorOutdoor
×
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.
5. Conclusions
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.
Acknowledgments
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).
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
is the number of channel paths,
and
are the complex value and delay of path
, respectively. The paths are assumed to be statistically independent, with normalized average power,
, where
is the average power of path
. The channel is time variant due to the motion of the mobile terminal (MT), but we will assume that the CIR is constant during one OFDM symbol. The time dependence of the CIR is not present in the notation for simplicity. Assuming that the insertion of a long enough cyclic prefix (CP) in the transmitter assures that the orthogonality of the subcarriers is maintained after transmission, the channel frequency response (CFR) can be expressed as
is the total number of subcarriers of the OFDM system.
transmit antennas depicted in Figure 1. The
vectors
hold the
-ary PSK or QAM coded data to be transmitted.
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
.
th element of the vector
is defined by
and
is the first pilot subcarrier.
, using an efficient
-points inverse fast Fourier transform (FFT) algorithm.
samples long guard interval, in the form of CP, is prefixed to vector
, resulting in the TD transmitted vector
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.
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.
th received signal sample of the
th symbol can be expressed as
is a sample of independent and identically distributed (iid) zero mean additive white Gaussian noise (AWGN) with variance
. Collecting the
samples of the symbol,
collects the noise samples that affect the
th symbol, the vector
represents the intersymbol interference (ISI) caused by the channel dispersion, and the matrix
is the
lower triangular Toeplitz channel convolution matrix with first column
(column
-vector with the discrete-time CIR (its elements are defined by (1)) padded with zeros). The
diagonal matrix that holds the phase rotation that affects each symbol sample is
is the matrix that removes the CP with
representing the
null matrix,
is the resulting TD noise column
-vector, and
is the common phase that affects all samples of the
th symbol. The last step in (8) was possible considering the structure of the matrices involved
is the
circulant matrix with circulant vector
and, due to the properties of the DFT,
, with the elements of
defined by (2).
channels that disturbed the transmission process. Both estimation algorithms will be introduced in the next section. Moreover, this block is also responsible for reducing the CFO, using the estimated CFO value
. This operation can be described by
, 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 [25] can be used to tackle this limitation, if proven necessary.
to FD with an efficient FFT operation. Assuming that the CFO is completely eliminated, the resulting FD column
-vector can be expressed as
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).
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
.
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.
are
is the
th element of
(complex data symbol conveyed by the
th subcarrier of the
transmit antenna path), and the elements of the TD pilot vector
are
and
hold the data-dependent and pilot-dependent components in
, respectively.
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
.
is applied to the input signal
, together with the TD equivalent of the FD multiband filter that selects the pilot subcarriers [24] (phase-shifted sum of the samples in the same relative position in all
segments of
samples). This operation can be described by
matrix
, the column
-vectors
and
hold the data-dependent and pilot-dependent components in
, respectively, and
is the resulting noise vector. The elements of
can be expressed by
maximum delay
(normalized to the system's sampling interval
) is short enough so that the adjacent CIR replicas in (19) to not overlap (fulfils the sampling theorem),
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.
can be expressed by
is the initial candidate frequency offset.
can be expressed by
for a system with 2 transmission antennas,
subcarriers and pilot separation
. The plots put in evidence that the CIRs energy is limited to 2 sets of samples and the data-dependent component spans the entire symbol duration.
.
(and the data-dependent component) is zero for
and
, the solution for (25) is
is the subcarrier separation. It should be noted that the solution in (26) presents a periodicity
and includes the condition when the CFO is completely eliminated
. A similar analysis reveals that the factor
has maximum magnitude for
has minimum values spread
Hz, with
maximum magnitude values in between, separated by
Hz.
-vector
that collects the samples of
with no CIRs energy (only data-dependent and noise; example depicted in Figure 2) and the cost function
as the energy in
:
increases (and the noise term tends to a floor in
). The elements in
may be obtained from one OFDM symbol or a set of symbols (with data and pilots) if higher accuracy on the estimate is required. From the previous analysis of the factor
, it is clear that the acquisition range of our cost function is
. The CFO estimate can be found by a line search within the acquisition range to find the minimum value of the cost function:
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.
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 [26] to find the global minimum.
Alamouti OFDM system with
subcarriers, sampling interval
nanoseconds and pilot separation
. The values in the plot were acquired using an
dB. In Figure 3(a), the separation of
390 kHz between consecutives minimum values is visible. Figure 3(b) shows in detail the interval around
. It is clear that it has a unique global minimum that is easy to find (no problem with local minimum values). In Figure 3(c), the
maximum values between consecutive minimum values are clearly visible. It also shows in detail that the separation of the maximum values around
is
195 kHz.
.
). The plots depict the standard deviation when the number of samples goes from 200 to 2000, in steps of 200 samples.
.
. The data-dependent component was eliminated from this vector, opening way to easily obtain an initial CIR estimate.
phase-shifted CIRs from
and removing the modulating exponential factor. The elements of the resulting vectors
can be expressed by
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.
and
.
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.