3.1. Properties of the Channel Correlation Matrix in Frequency Domain
The channel impulse response in time domain can be expressed as
where
is the complex gain of the
th path in the
th OFDM symbol period,
is the Kronecker delta function,
is the delay of the
th path in unit of sample point, and
is the number of resolvable paths. Assume that different paths
are independent from each other and the power of the
th path is
. The channel is normalized so that
The channel response in frequency domain
is the FFT of
and it is given by
where
denotes
points FFT operation. The channel autocorrelation matrix in frequency domain can be expressed as
where
denotes expectation. Denote the vector form of the channel autocorrelation matrix by
and we have
. It is easy to find that the matrix
is a circulant matrix. Therefore, as in [
20], the eigenvalues of
are given by
The formula (8) can be equivalently written as
We can easily obtain from (7) and (9) that the number of nonzero eigenvalues of
is equal to the total number of resolvable paths,
(see Appendix A)
. It is known by us that the rank of a square matrix is the number of its nonzero eigenvalues. Therefore the rank of
is
and
is a singular matrix since
. The matrix
does not have the inverse matrix and has only the Moore-Penrose inverse matrix. However, the rank of the matrix
is
N (see Appendix A), where
is an
by
identity matrix. Therefore, the matrix
is not singular and has the inverse matrix.
3.2. The Proposed Fast LMMSE Channel Estimation Algorithm
Let
denote the channel frequency response at pilot subcarriers of the
th OFDM symbol, and let
denote the vector of received signal at pilot subcarriers of the
th OFDM symbol after FFT. Denote the pilot signal of the
th OFDM symbol by
. The channel estimate at pilot subcarriers based on least square (LS) criterion is given by
The LMMSE estimator at pilot subcarriers is given by [
6]
where
is channel autocorrelation matrix at pilot subcarriers and is defined by
, where
denotes Hermitian transpose. It is easy to verify that the matrix
is circulant, the rank of
is equal to
and the rank of
is equal to
. The signal-to-noise ratio (SNR) is defined by
and
is a constant depending on the signal constellation. For 16QAM modulation
and for QPSK and BPSK modulation
If the channel autocorrelation matrix
and SNR are known in advance,
needs to be calculated only once. However, the autocorrelation matrix
and SNR are often unknown in advance and time varying. Therefore the LMMSE channel estimator becomes unavailable in practice. To solve the problem, we propose the fast LMMSE channel estimation algorithm. The algorithm can be divided into three steps. The first step is to obtain the estimate of channel autocorrelation matrices
and
. Firstly, we obtain the least square (LS) channel estimation at pilot subcarriers in time domain,
and it is given by
Secondly, the most significant taps (MSTs) algorithm [
21] has been proposed to obtain the refined channel estimation in time domain. The MST algorithm deals with each OFDM symbol by reserving the most significant
paths in terms of power and setting the other taps to be zero. The algorithm can reduce the influence of AWGN and other interference significantly, compared with the LS method. However, the algorithm may choose the wrong paths and omit the right paths because of the influence of AWGN and other interference. Thus, we will improve the algorithm of [
21] by processing several adjacent OFDM symbols jointly. We calculate the average power of each tap for
adjacent OFDM symbols,
and it is given by
Then we choose the
most significant taps from
and reserve the indeces of them into a set
. Finally, the refined channel estimation in time domain,
, is given by
Denote the first row of the matrix
by
. Then
can be given from (7) by
where
is a 1 by
vector with each entry
Since the matrix
is circulant,
can be acquired by circle shift of
. The second step is to obtain the estimate of SNR. The estimate of SNR,
, is given by
The third step is to obtain the estimate of the matrix
,
. We refer to the matrix
as the LMMSE matrix in this paper. Since
is a circulant matrix and
is a circulant matrix, the product of
and
is also a circulant matrix. Therefore, we need only to compute the estimate of the first row of the LMMSE matrix. Denote the first row of LMMSE matrix by
. The estimate of
,
, is given by (see Appendix B)
where
denotes
points IFFT operation. Therefore the estimated LMMSE matrix
can be obtained from circle shift of
. The channel estimation in frequency domain at pilot subcarriers for the
th OFDM symbol can be given by
The proposed fast LMMSE algorithm avoids the matrix inverse operation and can be very efficient since the algorithm only uses the FFT and circle shift operation. The proposed fast LMMSE algorithm can be summarized as follows.
Step 1.
Obtain the LS channel estimation of pilot signal in time domain,
, by formula (14).
Step 2.
Calculate the average power of each tap for
OFDM symbols,
, by formula (15). Then, we choose the
most significant taps from
and reserve it as
, by formula (18).
Step 3.
Obtain the estimate of SNR,
, by formula (19).
Step 4.
Obtain the estimate of the first row of the LMMSE matrix,
, by formula (20).
Step 5.
Obtain the estimation of the LMMSE matrix,
, by circle shift of
. Then, the channel estimation in frequency domain at pilot subcarriers can be obtained by formula (21).
It is noted that the estimation of the LMMSE matrix requires only
points FFT operation and circle shifting operation, which reduce the computational complexity significantly compared with the conventional LMMSE estimator since it requires the inverse operation of a large dimension matrix.