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Introduction
The LTE specifications are based on OFDM which operates on a broad-band channel; average bandwidths range from 5Mhz to 20Mhz. OFDM permits
us to up the channel into a large number of narrow-band channels which can be received and denoised individually be the receiver.
If we model the wireless channel as a multipath (Rayleigh) model, the channel impulse transfer function is given by
g(t) = \sum_{i=1}^L \alpha_i \delta_j (t - T_i) ,
where \delta() is the delta dirac function. Thus the channel is represented as a sum of L paths, each with an attenuation \alpha_i and a path delay T_i. In the frequency
domain, the corresponding transfer function G(f) = \sum_i \alpha_i e^{-2 \pi j f T_i} For a more detailed derivation, see LTE_channel_model
The figure above shows the channel transfer function corresponding to a 7 path channel mode. We can see that the attenuation of the channel varies with the frequency; however, if the number of sub-carriers are large enough i.e. each sub-carrier is sufficiently narrow-band, we can make the assumption that it is essentially constant within this sub-carrier. Thus, the simplest LTE channel estimation algorithm will estimate the channel transfer function G(f) = \{ G_1, G_2, G_3, \cdots, G_N \} where G_k represents the complex attenuation for each sub-carrier. The signal received can be decomposed into the signal for each sub-carrier, which can then be
corrected for the effect of the corresponding G().
This was one of the initial attractions of OFDM [Sari]; whereas older technologies would require
complex adaptive time domain equalization with a large number of taps, in OFDM this could be replaced by simple static frequency domain equalization
LTE channel estimation
While channel estimation can take various forms i.e. blind estimation, decision directed estimation, etc. we will focus, exclusively in
this article on pilot assisted estimation. This technique utilizes special symbols of known quantity, known as pilot symbols in the
data stream. Since the receiver at any given time knows what the correct value of the pilot symbols should be, it can compare the received
value vs the canonical value and back-calculate the effect of the channel. The trade-off is obviously between the density of the pilot
symbols (which incurs both spectrum as well as additional processing power at the receiver) and the error incurred in extending the
results of the pilot channel to the rest of the frame. Some of the techniques discussed below can be extended to a larger subset of the
frame as well, as shall be discussed below.
Channel model for OFDM LTE systems
For the rest of the section, we use the channel model as given in LTE_Channel_Model. We assume that a fixed set of pilot
signals \{p_1, p_2, \ldots, p_n\} are being transmitted by the transmitter. The pilots are evenly distributed amongst the other
transmissions. Our task is to estimate the channel conditions from the pilot sub-carriers and then apply it to the other sub-carriers. We
start by focusing on the pilot transmissions only. The channel model is given by
\vec{r} = P W h + n
which expands to where r is the received signal, P = diag(p) is a diagonal matrix containing the original vector \vec{p} on its diagonal, W is a FFT
matrix and h is the channel transform matrix given above, padded out with n - L zeroes. This equation thus expands to the
form given in below
\begin{eqnarray}
\left( \begin{matrix} r_1 \\ r_2 \\ \cdots \\ r_n \end{matrix} \right) =
\left( \begin{matrix}
p_1 & 0 & \ldots & 0 \\
0 & p_2 & \ldots & 0 \\
\cdots & \cdots & \ldots & \ldots & \\
0 & 0 & \ldots & p_n \\
\end{matrix} \right)
\left(
\begin{matrix}
W_N^{0,0} & \ldots & \ldots & W^{0,N-1}_N \\
W_N^{1,0} & \ldots & \ldots & W^{1,N-1}_N \\
\cdots & \cdots & \ldots & \ldots & \\
W_N^{N-1,0} & \ldots & \ldots & W^{N-1,N-1}_N \\
\end{matrix}
\right)
\left[ \begin{matrix} h_1 \\ h_2 \\ \cdots \\ h_l \\ 0 \\ 0 \\ \cdots \end{matrix} \right] \\
W_N^{k,l} = \frac{1}{\sqrt{N}}e^{-j 2 \pi \frac{kl}{N} }
\end{eqnarray}
The job of the estimation algorithm thus becomes as follows: given that we know the value of P, we have to come up with an estimate
for H, based on the received signal vector \vec{r}. The job would be absolutely simple if there was no noise term n involved.
However, this is not the case and choice of the estimation algorithm is a tradeoff between computational complexity and error.
Estimation techniques
Main article: Estimation?
Estimation algorithms are used when we have an underlying variable x which we are measuring through a second variable y = f(x),
which is a function of x but may also contain a noise term. The purpose is to create an estimate \hat{x}, which is, in some way a best guess of x, given the observation y; since we cannot directly verify the actual value, we typically minimize \mathcal{E}(f(\hat{x})
- f(x)), where \mathcal{E} is some error estimator.
There are multiple forms of estimators available, ranging from the simplest linear estimators, to advanced stochastic estimators.
Least Squares Estimator
Least Square estimation for a set of over determined linear equations Ax = b finds
that value of x which minimizes \Vert Ax - b \Vert. In other words, x_{LS} = \arg \min_{x} (b - Ax)^T (b - Ax). In our current
problem, we are trying to find an estimate for h given by \hat{h} = \arg \min_{h} \Vert r - h \Vert = \Vert Ph - n \Vert. By
substituting from the equation above we get
h_{LS} = P^{-1} r = \begin{bmatrix} \frac{r_1}{p_1} & \frac{r_2}{p_2} & \ldots & \frac{r_n}{p_n} \end{bmatrix}
The LS estimator in this situation is also known as the zero-forcing estimator, because it forces the residual error to be zero for each
pilot. It is computationally very simple, but it is not very accurate, since it completely ignores the known relationships between the
values of h, i.e. the covariance matrix.
Minimal Mean Square Estimator
MMSE estimator aims to estimate a random variable so that the mean square error over the entire domain is minimized. If x \in X is
a random variable which we are measuring through a second random variable y, the MMSE estimator of x = \hat{x}(y) is chosen
such that \mathbb{E}_X\left\{(\hat{x}(y) - x)^2\right\} is minimized. If we restrict ourselves to linear estimators i.e. \hat{x} = K.y
\mathbb{E}_X (\hat{x}\left\{(y) - x)^2\right\} = \mathbb{E} \left\{(K.y - x)^2\right\} = K^T \Sigma_{yy} K - \Sigma_{xx} - 2.K \Sigma_{xy}
where \Sigma_{ab} is the cross-covariance matrix for two vectors a and b and \Sigma_{aa} is the auto-correlation matrix
for a. We can then simplify the equation above by using the orthogonality principle which requires that \mathbb{E}[(\hat{x} - x)] =0 and
\mathbb{E}[(\hat{x} - x).y] = 0. By a little manipulation we get:
\begin{eqnarray}
K = \Sigma_{xy}\Sigma_{yy}^{-1} \\
\hat{x}(y) = \Sigma_{xy}\Sigma_{yy}^{-1}y
\end{eqnarray}
The Minimal Mean Square Error (MMSE) estimate for the OFDM channel given above is computed by combining the above equation with the original form for the received signal. (The following derivation is from \cite{deBeek95}).
\begin{eqnarray}
\hat{h_{MMSE}} = \Sigma_{hr}\Sigma_{rr}^{-1}y \\
\Sigma{hr^H} = \Sigma_{hh}W^H P^H \\
\Sigma{yy^H} = P W \Sigma_{hh}W^H P^H + \sigma^2_n I
\end{eqnarray}
where \Sigma_{hh} is the channel covariance matrix.
The MMSE estimator derived above is more accurate than the Least squares estimator; however it requires us to know \Sigma_{hh}, which requires us to estimate the value of \tau_{rms}. Also, the computation is complex, requiring the solution of a large linear estimation problem.
However, various simplifications are available, including the use of EM type algorithms for iterative computation, heuristic reduction of the problem size, etc.
Maximum likelihood estimation
If we have a random process
S which generates output as per a probability density function f(\theta) and we make an observation X = \{x_1, x_2, \ldots, x_n\}, we
can try and estimate the value of \theta so that the observation vector X is the most likely output of S(\theta). This estimate
of \theta is known as the maximum likelihood estimate. Mathematically, we can write this as \hat{\theta}_{MLE} = \arg \max_{\theta \in \Theta} Pr{X = S\vert \theta}.
In the frequency domain channel estimation problem, we have a channel which is characterized by the channel transform vector h. The
received signal r is an observation of the transform vector as is given in equation r = PWh + n = P_W h + n.
Note that the vector h has only L terms, so we can write it as
h = \left[ \begin{array}{cc} h & 0 \end{array} \right].
Corresponding to this, we also rewrite the FFT matrix as W = \left[ \begin{array}{cc} P_{W_L} & P_{W_z} \end{array}\right] The equation now becomes r = P_{W_L} h + n.
To get this to the form where we can use the maximum-likelihood result, we have to extract the estimator variable h. The matrix P_{W_L}
is strictly speaking not invertible, since it is a LxN matrix; however, it is possible to compute a pseudo inverse matrix P_{W_L}^+, which gives the solution \vec{x} = A^+\vec{c} for the over-determined least squares problem
A\vec{x} = \vec{c}. Using this pseudo-inverse, we can rewrite the previous equation as P_{W_L}^+ r = h + P_{W_L}^+ n = h + v, where v is a
transformation of the original noise vector, still with zero mean but with a covariance of \Sigma_{vv} = P_{W_L}^H \Sigma_{nn} P_{W_L}.
Substituting in the equation and taking the log of the probability function on the transformed received signal
\vec{P_{W_L}^+ r}, we get
\log(f(\vec{P_{W_L}^+ r}) = \log(f(h + v))
= \log \left[ (2 \pi)^{-\frac{N}{2}} \frac{1}{\sqrt{\det(\Sigma_{vv})}} e^{-\frac{1}{2}\left( (\vec{P_{W_L}^+r - h})^H \Sigma_{vv}^{-1} (\vec{P_{W_L}^+r - h}) \right) } \right]
= \text{constant}\ - \log(det(\Sigma_{vv})) - \left( (\vec{P_{W_L}^+r - h})^H \Sigma_{vv}^{-1}(\vec{P_{W_L}^+r - h}) \right)
The maximum likelihood estimator, \hat{h}_{MLE} is that value of h which maximizes the left hand size of the above equation. Note that h
only occurs in the last term on the right hand side. Since the term is negative, we maximize it by setting it to zero; thus \hat{h}_{MLE}
= P_{W_L}^+r. This is the orthogonal projection of \vec{r} on the column space of P_{W_L}.
Summary
For the pilot spectrum aided estimator in the frequency domain, we have derived three estimators, based on the received signal $r$, given
in the table below
| Method of estimation | Estimator | Comments |
| Least Squares | \left[r_1/p_i \right] |
| Minimum mean squared error | \Sigma_{hr} {\Sigma_{rr}}^{-1} p | Covariance matrix can be computed from below |
| Maximum likelihood | h = P_{W_L}^+ r | P_{W_L}^+ is the moore-penrose pseudo inverse of diag p W_L |
For a more detailed treatment of this subject, refer to Download:lte_channel_eq.pdf
References
{deBeek95}
Jan-Jaap~van de~Beek, Ove Edfors, and Magnus Sandel.
" On channel estimation in ofdm systems."
Vehicular Technology Conference, 1995 IEEE 45th, pages 1635 -- 1639, June 1995.
{channelEstML}
L.~Deneire, P.~Vandenameele, L.~van~der Perre, B.~Gyselinckx, and M.~Engels.
"A low-complexity ml channel estimator for ofdm."
Communications, IEEE Transactions on, 51(2):135 -- 140, feb
2003.
{channelEstSVD}
O. Edfors, M. Sandell, J.-J. van de Beek, S.K. Wilson, and P.O. Borjesson."Ofdm channel estimation by singular value decomposition." Communications, IEEE Transactions on, 46(7):931 --939, July
1998.
{kaiserpsam} P.Hoeher, S.Kaiser, and P.Robertson.
"Two-dimensional pilot-symbol-aided channel estimation by wiener filtering"
In Acoustics, Speech, and Signal Processing, 1997. ICASSP-97 ,1997 IEEE International Conference on, volume 3, pages 1845 --1848 vol.3,
April 1997.
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