Proc. of Int. Conf. on Control, Communication and Power Engineering 2010
where
Let p denotes first column of the DFT matrix k and
[ ]
Rhy = E hy H = RhhF H X H
[ ]
Rhh denotes upper left corner of Rhh .
R yy = E yy H = XFRhh F H X H + σ n 2 I N
Then the modified MMSE estimator becomes hˆ = PC ' PH X H y
Rhh is the auto-covariance matrix of h and σ n2 is the
{
2
mmse
}
where
noise variance E | n k | . Then the frequency domain MMSE estimate of hmmse is given by
mmse
(
⎡ ' C mmse = R hh ⎢ PX H XP ⎣
)
−1
⎤ ⎦
(
H H ' σ n2 + R hh ⎥ P X XP
(14)
)
−1
Although the LS estimator modifications do not have much impact, its performance in terms of mean square hˆmmse = Fhmmse error can be improved for arrange of SNR’s. LS estimator does not use the channel statistics. Since energy of h = F Rhy R yy −1 y outside the first L taps, excluding the low energy taps of −1 g will improve its performance , whereas the noise energy = F RhhF H X H XFRhhF H X H + σ N 2 I N Y is assumed to be constant over the entire range[11 13]. − 1 −1 −1 2 H H H H The modified LS estimator becomes, F X XF (13) = Rhh F X XF σ n + Rhh ' hˆLS = PC LS X HY The mean square error for MMSE can be calculated as −1 E = h − hˆmmse ' = ⎛ P H X H XP ⎞ where C LS ⎜ ⎟ 2 ⎝ ⎠ ⎛ ⎞ ⎜ h − hmmse ⎟ The complexity of the modified LS is high when MSE = mean⎜ ⎟ compared to the full LS but modified MMSE complexity h ⎜ ⎟ is equally complex to that of LS. ⎝ ⎠ Since MMSE is done based on the assumption that channel correlation & noise variance are known, but in practice they are to be taken as constant or found in an adaptive way. However under special conditions that the channel is T spaced & the energy will not leak outside [0 L-1]. In such case the modified MMSE equals the full MMSE, whereas the performance of modified LS dominates the LS estimate Eqn. (15).
( (
) )(
[(
)
](
)
)
Estimation error:
In this paper we will also evaluate the channel estimation error in the existing method and a new effective channel estimation approach with low estimation error as well. τ au is estimated by either using LS/MMSE methods.
Figure 3. General estimator structure
The MMSE channel estimator has the form shown in
ˆ Fig.3. If h is not Gaussian, then h mmse is not necessarily
The delay
a minimum mean square error estimator, however it will be a best linear estimator in the mean square sense. Both the estimators Eqn. (10) & Eqn. (13) have their drawbacks. The MMSE suffers from high complexity, whereas the Least Square has high mean square error.
τ au
is assumed to be equal to mTs. But in
practice this assumption is never satisfied [11]. The transmitted signal can be written as, x(t ) =
∑ x δ (τ − nTs ) n
n =0
C. Modified MMSE and LS The MMSE estimator requires the computation of N × N and C mmse matrix, which leads to complexity when N is large. One way of reducing the complexity is reducing the size of C mmse . It is observed that most of the energy in h is contained near the first few taps say L=To/T taps. Hence to reduce complexity only the taps with significant energy are alone considered and the remaining elements in the corresponding Rhh are
The channel estimation algorithms may have estimation error even in the absence of noise. In this section, we will discuss the methods to reduce this type of estimation error. Based on the excess delay, the channel can be classified into two classes. The first type system model where the channel excess delay is assumed to be equal to mTs/k. For the second model, excess delay is not a rational number. If a data sample xn is up sampled with a factor k, then
assumed to be zero. We have taken into account only the first L taps of h and setting Rhh (r , s ) = 0 for r , s ∈ [0, L − 1] .
xr be comes {xˆ n } where
{xˆ n } = 159 © 2009 ACEEE
N −1
⎧ x n if and only if k = nk ⎫ ⎨ ⎬ Otherwise ⎩0 ⎭