Part II Physical Layer for Downlink 121
8.4 Frequency-Domain Channel Estimation
In this section, we address the channel estimation problem over one OFDMA symbol (specifically a symbol containing reference symbols) to exploit the frequency-domain characteristics.
In the LTE context, as for any OFDM system with uniformly distributed reference symbols [18], the Channel Transfer Function (CTF) can be estimated using a maximum likelihood approach in the frequency domain at the REs containing the RSs by de-correlating the constant modulus RS. Using a matrix notation, the CTF estimatezp on reference symbol pcan be written as
zp=zp+zp=Fph+zp (8.12) for p∈(0, . . . , P) wherePis the number of available reference symbols andhis the L× 1 CIR vector. Fp is theP×Lmatrix obtained by selecting the rows corresponding to the reference symbol positions and the firstLcolumns of theN×NDiscrete Fourier Transform (DFT) matrix whereNis the FFT order.zpis aP×1 zero-mean complex circular white noise vector whoseP×Pcovariance matrix is given byCzp. The effective channel lengthL≤LCP is assumed to be known.
8.4.1 Channel Estimate Interpolation
8.4.1.1 Linear Interpolation Estimator
The natural approach to estimate the whole CTF is to interpolate its estimate between the reference symbol positions. In the general case, letAbe a generic interpolation filter; then the interpolated CTF estimate at subcarrier indexican be written as
zi=Azp (8.13)
Substituting Equation (8.12) into (8.13), the error of the interpolated CTF estimate is zi=z−zi=(FL−AFp)h−Azp (8.14) whereFLis theN×Lmatrix obtained by taking the firstLcolumns of the DFT matrix and z=FLh. In Equation (8.14), it can be seen that the channel estimation error is constituted of a bias term (itself dependent on the channel) and an error term.
The error covariance matrix is
Czi=(FL−AFp)Ch(FL−AFp)H+σ2zpAAH (8.15) whereCh=E[hhH] is the channel covariance matrix.
Recalling Equation (8.13), linear interpolation would be the intuitive choice. Such an estimator is deterministically biased, but unbiased from the Bayesian viewpoint regardless of the structure ofA.
8.4.1.2 IFFT Estimator
As a second straightforward approach, the CTF estimate over all subcarriers can be obtained by IFFT interpolation. In this case, the matrixAfrom Equation (8.13) becomes:
AIFFT= 1
PFLFHp (8.16)
The error of the IFFT-interpolated CTF estimate and its covariance matrix can be obtained by substituting Equation (8.16) into Equations (8.14) and (8.15).
With the approximation ofIL≈(1/P)FHpFp, whereILis theL×Lidentity matrix, it can immediately be seen that the bias term in Equation (8.14) would disappear, providing for better performance.
Given the LTE system parameters and the patterns of REs used for RSs, in practice (1/P)FHpFp is far from being a multiple of an identity matrix. The approximation becomes an equality whenK=N,N/W>LandN/W is an integer,9i.e. the system would have to be dimensioned without guard-bands and the RS would have to be positioned with a spacing which is an exact factor of the FFT orderN, namely a power of two.
In view of the other factors affecting the design of the RS RE patterns outlined above, such constraints are impractical.
8.4.2 General Approach to Linear Channel Estimation
Compared to the simplistic approaches presented in the previous section, more elaborate linear estimators derived from both deterministic and statistical viewpoints are proposed in [19–21]. Such approaches include Least Squares (LS), Regularized LS, Minimum Mean- Squared Error (MMSE) and Mismatched MMSE. These can all be expressed under the following general formulation:
Agen=B(GHG+R)−1GH (8.17)
where B, G and R are matrices that vary according to each estimator as expressed in Table 8.1, where0Lis the all-zerosL×Lmatrix.
The LS estimator discussed in [19] is theoretically unbiased. However, as shown in [21], it is not possible to apply the LS estimator to LTE directly, because the expression (FpFHp)−1is ill-conditioned due to the unused portion of the spectrum corresponding to the unmodulated subcarriers.
To counter this problem, the classical robust regularized LS estimator can be used instead, so as to yield a better conditioning of the matrix to be inverted. A regularization matrixαILis introduced [22] whereαis a constant (computed off-line) chosen to optimize the performance of the estimator in a given Signal-to-Noise Ratio (SNR) working range.
The MMSE estimator belongs to the class of statistical estimators. Unlike deterministic LS and its derivations, statistical estimators need knowledge of the second-order statistics (PDP and noise variance) of the channel in order to perform the estimation process, normally with much better performance compared to deterministic estimators. However, second-order statistics vary as the propagation conditions change and therefore need appropriate re- estimation regularly. For this reason statistical estimators are, in general, more complex due
9Wis the spacing (in terms of number of subcarriers) between reference symbols.
Table 8.1: Linear estimators.
Components of interpolation filter (see Equation (8.17))
Interpolation method B G R
Simple interpolator A IP 0L
FFT P1FLFHp IP 0L
LS FL Fp 0L
Regularized LS FL Fp αIL
MMSE FL Fp σ2zpCh−1
Mismatched MMSE FL Fp σ2zp/σ2hãIL
to the additional burden of estimating the second-order statistics and computing the filter coefficients.
A mismatched MMSE estimator avoids the estimation of the second-order channel statistics and the consequent on-line inversion of an L×L matrix (as required in the straightforward application of MMSE) by assuming that the channel PDP is uniform10[20].
This estimator is, in practice, equivalent to the regularized LS estimator where the only difference lies in the fact thatα=σ2zp/σ2his estimated and therefore adapted. The mismatched MMSE formulation offers the advantage that the filter coefficients can be computed to be real numbers because the uniform PDP is symmetric. Moreover, since the length of the CIR is small compared to the FFT size, the matrix Agencan be considered to be ‘low-density’, so storing only the significant coefficients can reduce the complexity.
However, LTE does not use an exactly uniform pattern of reference symbols; for the cell- specific RSs this is the case around the d.c. subcarrier which is not transmitted. This implies that a larger number of coefficients needs to be stored.
8.4.3 Performance Comparison
For the sake of comparison between the performance of the different classes of estimator, we introduce the Truncated Normalized Mean Squared Error (TNMSE).
For each estimator, the TNMSE is computed from its covariance matrixCzand the true CTFzas follows:
TNMSEz= Ttr(Cz)
Ttr(FLChFHL) (8.18)
where Ttr{ã}denotes the truncated trace operator, where the truncation is such that only the Kused subcarriers are included.
Figure 8.9 shows the performance of an LTE FDD downlink with 10 MHz transmission bandwidth (N=1024) and Spatial Channel Model-A (SCM-A – see Section 20.3.4).
It can be seen that the IFFT and linear interpolation methods yield the lowest performance.
The regularized LS and the mismatched MMSE perform equally and the curve of the latter is therefore omitted. As expected, the optimal MMSE estimator outperforms any other estimator.
10This results in the second-order statistics of the channel having the structure of an identity matrix.
Figure 8.9: Frequency-domain channel estimation performance.
The TNMSE computed over all subcarriers actually hides the behaviour of each estimator in relation to a well-known problem of frequency-domain channel estimation techniques:
the band-edge effect. This can be represented by the Gibbs [23] phenomenon in a finite- length Fourier series approximation; following this approach, Figure 8.10 shows that MMSE- based channel estimation suffers the least band-edge degradation, while all the other methods presented are highly adversely affected.