One-Dimensional Channel Estimators

By one-dimensional channel estimation, we mean that we only use channel information along the frequency domain. In other words, we use the channel information at pilot subcarriers obtained by the LS estimator to estimate the channel information at data subcarriers via interpolation or extrapolation. The material in this section is largely taken from [10].

3.2.1 Polynomial Interpolation and Extrapolation

There are N terms, each a polynomial of degree N − 1 and each constructed to be zero at all of the xi except one, at which it is constructed to be yi.

A better method for constructing the interpolating polynomial is Neville’s algorithm as follows: Let P₁ be the value at x of the unique polynomial of degree zero passing through the point (x₁, y₁); so P₁ = y₁. Likewise define P₂, P₃, . . . , P_N. Now let P₁₂ be the value at x of the unique polynomial of degree one passing through both (x1, y1) and (x2, y2). Likewise P23, P34, . . . , P(N −1)N. Similarly, for higher-order polynomials, compute up to P123...N, which is the value of the unique interpolating polynomial through all N points, i.e., the desired answer. The various P s form a “tableau” on the left lading to a single “descendent” at the extreme right. For example, with N = 4,

x₁ :

Neville’s algorithm is a recursive way of filling in the numbers in the tableau a column at a time, from left to right. It is based on the relationship between a “daughter” P and its two “parents” as

Pi(i+1)...(i+m)= (x − xi+m)Pi(i+1)...(i+m−1)+ (xi− x)P(i+1)(i+2)...(i+m)

x − x . (3.12)

This recurrence works because the two parents already agree at points xi+1, . . . , xi+m−1. An improvement on the recurrence (3.12) is to keep track of the small differences between parents and daughters, namely, to define (for m = 1, 2, . . . , N − 1),

C_m,i ≡ P_i...(i+m)− Pi...(i+m−1), D_m,i ≡ P_i...(i+m)− P(i+1)...(i+m).

(3.13)

Then one can easily derive from (3.12) the relations

D_m+1,i = (x_i+m+1− x)(C_m,i+1− D_m,i) x_i− x_i+m+1 , C_m+1,i = (x_i− x)(C_m,i+1− D_m,i)

x_i− x_i+m+1 .

(3.14)

At each level m, the Cs and Ds are the corrections that make the interpolation one order higher. The final answer P_1...N is equal to the sum of any y_i plus a set of Cs and/or Ds that form a path through the family tree to the rightmost daughter.

Usually, linear and second-order interpolations are employed due to the consideration of complexity, as discussed in [11], [12] and [13]. The mathematical expression of linear and second order interpolations are given below.

3.2.1.1 Linear interpolation

The linear interpolation is given by

H_e(k) = H_e(m + l) = (H_p(m + 1) − H_p(m))l

L + H_p(m) (3.15)

where Hp(k), k = 0, 1, · · · , Np, are the channel frequency responses at pilot subcarriers, L is the distance between the two given data, that is, the pilot subcarriers spacing, and 0 ≤ l < L.

3.2.1.2 Second order interpolation

The second-order interpolation is given by

H_e(k) = H_e(m + l)

= c₁H_p(m − 1) + c₀H_p(m) + c₋₁H_p(m + 1) (3.16)

where

c₁ = α(α − 1)

2 ,

c₀ = −(α − 1)(α + 1), c₋₁ = α(α + 1)

2 ,

α = l L.

Other notations are the same as in linear interpolation.

3.2.2 Rational Function Interpolation and Extrapolation

Some functions are not well approximated by polynomials, but are well approximated by rational functions. We denote by Ri(i+1)...(i+m) a rational function passing through the m + 1 points (x_i, y_i), . . . , (x_i+m, y_i+m). Suppose

Ri(i+1)...(i+m)= P_µ(x)

Q_ν(x) = p₀+ p₁x + · · · + p_µx^µ

q₀+ q₁x + · · · + q_νx^ν . (3.17) Since there are µ + ν + 1 unknown µs and νs (q₀ being arbitrary), we must have m + 1 = µ + ν + 1.

Rational functions are sometimes superior to polynomials because of their ability to model functions with poles, that is, zeros of the denominator of (3.17). These poles might occur for real values of x, if the function to be interpolated itself has poles. More often, the function f (x) is finite for all finite real x, but has an analytic continuation with poles in the complex x-plane. Such poles can ruin a polynomial approximation, especially those at real values of x.

Bulirsch and Stoer found an algorithm of the Neville type which performs rational function extrapolation on tabulated data. The algorithm is summarized by a recurrence relation:

Ri(i+1)...(i+m) = R(i+1)...(i+m)+ R(i+1)...(i+m)− Ri...(i+m−1)

This recurrence generates the rational functions through m + 1 points from the ones through m and (the term R(i+1)...(i+m−1)) m − 1 points. It is started with

R_i = y_i and R ≡ [Ri(i+1)...(i+m) with m = −1] = 0.

Now, we can convert the recurrence (3.18) to one involving only the small differences Cm,i ≡ Ri...(i+m)− Ri...(i+m−1),

D_m,i ≡ R_i...(i+m)− R(i+1)...(i+m).

(3.19)

Note that these satisfy the relation

C_m+1,i − D_m+1,i = C_m,i+1− D_m,i (3.20)

which is useful in proving the recurrences

Dm+1,i = ¡Cm,i+1_x−x (Cm,i+1− Dm,i)

3.2.3 Cubic Spline Interpolation [10], [14], [15]

Cubic spline is one very effective, well-behaved, computationally efficient interpolation. The approach is to fit cubic polynomials to adjacent pairs of points and choose the values of the two remaining parameters associated with each polynomial such that the polynomials covering adjacent intervals agree with one another in both slope and curvature at their common endpoint. The cubic spline interpolation is developed in the following.

Given a tabulated function yi = y(xi) and its second order derivative y⁰⁰, i = 1, . . . , N, let us focus our attention on one particular interval, say between xj and xj+1. The goal of cubic spline interpolation is to get an interpolation formula that is smooth in the first deriv-ative and continuous in the second derivderiv-ative, both within the interval and at its boundaries.

A little calculation shows that there is only one way to arrange this construction, that is, y = Ay_i+ By_i+1+ cy_j⁰⁰+ Dy⁰⁰_j+1 (3.22)

Combined with (3.23), we take the derivatives of (3.22) with respect to x, yielding dy

dx = y_j+1− y_j

x_j+1− x_j −3A²− 1

6 (xj+1− xj)y_j⁰⁰+3B²− 1

6 (xj+1− xj)y_j+1⁰⁰ (3.24) for the first derivative and

d²y

dx² = Ay⁰⁰_j + By_j+1⁰⁰ (3.25)

for the second derivative. Since A = 1 at x_j, A = 0 at x_j+1, while B is just the other way around, (3.25) shows that y⁰⁰ is just the tabulated second derivative, and also that the second derivative will be continuous across the boundary.

The only problem now is that we supposed the y⁰⁰_i’s to be known, when actually they are not. The key idea of a cubic spline is to require the continuity of the first derivative and to use it to get equations for the second derivatives y⁰⁰_i.

We set (3.24) evaluated for x = xj in the interval (xj−1, xj) equal to the same equation evaluated for x = x_j but in the interval (x_j, x_j+1). With some arrangement, this gives, for j = 2, . . . , N − 1,

These are N − 2 linear equations in the N unknowns y_i⁰⁰, i = 1, . . . , N. Therefore there is a two-parameter family of possible solutions.

For a unique solution, we need to specify two further conditions, typically taken as boundary conditions at x₁ and x_N. The most common ways of doing this are either

• set one or both y⁰⁰₁ and y⁰⁰_N equal to zero, giving the so-called natural cubic spline, or

• set either of y₁⁰⁰ and y_N⁰⁰ to some values so as to make the first derivative of the interpo-lating function have a specified value on either or both boundaries.

3.2.4 The Maximum Likelihood Channel Estimator

As mentioned before, the LMMSE estimator exploits channel correlations in time and fre-quency domains. It needs knowledge of the channel statistics and the operating SNR. As indicated in [8], although it can work with mismatched conditions on parameter values, its performance degrades if the assumed Doppler frequencies and the delay spread are smaller than the true ones.

The LMMSE estimator regards the channel impulse response as a random vector whose particular realization is to be estimated. On the contrary, in maximum likelihood estimation (MLE), the channel impulse response is viewed as a deterministic but unknown vector and no information on the channel statistics or the operating SNR is required in this scheme.

The MLE of h is give by [16]

hbMLE = D⁻¹B^HHbp,LS (3.27)

where D is a square matrix D = B^HB whose entries are given by

£B¤

n,k = e^−j2πkin/N, 0 ≤ n ≤ N_p− 1, 0 ≤ k ≤ L − 1, (3.28)

£D¤

n,k =

NXp−1 m=0

e^{j2π(n−k)im/N}, 0 ≤ n, k ≤ L − 1, (3.29)

where bH_p,LS is given by (3.5), i_n are pilot locations, N_p is the number of pilots and L is channel length.

Equation (3.27) indicates that MLE requires the invertibility of D. Such a condition is met if and only if B is full rank and N_p ≥ L. this means that the number of pilots must be not smaller than the number of channel taps.