This chapter further explore maximum-likelihood (ML) CFO estimates of TDD-OFDMA WiMAX downlink system that uses three of identical fractional-period OFDM blocks obtained from interpolations. An efficient algorithm is provided to solve the associated highly nonlinear ML equation. We convert the problem of obtaining the ML solution from searching exhaustively over the entire uncertainty range to that of solving a spectrum polynomial of degree 4, thereby greatly reducing the computational load. By properly truncating the polynomial, we obtain a closed-form expression for the corresponding zeros so that the root-searching procedure is greatly simplified.
5.1 Maximum-Likelihood Estimation of Carrier Fre-quency Offset
Let s(p, q) be the qth sample of the pth(time-domain) short pilot symbol and assume that the preamble part of a transmitted package consists of three identical short pilot symbols obtained from interpolations, as shown in Fig. 2.6. We thus have the relation s(p, q) = r((p − 1)N + q) for p = 1, 2, 3 and q = 1, ..., N where r(·) is the interpolated preamble samples. Consider a frequency-selective channel with a maximum delay spread
shorter than a CP duration. After discarding the CP, the remaining received three pilot symbols y(p, q) can be represented as
y(p, q) = ej2πε((p−1)N +q)
3N x(p, q) + v(p, q) (5.1)
for p = 1, 2, 3, and q = 1, ..., N , where x(p, q) = xint((p − 1)N + q) is the interpolated channel output corresponding to the transmitted pilot symbol s(p, q), y(p, q) = yint((p − 1)N + q) and v(p, q) = w((p − 1)N + q).
Define the two vectors
Y(q) =£
where (·)T denotes the matrix transpose. Then, we have
Y(q) = A(ε)ej2πεq3N x(1, q) + V(q)
= A(ε)x(q) + V(q), q = 1, ..., N (5.4)
where x(q) = ej2πεq/3Nx(1, q), and V(q) = [v(1, q) v(2, q) v(3, q)]T. The received sam-ples can thus be expressed compactly as
Y = A(ε)X + V (5.5)
where Y = [Y(1), ..., Y(N)], X = [x(1), ..., x(N)], and V = [V(1), ..., V(N)]. Hence, given the received sample vectors Y, we have to estimate ε through the deterministic vector A(ε).
Since the noise is temporally white Gaussian, Y(q) is a multivariate Gaussian dis-tributed random vector with covariance matrix σv2I, where I is the identity matrix. The joint ML estimates of A and X, treating X as a deterministic unknown vector, are
obtained by minimizing the joint probability density function
The corresponding log-likelihood function, after dropping constant and unrelated terms, is given by
Λ(A, x(q)) = XN
q=1
kY(q) − Ax(q)k2 (5.7)
For a given A, setting ∇x(q)kY(q) − Ax(q)k2 = 0, where ∇x(q) denotes complex gradient operation with respect to x(q), we obtain the conditional ML estimate ˆx(q) = xLS(q) = A+Y(q), where A+= AH/K and H denotes the Hermitian operation. Substituting the least-square solution xLS(q) into (5.7), we obtain
Λ(A) = is the (time-averaged) autocorrelation matrix of the received sample vectors Y(q), it is a Hermitian matrix such that ˆRY Y(i, j) = ˆR∗Y Y(j, i), where (∗) denotes the complex conjugate. The CFO estimate is then given by
ˆ
Although (5.9) gives a compact representation of the ML CFO estimate, it requires an exhaustive search over the entire uncertainty range. The resulting complexity may make its implementation infeasible.
We observe, however, that A has a special structure that can be of use to reduce the complexity of searching the desired CFO solution of (5.9). Invoking an approach similar to that used by the MUSIC algorithm, we set z = ej2πεN/3N and define the parameter vector of this important observation, we restate it in the form of the following proposition.
Proposition 1: The log-likelihood function for a candidate CFO ε is given by
Λ(ε) = X2 n=−2
s(n)ej2πnεN3N (5.12)
Some remarks about this proposition are in order.
Remarks:
R1. s(n) is the summation of diagonal entries of ˆRY Y and is also equivalent to the aperiodic autocorrelation value of the waveform {y(i, j), 1 ≤ i ≤ 3, 1 ≤ j ≤ N}, i.e., s(n) =P3−n
m=1
PN
j=1y∗(m + n, j)y(m, j).
R2. It can be shown that, in the absence of noise
s(n) = ˆσ2x(K − |n|)e−jnθo = ∆(n) (5.13) where θo = 2πεN/3N, and ˆσx2 = (1/N)PN
n=1|x(n)|2. When noise is present, the mean value of s(n) is the same as its noiseless value except for n = 0; more
specifically, E[s(n)] = ∆(n)+Kσw2δ(n), where δ(·) is the Kronecker delta function.
Evaluating (5.11) at the unit circle z = ej2πεN/3N = ejθ, we obtain the DTFT of the sequence s(n), which has an envelope similar to sinc2(ε) whose maximum value is at the correct “modified” frequency ε0 = ε(N/3N).
R3. Due to the Hermitian nature of ˆRY Y, s(n) is a conjugate symmetric sequence of length 5. The symmetric property of s(n) guarantees that its Fourier transform Λ(ej2πε) is real and nonnegative. This also follows from the semi-positive defi-niteness of the quadratic form AHRˆY YA. Because s(n) and the log-likelihood function constitute a Fourier transform pair, we will henceforth refer to Λ(ej2πε) as the log-likelihood spectrum or spectrum, for short, and the polynomial defined by (5.11) the spectrum polynomial.
R4. {s(n), n = 0, 1, 2} = C constitutes a set of sufficient statistic for estimating ε.
Almost all previous correlation-based algorithms use only a subset of C. It is expected that an algorithm that uses the sufficient statistic would outperform those that use only a part of the sufficient statistic.
R5. Computing the desired CFO estimate through (5.11) is equivalent to searching for the peak of the candidate spectrum Λ(ej2πε). Hence, the spectrum can be computed using a DFT, but the resolution of the CFO estimate ˆε depends on the size of the DFT. Padding more zeros in the sequence s(n) results in higher resolution at the expense of inducing higher computation complexity.
As the spectrum is a real smooth function of θ, taking a derivative of Λ(ejθ) with respect to θ and setting ∂Λ(ejθ)/∂θ = ˙Λ(θ) = 0, we obtain
F (z) = F∗(z) (5.14)
where F (z) = P2
n=1ns(n)zn is a polynomial of order 2. As mentioned before, in a noiseless environment, Λ(ej2πε), the Fourier transform of {s(n)}, is a scaled version of
the function sinc2(ε), and all roots of ˙Λ(z) = ˙Λ(θ)|ejθ=z = j(F (z) − F∗(z)) are on the unit circle.
For simplicity, we shall use the first approach, i.e., the desired estimate is to be obtained by
Note that we have converted the exhaustive search problem of (5.9) to a root-finding problem, reducing the candidate solution number from infinity to at most 4.
We summarize the procedure leading to (5.16) as follows.
1) Collect three received symbols obtained from interpolations and construct the sample correlation matrix ˆRY Y.
2) Calculate the coefficients of F (z) based on ˆRY Y. 3) Find the nonzero unit-magnitude roots of (5.14).
4) Obtain the CFO estimate from (5.15) and (5.16).
We notice that the solutions of (5.14) are the nonzero roots of the polynomial
G(z) = z2(F (z) − F∗(z)) (5.17)
On the other hand, (5.14) implies that the roots of ˙Λ(z) satisfy the equation Im{F (z)} = 0, where Im{F (z)} is the imaginary part of F (z). This observation indicates that the nonzero roots of F1(z) = z−1F (z) = 0 (the root z = 0 of F (z) is undesired) are a subset of the roots of Im{F (z)} = 0. When F (z) is an arbitrary polynomial, its roots are not necessarily a subset of those of the corresponding G(z) defined by (5.17).
Proposition 2: In the absence of noise, the polynomial defined by (5.17), G(z), can be decomposed into
G(z) = F1(z)Q(z) (5.18)
where the desired CFO estimate is one of the roots of Q(z) defined by Q(z) = z3− s(−1)
2 × s(2) = z3− d (5.19)
where d = s(−1)/(2 × s(2)).
When noise is present, the above equality becomes an approximation only. Never-theless, the desired CFO estimate can still be derived immediately from taking the 3th root of d. The global maximum that collocated with a root of Q(z) corresponds to the desired CFO estimate while the remaining roots of Q(z) locate at a local minimum (null) of the spectrum. On the other hand, the roots of F1(z) are at the local sidelobe peaks of the spectrum. It is clear that the union of the roots of Q(z) and F1(z) is the set of the roots of G(z). Hence, the complexity of extracting the roots is significantly reduced, for we only have to solve the equation Q(z) = 0, which happens to have a closed-form expression for its roots. The above discussion suggests the following simplified CFO estimate algorithm.
1) Collect three received symbols obtained from interpolations and construct the sample correlation matrix ˆRY Y.
2) Compute the coefficients d based on two correlation values s(−1) and s(2).
3) Solve Q(z) = 0 for the three unit-magnitude roots of d, {zi}.
4) Find the estimate from (5.15) and (5.16).
5.2 Simulation Results and Discussions
Numerical examples are provided in this section to examine the behavior of the proposed CFO estimation technique. We utilize three short training symbols which are
the same as those used in the IEEE 802.16e downlink preamble. CFO is normalized by subcarrier spacing and the mean values and mean-squared errors (MSE) of various estimates are computed by 104 independent trials.
0 5 10 15 20 25 30 35 40
10−7 10−6 10−5 10−4 10−3 10−2
SNR (dB)
MSE of the Carrier Frequency Offset
FFT size = 128, for IDcell 0 in segment 0 via maximum−likelihood estimation
Linear interpolation Sinc interpolation
Figure 5.1: MSE of normalized CFO ˆε estimate as a function of the SNR in AWGN channel using IDcell 0 preamble in segment 0 via maximum-likelihood estimation.
0 5 10 15 20 25 30 35 40
MSE of the Carrier Frequency Offset
FFT size = 128, for IDcell 0 in segment 0 via maximum−likelihood estimation
Linear interpolation Sinc interpolation
Figure 5.2: MSE of normalized CFO ˆε estimate as a function of the SNR in multipath fading channel using IDcell 0 preamble in segment 0 via maximum-likelihood estimation.
0 5 10 15 20 25 30 35 40
MSE of the Carrier Frequency Offset
FFT size = 128, for IDcell 0 in segment 0 via maximum−likelihood estimation
Linear interpolation Sinc interpolation
Figure 5.3: MSE of normalized CFO ˆε estimate as a function of the SNR in AWGN channel using IDcell 0 preamble in segment 1 via maximum-likelihood estimation.
0 5 10 15 20 25 30 35 40
MSE of the Carrier Frequency Offset
FFT size = 128, for IDcell 0 in segment 0 via maximum−likelihood estimation
Linear interpolation Sinc interpolation
Figure 5.4: MSE of normalized CFO ˆε estimate as a function of the SNR in multipath fading channel using IDcell 0 preamble in segment 1 via maximum-likelihood estimation.
0 5 10 15 20 25 30 35 40
MSE of the Carrier Frequency Offset
FFT size = 128, for IDcell 0 in segment 0 via maximum−likelihood estimation
Linear interpolation Sinc interpolation
Figure 5.5: MSE of normalized CFO ˆε estimate as a function of the SNR in AWGN channel using IDcell 0 preamble in segment 2 via maximum-likelihood estimation.
0 5 10 15 20 25 30 35 40 10−7
10−6 10−5 10−4 10−3 10−2 10−1
SNR (dB)
MSE of the Carrier Frequency Offset
FFT size = 128, for IDcell 0 in segment 0 via maximum−likelihood estimation
Linear interpolation Sinc interpolation
Figure 5.6: MSE of normalized CFO ˆε estimate as a function of the SNR in multipath fading channel using IDcell 0 preamble in segment 2 via maximum-likelihood estimation.