CFO Estimation





H₀^dC₀^d H₁^dC₁^d · · · H_{N −1}^d C_{N −1}^d H₁^dC₁^d H₂^dC₂^d · · · H₀^dC₀^d

... ... . .. ...

H_{N −1}^d C_{N −1}^d H₀^dC₀^d · · · H_{N −2}^d C_{N −2}^d





. (3.19)

For the special case of no fading, we have

Y_i = eS_i^dC^d_WWs(²_d) + M U I + AW GN (3.20)

where C^d_W is N × N spreading matrix for desired user and given by

C^d_W =







C₀^d C₁^d · · · C_{N −1}^d C₁^d C₂^d · · · C₀^d

... ... ... ...

C_{N −1}^d C₀^d · · · C_{N −2}^d





. (3.21)

3.3 CFO Estimation

Linear-filtering the frequency domain samples Yi by the regression vector V, we obtain the associated output energy

E[|V^HY_i|²] = V^HE[YiY_i^H]V = V^HR_{Y Y}V (3.22)

where RY Y = E[YiY^H_i ] and (·)^H denote conjugate transpose (Hermitian) of the argu-ment. The following constrained optimization problem

V_opt(²) = arg min

V E[|V^HY_i|²] (3.23)

subject to the constrain

V^HC^d_HWs(²) = 1 (3.24)

can be easily solved by invoking Lagrange multiplier method to obtain

∇^V(V^HR_{Y Y}V) − λ∇^V(V^HC^d_HWs(²) − 1) = 0, (3.25)

which leads to

2R_{Y Y}V = λ2C^d_HWs(²). (3.26)

Assume RY Y is nonsingular, we have

V = λR⁻¹_{Y Y}C^d_HWs(²) (3.27)

where

λ = 1

(C^d_HWs(²))^HR⁻¹_{Y Y}C^d_HWs(²). (3.28) Therefore,

V_opt(²) = R⁻¹_{Y Y}C^d_HWs(²)

(C^d_HWs(²))^HR⁻¹_{Y Y}C^d_HWs(²)

= R⁻¹_{Y Y}C^d_HWs(²)

s^H(²)W^HC^dH_H R⁻¹_{Y Y}C^d_HWs(²) (3.29) and the minimum output energy (MOE) is given by

M OE(²) = E[|V^H_opt(²)Y_i|²]

= V^H_opt(²)RY YV_opt(²)

= (R⁻¹_{Y Y}C^d_HWs(²))^HR_{Y Y}R⁻¹_{Y Y}C^d_HWs(²) (s^H(²)W^HC^dH_H R⁻¹_{Y Y}C^d_HWs(²))²

= 1

s^H(²)W^HC^dH_H R⁻¹_{Y Y}C^d_HWs(²). (3.30) Since the weighting vector Vopt minimizes the filter output energy with a unit gain at the given ², MOE as a function of the CFO ² should exhibit a peak in the neighborhood of the correct CFO ²d of desired user. Fig. 3.8 plots a typical normalized MOE(²) as a function of the normalized CFO.

Hence, the desired CFO estimate is then given by

ˆ²d = arg max

² {E[|V^H_opt(²)Yi|²]}

= arg min

² {s^H(²)W^HC^dH_H R⁻¹_{Y Y}C^d_HWs(²)}. (3.31)

−10 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.1

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Normalized CFO

Normalized MOE

MOE (SNR=10dB, N_s=100, K=10, perfect channel info) CFO of Desired User

Figure 3.8: Normalized MOE vs Normalized CFO where MOE is normalized by its peak value and the true CFO = 0.7 subcarrier spacings.

Using the correlation ergodicity assumption

Rb_{Y Y} → RY Y, Ns → ∞, (3.32)

the data correlation matrix can be approximated by time-averaging over Ns snapshots Rb_{Y Y} = 1

Ns Ns

X

i=1

Y_iY^H_i . (3.33)

However, the CFO estimate requires an exhaustive search over the entire uncertainty range. The resulting complexity may make its implementation infeasible. We observe that s(²) has a special structure that can be of use to reduce the complexity of searching the desired CFO solution. Define

Rb_ZZ = W^HC^dH_H Rb⁻¹_{Y Y}C^d_HW (3.34)

then

ˆ²d = arg min

² {s^H(²) bR_ZZs(²)}. (3.35)

The Root MUSIC algorithm, [16], [17] suggests that we set z = e^j2π²/N and define the parametric vector

s(z) = £

1, z, z², · · · , z^{N −1}¤T

(3.36) so that s(z)^HRb_ZZs(z) can be expressed as a polynomial of order 2N − 1,

Λ(z)^def= s(z)^HRb_ZZs(z) =

N −1X

n=−(N −1)

s(n)zⁿ (3.37)

where s(n) = P

i,jRˆZZ(i, j), for n = j − i, and n = −N + 1, · · · , N − 1. As Λ(z) is a real smooth function of ², taking derivative of Λ(e^j2π²/N) with respect to ² and setting

∂Λ(e^j2π²/N)/∂²^def= ˙Λ(²) = 0, we obtain

F (z) − F^∗(z) = 0, (3.38)

where F (z) = PN −1

n=1 ns(n)zⁿ is a polynomial of order N − 1. Let Z ^def= {zi} be the nonzero complex roots of F (z) − F^∗(z) = 0, then the desired estimate is given by

ˆ²d= N

j2π ln ˆz (3.39)

where

ˆ

z = arg min

zi∈Z{s(z)^HRb_ZZs(z)}. (3.40) The range of estimator is ±^N₂ subcarrier spacing. Fig. 3.9 shows the locations of the normalized roots of the polynomial, including the desired CFO estimate.

We summarize our CFO estimation algorithm as following:

1. Collect Ns received blocks and construct the sample correlation matrix ˆRY Y, b

R_{Y Y} = _N¹

s

PNs

i=1Y_iY^H_i and compute bR_ZZ = W^HC^dH_H Rb⁻¹_{Y Y}C^d_HW with channel information or bR_ZZ = W^HC^dH_WRb⁻¹_{Y Y}C^d_WW without channel information.

2. Calculate the coefficients of F (z) based on ˆRZZ where F (z) = PN −1

n=1 ns(n)zⁿ, s(n) =P

i,jRˆZZ(i, j), for n = j − i.

−10 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 0.2

0.4 0.6 0.8 1

Normalized CFO

Normalized 1 / MOE

1/MOE (SNR=10dB, N_s=100, K=10, perfect channel info) Candidate of CFO

CFO of Desired User

Figure 3.9: Normalized 1/MOE vs Normalized CFO where 1/MOE is normalized by its peak value and the associated root distribution; true CFO = 0.7 subcarrier spacings.

3. Find the nonzero roots of F (z) − F^∗(z) = 0 and discard undesired roots which outside of CFO estimate range.

4. Obtain the CFO estimate from ˆ²d = _j2π^N ln ˆz and ˆz = arg minzi{s(z)^HRb_ZZs(z)}, z = e^j2π²/N.

3.4 Simulation Results and Discussion

We use the data format of the IEEE 802.11a standard with a sample interval of 50 ns in our simulations. As mentioned before, CFO is normalized by subcarrier spacing.

Two static frequency-selective fading channel whose power delay profiles have sixteen exponentially decaying paths and rms delay spreads of 50 ns and 150 ns, respectively, are considered. The former channel is referred to as Model A while the latter is referred to as Model B. Typical magnitude squared frequency responses for these two classes of channels are depicted in Figs. 3.10 and 3.11, respectively. Obviously, Channel B is more selective than Channel A. Table I lists the system and channel parameters used in the

simulation. The signal-to-noise (SNR) ratio is defined as the ratio between the desired signal power and the noise power. Perfect power is assumed such that all received user signal powers are the same.

Table I

System and Channel Simulation Parameters

Parameters Values

Number of subcarriers 64

Length of cyclic prefix 16 samples

OFDM symbol period 64+16 samples

Modulation QPSK

Spreading code Random sequence {-1,1}

Length of spreading Code 64

Channel model (i) AWGN

(ii) Frequency selective fading channel 16-path with rms delay spreads of 50 ns (Model A) 16-path with rms delay spreads of 150 ns (Model B)

Power delay profile Exponential decay

Number of data blocks (N_s) 50, 100, 200

Number of users (K) 10, 30, 50

All received user signal powers are the same Fig. 3.12 shows the mean-squared error (MSE) performance of the CFO estimate under various channel conditions. A 10-user asynchronous OFDM-CDMA system is con-sidered and each computer run consists of 100 data blocks. When channel information {H_i^d} is not available, the channel estimate is obtained by assuming {H_i^d = 1}. The performance in channel A without channel information is better than that in channel B because high frequency selective implies low subcarrier correlation and makes it more difficult to distinguish the desired user from the other users via the spreading code. And if the receiver can not distinguish the desired signal from interference, the algorithm tends to cancel the desired signal, eliminating both MUI and part of the desired signal.

On the other hand, as expected, when channel information is available the performance under both circumstances improve. The performance in channel B with channel infor-mation is better than that in channel A because the more selective channel fading is the larger the distance between the effective channel spreading matrices associated with

different users. In AWGN channels, only the spreading code is available to distinguish different user signals. Fig. 3.13 compares the MSE performance with different number of data blocks. When the data correlation matrix is unknown, we replace the ensemble averages, RY Y, for the time averages, bR_{Y Y}. As the number of samples increases, bR_{Y Y} becomes “closer” to R_{Y Y}, i.e., if one uses more data blocks for estimating the correlation matrix, the resulting performance will be “closer” to the optimal performance using the true ensemble correlation matrix. The impact of the number of users is plotted in Fig.

3.14 where, as has been expected, it is shown that the MUI power is proportional to the number of system users whose presence degrade the frequency estimate’s performance accordingly. Performance of CFO estimate depends mainly on multiuser interference and noise. These performance curves do confirm that the proposed estimator is able to offer reasonable good performance even in a heavily-loaded system. Fig. 3.15 compares the averaged estimate CFO with the true CFO at SNR = 10 dB and shows that as long as the true CFO is within ±32 (i.e. ±N/2 ) subcarrier spacings, good estimated values can be obtained.

0 10 20 30 40 50 60 70 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

2 Channel A

Square Magnitude of Transform Function

Normalized Frequency

Figure 3.10: Magnitude squared of Channel(Model) A’s frequency response.

0 10 20 30 40 50 60 70

0 0.5 1 1.5 2 2.5 3

3.5 Channel B

Square Magnitude of Transform Function

Normalized Frequency

Figure 3.11: Magnitude squared of Channel(Model) B’s frequency response.

0 2 4 6 8 10 -50

-40 -30 -20 -10 0

MSE(dB)

SNR(dB)

Channel(Model) B without channel info Channel(Model) A without channel info Channel(Model) A with perfect channel info Channel(Model) B with perfect channel info AWGN

Figure 3.12: MSE performance of CFO estimates for different channel conditions; K = 10, N_s= 100, true CFO=0.7 subcarrier spacings.

0 2 4 6 8 10

-55 -50 -45 -40 -35 -30 -25 -20 -15 -10

MSE(dB)

SNR(dB)

Channel A (N_s=50) Channel A (N_s=100) Channel A (N_s=200) Channel B (N_s=50) Channel B (N_s=100) Channel B (N_s=200) AWGN (N_s=50) AWGN (N_s=100) AWGN (N_s=200)

Figure 3.13: MSE performance of CFO estimates for different number of data samples, Ns; K = 10, true CFO=0.7 subcarrier spacings, with perfect channel information.

0 2 4 6 8 10

Figure 3.14: MSE performance of CFO estimates for different number of users, K;

Ns = 100, true CFO=0.7 subcarrier spacings, with perfect channel information.

−40 −30 −20 −10 0 10 20 30 40

Figure 3.15: Relative CFO estimate versus relative CFO; K = 10, Ns = 500, with perfect channel information.

在文檔中 MIMO-OFDM 與 OFDM-CDMA 系統之頻率估計 (頁 36-46)