Time-Varying Doppler Frequency Offset Estimation Method for LTE-TDD Uplink with Multi-user in HST Scenario

Abstract

A time-varying Doppler frequency offset estimation method is proposed for the high-speed-train travelling over 350 km/h ground speed in the long term evolution-time division duplex (LTE-TDD) uplink with the multi-user. Based on the separated pilot symbols of each user, the single pilot symbol in one subframe is firstly segmented into halves to enlarge the estimation range, and then the cross correlation of the two pilot symbols in one subframe is employed to improve the estimation accuracy. To further improve the estimation accuracy, the autoregressive model is used to track the variation of the Doppler frequency offsets between the subframes. Simulation results show that the performance of the proposed method is much better than those of the available methods for the LTE-TDD uplink with the high-speed mobile multi-user, and it has a low computational complexity.

Appendix

In the section, the derivation of the MSE lower bound is given by [18, 19]. For received signal via the AWGN channel with the time-varying frequency offset, one can define $B^{(u,q)}_{j}(n)$ as

$$\begin{aligned} B^{(u,q)}_{j}(n)&= \bar{\bar{\gamma }}^{(u)*}_{j}(3,n) \bar{\bar{\gamma }}^{(u)}_{j}(10,n) \nonumber \\&= e^{j14\pi \triangle \varepsilon ^{(u)}_{j}/N_{D}}p^{ (u)*}_{j}(3,n)p^{(u)}_{j}(10,n)+\dot{w}^{(u)}_{j}(n) \nonumber \\&= e^{j14\pi \triangle \varepsilon ^{(u)}_{j}/N_{D}}|p^{(u)*}_{j}(3,n)|^{2}+\dot{w}^{(u)}_{j}(n) \end{aligned}$$

(28)

where $n=0,\ldots ,N_{D}-1$, and assume that the two pilot symbols are the same, i.e, $p^{(u)}_{j}(3,n)=p^{(u)}_{j}(10,n)$. The products of two noise terms in (28) can be negligible, so the $\dot{w}^{(u)}_{j}(n)$ can be written as

$$\begin{aligned} \dot{w}_j^{(u)}(n)&= \tilde{w}_j^{(u)*}(3,n)p_j^{(u)}(10,n){e^{j2\pi \vartriangle \varepsilon _j^{(u)}(10N - {N_D} + n)/{N_D}}} \nonumber \\&\quad + \tilde{w}_j^{(u)}(10,n)p_j^{(u)*}(3,n){e^{ - j2\pi \vartriangle \varepsilon _j^{(u)}(3N - {N_D} + n)/{N_D}}} \end{aligned}$$

(29)

From (28), denoting $\beta =exp[-j14\pi \triangle \hat{\varepsilon }^{(u)}_{j,2}/N_{D}]$, we obtain

$$\begin{aligned} tan\biggl [\frac{14\pi }{N_{D}}(\triangle \varepsilon ^{(u)}_{j}-\triangle \hat{\varepsilon }^{(u)}_{j,2})\biggr ]&\approx \frac{\mathfrak {I}\biggl (B^{(u,q)}_{j}(n)\beta \biggr )}{\mathfrak {R}\biggl (B^{(u,q)} _{j}(n)\beta \biggr )}\nonumber \\&\approx \frac{14\pi }{N_{D}}\biggl (\triangle \varepsilon ^{(u)}_{j} -\triangle \hat{\varepsilon }^{(u)}_{j,2}\biggr ) \end{aligned}$$

(30)

Substituting (28) into (30), it follows that

$$\begin{aligned} e^{(u)}_{j}&= \triangle \varepsilon ^{(u)}_{j}-\triangle \hat{\varepsilon }^{(u)} _{j,2}\nonumber \\&= \frac{N_{D}}{14\pi N}\frac{\mathfrak {I}\biggl (B^{(u,q)}_{j}(n)\beta \biggr )}{\mathfrak {R}\biggl (B^{(u,q)}_{j}(n)\beta \biggr )} \nonumber \\&= \frac{N_{D}}{14\pi N}\frac{\mathfrak {I}\biggl (\dot{w}^{(u)}_{j}(n)\beta \biggr )}{|p^{(u)}_{j}(3,n)|^{2}+\mathfrak {R}\biggl (\dot{w}^{(u)}_{j}(n) \beta \biggr )} \end{aligned}$$

(31)

At the high signal-to-noise ratios, a condition compatible with successful communications signaling, (31) may be approximated by

$$\begin{aligned} e^{(u)}_{j}\approx \frac{N_{D}}{14\pi N}\frac{\mathfrak {I}\biggl (\dot{w}^{(u)}_{j}(n)\beta \biggr )}{|p^{(u)}_{j}(3,n)|^{2}} \end{aligned}$$

(32)

For the separated user with multi-user interference, only $N_{ca}$ subcarriers is used to transmit the signal in the frequency domain, so the average power of the pilot in the time-domain, i.e., $\sum _{n=0}^{N_{D}-1}|p^{(u)}_{j}(i,n)|^{2},i=3,10$, is approximately $N_{ca}$. Therefore, the variance of fine estimation can be obtained after some straightforward calculations

$$\begin{aligned} var\biggl (\triangle \hat{\varepsilon }^{(u)}_{j,2}\biggr )\approx \biggl (\frac{N_{D}}{14\pi N}\biggr )^{2}\frac{1}{N_{ca}SNR} \end{aligned}$$

(33)

where $\textit{SNR}=1/\sigma ^{2}_{w}$.

From (32), after some manipulations, one can find that the frequency offset estimation noise $e^{(u)}_{j}=\triangle \varepsilon ^{(u)}_{j}-\triangle \hat{\varepsilon }^{(u)}_{j,2}$ has the following autocorrelation

$$\begin{aligned} R^{(u,e)}_{j}(l)&= E[e^{(u)}_{j}e^{(u)}_{j+l}] \nonumber \\&= \left\{ \begin{array}{l@{\quad }l} \biggl (\frac{N_{D}}{14\pi N}\biggr )^{2}\frac{1}{N_{ca}SNR},&{}\quad l=0 \\ 0,&{}\quad \textit{otherwise} \end{array} \right. \end{aligned}$$

(34)

Thus, the power spectral density (PSD) of loop noise is given by

$$\begin{aligned} P(f)&= T\sum _{l}R^{(u,e)}_{j}(l)e^{-j2\pi lfT} \nonumber \\&= \biggl (\frac{N_{D}}{14\pi N}\biggr )^{2}\frac{T}{N_{ca}SNR} \end{aligned}$$

(35)

Based on a linearized equivalent model [20], we can derive the tracking performance as

$$\begin{aligned} \textit{MSE}=\int _{-\frac{1}{2T}}^{\frac{1}{2T}}P^{(u)}_{j}(f)|G(f)|^{2}df \end{aligned}$$

(36)

where $G(f)$ is the closed-loop transfer function given by

$$\begin{aligned} G(f)=-\frac{\rho }{e^{j2\pi fT}-(1-\rho )} \end{aligned}$$

(37)

where $T=T_{g}+T_{u}$, $T_{g}$ and $T_{u}$ are the length of the CP and FFT respectively. Therefore, the MSE is

$$\begin{aligned} \textit{MSE}&= \biggl (\frac{N_{D}}{14\pi N}\biggr )^{2}\frac{T}{N_{ca}SNR} \int _{-\frac{1}{2T}}^{\frac{1}{2T}}|-\frac{\rho }{e^{j2\pi fT}-(1-\alpha )}|^{2}df \nonumber \\&= \biggl (\frac{N_{D}}{14\pi N}\biggr )^{2}\frac{T}{N_{ca}SNR}\int _{-\frac{1}{2T}}^ {\frac{1}{2T}}\frac{1}{(1-\rho )^2+1-2(1-\rho )cos2\pi fT}df \nonumber \\&= \biggl (\frac{N_{D}}{14\pi N}\biggr )^{2}\frac{T}{N_{ca}SNR}\int _{-\frac{1}{2T}}^{\frac{1}{2T}} \frac{1}{2(1-cos2\pi fT)(1-\rho )+\rho ^{2}}df \nonumber \\&= \biggl (\frac{N_{D}}{14\pi N}\biggr )^{2}\frac{T}{N_{ca}SNR} \int _{-\frac{1}{2\pi }}^{\frac{1}{2\pi }}\frac{1}{tan^{2}\theta +(\frac{\rho }{2-\rho })^2}d\theta \nonumber \\&= \biggl (\frac{N_{D}}{14\pi N}\biggr )^{2}\frac{\frac{\rho }{(2-\rho )}}{N_{ca}SNR} \end{aligned}$$

(38)