Doppler Spread Estimation for Broadband Wireless OFDM Systems Over Rician Fading Channels

Abstract

In this paper, we present a new Doppler spread estimation algorithm for broadband wireless orthogonal frequency division multiplexing (OFDM) systems with fast time-varying and frequency-selective Rayleigh or Rician fading channels. The new algorithm is developed by analyzing the statistical properties of the power of the received OFDM signal in the time domain, thus it is not affected by the influence of frequency-domain inter-carrier interference (ICI) introduced by channel variation within one OFDM symbol. The operation of the algorithm doesn’t require the knowledge of fading channel coefficients, transmitted data, or signal-to-noise ratio (SNR) at the receiver. It is robust against additive noise, and can provide accurate Doppler spread estimation with SNR as low as 0 dB. Moreover, unlike existing algorithms, the proposed algorithm takes into account the inter-tap correlation of the discrete-time channel representation, as is the case in practical systems. Simulation results demonstrate that this new algorithm can accurately estimate a wide range of Doppler spread with low estimation latency and high computational efficiency.

Appendix

1.1 Proof of (8) and (9)

The auto-correlation function of the received time-domain OFDM signal is calculated as

$$ \begin{aligned} R_{yy}(s,u,n,m)=& {\mathbb E}\left\{y^{(s+u)}(n+m)\left[y^{(s)}(n)\right]^*\right\} \cr =&{\mathbb E}\left\{\left[{\frac{1} {\sqrt{1+K}}}\sum_{l_1=-L_1}^{L_2}h^{(s+u)}(n+m,l_1)x^{(s+u)}(n+m-l_1)+ \sqrt{{\frac{K} {1+K}}}h^{(s+u)}_{{LOS}}(n+m)\sum_{p_1=-P_1}^{P_2} {\sigma}_{p_1}x^{(s+u)}(n+m-p_1) + v^{(s+u)}(n+m)\ \right. \right] \cr & \left. \times \left[{\frac{1}{\sqrt{1+K}}} \sum_{l_2=-L_1}^{L_2}h^{(s)}(n,l_2)x^{(s)}(n-l_2)+\sqrt{{\frac{K} {1+K}}}h^{(s)}_{{LOS}}(n) \sum_{p_2=-P_1}^{P_2}{\sigma}_{p_2} x^{(s)}(n-p_2) + v^{(s)}(n)\right]^*\right\} \cr =&{\frac{1} {1+K}}\sum_{l_1=-L_1}^{L_2}\sum_{l_2=-L_1}^{L_2}{\mathbb E}\left\{h^{(s+u)}(n+m,l_1) \left[h^{(s)}(n,l_2)\right]^*x^{(s+u)}(n+m-l_1)\left[x^{(s)}(n-l_2)\right]^*\right\} \cr &+{\frac{K} {1+K}}\sum_{p_1=-P_1}^{P_2}\sum_{p_2=-P_1}^{P_2}\sigma_{p_1}\sigma_{p_2}^*{\mathbb E}\left\{h^{(s+u)}_{{LOS}}(n+m) \left[h^{(s)}_{{LOS}}(n)\right]^*x^{(s+u)}(n+m-p_1)\left[x^{(s)}(n-p_2)\right]^*\right\} \cr &+{\mathbb E}\left\{v^{(s+u)}(n+m)\left[v^{(s)}(n)\right]^*\right\} \cr =&\left[{\frac{1} {N(1+K)}}\sum_{l_1=-L_1}^{L_2}\sum_{l_2=-L_1}^{L_2}\sum_{k=0}^{N-1} C_{l_1,l_2}J_0(2\pi f_dmT_s)e^{j{\frac{2\pi (l_2-l_1+m)k}{N}}} +{\frac{K} {N(1+K)}}\sum_{p_1=-P_1}^{P_2}\sum_{p_2=-P_1}^{P_2}\sum_{k=0}^{N-1} \sigma_{p_1}\sigma_{p_2}^*e^{j2\pi f_dmT_s\cos\theta_0}e^{j{\frac{2\pi (p_2-p_1+m)k}{N}}} +\sigma ^2\delta (m)\right]\delta(u) \end{aligned} $$

(32)

where x ⁽ⁱ⁾(n) defined in (1) is used in the derivation of the last equality, thus (8) is proved.

The proof of Eq. 9 relies on the following two identities: ${\mathbb E}\{v_1v_2v_3\}=0$ and ${\mathbb E}\{v_1v_2v_3v_4\}={\mathbb E}\{v_1v_2\}{\mathbb E}\{v_3v_4\}+{\mathbb E}\{v_1v_3\}{\mathbb E}\{v_2v_4\}+{\mathbb E}\{v_1v_4\}{\mathbb E}\{v_2v_3\},$ where $v_1, v_2, v_3$ and v ₄ are zero-mean Gaussian random variables. With the above identities, (9) can be derived based on the definition of auto-covariance function. The derivation process is extremely lengthy and tedious, but straightforward, thus the details are omitted here for brevity.