An Effective Time Delay Estimation Method Based on Frequency Difference Compensation for Narrowband RF Signals

Abstract

This paper addresses a novel time delay estimation (TDE) method for narrowband RF signals with intermediate frequency difference (IFD). Since IFD is the main factor which affects the TDE precision, we present the frequency detection TDE model with IFD and discuss the effects of the IFD on the TDE. Then, a frequency detection TDE method based on IFD compensation is proposed. Especially, an improved frequency-domain cross-correlation method is adopted to estimate the IFD. The effectiveness of the algorithm is verified by MATLAB simulation. Simulation results demonstrate that the proposed TDE method shows a significant performance improvement over the classical TDE method based on Hilbert transform. Furthermore, the proposed method is also used in the actual system, and its estimation value is very close to the true value, which presents an enormous engineering practicality.

Appendices

Appendix 1

The derivation of (11) is following:

$$\begin{aligned} \hbox {DFT}(s(n))= & {} S(k) \end{aligned}$$

(36)

$$\begin{aligned} \hbox {DFT}(s(n)e^{{j2\pi nD_f }/N})= & {} S(k-D_f )\end{aligned}$$

(37)

$$\begin{aligned} \hbox {DFT}(s(n-D_t ))= & {} S(k)e^{{-j2\pi kD_t }/N} \end{aligned}$$

(38)

Define ${n}'=n-D_t$

$$\begin{aligned} \hbox {DFT}(s({n}'))={S}'(k)=S(k)e^{{-j2\pi kD_t }/N} \end{aligned}$$

(39)

Based on the above results, it leads to

$$\begin{aligned} \hbox {DFT}(s(n-D_t )e^{{j2\pi nD_f }/N})= & {} \hbox {DFT}(s({n}')e^{{j2\pi {n}'D_f }/N}e^{{j2\pi D_t D_f }/N})\nonumber \\= & {} e^{{j2\pi D_t D_f }/N}\hbox {DFT}(s({n}')e^{{j2\pi {n}'D_f }/N}) \nonumber \\= & {} e^{{j2\pi D_t D_f }/N}{S}'(k-D_f )\nonumber \\= & {} e^{{j2\pi D_t D_f }/N}S(k-D_f )e^{{-j2\pi (k-D_f )D_t }/N} \nonumber \\= & {} S(k-D_f )e^{{-j2\pi (k-2D_f )D_t }/N} \end{aligned}$$

(40)

The cross-power spectrum density between the two signals is

$$\begin{aligned} P_{xy} (k)=\lambda S^{{*}}(k)S(k-D_f )e^{{-j2\pi D_t (k-2D_f )}/N}+W_p (k),\;D_f ={\Delta f_0 N}/{f_s }, \end{aligned}$$

(41)

where $N$ denotes the length of $P_{xy} (k)$ and $W_p (k)=\lambda S(k-D_f )e^{{-j2\pi D_t (k-2D_f )}/N}W_1^{*} (k)+S^{{*}}(k)W_2 (k)+W_1^{*} (k)W_2 (k)$.

Appendix 2

We define $A_c (n)=a(n)\cos \varphi _1 (n)$ and $A_s (n)=a(n)\sin \varphi _1 (n)$

$$\begin{aligned} H\left[ {A_c (n)\cos (\omega _0 n)} \right] =A_c (n)\sin (\omega _0 n) \end{aligned}$$

(42)

$$\begin{aligned} H\left[ {A_s (n)\sin (\omega _0 n)} \right]= & {} -A_s (n)\cos (\omega _0 n) \end{aligned}$$

(43)

$$\begin{aligned} x(n)= & {} a(n)\cos (\omega _0 n+\varphi _1 (n))+w_1 (n) \nonumber \\= & {} a(n)\cos (\varphi _1 (n))\cos (\omega _0 n)\!-\!a(n)\sin (\varphi _1 (n))\sin (\omega _0 n)\!+\!w_1 (n) \nonumber \\= & {} A_c (n)\cos (\omega _0 n)-A_s (n)\sin (\omega _0 n)+w_1 (n) \end{aligned}$$

(44)

$$\begin{aligned} H[x(n)]= & {} H\left[ {A_c (n)\cos (\omega _0 n)} \right] -H\left[ {A_s (n)\sin (\omega _0 n)} \right] +H\left[ {w_1 (n)} \right] \nonumber \\= & {} A_c (n)\sin \omega _0 n+A_s (n)\cos \omega _0 n+H[w_1 (n)] \nonumber \\= & {} a(n)\sin (\omega _0 n+\varphi _1 (n))+H[w_1 (n)], \end{aligned}$$

(45)

where $H[\cdot ]$ stands for Hilbert Transform. Based on the above results, it leads to

$$\begin{aligned} a(n)=\sqrt{x^{2}(n)+H^{2}[x(n)]-w_x (n)}, \end{aligned}$$

(46)

where $w_x (n)=w_1^2 (n)+H^{2}[w_1 (n)]+2\left\{ {w_1 (n)s_1 (n)+H[w_1 (n)]H[s_1 (n)]} \right\}$. Similarly, we have

$$\begin{aligned} a(n-D_t )=\sqrt{y^{2}(n)+H^{2}[y(n)]-w_y (n)}, \end{aligned}$$

(47)

where $w_y (n)=w_2^2 (n)+H^{2}[w_2 (n)]+2\left\{ {w_2 (n)s_2 (n)+H[w_2 (n)]H[s_2 (n)]} \right\}$.