Continuous Wave Interference Effects on Ranging Performance of Spread Spectrum Receivers

Abstract

Continuous wave interference (CWI) can induce significant ranging error for a spread spectrum receiver. Since both the spread spectrum signal and CWI are deterministic signals but truncated by the integrate and dump filters, deterministic signal models are built, and then their continuous frequency spectrums and specific signal forms in the processing flow are analyzed. This paper provides analytical expressions for code tracking performance of ranging receivers using coherent early minus late discriminators. The impacts of interference to signal power ratio (ISR), front-end bandwidth, integration time and early-late spacing on ranging bias are taken into account. According to the property of pseudo-random noise (PRN) code, the average magnitude of the code frequency spectrum is presented to investigate the general properties of ranging bias, and the magnitude fluctuations around the average are considered to analyze the properties of ranging bias with specific PRN codes. It is shown that the ranging bias induced by CWI depends on ISR, interference frequency, initial phases, code properties, receiver parameters, etc. Meanwhile, the envelope magnitude of the code frequency spectrum is provided for an easy way of estimating the upper bound of ranging bias. The 32 GPS C/A codes are taken as examples to show their worst cases of ranging performance under CWI.

Appendices

Appendix 1

Rewrite Eq. (19),

$$\begin{aligned} R_{Ek} (\varepsilon _k )&= \frac{1}{T}\int _{(k-1)T}^{kT} {r(t)s_0 (t\!+\!\varepsilon _k \!+\!\tau /2)dt} \!=\!\frac{1}{T}\int _0^T {r\left[ {t+(k-1)T} \right] s_0 (t+\varepsilon _k \!+\!\tau /2)dt} \nonumber \\&= \frac{1}{T}\int _0^T {r_{Tk} (t)s_0 (t\!+\!\varepsilon _k \!+\!\tau /2)dt} \end{aligned}$$

(37)

With decomposing $r_{Tk} (t)$ into the sum of desired signal $s_{Tk} (t)$ and SWI $\iota _k (t),\,R_{Ek} (\varepsilon _k )$ can be investigated by the two terms separately.

The first term, the desired signal segment, is

$$\begin{aligned} R_{Ek1} (\varepsilon _k )&= \frac{1}{T}\int _0^T {s_{Tk} (t)s_0 \left( t+\varepsilon _k +\frac{\tau }{2}\right) dt} \nonumber \\&= \frac{1}{T}\int _0^T {\int _{-\infty }^\infty {S_{Tk} (f)e^{j2\pi ft}df} s_0 \left( t+\varepsilon _k +\frac{\tau }{2}\right) dt} \nonumber \\&= \frac{1}{T}\int _{-\infty }^\infty {S_{Tk} (f)\int _0^T {s_0 \left( t+\varepsilon _k +\frac{\tau }{2}\right) e^{j2\pi ft}dt} df} \nonumber \\&\approx \frac{1}{T}\int _{-\infty }^\infty {S_{Tk} (f)S_T^{*} (f)e^{-j2\pi f\left( \varepsilon _k +\frac{\tau }{2}\right) }df} \end{aligned}$$

(38)

In (38), the assumptions that $\varepsilon _k$ is small and $\tau$ is only a fraction of chip period, are used to ignore the shift by $\varepsilon _k +\tau /2$ in the limits of integration to yield the approximation.

Substituting (16) into (38) yields

$$\begin{aligned} R_{Ek1} (\varepsilon _k )\approx \int _{-B/2}^{B/2} {\frac{\sqrt{2C_s }}{2}\frac{\left| {S_T (f)} \right| ^{2}}{T}e^{j\Delta \theta _k }e^{-j2\pi f\left( \varepsilon _k +\frac{\tau }{2}\right) }df} \end{aligned}$$

(39)

Since the integral bandwidth is wide enough to average the discrete spectrum-like shape to the smooth spectrum level, we can replace the spectrum term in (39) with ideal random code spectrum, which has been widely used in partialband and wideband signal analysis [19, 20, 28]. So (39) can be expressed

$$\begin{aligned} R_{Ek1} (\varepsilon _k )\approx \int _{-B/2}^{B/2} {\frac{\sqrt{2C_s }}{2}G_s (f)e^{j\Delta \theta _k }e^{-j2\pi f\left( \varepsilon _k +\frac{\tau }{2}\right) }df} \end{aligned}$$

(40)

using random code spectrum $G_s (f)=T_c \hbox {sinc}^{2}(\pi fT_c )$ for BPSK modulation.

Meanwhile, by the same progress and assumptions, the second term, the interference segment becomes

$$\begin{aligned} R_{Ek2} (\varepsilon _k )&= \frac{1}{T}\int _0^T {\iota _k (t)s_0 \left( t\!+\!\varepsilon _k \!+\!\frac{\tau }{2}\right) dt} \!=\!\frac{1}{T}\int _0^T {\int _{-B/2}^{B/2} {l_k (f)e^{j2\pi ft}df} s_0 \left( t\!+\!\varepsilon _k \!+\!\frac{\tau }{2}\right) dt} \nonumber \\&= \frac{1}{T}\int _{-B/2}^{B/2} {l_k (f)\int _0^T {s_0 \left( t+\varepsilon _k +\frac{\tau }{2}\right) e^{j2\pi ft}dt} df}\nonumber \\&\approx \frac{1}{T}\int _{-B/2}^{B/2} {l_k (f)S_T^{*} (f)e^{-j2\pi f\left( \varepsilon _k +\frac{\tau }{2}\right) }df} \end{aligned}$$

(41)

Appendix 2

In the presence of CWI, the ranging bias is closely related to the phases:$\alpha _n ,\,\theta _0$ and $\theta _\iota$. Since $\alpha _n$ is known for a specific PRN code, we turn our attention to the difference of $\theta _0$ and $\theta _\iota$, and find the maximum ranging bias.

To make the analysis simpler, some abbreviations are introduced:

$$\begin{aligned} \theta&= \theta _\iota -\theta _0 \end{aligned}$$

(42)

$$\begin{aligned} a&= \sqrt{\frac{C_\iota }{C_s }}\frac{S_{0n} }{T_0 }\sin \left( \pi \frac{n}{T_0 }\tau \right) \end{aligned}$$

(43)

$$\begin{aligned} b&= \int _{-B/2}^{B/2} {G_s (f)\sin (\pi f\tau )2\pi {\textit{fdf}}} \end{aligned}$$

(44)

$$\begin{aligned} c&= \sqrt{\frac{C_\iota }{C_s }}\frac{2\pi nS_{0n} }{T_0^2 }\sin \left( \pi \frac{n}{T_0 }\tau \right) \end{aligned}$$

(45)

Then, from (34),

$$\begin{aligned} \varepsilon _{\mathrm{bias},n} =\frac{a\sin (\theta -\alpha _n )}{b+c\cos (\theta -\alpha _n )} \end{aligned}$$

(46)

It is noted that the denominator of (46) is the slope of discriminator. It is reasonable to assume that $\left| b \right| >\left| c \right|$ when the loop keeps steady.

There are two cases: Firstly, if $\sin ({\pi \tau n}/{T_0 })=0$, then $a=0$ and $\varepsilon _{\mathrm{bias},n}$ is zero constantly. Secondly, $\sin ({\pi \tau n}/{T_0 })\ne 0$ and $a\ne 0$. In the latter case, the first and second derivative of the bias versus phase $\theta$ is needed to find the maximum value of $\varepsilon _{bias,n}$.

The first derivative of the bias is

$$\begin{aligned} \frac{\partial \varepsilon _{\mathrm{bias},n} }{\partial \theta }=\frac{ab\cos (\theta -\alpha _n )+ac}{\left[ {b+c\cos (\theta -\alpha _n )} \right] ^{2}} \end{aligned}$$

(47)

which indicates that only if $\cos (\theta -\alpha _n )=-c/b,\,{\partial \varepsilon _{\mathrm{bias},n} }/{\partial \theta }=0$.

The second derivative is

$$\begin{aligned} \frac{\partial ^{2}\varepsilon _{\mathrm{bias},n} }{\partial \theta ^{2}}=\frac{a\sin (\theta -\alpha _n )\left[ {bc(2a-1)\cos (\theta -\alpha _n )-b^{2}+2ac^{2}} \right] }{\left[ {b+c\cos (\theta -\alpha _n )} \right] ^{3}} \end{aligned}$$

(48)

Under the assumption that $\left| b \right| >\left| c \right|$ and $a\ne 0,\,\cos (\theta -\alpha _n )=-c/b\ne 1$ or $-1$, then the first product term of the numerator of (48), $a\sin (\theta -\alpha _n )$, is nonzero. Meanwhile, when $\cos (\theta -\alpha _n )=-c/b$, the second product term of the numerator can be simplified to $c^{2}-b^{2}$, and it is also nonzero. In a word, when ${\partial \varepsilon _{\mathrm{bias},n} }/{\partial \theta }=0,\,{\partial ^{2}\varepsilon _{\mathrm{bias},n} }/{\partial \theta ^{2}}\ne 0$.

Then we can conclude that, if

$$\begin{aligned} \cos (\theta -\alpha _n )=-\frac{c}{b}=-\frac{\sqrt{\frac{C_\iota }{C_s }}\frac{2\pi nS_{0n} }{T_0^2 }\sin (\pi \frac{n}{T_0 }\tau )}{\int _{-B/2}^{B/2} {G_s (f)\sin (\pi f\tau )2\pi {\textit{fdf}}} } \end{aligned}$$

(49)

the maximum or minimum values of the ranging bias is

$$\begin{aligned} \varepsilon _{\mathrm{bias},n}^{\max \,\mathrm{or}\,\min }&= \pm \frac{a}{\sqrt{b^{2}-c^{2}}} \nonumber \\&= \pm \frac{\sqrt{\frac{C_\iota }{C_s }}\frac{S_{0n} }{T_0 }\sin \left( \pi \frac{n}{T_0 }\tau \right) }{\sqrt{\left[ {\int _{-B/2}^{B/2} {G_s (f)\sin (\pi f\tau )2\pi {\textit{fdf}}} } \right] ^{2}-\left[ {\sqrt{\frac{C_\iota }{C_s }}\frac{2\pi nS_{0n} }{T_0^2 }\sin \left( \pi \frac{n}{T_0 }\tau \right) } \right] ^{2}}} \end{aligned}$$

(50)

whose sign depends on the sign of $a\sin (\theta -\alpha _n )$. The maximum value and the minimum value are symmetrical about zero, consequently, in this paper, we only consider the maximum value, absolute value of (50).

In addition, the first case can be analyzed as a special case of the second, using (49) and (50). As a result, they will not be treated differently.