High Accuracy Velocity Measurement Based on Keystone Transform Using Entropy Minimization

Velocity measurement is a basic task of radars. The target velocity is usually estimated according to the Doppler frequency shift. While traditional Doppler methods are unsuitable for high-speed targets, since the serious range migration between adjacent echoes causes phase wrapping. The serious range migration also interferes the coherent integration to improve the accuracy of the velocity estimation. A velocity measurement method based on Keystone transform using entropy minimization is studied to solve this problem. This method applies Keystone transform to the echo to calculate the ambiguity degree with the help of entropy minimization. The proposed algorithm estimates the ambiguity degree with no error at a wider range of SNR than the traditional method. The ambiguous Doppler frequency is obtained according to the slow time. Theoretical analyses and simulations show that this method has very high precision.

SECTION I. Introduction

The Millimeter wave (MMW) radar has been widely used because of the advantage of higher Doppler velocity resolution compared with centimeter wave radar [1]–​[3]. However with the development of aerospace technology, the target velocity has become higher and higher [4]–​[6]. The method of continuous wave fails to measure the target Doppler frequency with enough accuracy. The present methods in Pulse doppler (PD) radar for measuring the velocity do not work well, since high-speed targets will cause serious range migration which makes coherent integration invalid. Therefore, the range migration must be compensated in PD radar [7]–​[9].

Keystone transform has already been widely used in radar signal processing. Perry et al.[10] firstly applied Keystone transform in Synthetic aperture radar (SAR) imaging for moving targets. Zhang et al. [11] applied Keystone transform in dim target detection. Yuan et al.[7] adopted Keystone transform into PD radar to detect weak targets. The above examples show that Keystone transform can compensate the range migration.

In Keystone transform, the key is to calculate ambiguity degree $K$. As in Refs.[7], [9], the traditional method finds the ambiguity degree $K$ through maximizing the peak of an echo. It is simple but performs badly at low SNR. Actually, the essence of maximizing echo peak is finding the sharpest one. Entropy minimization is more suitable for the degree of the sharpness[12].

Entropy minimization has been widely used in many fields. In Electroencephalogram (EEG) signals, it is a measure of the degree of order/disorder of the signal[13]. It is used in Inverse synthetic aperture radar (ISAR) imaging which believes that the smaller the image entropy is, the better the image is focused[12]. Entropy minimization is also used in Doppler beam sharpening (DBS) imaging which believes that the image entropy increases with the image becoming blurred[14].

In this paper, we propose the velocity measurement based on Keystone transform, in which the entropy minimization is used to find the ambiguity degree $K$. Simulations show that this method performs better than the traditional methods. The remainder of this paper is organized as follow. In Section II, velocity measurement based on Keystone transform using entropy minimization is proposed. Section III gives the simulation results and verifies the validity of this algorithm. Finally, this paper is concluded in Section IV.

SECTION II. Algorithm Descrition

The proposed algorithm has three main steps: Keystone transform, calculation of ambiguity degree $K$ and velocity estimation of high precision. Firstly, the radar signal model is introduced.

1. Signal Model

In Refs.[7], [9], the radar transmits LFM signals and the $n^{\text{th}}$ echo is expressed as $$\begin{align*}&s(\widehat{t},n)= \text{rect} [\frac{\widehat{t}}{T_{r}}]\mathrm{e}^{\mathrm{j}(2\pi f_{c}t+k\pi\widehat{t}^{2})},\\ &\widehat{t}=t-nPRT, n=0,1, \cdots, N-1\tag{1}\end{align*}$$

In Eq.(1), symbols involved are explained as follow: rect $[\bullet]$: the unit rectangular function,

$N$:	the total number of echoes,
$M$:	the total number of samples in each echo,
$f_{c}$:	the carrier frequency,
$k$:	the chirp rate,
$T_{r}$:	time width of the chirp pulse,
$B$:	the signal bandwidth,
$PRT$:	pulse repetition interval,
$\widehat{t}$:	the fast time (range directions),
$nPRT$:	the slow time (azimuth directions).

The fast time and slow time are illustrated in Fig. 1 [15].

Fig. 1.

Two-dimensional matrix of echoes data

Assume that the radial speed of the moving target is $v_{0}$, the original range between radar and target is $r_{0}$. Then the delay time of the $n^{\text{th}}$ echo is $$\begin{align*}\tau_{n} &=2\frac{r_{0}+ v_{0}nPRT}{\mathrm{c}}\\ &= \tau_{0}+(2 v_{0}/\mathrm{c})nPRT\tag{2}\end{align*}$$ where $\tau_{0}=2r_{0}/\mathrm{c},\ n=0,1,2, \cdots, N-1$. The $n^{\text{th}}$ echo after downconversion is $$\begin{equation*}s_{r}(\widehat{t},n)= \text{rect}[\frac{\widehat{t}-\tau_{n}}{T_{r}}]\mathrm{e}^{\mathrm{j}k\pi(\widehat{t}-\tau_{n})^{2}}\mathrm{e}^{-\mathrm{j}2\pi f_{c}\tau_{n}}\mathrm{e}^{\mathrm{j}2\pi f_{d}\widehat{t}}\tag{3}\end{equation*}$$ where $f_{d}=2v_{0}f_{c}/\mathrm{c}$ is the Doppler frequency.

According to Ref.[7], the FFT of Eq.(3) over fast time $\widehat{t}$ is shown as follow $$\begin{align*}S_{r}(f,n)=&\frac{1}{\sqrt{k}} \text{rect} [\frac{f- f_{d}}{B}]\mathrm{e}^{-\mathrm{j}\frac{\pi}{k}(f- f_{d})^{2}}\\ &\times \mathrm{e}^{-\mathrm{j} 2\pi f_{c}\tau_{n}}\mathrm{e}^{-\mathrm{j} 2\pi(f- f_{d})\tau_{n}}\tag{4}\end{align*}$$

The FFT of the impulse response of the matched filter over fast time is $$\begin{equation*}H(f)= \frac{1}{\sqrt{k}} \text{rect} [ \frac{f}{B}]\mathrm{e}^{\mathrm{j}\frac{\pi}{k}f^{2}}\tag{5}\end{equation*}$$

Multiplying Eq.(4) by Eq.(5), we can get the output of the matched filter $$\begin{align*}S_{o}(f,n)&=\frac{1}{k} \text{rect} [\frac{f- f_{d}/2}{B-\vert \mathrm{f}_{d}\vert }]\mathrm{e}^{-\mathrm{j}\frac{\pi}{k} f_{d}^{{2}}}\\ &\times \mathrm{e}^{-\mathrm{j} 2\pi f(\tau_{n}- f_{d}/k)}\mathrm{e}^{-\mathrm{j} 2\pi(f_{c}- f_{d})\tau_{n}}\tag{6}\end{align*}$$

Taking IFFT to Eq.(6) over frequency-domain f, we can obtain the expression in time-domain after the matched filter. $$\begin{align*}s_{o}(\widehat{t}, n) & =\text{sinc}[(B- f_{d})(\widehat{t}- \tau_{n}+\frac{f_{d}}{k})]\\ & \times \mathrm{e}^{\mathrm{j} \pi f_{d} \widehat{t}} \mathrm{e}^{\mathrm{j} \pi f_{d} \tau_{n}} \mathrm{e}^{\mathrm{j} 2 \pi f_{d} n P R T} \mathrm{e}^{-\mathrm{j} 2 \pi f_{c} \tau_{0}}\tag{7}\end{align*}$$

According to nature of $\text{sinc} ()$ function, the peak of every echo is at $\tau_{n}-f_{d}/k$ which is variant. Fig. 2 shows the sketch map of the range migration between echoes. The Migration through resolution cell (MTRC) will be more than 1 range cell when the target velocity is high enough.

Fig. 2.

The range migration

2. Keystone Transform

It can be seen from Eq.(6) that the coupling between $f$ and $n$ leads to the range migration which can be handled by Keystone transform.

Let $$\begin{equation*}n=\frac{f_{c}}{f+f_{c}}l\tag{8}\end{equation*}$$ where $l$ is the virtual slow time. Similar to method in Refs.[7], [9], we take Eq.(8) into Eq.(6), $$\begin{align*}S_{o\_k}(f, l)= & \text{rect}[\frac{f}{B-\vert f_{d}\vert }] \mathrm{e}^{-\mathrm{j} \frac{\pi}{k} f_{d}{}^{2}} \mathrm{e}^{-\mathrm{j} 2 \pi f\left(\tau_{0}-\frac{f_{d}}{k}\right)}\\ & \times \mathrm{e}^{-\mathrm{j} 2 \pi(f_{c}- f_{d}) \tau_{0}} \mathrm{e}^{\mathrm{j} 2 \pi f_{d} P R T l}\tag{9}\end{align*}$$

Take IFFT of Eq.(9) over $f$ $$\begin{align*}s_{o\_ k}(\hat{t}, l)= & \text{sinc}\left[(B-\vert f_{d}\vert)(\widehat{t}- \tau_{0}+\frac{f_{d}}{k})\right]\tag{10}\\ & \times \mathrm{e}^{\mathrm{j} \pi f_{d} \widehat{t}} \mathrm{e}^{-\mathrm{j} 2 \pi(f_{c}- f_{d}) \tau_{0}} \mathrm{e}^{\mathrm{j} 2 \pi f_{d}lPRT}\end{align*}$$

In Eq.(10), the peak of every echo is at $\tau_{0}-f_{d}/k$ which is a constant value after Keystone transform.

Sinc interpolation is implemented to achieve the transform in project [15], [16] $$\begin{equation*}S(f,l)=\sum\limits_{n=0}^{N-1}S_{O}(f,n) \text{sinc} [\frac{lf_{c}}{f_{c}+f}-n]\tag{11}\end{equation*}$$ where $l=0,1,2, \cdots, N-1$.

Also, it will compensate the ambiguity degree $K$ when the target's velocity is high and the PRF is low. The definition of ambiguity degree $K$ and ambiguous Doppler frequency $\widetilde{f}$ is shown as Eq.(12) $$\begin{equation*}f_{d}=\widetilde{f}+K\cdot PRF\tag{12}\end{equation*}$$

The ambiguity degree $K$ must be an integer. The ambiguous Doppler frequency $\widetilde{f}$ must satisfy Eq.(13) $$\begin{equation*}\vert \widetilde{f}\vert < PRF/2\tag{13}\end{equation*}$$

So the modified Eq.(11) is as follow: $$\begin{equation*}S(f,l)=\sum\limits_{n=0}^{N-1}S_{O}(f,n)\text{sinc}[\frac{lf_{c}}{f_{c}+f}-n]e^{\mathrm{j}2\pi Kl\frac{f_{c}}{f_{c}+f}}\tag{14}\end{equation*}$$

IFFT is applied to Eq.(14) to transform it to time domain, which is represented as $$\begin{equation*}s_{k}(\widehat{t},l)= \text{IFFT} [S(f, l)]\tag{15}\end{equation*}$$

It is shown from Eq.(14) that the ambiguity degree $K$ is unknown. Therefore, calculating ambiguity degree $K$ is a key to Keystone transform.

3. Ambiguity Degree $\boldsymbol{K}$ Estimation Based on Entropy Minimization and Velocity Estimation

In order to estimate integer $K$, the traditional method substitutes different values of $K$ in Eq.(14) and then finds the maximum peak of the formula $$\begin{equation*}s_{k\_sum}(\widehat{t})=\sum\limits_{l=0}^{N-1}s_{k}(\widehat{t},l)\tag{16}\end{equation*}$$ which has the maximum peak when the range migration is compensated only at the correct $K$.

Actually, the sum of echoes has waveform passivation. In statistical language, people always use the entropy to describe the degree of sharpening. The higher the degree of sharpening is, the smaller the entropy is. The entropy makes use of the data around the peak to measure the sharpness, which makes it more effective to describe the migration than determining the maximum only. The entropy is defined as[17], [18] $$\begin{equation*}H(K)=-\sum\limits_{\widehat{t}=0}^{M-1}s_{k}{{}^{\prime}}(\widehat{t})\ln s_{k}{{}^{\prime}}(\widehat{t})\tag{17}\end{equation*}$$ where $$\begin{equation*}s_{k}^{\prime}(\widehat{t})=\frac{\sum\limits_{l=0}^{N-1} s_{k}(\widehat{t}, l)}{\sum\limits_{\widehat{t}=0}^{M-1} \sum\limits_{l=0}^{N-1} s_{k}(\widehat{t}, l)}\tag{18}\end{equation*}$$

The method based on entropy minimization calculates the entropy of Eq.(15) according to Eq.(17). And then find the minimum entropy.

From the Eq.(17) we can see that the method of entropy minimization uses all of the data, while the traditional method just uses one point which is the maximum. Also, when the SNR is low, the maximum is easily affected in traditional methods while the entropy is harder to be affected in the method of entropy minimization since this method makes use of all the data sufficiently.

According to Eq.(9), the ambiguous Doppler frequency $\widetilde{f}$ can be obtained through FFT over slow time. The slow time can be represented as $l\cdot PRT$ $$\begin{align*}S_{o \_{k}}(f, w)= & 2 \pi r e c t[\frac{f}{B-\vert f_{d}\vert }] \mathrm{e}^{-j \frac{\pi}{k} f_{d}^{2}} \mathrm{e}^{-j 2 \pi f\left(\tau_{0}-\frac{f_{d}}{k}\right)}\\ & \times \mathrm{e}^{-j 2 \pi(f_{c}- f_{d}) \tau_{0}} \delta(w-2 \pi f_{d})\tag{19}\end{align*}$$

The sampling frequency is PRF in slow time. So the ambiguous Doppler frequency is $$\begin{equation*}\widetilde{f}=f_{d}-K\cdot PRF\tag{20}\end{equation*}$$ where $K$ is the estimated ambiguity degree.

So the Doppler frequency $f_{d}$ can be ascertained according to Eq.(12). The velocity $v_{0}$ can be easily obtained by $$\begin{equation*}v_{0}=cf_{d}/2f_{0}\tag{21}\end{equation*}$$

SECTION III. Simulation and Analysis

In practice, the warning radar will find targets and estimate the speed range at first. Based on the speed range, the range of ambiguity degree $K$ may be calculated. The proposed method works with the knowledge provided by the warning radars.

Table 1 lists the simulation parameters (the following text simulates with the same parameters without otherwise specified). Fig. 3 reveals the range migration obtained by simulation experiments.

In Table 1, the Doppler frequency $f_{d}$ is equal to 3266.6667 KHz. According to the Eq.(12), $\mathrm{K}$ and $\widetilde{f}$ are 653 and 1666.7 Hz, respectively. Fig. 4 shows the calculation of ambiguity degree based on entropy minimization, according to which, this method can calculate the ambiguity degree accurately. In Fig. 4, the ordinate represents the entropy of different ambiguity degree k.

Fig. 3.

The range migration before keystone transform

Table 1. Simulation parameters

Fig. 5 demonstrates the echoes which have been processed by Keystone transform. Compared with Fig. 3, the range migration is compensated by Keystone transform.

Fig. 4.

Calculating ambiguity degree based on entropy minimization

Fig. 5.

The compensation after keystone transform

The ambiguous Doppler frequency can be obtained by FFT over slow-time which is shown as Fig. 6. In Fig. 4, the ambiguity degree $K$ is 653. In Fig. 6, the ambiguous Doppler frequency is 1660Hz. Then the Doppler frequency is $f_{d}=K\cdot PRF+\widetilde{f}=3266.660\text{KHz}$. The Doppler frequency error is 6.7Hz which is caused by frequency quantization error. Hence the velocity error is 0.0287m/s.

In order to compare the method of traditional and entropy minimization, 1000 Monte-Carlo trials were carried out under the different SNR. Fig. 7 shows the error rate varying with SNR based on both traditional method and entropy minimization. It can be seen that the traditional method calculates the ambiguity degree accurately only when SNR exceed 0 dB, while the method of entropy minimization only requires SNR to exceed -12dB. So the method of entropy minimization performs better than the traditional method.

Fig. 6.

Ambiguous doppler frequency

Fig. 7.

Calculating ambiguity degree based on the entropy minimization and traditional methods

Fig. 8 shows how velocity error varies with SNR based on entropy minimization of Keystone transform. From Fig. 8, it can be seen that for SNR lower than -15.5dB, the velocity error is about 20m/s because the ambiguity degree $\mathrm{K}$ cannot be calculated correctly. For SNR between -15.5dB and -14.5dB, the velocity error descends quickly. For SNR above -14dB, the velocity error is about 0.03m/s and the velocity error curve is a horizontal line. This is because that the ambiguity degree $K$ and the ambiguous Doppler frequency are estimated correctly when SNR exceed -14dB. Also the fixed velocity error is caused by frequency quantization error. The maximum frequency quantization error is $\Delta f_{\max}=\frac{PRF}{2I}=9.765\text{Hz}$ and the corresponding maximum velocity error is $\Delta v_{\max}= \frac{\Delta f_{\max}c}{2f_{0}}=0.042\mathrm{m}/\mathrm{s}$. So the velocity error of this method is less than 0.042m/s when SNR exceeds -14dB.

Finally, to confirm the effectiveness of the proposed algorithm, comparison experiments on the relation between velocity error and SNR were carried out using phase difference of received signals[19]. Fig. 9 shows the relation between velocity error and SNR based on the phase difference of received signals. From Fig. 9, it can be seen that the velocity error is about 0.1m/s when SNR exceeds 8dB, so the proposed method in this paper gets great advantages in the aspects of both SNR and velocity error compared with the phase difference of received signals.

Fig. 8.

Relation between velocity error and SNR based on entropy minimization of keystone transform

Fig. 9.

Relation between velocity error and SNR based on phase difference of received signals

SECTION IV. Conclusion

This paper proposes a velocity measurement based on Keystone transform which can solve the range migration effectively. This method applies entropy minimization which estimates the ambiguity degree with no error at a wider range of SNR than the traditional method. The simulations and analyses show that the presented method has very high accuracy in velocity measurement.