CC BY
9ТЕХНИЧЕСКИЕ НАУКИ
Nguyen Huu Dong
MA, Military Institute of Science and Technology, Vietnam Republic
Nguyen Trung Thanh
Ph-D, Military Technical Academy, Vietnam Republic
INVESTIGATE THE APPLICATION OF THE SPECTRUM PEAK LOCATION TECHNIQUE IN SPEED ESTIMATION IN POLICE ENFORCEMENT RADARS
Abstract: The Doppler effect has been applied in police enforcement radars for many years to estimate car speed. In this application, the difference of the received and the transmitted frequencies, the Doppler frequency shift, is measured to calculate the speed of the car. With the transmitted frequency is normally known, this process becomes the estimation of the received frequency.
With the rapid development of the digital signal processing recently, the received frequency can be estimated by using the Fast Fourier Transform. The drawback of this solutions is that the computation complexity is increased. A more efficient method is the interpolation approach. This approach utilises several FFT samples near the peak of the spectrum for interpolating the received frequency.
Simulation results indicate that the Spectrum Peak Location technique can achieve higher accuracy than the traditional FFT technique and therefore shows a potential application in the police enforcement radars.
Key words: Police speed radar, Doppler frequency, FFT
1. Introdution
In 1842, Austrian mathematician and physicist Christian Doppler discovered a phenomenon relating to changes in the color of a light source depending on its movement [1]. If the light source is moving toward the observer, the color of the light appears bluer. On the other hand, the light appears redder when the light source is moving away from the observer. This phenomenon, known as the Doppler effect, has wide applications, not only in physics but other engineering aspects, including police enforcement radar.
Radar transmits electromagnetic signals and processes the reflected signals from the target. If the target is in radial motion relative to the radar, the frequency of the received signal is different from the transmitted frequency due to the Doppler effect. The change in frequency is the Doppler frequency shift and can be calculated as [2]:
f _ 2 vr_2fcVr ,,,
>d - ~ T (1)
where: fd is the Doppler frequency shift, fc is the carier frequency of the radar, vr is the radial velocity of the target, and c = 3.108 m/s is the speed of radio propagation.
From equation (1), if the Doppler frequency shift, fd, can be measured, the target radial velocity can be deduced.
There have been many attempts to measure Dop-pler frequency shift in police enforcement radars, using different methods [3],[4]. The Doppler frequency shift was measured directly in early CW radars, by mixing the transmitted (Tx) and received (Rx) signals [5]. With the rapid development of DSP and FPGA technology, one recent trend is the utilisation of the Fast Fourier Transform (FFT) to produce the Rx signal spectrum, from which the peak is located and, assumed to be the Rx frequency. With a given Tx frequency, that means Doppler frequency and target velocity can be deduced. The actual peak of spectrum in most instances, however, lies between two FFT samples. The bigger the gap
between these two samples, hereafter known as the frequency bin width of the FFT, the larger the error in the frequency estimation. The common solutions to achieve higher estimation accuracy are either collecting more data samples or zero-padding the received signal [6], thereby reducing the bin width between FFT samples. The drawback of these solutions is that the computation complexity is increased due to the increase of the FFT size.
A more efficient method is the interpolation approach [7]. This approach utilises several FFT samples near the peak for interpolating the carrier frequency and consists of two steps. In the first, the maximum value of the FFT output samples is located. This value is the coarse estimation, which is the starting point for the subsequent fine estimation. In the second step, several FFT output samples around the maximum sample are taken for interpolating the actual peak frequency. In [8], Provencher used only two samples for interpolation, but three samples are more frequently used, as suggested in [9-12]. In [13] and more recently in [14], it was shown that the interpolation algorithm, hereafter known as the Spectrum Peak Location (SPL) technique, using three samples, closely follows the Cramer-Rao lower bound in high SNR regions.
The aim of this paper is to investigate the potential use of the SPL technique in the police enforcement radar by comparing the new SPL and the traditional FFT techniques in the application, in terms of both the accuracy and the computational complexity. The remainder of this paper is organized as follows: Section II presents the FFT technique and its accuracy in estimating the velocity in police enforcement radar. Sections III presents the SPL technique and its accuracy in the same situation. A comparison of the two technique is followed in section IV and the conclusion is provided in Section V.
2. The FFT technique and its accuracy
2.1. The FFT technique for estimating Doppler frequency
With the Tx frequency is known, the task of estimating the Doppler frequency becomes determining the Rx frequency. This can be done by implementing the
Fc(Tx)
<-
Fc + Fd (Rx) -►
FFT of the Rx signal and locating the peak of the FFT samples. The block diagram of the FFT technique is shown in Figure 1.
Power Amp
Antenna
and Circulator
Fc
Figure 1. The block diagram of the FFT technique
The process of estimating the Doppler frequency fd using the FFT technique, therefore, consists of the following steps:
1. Produce the digital spectrum of the Rx pulse by performing the FFT
2. Scan the spectrum to locate the peak and, therefore, the index k
3. The Rx frequency can be estimated as follow:
fc fs
fpeak N
(2)
and 20 KHz ADC, there are 2000 samples in one range gate. If the FFT technique processes data in one range gate at a time, in other word, if the FFT size is 2000 points, the frequency resolution (frequency bin) of the FFT is:
A= .20000 = 10 Hz
Jres N 2000
(3)
where: fs is the sampling frequency and N is the FFT size
Because the real peak of the Rx frequency could be anywhere between 2 frequency samples of the digital spectrum, this step potentially introduces error. The error is maximum when the real peak is exactly in the middle of 2 consecutive frequency samples and minimum when the peak coincides with one of the frequency samples.
2.2. Accuracy of the FFT technique
The term "accuracy" in this paper is refer to as the accuracy of the "digital domain", which is related to the error of all the digital processing after the ADC. All the error prior to the ADC is out of the scope of this paper.
The accuracy of the FFT technique is simulated in Matlab with similar parameters to the working systems with the configuration as follow: Fc = 10.5 GHz, pulse mode, pulse width 100 ms [3].
With the car speed from 20 to 200 km/h, the Dop-pler frequency ranges from 390 Hz to 3900 Hz, and therefore the ADC speed of 20 KHz is high enough for the application. With a given pulse width of 100 ms,
The accuracy of the FFT technique is tested with random Rx signals generated in MATLAB. One data sets of 10,000 random Rx signals, which is considered as statistically sufficient, is generated for the simulation. For a single random Rx signal generated, its magnitude spectrum is calculated by the FFT, and the Rx frequency is estimated by locating the peak in the frequency samples after the FFT. This estimated Rx frequency is then compared to the input frequency for absolute error. At the end, 10,000 errors are produced and from them, two variables, the mean error (ME) and the root mean squared error (RMSE) are calculated. They represent the bias and mean deviation of the estimation using the FFT technique, therefore, both are taken as the quality benchmark for the FFT technique.
The first simulation, however, is a run in a noise-free environment where no noise is added to the simulated Rx signal. The purpose of this simulation is to evaluate the accuracy limit of the FFT technique.
Next, the simulations with different SNRs are run. In the application of radar speed gun where targets are close and noise is limited, the SNR of 20dB to 0 dB are simulated to represent the typical situations in the case. Several simulation results are shown in Fig 2 and Table 1.
Absolute Error Distribution, Mean Error = 0.024246, RMSE = 2.8615 1500 f-f-f-f-f-F-
Absolute Error Distribution, Mean Error = 0.029246, RMSE = 2.8647 1500 f-f-f-f-f-F-
-6 -4 -2 0 2 4 6
FFT technique
Figure 2: Absolute error distribution of the FFT technique
Table 1: Simulation results for the accuracy of the FFT technique
SNR Noise free 20 dB 15 dB 10 dB 5dB 0dB
Mean Error (Hz) 0.0242 0.0202 0.0282 0.0232 0.0302 0.0292
RMSE (Hz) 2.8615 2.8615 2.8616 2.8617 2.8624 2.8647
3. The SPL technique and its accuracy The block diagram of the SPL technique is similar
3.1. The SPL technique to that of the FFT technique, the differences come after
the FFT as shown in Fig 3.
Figure 3. The block diagram of the SPL technique
The SPL technique firstly locates the peak in the Then a parabolic (second-order polynomial) fit is im-FFT output magnitude samples Y (k)|, the index k of plemented over the peak and the two adjacent FFT sam-the peak gives a coarse estimation of the Rx frequency. ples (one for each side) as shown in Fig 4. This process
calculates the correction factor M for the fine frequency estimation.
|Y|
A
k-1 k k' k+1 Sample
0
Figure 4: Parabol fit for 3 samples in the SPL technique.
The mathematical derivation of the SPL technique is shown in [15], and the correction factor Ak can be calculated as:
Ak =■
|y (fc + i)| - jy (fc - i)| 4|y (fc)| - 2jy (fc + 1)| - 2jy (fc - 1)|
(4)
where Ak is the estimated location of the peak relative to sample kth. The frequency of the Rx signal, therefore, can be deduced as:
_ (fc+ Afc) /s Tpeafc N (5)
where fs is the sampling frequency and N is the size of the FFT.
The process of determining fd using the SPL technique, therefore, consists of the following steps: 1. Calculate the spectrum of the Rx pulse by FFT
2. Scan the spectrum to locate the peak
3. Use equation (4) to find the correction factor and then use equation (5) to calculate the Rx frequency
4. Subtract the Tx from the Rx frequency to find fd The computational complexity of the SPL technique is therefore almost exclusively occupied by the FFT.
3.2. The accuracy of the SPL technique Simulation of the SPL technique is similar to that of the FFT technique in the previous section, only minor differences after locating the peak of the FFT. Those differences are the calculations of the equations (4) and (5).
The same data set of 10,000 random Rx signals is applied to the simulation and several results are presented in Fig 5 and Table 2.
Absolute Error Distribution, Mean Error = 0.024275, RMSE = 1.6626
3000
2000
1000
0^ -2.5
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 Absolute Error Distribution, Mean Error = 0.033508, RMSE = 1.6696
2000
m 1500
T3
I I 1000
tr: z
CO
500
0
-3 -2 -1 0 1 2
SPL technique
Figure 5: Absolute error distribution of the SPL technique
Table 2: Simulation results for the accuracy of the SPL technique
SNR Noise free 20 dB 15 dB 10 dB 5dB 0dB
Mean Error (Hz) 0.0243 0.0241 0.0250 0.0253 0.0250 0.0335
RMSE (Hz) 1.6626 1.6626 1.6633 1.6629 1.6661 1.6696
4. Comparison between the FFT and the SPL techniques
4.1. Computational complexity
Both the 2 techniques need to perform the FFT. After the FFT, the FFT technique only needs to scan for the peak and the calculation of the equation (2). On the other hand, after the FFT, the SPL technique requires further computation, including scan for the peak, and the calculation of the equations (4) and (5). Therefore, the SPL techique requires more computation than the FFT techique.
However, most of the computation in both techniques belong to the calculation of the FFT. The remaining computation are both negligible in comparison to that required by the FFT. Therefore, the two techniques can be considered as comparable in terms of computational complexity.
4.2. Accuracy
To further test and compare FFT and SPL techniques, 10 data sets, each has 10000 random Rx signals are generated for the purpose. For each data set, two variables RMSE and ME are produces for each techniques. Results for SNR = 20 dB are presented in Fig 6 and Fig 7.
In Figure 6, both two techniques show that the mean value of the error fluctuates closely around zero, indicating unbiased estimations. In Figure 7, the SPL technique is superior compared to the FFT technique in terms of RMSE. If the SPL is applied where the SNR=20 dB, an estimated accuracy of approximatelly 1.66 Hz can be achieved in comparison to about 2.86 Hz of the FFT technique. The SPL technique, therefore, outperforms the FFT technique in terms of accuracy.
Mean Value of Error,SNR=20
0.08
0.06
0.04
0
3 0.02
"ÖÖ >
C TO 0
0
-0.02 -0.04 -0.06 -0.08
F FFT SPL ,
/ \ //
— /__ ...... \ //
/ X //
//
/
3
4
7
5 6
Data Set
Figure 6: Mean value of error, SNR = 20 dB
10
1
2
8
9
Ul
2.8
0
2.6
(0 CT
C/) 2.4
C (0
S 2.2
O
O 2
1.8
1.6
Root Mean Square of Error,SNR=20
12 3 4
5 6 7
Data Set
Figure 7: Root mean square of error, SNR = 20 dB
SDF SPL
--
8 9
10
3
5. Conclusion
The paper proposes the use of the SPL technique in the police enforcement radar. Simulation results show that the SPL technique requies as comparable computation as the traditional FFT technique but achieves much higher accuracy. The SPL technique, therefore, shows the potential application in the field.
References
[1] A. Eden, "The Search for Christian Doppler", Springer-Verlag, Berlin,Germany, 1992.
[2] M. Skolnik, "Introduction to radar systems", chapter 3, pp 104-148, McGraw-Hill, Inc., New York,USA, 2001.
[3] Donald Sawicki, "Police traffic speed radar handbook", CorpRadar.com, 2011
[4] Kevin M.Morrison, "The complete book on speed enforcement", Charles C Thomas Publisher, 2012
[5] M. Skolnik, "Radar handbook", chapter 1, McGraw-Hill, Inc., USA, 1990.
[6] John G. Proakis and Dimitris G. Mannolakis, "Digital signal processing", ISBN-10: 0131873741, Pearson; 4 edition, 2006
[7] Richard G. Lyons, "Understanding digital signal processing", chapter 13, pp 730-734, Prentice-Hall, Inc., USA, 2011.
[8] S. Provencher. Estimation of Complex SingleTone Parameters in the DFT Domain. Signal Processing, IEEE Transactions on, 58(7):3879- 3883, 2010
[9] M.D. Macleod, "Fast nearly ML estimation of the parameters of real or complex single tones or resolved multiple tones", Signal Processing, IEEE Transactions on, 46(1):141—148, 1998.
[10] B.G. Quinn, "Estimating frequency by interpolation using Fourier coefficients", Signal Processing, IEEE Transactions on, 42(5): 1264-1268, 1994.
[11] B.G. Quinn, "Estimation of frequency, amplitude, and phase from the DFT of a time series", Signal Processing, IEEE Transactions on, 45(3):814- 817, 1997.
[12] E. Jacobsen and P. Kootsookos, "Fast, Accurate Frequency Estimators", Signal Processing Magazine, IEEE, 24(3):123-125, 2007.
[13] C. Candan, "A Method For Fine Resolution Frequency Estimation From Three DFT Samples", Signal Processing Letters, IEEE, 18(6):351- 354, June 2011.
[14] C. Candan, "Analysis and Further Improvement of Fine Resolution Frequency Estimation Method From Three DFT Samples", Signal Processing Letters, IEEE, 20(9):913-916, Sept 2013.
[15] M.A Richards. "Fundamentals of radar signal processing", chapter 5, pp 264-270, McGraw-Hill, Inc, NewYork, USA, 2005
