# An LFMCW detector with new structure and FRFT based differential distance estimation method

- Kai Yue
^{1}, - Xinhong Hao
^{1}Email authorView ORCID ID profile and - Ping Li
^{1}

**Received: **4 April 2016

**Accepted: **17 June 2016

**Published: **29 June 2016

## Abstract

This paper describes a linear frequency modulated continuous wave (LFMCW) detector which is designed for a collision avoidance radar. This detector can estimate distance between the detector and pedestrians or vehicles, thereby it will help to reduce the likelihood of traffic accidents. The detector consists of a transceiver and a signal processor. A novel structure based on the intermediate frequency signal (IFS) is designed for the transceiver which is different from the traditional LFMCW transceiver using the beat frequency signal (BFS) based structure. In the signal processor, a novel fractional Fourier transform (FRFT) based differential distance estimation (DDE) method is used to detect the distance. The new IFS based structure is beneficial for the FRFT based DDE method to reduce the computation complexity, because it does not need the scan of the optimal FRFT order. Low computation complexity ensures the feasibility of practical applications. Simulations are carried out and results demonstrate the efficiency of the detector designed in this paper.

## Keywords

## Background

According to the ‘Global Road Safety Partnership’ site, “Every day, more than 3000 people around the world lose their lives due to road crashes” (Savannah 2011). Many accidents happen as results of driver inattention or bad weather conditions causing poor visibility, as well as pedestrian inattentiveness to their surroundings (Gandhi and Trivedi 2007). So some advanced systems like collision avoidance radars have been designed to control the vehicle in an emergency situation (Lu et al. 2010). In this paper, a LFMCW detector is designed for a collision avoidance radar to estimate the distance between the radar and the vehicle or pedestrian.

Usually, distance between the radar and the vehicle or pedestrian can be obtained by estimating the time delay between the transmit signal and its echo signal. Some common time delay estimation (TDE) have been investigated to detect the distance, which are implemented by finding the peak of cross-correlation between the transmit signal and its echo (Knapp and Carter 1976) or by using adaptive filtering, Hilbert transform and fractional Fourier transform (FRFT) (Grennberg and Sandell 1994; Lin and Chern 1998; Sharma and Joshi 2007; Zhou et al. 2014). Because of the advantage of the FRFT in terms of processing linear frequency modulation (LFM) signals, FRFT are attracting more and more attention in the field of LFM signal processing. However, high computation complexity coming from the scan of the optimal FRFT order restricts its practical applications. Therefore, to utilize FRFT solving FM signal problems in practical applications, a novel intermediate frequency signal (IFS) based structure is designed for the transceiver and a related differential distance estimation (DDE) method based on FRFT is implemented in the signal processor. The LFMCW detector made up of this transceiver and signal processor can estimate the distance effectively.

The rest of this paper is organized as follows. System model is given in “Preliminaries” section. “Proposed LFMCW detector” section describes the proposed IFS based structure and DDE method based on FRFT in detail. In “Simulation and results” section, the simulation results are presented. Finally, the conclusion is drew in the fifth section.

## Preliminaries

### Fractional Fourier transform

As a generalization of conventional Fourier transform (FT), the FRFT has been shown to be more effective than the FT in many signal processing areas. Several signal processing methods based on FRFT have been proposed and their properties also have been derived and discussed, such as convolution, filtering, correlation and so on (Ozaktas and Barshan 1994; Almeida 1994; Ozaktas et al. 2001). To make the FRFT come true, some fast discrete fractional Fourier transform (DFRFT) have been done (Ozaktas et al. 1996; Pei and Yeh 1999; Candan and Kutay 2000; Ran et al. 2001). Among them, the Ozaktas’ DFRFT fast algorithm is widely regarded as an efficient one (Ozaktas et al. 1996), which has high precision, fast speed and the computation complexity matched with FFT.

*α*of signal

*f*(

*t*), denoted as

*F*

_{ α }(

*u*), is defined as (Ozaktas et al. 2001):

*K*

_{ α }(

*u*,

*t*) is given as:

*t*is time, and

*u*is the coordinate in fractional Fourier domain (FRFD). Meanwhile

*j*is the imaginary unit and the function exp() is the exponential function.

*α*is the rotation angle of FRFT. When

*α*increases from 0 to

*π*/2, the FRFT transforms a continuous signal from its time domain to the Fourier image. If

*α*or

*α*+

*π*is multiples of 2

*π*, the kernel

*K*

_{ α }(

*u*,

*t*) is simplified as

*δ*(

*t*−

*u*) or

*δ*(

*t*+

*u*), respectively. FRFT with rotation angle

*α*defined in the time–frequency domain is shown in Fig. 4. In this figure,

*t*is time and

*f*is frequency. Meanwhile

*u*is the coordinate in FRFD and

*v*is the amplitude in FRFD.

*f*

_{ if1}(

*t*) and

*f*

_{ if2}(

*t*) are instantaneous frequency of two signals with

*IF*1 and

*IF*2 as their initial frequency (IF).

*u*

_{ p1}and

*u*

_{ p2}are peak positions of the two signals in FRFD after FRFT. It is noticed that with the change of the rotation angle

*α*, the axis

*u*will perpendicular to the instantaneous frequency

*f*

_{ if1}(

*t*) and

*f*

_{ if2}(

*t*) when the rotation angle

*α*=

*β*+

*π*/2. Under this condition, the relationship between the peak positions in FRFD and the initial frequency of the signal can be expressed as

*γ*= cot

*α*·\(\frac{1}{2\Delta x}\) is the normalized sampling interval in both time domain and FRFD. This DFRFT can be realized by three steps:

- (1)
Multiply a chirp signal in the time domain;

- (2)
Carry out FFT to the signal after first step.

- (3)
Multiply another chirp signal and we get the DFRFT of the original signal.

We defined that multiplication times of an algorithm is the computation complexity. As a result, the computation complexity of this DFRFT with *N* points are 2*N* + *N*log_{2}
*N*/2, which is not so more than FFT.

## Proposed LFMCW detector

According to Figs. 2 and 3, it can be noticed that the structure of this detector is different from the traditional one. Traditional LFMCW detectors use the beat frequency signal (BFS) to gain the wanted information, whereas in the proposed detector information is acquired from the IFS. The mixer puts out BFS by mixing the echo signal with the transmit signal. However, IFS is obtained by mixing the echo signal with the reference carrier signal. Compared to BFS, the frequency modulation slope of the transmit signal is remained in the IFS. Because of the knowledge of the frequency modulation slope, FRFT can be carried out without hunting for the optimal order. As a result, the computation complexity of the FRFT based DDE method is significantly low.

*s*

_{1}(

*t*) with amplitude

*a*

_{1}, carrier frequency

*f*

_{1}and the frequency modulation slope

*K*

_{1}is expressed as

*s*

_{2}(

*t*) which can be expressed as

*τ*(

*t*) is the time delay and

*λ*

_{1}is the amplitude factor. Then we get the BFS

*s*

_{ bf }(

*t*) and it can be formulated as

*R*

_{0}is the initial distance and

*v*

_{0}is the relative speed.

*c*is the velocity of electromagnetic wave) and the

*s*

_{ bf }(

*t*) can be expressed as

*a*

_{ bf }is the amplitude. We can find that the frequency modulation slope of

*s*

_{ bf }(

*t*) is \(\frac{{4K_{1} v_{0} c + 2v_{0}^{2} }}{{c^{2} }}\) and it is unknown. Consequently, we have to make search for the optimal rotation angle to determine it.

*s*

_{0}(

*t*) = cos2

*π*(

*f*

_{1}

*t*), and we obtain the IFS

*s*

_{ if }(

*t*) which can be expressed as

*a*

_{ if }is the amplitude. Considering \(\tau (t) = \frac{{2(R_{0} - v_{0} t)}}{c}\), the equation can be rewritten as

Because of the relationship *v*
_{0} ≪ *c* in practice, the frequency modulation slope \(\frac{{K_{1} c^{2} + 4K_{1} v_{0} c + v_{0}^{2} }}{{c^{2} }}\) of *s*
_{
if
}(*t*) is approximate *K*
_{1}. Therefore, the frequency modulation slope of *s*
_{
if
}(*t*) is known for us and the FRFT with the optimal rotation angle can be carried out without scan for the optimal rotation angle.

*τ*(

*t*) can be regarded as a constant

*τ*for each computation. So the equation can also be formulated as

Normally, this equation is utilized for distance estimation. From this equation, we can also find that the frequency modulation slope of *s*
_{
if
}(*t*) is known for us and the FRFT with the optimal rotation angle can be carried out without scan for the optimal rotation angle.

### Derivation of the DDE method

*f*

_{ if }(

*t*) as shown in Fig. 6 can be denoted as:

*f*

_{0}is the carrier frequency and

*T*is the period of the modulated signal.

*K*=

*B*/

*T*is the frequency modulation slope and

*B*is the modulation bandwidth. Parameter

*n*is the cycle times. So the transmit signal within a modulation period can be expressed as:

*n*(

*n*+ 1)

*KT*

^{2}/2 + Φ

_{0}(Φ

_{0}is the initial phase), and

*a*is the amplitude of the transmit signal.

_{ τ }= (

*Kτ*

^{2}− 2

*f*

_{0}

*τ*− 2

*nKTτ*)/2 +

*n*(

*n*+ 1)

*KT*

^{2}/2 + Φ

_{1},

*λ*is the amplitude factor of the echo signal, and the parameter

*τ*is the time delay.

*x*(

*t*) = cos 2

*πf*

_{0}

*t*in two separate mixers respectively, then the IFSs can be derived as:

It’s noticed that the frequency modulation slope *K* is remained in the IFSs mentioned as above. So FRFT will be carried out to the IFSs with angle *α* = arctan(*K*) + *π*/2, without hunting for the optimal rotation angle.

*f*

_{ it }and

*f*

_{ ir }of the IFSs can be simply obtained as the Eq. (16).

Obviously, the time delay *τ* can be estimated through the difference between the *f*
_{
it
} and *f*
_{
ir
}.

*α*= arctan(

*K*) +

*π*/2) is optimal, the relationship between

*f*

_{ ir }(

*f*

_{ it }) and the peak position

*u*

_{ pr }(

*u*

_{ pt }) in FRFD of the IFSs can be written as

*u*

_{ pr }(

*u*

_{ pt }), the time delay estimation \(\widetilde{\tau }\) is achieved and expressed as:

*τ*= 2

*R*/

*c*(

*R*is the distance and

*c*is the velocity of electromagnetic wave), the distance estimation value can be formulated as:

### Distance estimation precision and computation complexity

*u*

_{ pr }(

*u*

_{ pt }). The estimation accuracy of

*u*

_{ pr }(

*u*

_{ pt }) is equal to the sampling interval in the fractional domain, and can be expressed as \(\Delta u\; = \;{1 \mathord{\left/ {\vphantom {1 {\sqrt N }}} \right.} {\sqrt N }}\) after normalization (Ozaktas et al. 1996). According to the Ozaktas et al. (1996), the real sampling interval Δ

*u*

_{ r }can be expressed as

*f*

_{ s }is the sampling frequency in time domain and

*N*is the computation point for each processing. Therefore, the distance estimation precision Δ

*R*is

*f*

_{ s }= 10

*B*is reasonable. Meanwhile, the frequency modulation slope

*K*=

*B*/

*T*. Consequently, the distance estimation precision can be rewrite as

*c*and

*α*are constants, the period of the modulated signal

*T*and sampling points are the main influencing factors for the distance estimation precision. Improvement of the distance estimation precision depends on

*T*and

*N*, but not the modulation bandwidth. It is beneficial for engineering applications. Figure 7 shows the relationships between the distance estimation precision and the period of the modulated signal or the sampling points.

*M*times scan can be written as

Compared with Eq. (24), the computation complexity of the proposed DDE method described as Eq. (23) decreases obviously.

## Simulation and results

In this section, simulations are carried out to validate the performance of the detector proposed in this paper. The carrier frequency is 3.0 GHz, and the modulation bandwidth is 30 MHz. The modulation period is 3.3 × 10^{−6} s.

## Conclusion

In this paper, a novel FRFT based LFMCW detector is proposed to achieve distance measurement for a collision avoidance radar. On the one hand, an FRFT based DDE method is proposed for the detector to realize distance estimation. On the other hand, a new IFS based structure transceiver is designed for the detector to reduce computation complexity of FRFT by taking advantage of the knowledge of frequency modulation slope. Benefited from the FRFT based DDE method and structure, the detector can estimate the distance effectively and make the FRFT feasibly applied in practice rather than merely researched theoretically. Besides, practical FRFT applications in LFM systems have important significance for improving the performance of LFM systems.

## Declarations

### Authors’ contributions

KY and XH made substantial contributions to conception and design of the advanced strategy of the algorithm proposed. PL revised the manuscript critically to meet the expected standards in scientific publishing. All authors read and approved the final manuscript.

### Authors information

Kai Yue was born in Hebei Province, China, in 1986. He received the B.S. degree in mechatronic engineering from Beijing Institute of Technology, Beijing, China, in 2011. Presently, he is a Ph.D. candidate in the School of Mechatronical Engineering, Beijing Institute of Technology. His research interests include digital signal processing, time–frequency analysis and signal detection and estimation.

Xinhong Hao, was born in Henan Province, China, in 1974. She received her B.S. degree, M.S. degree and the Ph.D. degree in mechatronic engineering from Beijing Institute of Technology, in 1996, 1999 and 2007, respectively. She is now an Associate Professor in Beijing Institute of Technology. Her main research interests include target detection theory of radio sensor and signal processing, real-time signal processing and time–frequency analysis.

Ping Li received the B.S. and M.S. degree in Mechantronical Engineering from Dalian Jiaotong University, Dalian, China, in 1985 and 1987, respectively, and the Ph.D. degrees from the Beijing Institute of Technology, China, in 1995. In Sept. 1996, she joined School of Mechatronical Engineering, Beijing Institute of Technology, China. Her research interests include information countermeasure in wireless systems, information security, terminal information resistance and information security in wireless communication.

### Acknowledgements

This work is supported by the State Key Program of Basic Research of China under Grant No. 613196, and the authors acknowledge the suggestions of Li Ziqiang.

### Competing interests

The authors declare that they have no competing interests.

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

## Authors’ Affiliations

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