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Nonlinear error correction for Terahertz FMCW System by a new beat frequency estimation method

Cheng-Wu You, Shi-Tao Chen, Tian-Yi Wang, Jin-Song Liu, Ke-Jia Wang, and Zhen-Gang Yang

Open Access Open Access

Abstract

Terahertz (THz) frequency modulated continuous wave (FMCW) technology is a means of nondestructive testing. The signal’s nonlinearity is an unavoidable problem in the daily application of THz FMCW technology. The signal’s nonlinearity will lead to the spectrum broadening of the FMCW’s beat frequency (BF) signal, which degrades the range resolution and result in distance-measuring error. Traditional methods require additional hardware or require a lot of computation, which are not conducive to the miniaturization of the system and real-time measurement. A novel method for correcting the nonlinear error of THz FMCW technology has been proposed and demonstrated in this article. In the proposed method, the windowed Fourier transform (WFT) is introduced to estimate the BF corresponding to the measured target, according to the linearity distribution of voltage-controlled oscillator (VCO). In this way, the measured target's BF can be accurately estimated from the unprocessed BF signal with a poor linearity. From the estimated BF of the reference target, the non-linear compensation coefficients are calculated. With the non-linear compensation coefficients, the non-linearity of the output BF signal can be calibrated. The results of simulations and experiments show that the proposed method allows the range resolution of an FMCW system to reach the theoretical limit.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

THz waves are defined as electromagnetic waves ranging from 0.1 THz to 10 THz [1]. THz technology is an important technique in nondestructive testing (NDT). Terahertz technology is widely used in quality control and authentication of packaged integrated circuits [2], counterfeit electronic components detection [3], aviation congruent material inspection [4], art painting analysis [5, 6], inspection of buildings and architectural art [7], biological detection [8] and security inspection [9, 10] et al. THz FMCW technology is a common technology used in THz daily industrial applications [11]. In contrast with pulsed time of flight system, THz FMCW provides both high spatial resolution and high signal to noise ratio (SNR) [12]. This stems from the fact that the measurement of time is converted to the measurement of BF in THz FMCW distance-measuring [13]. These advantages make THz FMCW widely used in THz NDT [14, 15].

In practical application, the nonlinearity of FMCW signal is an unavoidable problem because of the frequency response’s fluctuation and the noise of the modulating circuit [16]. Due to the nonlinearity of the FMCW generating circuit, the frequency of the obtained frequency modulated signal does not change linearly with time in a sweep period, which makes the phase shift of the signal deviate from the theoretical case. Thus, the frequency of the BF output signal is not a time invariant constant. The more serious the signal’s nonlinearity is, the more serious the fluctuation of BF signal frequency with time. When a THz FMCW system ranges a target, the single peak representing the target that should theoretically appear on the spectrum of the output BF signal becomes multiple peaks. This makes the location of the target indistinguishable on the distance spectrum. From what has been discussed above, the non-linearity of the frequency modulated signal results in the spectrum of the BF output signal being a broadened spectral line rather than a single main lobe, which decreases the range performance of THz FMCW system [17]. In order to improve the range performance of THz FMCW, it is necessary to correct the nonlinearity of FMCW signal.

Since the 1990s, many research reports related to the nonlinear correction of FMCW signal have been published. In 1994, A. Dieckmann used Mach-Zehnder interference technology to correct the nonlinear tuning of the laser, and successfully realized the short-range ranging of FMCW lidar [18]. In 1996, Iiyama realized the linearization of the triangular wave frequency modulation within the frequency modulation range of 100GHz, by using a photoelectric negative feedback loop with reference interferometer and phase comparator [19]. In 1999, Christer J. Karlsson et al. studied the laser tuning nonlinearity in FMCW lidar, and discussed a method of nonlinear correction by compensation [20]. In 2001, Richard Schneider et al. proposed a method of compensating the frequency modulation nonlinear error by using the calibrated interference signal of a continuous wave laser with two antisymmetric tuned curves [21]. In 2009, Peter A. Roos et al. completed the nonlinear correction by using autoheterodyne technology to lock the beat frequency of the feedback loop to a reference frequency [22]. In 2010, Zebw. Barber et al. proposed an active chirp linearization method for linearization feedback compensation of the FMCW radar [23]. In 2012, Ningfang Song et al. obtained better ranging accuracy by correcting the quadratic and cubic tuning nonlinear coefficients of the 1.55μm FMCW lidar [24]. In 2013, Esther Bauman et al. built a FMCW laser ranging system and calibrated the tunable laser using an optical frequency comb [25]. In 2018, Fumin Zhang et al. proposed a new amplitude modulation method to correct the nonlinear error of FMCW [26]. In 2019, Onur Toker and Marius Brinkmann developed an optimal upsampling theory based on almost-causal finite impulse response (FIR) filters [27]. Recently, an FMCW range measuring system based on a micro-resonant soliton comb was proposed by Linhua Jia et al [28]. The nonlinearity of the system is corrected by sampling at equal intervals at a certain frequency.

Most of the above methods require the use of an additional reference interferometer or an additional soliton comb to handle nonlinear modulation, which will increase the volume of FMCW system. That is not conducive to the daily application of THz FMCW technology, because THz FMCW sources generally need to be placed on a raster scanning system scanning seat in daily applications. In this paper, a nonlinear distortion model is presented by analyzing the source of the nonlinear distortion. According to the model, a windowed Fourier transform (WFT) method is proposed to estimate the beat frequency of the reference target. Combining the estimated BF of the reference target, the nonlinear compensation coefficients can be computed. And then, nonlinearity of an FMCW signal can be corrected with the compensation coefficients. In this paper, the effectiveness of the proposed method is verified by simulations and experiments. The experimental results show that the proposed method allows an FMCW system’s range resolution to reach the theoretical limit. Our method is efficient and easy to operate, which is beneficial to FMCW’s daily application.

2. Theory

2.1 FMCW ranging system

Schematic of the THz FMCW source is shown in Fig. 1. Driven by the input ramp signal, the VCO generates a sweep signal. Then the frequency multiplier (FM) multiplies the frequency of the sweep signal to the range at work. After the sweep signal passes through the FM, one part is transmitted by the transmitter as the detection signal, and the other part will be injected into the mixer as the reference signal. The echo signal is received by the receiver and then injected into the frequency mixer. The echo signal and the reference signal are multiplied in the frequency mixer. After passing through the band pass filter and analog-to-digital converter, the signal from the mixer is output as the output BF signal.

 figure: Fig. 1.

Fig. 1. Schematic of the THz FMCW source.

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For an FMCW source with sweep period T, sweep band width B and sweep starting frequency f0, the reference signal can be written as

$${I_{ref}} = exp \left[ {j\left( {\frac{{\pi B}}{T}{t^2} + 2\pi {f_0}t + {\varphi_0}} \right)} \right]$$
where t is the time and φ0 is the initial phase. Assuming that the time delay of the echo signal is τ, the receive signal can be written as
$${I_{rec}} = exp \left\{ {j\left[ {\frac{{\pi B}}{T}{{({t - \tau } )}^2} + 2\pi {f_0}({t - \tau } )+ {\varphi_0}} \right]} \right\}.$$
Then the BF output signal can be written as [29, 30]
$${I_{BF}} = exp \left[ {j\left( {\frac{{2\pi B\tau }}{T}t\textrm{ - }\frac{{\pi B}}{T}{\tau^2} + 2\pi {f_0}\tau } \right)} \right].$$

2.2 Mathematical model of FMCW’s nonlinearity and Method for estimating the beat frequency of target

In many cases, the nonlinearity of the FMCW is caused by the fluctuation of the VCO’s response frequency. In general, the linearity of a VCO becomes worse from the middle of the tuning region to the sides [31]. For an FMCW source with linearity η, sweep period T, sweep band width B and sweep starting frequency f0, its second-order relative distortion model of frequency sweep in one period can be expressed as

$$f = \frac{B}{T}t + {f_0} + \xi (t )\frac{{4\eta B}}{{{T^2}}}{\left( {t - \frac{T}{2}} \right)^2}$$
where t is the time, ξ(t) is the adjustment factor and its value is between -1 and 1. Based on the previous discussion, the schematic diagram of FMCW ranging can be drawn as Fig. 2. As shown in Fig. 2, the frequency of the beat signal is close to a constant value in a certain time domain. The constant value is close to the theoretical beat frequency corresponding to the target. Therefore, as long as the rectangular window range is properly selected, the beat frequency corresponding to the target can be estimated accurately by the windowed Fourier transform. The range of the rectangular window can be found by experiment or according to the voltage-frequency corresponding curve of the VCO.

2.3 Methods of nonlinear correction

Assume that the nonlinear term of FMCW is ε(t), depending on Section 2.1, the reference signal and the receive signal can be rewritten as

$$\left\{ \begin{array}{l} I_{ref}^{\prime} = exp \left[ {j\left( {\frac{{\pi B}}{T}{t^2} + 2\pi {f_0}t + \varepsilon (t )+ {\varphi_0}} \right)} \right]\\ I_{rec}^{\prime} = exp \left\{ {j\left[ {\frac{{\pi B}}{T}{{({t - \tau } )}^2} + 2\pi {f_0}({t - \tau } )+ \varepsilon ({t - \tau } )+ {\varphi_0}} \right]} \right\} \end{array} \right..$$
Accordingly, the BF output signal can be expressed as
$$I_{BF}^{\prime} = exp \left\{ {j2\pi \left[ {\frac{{B\tau }}{T}t - \frac{B}{{2T}}{\tau^2} + {f_0}\tau + \varepsilon (t )- \varepsilon ({t - \tau } )} \right]} \right\}.$$

 figure: Fig. 2.

Fig. 2. Schematic diagram of THz FMCW ranging.

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Let’s call ‘f0τ+{ε(t)-ε(t-τ)}’as ‘τ·σ(t)’. Then Eq.(6) can be written as

$$I_{BF}^{\prime} = exp \left\{ {j2\pi \left[ {\frac{{B\tau }}{T}t - \frac{B}{{2T}}{\tau^2} + \tau \sigma (t )} \right]} \right\} = exp \left\{ {j2\pi \left[ {{f_{BF}}t - \frac{T}{{2B}}f_{BF}^2 + {f_{BF}}\frac{T}{B}\sigma (t )} \right]} \right\}$$
where fBF is the output BF corresponding to the target 0.5cτ away from the FMCW source. The nonlinear compensation coefficients C(t)=exp{-j2πfBFTσ(t)B-1} can be solved by finding exp(j2πfBF t-πTfBF2B-1). Since fBF can be estimated by the method described in Section 2.2, the nonlinear compensation coefficients C(t) can be calculated as
$$C(t )= exp \left[ { - j2\pi {f_{BF}}\frac{T}{B}\sigma (t )} \right] = {({I_{BF}^{\prime}} )^{ - 1}}exp \left[ {2\pi j\left( {{f_{BF}}t - \frac{T}{{2B}}f_{BF}^2} \right)} \right].$$
When the distance between a target and the THz FMCW source is close to 0.5cτ, σ(t) can be approximated as constants. In this case, the BF is fBF1. Then the BF output signal can be expressed as
$$I_{BF1}^{\prime} = exp \left\{ {j2\pi \left[ {{f_{BF1}}t - \frac{T}{{2B}}f_{BF1}^2 + {f_{BF1}}\frac{T}{B}\sigma (t )} \right]} \right\}.$$
Depending on Eq.(8) and Eq.(9), the I’BF1 can be corrected as
$${\textrm{I}_{corrected}} = I_{BF1}^{\prime}{[{C(t )} ]^{\frac{{f_{BF1}^{\prime}}}{{{f_{BF}}}}}}$$
where f’BF1 can be estimated by the proposed WFT.

In practice, part of the signal from the transmitter will leak into the frequency mixer. This additive interference caused by the leaked signal needs to be removed. In view of this, a non-reflective reference signal S0 which is regard as a baseline for subtraction, should be measured at first. And then a BF signal S, before being processed, should be converted as

$$I = ({S - {S_0}} )+ jH({S - {S_0}} )$$
where ‘H’ denotes the Hilbert transform.

To sum up the above discussion, the whole nonlinear correction procedure can be illustrated in Fig. 3.

 figure: Fig. 3.

Fig. 3. The flow-chart for the nonlinear correction procedure.

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3. System simulation

3.1 Simulation of single target ranging

In this simulation model, the starting frequency of the FMCW source is set to 276 Gigahertz (GHz), the sweep band width B is set to 50 GHz, the sweep period T is set to 250 microseconds (μs) and the linearity is set to 0.05. Thus, the BF spectral resolution of this FMCW source is equal to T-1=4 Kiloherts (kHz). The distance d between the reference target and the source is set at 540 millimeters (mm). According to the FMCW BF-distance formula: ‘fBF=2Bd(CT)-1’, the beat frequency corresponding to the reference target should be 720 kHz. The frequency sweep f(t) of the FMCW source is set as

$$f(t) = [{2 \times {{10}^5}t + 276 + 1.6 \times {{10}^8}{{({t - 1.25 \times {{10}^{ - 4}}} )}^2}\xi (t )} ]GHz$$
where t is the time, ξ(t) is a random function and its value is between -1 and 1. Thus, the output BF signal Sref in this situation can be expressed as
$${\textrm{S}_{ref}} = \cos \left\{ {\int {\left[ {f(t )- f\left( {t - \frac{{2 \times 0.54}}{{3 \times {{10}^8}}}s} \right)} \right]} dt} \right\} = \cos \left\{ {\int {[{f(t )- f({t - 3.6 \times {{10}^{ - 9}}s} )} ]dt} } \right\}$$

The spectrum of Sref is shown in Fig. 4(a). As shown in Fig. 4(a), the BF can hardly be estimated due to spectrum broadening. Figure 4(b) is the windowed Fourier transform result of Sref. According to Fig. 4(b), the BF of Sref is 720 kHz which is equal to the result of theoretical calculation. It can be seen that the windowed Fourier transform can estimate the beat frequency accurately.

 figure: Fig. 4.

Fig. 4. (a) The spectrum and (b) the WFT result of Sref.

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With the estimated value of the BF, the nonlinear compensation coefficients C(t) can be calculated as

$$\begin{array}{l} C(t )= {[{{S_{ref}} + jH({{S_{ref}}} )} ]^{ - 1}}exp \left[ {j2\pi \left( {{f_{BF}}t - \frac{T}{{2B}}f_{BF}^2} \right)} \right]\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = {[{{S_{ref}} + jH({{S_{ref}}} )} ]^{ - 1}}exp [{j2\pi ({7.2 \times {{10}^5}t - 1.296 \times {{10}^{ - 3}}} )} ]\end{array}.$$
When there is a target to be measured at a distance of 570 mm from the FMCW source, the output BF signal Starget in this situation can be expressed as
$${\textrm{S}_{t arg et}} = \cos \left\{ {\int {\left[ {f(t )- f\left( {t - \frac{{2 \times 0.57}}{{3 \times {{10}^8}}}s} \right)} \right]} dt} \right\} = \cos \left\{ {\int {[{f(t )- f({t - 3.8 \times {{10}^{ - 9}}s} )} ]dt} } \right\}.$$

As shown in Fig. 5(b), the BF of Starget is 760 kHz. Thus, the nonlinear correction result of Starget can be calculated as

$${S_{corrected}} = Re \left\{ {[{{S_{t\arg et}} + jH({{S_{t\arg et}}} )} ]{{[{C(t )} ]}^{\frac{{760}}{{720}}}}} \right\}$$
where Re denotes taking the real part.

 figure: Fig. 5.

Fig. 5. (a) The frequency spectrum, (b) the WFT result and (c) the frequency spectrum after nonlinear correction of Starget.

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The frequency spectrum of the nonlinear corrected signal is shown in Fig. 5(c). As can be seen from Fig. 5(c), the full width at half maximum (FWHM) of the main lobe is 4 kHz. This is equal to the spectral resolution limit of the FMCW source’s BF signal.

The above discussion shows that the resolution of the FMCW source can reach the theoretical limit after nonlinear correction with the proposed method.

3.2 Simulation of multi-target ranging

In this section’s simulation model, the parameters of the FMCW source and the position of the reference target are the same as in Section 3.1. There are three targets to be measured in the emission direction of the FMCW source, and the distance between them and the FMCW source is 543 mm, 549 mm, 576 mm, respectively. The beat frequencies corresponding to the above three distances are 724 kHz, 732 kHz and 768 kHz respectively. Thus, the output BF signal S3target can be expressed as

$$\begin{array}{l} {\textrm{S}_{3t arg et}} = cos \left\{ {\int {[{f(t )- f({t - 3.62 \times {{10}^{ - 9}}s} )} ]dt} } \right\} + \ldots \\ \ldots cos \left\{ {\int {[{f(t )- f({t - 3.66 \times {{10}^{ - 9}}s} )} ]dt} } \right\} + cos \left\{ {\int {[{f(t )- f({t - 3.84 \times {{10}^{ - 9}}s} )} ]dt} } \right\} \end{array}.$$

As shown in Fig. 6(b), the BF of Starget is 728 kHz. Thus, the nonlinear correction result of S3target can be calculated as

$${S_{3corrected}} = Re \left\{ {[{{S_{3t\arg et}} + jH({{S_{3t\arg et}}} )} ]{{[{C(t )} ]}^{\frac{{728}}{{720}}}}} \right\}.$$
The frequency spectrum of S3target after nonlinear correction is shown in Fig. 6(c). In Fig. 6(c), it can be found that there are three targets in the emission direction of the FMCW source. The values of frequency peaks denoting the three targets are 724 kHz, 732 kHz and 768 kHz respectively. The three frequencies correspond to distances of 543 mm, 549 mm and 576 mm, respectively.

 figure: Fig. 6.

Fig. 6. (a) The frequency spectrum, (b) the WFT result and (c) the frequency spectrum after nonlinear correction of S3target.

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The above result demonstrates that the frequency spectrum, corrected with the proposed method, can effectively and accurately denote more than one target.

4. Experiments

4.1 FMCW source

The FMCW source used in our experiments has a starting frequency of 276 GHz, a sweep band width of 50 GHz and a sweep period of 250 μs. According to the above parameters, it can be calculated that the frequency spectral resolution of the source is 4 kHz. That is, the range spectral resolution is 3 mm.

4.2 Target ranging experiment

In the experiment of this section, an absorbing material is used as a non-reflective target and a metal mirror is used as a reflective target. The photograph of the experiment is shown in Fig. 7. The non-reflective target is first measured as shown in Fig. 7(a) and the output BF signal Szero is recorded. Then the reflective target is placed 500 mm in front of the FMCW source and the output BF signal Sref is recorded.

 figure: Fig. 7.

Fig. 7. (a) Photograph of the experiment measuring the non-reflective target; (b) photograph of the experiment measuring the reflective target.

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The time domain spectrum of Szero and Sref are shown in Fig. 8. According to Fig. 8(c), it can be estimated that the BF of Sref is 788 kHz. Then the output BF signal of the target whose distance from the source is around 500 mm can be calibrated depending on Eq.(8), Eq.(10) and Eq.(11).

 figure: Fig. 8.

Fig. 8. (a) The time domain spectrum of Szero; (b) the time domain spectrum of Sref; (c) the WFT result of (Sref-Szero); (d) The frequency domain spectrum of Sref; (e) The frequency domain spectrum of of (Sref-Szero); (f) the frequency spectrum after nonlinear correction of (Sref-Szero).

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By comparing Fig. 8(d) and Fig. 8(e), it can be seen that there is a signal leak at transmitter when the FMCW source is working in practice, which results in serious low-frequency interference in the output BF signal’s spectrum. Therefore, it is necessary to subtract a zero reference before nonlinear correction. The comparison between Fig. 8(e) and Fig. 8(f) shows that the spectrum broadening is well suppressed after nonlinear correction with our method.

According to the nonlinear compensation coefficient calculated by Sref and Eq.(9), we carried out a series of ranging experiments within the range of 473 mm to 530 mm from the source at an interval of 3 mm. The nonlinear correction results of the above measured BF signals are presented in Fig. 9. According to the formula of frequency and distance: ‘d = cT(2B)-1’, 4 kHz frequency difference denotes the actual distance of 3mm. Therefore, it can be seen from Fig. 9 that within the range 473 mm to 530 mm ahead of the FMCW source, the spectrum of the corrected BF signal has the ability to accurately denote a target at the accuracy of the FMCW’s range spectral resolution. Besides, the FWHM of the main lobes in Fig. 9 are all 4 kHz, which indicates that the proposed method allows the FMCW system to distinguish two targets with a distance of 3mm.

 figure: Fig. 9.

Fig. 9. (a)∼(t) are the calibrated BF signals’ spectrum of the target with a distance of 473mm∼530mm from the FMCW source respectively.

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4.3 Imaging experiment on a sample with multilayer structure

The photograph of the imaging experiment is shown in Fig. 10. The sample is made up of a 12 mm thick resin frame and two pieces of cardboard. There are metal letters on the inside of both pieces of cardboard. In the experiment, the sample is fixed on the motorized positioning system. Controlling the movement of the motorized positioning system, the scanning of the sample can be realized.

 figure: Fig. 10.

Fig. 10. (a) Photograph of the imaging experiment; (b) the metal letter ‘T’ on the inside of the front board of the sample; (c) the metal letter ‘Z’ on the inside of the back board of the sample.

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The imaging result after correction with the proposed method is shown in Fig. 11. As can be seen from Fig. 11(b), the information on the front board of the sample can be effectively described by the imaging result, and the contour of the metal letter ‘T’ on the inside can be clearly identified. As can be seen from Fig. 11(c), the information on the back board of the sample can be effectively described by the imaging result, and the contour of the metal letter ‘Z’ on the inside can be clearly identified. According to Fig. 11(d) and Fig. 11(e), the distance between the front board of the sample and the back board of the sample in the imaging result is 597mm-585mm=12mm. This is consistent with the actual thickness of the resin frame.

 figure: Fig. 11.

Fig. 11. Imaging result of the sample. (a) three-dimensional tomography of the sample;(b) Two-dimensional information of the layer where point A is;(c) Two-dimensional information of the layer where point B is; (d) the distance spectrum corresponding to point A; (e) the distance spectrum corresponding to point B.

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According to the above discussion results, it can be concluded that the proposed method makes the FMCW system effectively reflect the information of each layer of the multi-layer sample.

5. Conclusion

In this article, a novel nonlinear correction method for THz FMCW is proposed. When calculating the nonlinear compensation coefficients, WFT is used to estimate the BF in the proposed method. Compared with the traditional method of converting BF by actual distance, the proposed method makes the estimation of the nonlinear compensation coefficients more accurate and the elimination of the output BF signal’s nonlinearity more thorough. The simulations and experiments in this paper prove that the proposed method can effectively correct the output BF signal’s nonlinearity of FMCW system. After the BF signal is calibrated by the method in this paper, the FWHM of the main lobe in the spectrum is equal to the resolution of the spectrum. That is to say, the proposed method allows an FMCW system’s range resolution to reach the theoretical limit. In addition, the imaging experiment of a sample with multi-layer structure proves that the proposed method makes an FMCW imaging system effectively reflect each layer’s information of the sample. Compared with common methods such as introducing a reference interferometer or a soliton comb to calibrate the nonlinearity, the proposed method does not require any additional hardware. This is beneficial to the miniaturization of the THz FMCW detector. The proposed method is more suitable for the daily testing situation where the THz FMCW detector needs to be placed on the scanning system and move together with the scanning system.

Acknowledgment

The authors thank Ruo-Wen Xu for helpful discussions.

Disclosures

The authors declare that there are no conflicts of interest related to this article.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (11)
Fig. 1.
Fig. 1. Schematic of the THz FMCW source.

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Fig. 2.
Fig. 2. Schematic diagram of THz FMCW ranging.

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Fig. 3.
Fig. 3. The flow-chart for the nonlinear correction procedure.

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Fig. 4.
Fig. 4. (a) The spectrum and (b) the WFT result of Sref.

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Fig. 5.
Fig. 5. (a) The frequency spectrum, (b) the WFT result and (c) the frequency spectrum after nonlinear correction of Starget.

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Fig. 6.
Fig. 6. (a) The frequency spectrum, (b) the WFT result and (c) the frequency spectrum after nonlinear correction of S3target.

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Fig. 7.
Fig. 7. (a) Photograph of the experiment measuring the non-reflective target; (b) photograph of the experiment measuring the reflective target.

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Fig. 8.
Fig. 8. (a) The time domain spectrum of Szero; (b) the time domain spectrum of Sref; (c) the WFT result of (Sref-Szero); (d) The frequency domain spectrum of Sref; (e) The frequency domain spectrum of of (Sref-Szero); (f) the frequency spectrum after nonlinear correction of (Sref-Szero).

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Fig. 9.
Fig. 9. (a)∼(t) are the calibrated BF signals’ spectrum of the target with a distance of 473mm∼530mm from the FMCW source respectively.

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Fig. 10.
Fig. 10. (a) Photograph of the imaging experiment; (b) the metal letter ‘T’ on the inside of the front board of the sample; (c) the metal letter ‘Z’ on the inside of the back board of the sample.

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Fig. 11.
Fig. 11. Imaging result of the sample. (a) three-dimensional tomography of the sample;(b) Two-dimensional information of the layer where point A is;(c) Two-dimensional information of the layer where point B is; (d) the distance spectrum corresponding to point A; (e) the distance spectrum corresponding to point B.

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Equations (18)

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I r e f = e x p [ j ( π B T t 2 + 2 π f 0 t + φ 0 ) ]
I r e c = e x p { j [ π B T ( t − τ ) 2 + 2 π f 0 ( t − τ ) + φ 0 ] } .
I B F = e x p [ j ( 2 π B τ T t  -  π B T τ 2 + 2 π f 0 τ ) ] .
f = B T t + f 0 + ξ ( t ) 4 η B T 2 ( t − T 2 ) 2
{ I r e f ′ = e x p [ j ( π B T t 2 + 2 π f 0 t + ε ( t ) + φ 0 ) ] I r e c ′ = e x p { j [ π B T ( t − τ ) 2 + 2 π f 0 ( t − τ ) + ε ( t − τ ) + φ 0 ] } .
I B F ′ = e x p { j 2 π [ B τ T t − B 2 T τ 2 + f 0 τ + ε ( t ) − ε ( t − τ ) ] } .
I B F ′ = e x p { j 2 π [ B τ T t − B 2 T τ 2 + τ σ ( t ) ] } = e x p { j 2 π [ f B F t − T 2 B f B F 2 + f B F T B σ ( t ) ] }
C ( t ) = e x p [ − j 2 π f B F T B σ ( t ) ] = ( I B F ′ ) − 1 e x p [ 2 π j ( f B F t − T 2 B f B F 2 ) ] .
I B F 1 ′ = e x p { j 2 π [ f B F 1 t − T 2 B f B F 1 2 + f B F 1 T B σ ( t ) ] } .
I c o r r e c t e d = I B F 1 ′ [ C ( t ) ] f B F 1 ′ f B F
I = ( S − S 0 ) + j H ( S − S 0 )
f ( t ) = [ 2 × 10 5 t + 276 + 1.6 × 10 8 ( t − 1.25 × 10 − 4 ) 2 ξ ( t ) ] G H z
S r e f = cos ⁡ { ∫ [ f ( t ) − f ( t − 2 × 0.54 3 × 10 8 s ) ] d t } = cos ⁡ { ∫ [ f ( t ) − f ( t − 3.6 × 10 − 9 s ) ] d t }
C ( t ) = [ S r e f + j H ( S r e f ) ] − 1 e x p [ j 2 π ( f B F t − T 2 B f B F 2 ) ] = [ S r e f + j H ( S r e f ) ] − 1 e x p [ j 2 π ( 7.2 × 10 5 t − 1.296 × 10 − 3 ) ] .
S t a r g e t = cos ⁡ { ∫ [ f ( t ) − f ( t − 2 × 0.57 3 × 10 8 s ) ] d t } = cos ⁡ { ∫ [ f ( t ) − f ( t − 3.8 × 10 − 9 s ) ] d t } .
S c o r r e c t e d = R e { [ S t arg ⁡ e t + j H ( S t arg ⁡ e t ) ] [ C ( t ) ] 760 720 }
S 3 t a r g e t = c o s { ∫ [ f ( t ) − f ( t − 3.62 × 10 − 9 s ) ] d t } + … … c o s { ∫ [ f ( t ) − f ( t − 3.66 × 10 − 9 s ) ] d t } + c o s { ∫ [ f ( t ) − f ( t − 3.84 × 10 − 9 s ) ] d t } .
S 3 c o r r e c t e d = R e { [ S 3 t arg ⁡ e t + j H ( S 3 t arg ⁡ e t ) ] [ C ( t ) ] 728 720 } .
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