# Improving FMCW GPR Precision through the CZT Algorithm for Pavement Thickness Measurements

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## Abstract

**:**

## 1. Introduction

## 2. Principle and System Scheme

#### 2.1. FMCW GRP Measurement System

_{C}, S, and B, respectively, is connected to the mixer and the transmit antenna through the power divider. The emission signal is divided into two parts, T

_{1}and T

_{2}, in which T

_{1}reaches the upper surface of the tested sample, and the reflected signal is R

_{1}. After T

_{2}passes through the sample to be tested, it is reflected on the lower surface, and the reflected signal is R

_{2}. R

_{1}and R

_{2}are two sinusoidal signals of different frequencies, which are superimposed and mixed with the transmitted signal, and theoretically generate down-converted signals of two frequencies. After spectrum analysis, the lower frequency is f

_{1}, the higher frequency is f

_{2}, and the frequency difference is Δf = f

_{2}− f

_{1}.

_{C}is the period of the chirp pulse, Δf is the difference between two frequency peaks, c is the speed of light in free space, ε is the sample relative permittivity, and B is the bandwidth of the chirp. If the thickness is known, the dielectric constant of the sample can also be obtained by conversion.

_{0}is the relative permittivity of air. Formula (3) can calculate the distribution of the surface layer of the measured object.

#### 2.2. Chirp-z Transformation

_{c}are both constants. Therefore, the accuracy of the measurement is related to the frequency difference Δf, and the two down-conversion frequencies of the Intermediate Frequency (IF) signal are the key parameters of the measurement. FFT is a commonly used algorithm for calculating frequency, which can quickly calculate all N-point discrete Fourier transform (DFT) values, that is, all equally spaced sample values of the Z transform on the unit circle of the Z plane. It is well known that the DFT of the sequence is the Z transform of X(n) sampling at N points on the unit circle at equal intervals, and the sampling interval is ${e}^{j\frac{2\pi}{N}}$. Additionally, DFT is the spectrum uniformly distributed over N points on the unit circle of the Z plane. When X(n) is a short-time sequence, the frequency resolution $\frac{2\pi}{N}$ obtained by DFT will be extremely low. Nevertheless, high sampling points will lead to an increase in the amount of computation and cost. However, FMCW radars often only require a part of the spectrum rather than calculating Z-transformed samples for the entire unit circle. The CZT transform is a spiral sampling calculation method that does not sample on the unit circle. The frequency resolution will be greatly improved by calculating the part of the spectrum of interest. The CZT transform of the sequence formula can be expressed as:

_{0}, W

_{0}, θ, and ϕ are all constants defined by computational requirements. When

#### 2.3. Numerical Simulation

## 3. Experiments

## 4. Results and Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 2.**FFT and CZT algorithm simulation: (

**a**) An IF signal with frequencies of 6 and 10 Hz. (

**b**) Spectral analysis of IF signals by FFT and CZT algorithms. (

**c**) FFT and CZT calculations for samples with thicknesses of 20, 22, and 24 cm. (

**d**) CZT calculates samples with a linear thickness growth from 22 to 23 cm.

**Figure 3.**(

**a**) Concrete model with three layers of thickness. (

**b**) The proposed FMCW radar is used for model thickness measurement.

**Figure 5.**(

**a**) The IF signal diagram after detecting a wall. (

**b**) The spectrum diagram of the IF signal obtained by the CZT algorithm.

**Figure 6.**(

**a**) Three spectrogram datasets calculated by CZT (

**b**) Two-dimensional imaging of asphalt pavement.

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**MDPI and ACS Style**

Huang, T.; Zhang, C.; Lu, D.; Zeng, Q.; Fu, W.; Yan, Y.
Improving FMCW GPR Precision through the CZT Algorithm for Pavement Thickness Measurements. *Electronics* **2022**, *11*, 3524.
https://doi.org/10.3390/electronics11213524

**AMA Style**

Huang T, Zhang C, Lu D, Zeng Q, Fu W, Yan Y.
Improving FMCW GPR Precision through the CZT Algorithm for Pavement Thickness Measurements. *Electronics*. 2022; 11(21):3524.
https://doi.org/10.3390/electronics11213524

**Chicago/Turabian Style**

Huang, Tongxing, Chaoyang Zhang, Dun Lu, Qiuyu Zeng, Wenjie Fu, and Yang Yan.
2022. "Improving FMCW GPR Precision through the CZT Algorithm for Pavement Thickness Measurements" *Electronics* 11, no. 21: 3524.
https://doi.org/10.3390/electronics11213524