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University of Pisa
High Precision Displacement Measurements via
differential phase estimation using a
Photonics-based Dual-band Radar System
Milad Khosravanian
Advisor:
Prof. Antonella Bogoni
Scuola Superiore Sant'Anna Istituto di Tecnologie della Comunicazione, dell'Informazione e della Percezione “TeCIP”
Pisa, Italy
Tutor:
Prof. Filippo Giannetti - Prof. Marco Luise
Università di Pisa Dipartimento di Ingegneria
dell'Informazione Pisa, Italy
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Master Thesis
Title: High Precision Displacement Measurements via Differential Phase estimation using a Photonics-based Dual-band Radar System
Author: Milad Khosravanian
Advisor: Prof. Antonella Bogoni
Tutor: Prof. Filippo Giannetti, Prof. Marco Luise
Centre: Consiglio Nazionale delle Ricerche,Scuola Superiore Sant'Anna, Istituto di Tecnologie della Comunicazione,dell'Informazione e della Percezione “TeCIP” April 2017
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Acknowledgements
This thesis includes some of the most important results of the research that I have carried out in the framework of my six months lasting activities at the TeCIP Institute (Technologies Communication Information and Perception which belongs to the National Research Council (CNR)).
I started my activity in TeCIP in September 2016, for the developing of data analysis algorithms and for the creation of tools for the deformation and displacement measurement using Dual-Band Software-Defined coherent Radar, based on single Photonic Transceiver.
Most of the lessons learned from Photonic Transceiver are revealed to our ground based Photonic Radar to measure the submillimeter movements in the Line-of-sight “LOS”, which is the subject of this thesis.
This research started more than a year ago, which presented us with the accurate calculation respectful as other techniques, GB-SAR and Terrestrial Laser Scanning.
I was fully involved in this project, which could be useful for assessing the potential of the GB-SAR technique to measure deformation. The IBIS instrument is the first GB-SAR system available on the worldwide market, developed by the Systems Engineering SpA (IDS), one of the industrial company that will be interesting for this project’s context.
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This work would not have been possible without the support and commitment of my advisors, Prof. Giannetti, Prof. Bogoni and Prof. Luise and Dr. Sergio Pinna, Dr. Salvatore Maresca. I would like to express my gratitude to them for their advice, their patience, their comments and lengthy discussions, and why not, for their kind time pressure and the trust that they put on me. Also I would like to thank my colleague Suzzane Melo, PHD. Student of TeCIP, for her unconditional support and for her advice, because she has always been there and she has always performed in our experiments. Special thanks to my girlfriend, my family and my friends for their valuable support.
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Abstract
This thesis is written in four chapters. In the first chapter, the relevance of this study in our present life and in our environment will be talked about. This system can be used to protect humanity from unforeseen events.
In the second chapter, a photonics-based architecture of a multi-band coherent radar system is proposed and validated. The precision and flexibility of photonic technologies are exploited for generating and simultaneously detecting multiple radar signals in an extremely wide frequency range. Moreover, the fully digital approach enables the software-defined radio paradigm, allowing the flexible use of several advanced radar techniques such as waveform diversity or frequency hopping. The proposed architecture is therefore promising for future radar systems that need to adapt to different scenarios for improved situation awareness.The proposed system exploits a single laser unit for the multiband transmitter and receiver sections, reducing the architectural complexity with potential benefits on system dimensions, cost, and reliability.
In this chapter, we alsodetailing the principle of operation of the proposed
multi-band coherent radar system, and we describe the implementation of a proof-of-concept dual-band transceiver operating in the X- and S-bands simultaneously and independently. The results from the characterization of the transceiver are presented.
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The system validation through the coherent detection of moving targets confirms the suitability of the proposed solution, laying the basis for a new paradigm of radar systems.
In the third chapter, the coherence phase among carriers generated on different frequency bands by the photonics-based radar is exploited to perform differential phase estimation for enhanced displacement measures.
The system employs stepped frequency continuous waves simultaneously in the S- and X-band, measuring the differential phase over a synthetic band up to 7.4GHz. The experimental results demonstrate a precision < 200µm without using correction algorithms.
At the middle of this chapter, the same dual-band radar system is used for precise target displacement measures in a multi-target scenario. The radar has been designed for the monitoring and the prevention of possible structural failures of buildings and landslides.
There we present a preliminary numerical and experimental evaluation of the displacement accuracy in the case of a multi-target scenario. Our study is focused to understand the impact of multiple scatterers in the target displacement estimation. Simulation results show a typical displacement accuracy <0.2mm for considered distances up to 400m, in presence of five scatterers. These results are confirmed by preliminary experiment outcomes.
At the end, we show the manually unwrapping phase with numerical result of a simple experiment with the new SFCW parameter.
The fourth chapter is dedicated to conclusion and the next feature on the project.
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Key words: Coherent radar, down-converter, mode-locked laser (MLL), photonic transceiver, software-defined radio (SDR), up-converter, differential phase estimation, dual-band radar, microwave photonics.
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Content
1. Chapter 1. The project’s goal and its relevance in our life………... 12
2. Chapter 2 ………. 15
2-1: Introduction ………...……….. 16
2-2: The Principles of System Design and Signal Processing ……….. 19
2-3: Developed Proof-of-Concept Photonics-Based Transceiver ………... 24
2-4: Radar Demonstration ………... 36
3. Chapter 3………40
3-1: Operation Principle and photonics-based radar characteristic………. 41
3-2: One Target Displacement Measurement……….. 45
3-3: Multi-Target simulation Displacement Measurement……….. 50
3-4: Manual phase unwrapping process………... 57
4. Chapter 4, Conclusion………..………. 62
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List of figures
Figure 2-1 Layout of origin of the multi-band photonic transceiver…………...……….….. 20
Figure 2-2 Block diagram of the implemented DSP and spectra in the different state……….….. 22
Figure 2-3 Photograph of the developed photonic transceiver………... 24
Figure 2-4 Block diagram of the digital waveform generation and……… 26
Figure 2-5 Sampling oscilloscope trace of the generated signals at the BPFs outputs………... 28
Figure 2-6 Sampling oscilloscope trace of the generated signals at the BPFs outputs………... 28
Figure 2-7 Electrical spectrum of the signal at the WBA output………... 29
Figure 2-8 Phase noise of the MLL, at its two IFs, and of the two optically generated carriers…...……. 30
Figure 2-9 Two-tone SFDR of the photonic receiver………...……….. 31
Figure 2-10 FFT of the received signal sampled at 400 MS/s……… 32
Figure 2-11 IFFT of the signal filtered at 75 MHz and decimated to 100 MS/s……….……33
Figure 2-12 IFFT of the signal filtered at 125 MHz and decimated to 100 MS/s……….. 33
Figure 2-13 Autocorrelation function of the signal at 75 and125 IF……….. 34
Figure 2-14 Experimental setup of the moving targets' detection demonstration……….. 37
Figure 2-15 Measured Doppler spectra for the S-band and X-band carriers……….. 38
Figure 2-16 Velocity of the target as detected by the two carriers………. 39
Figure 3-1 Scheme of the developed radar experiment……….. 42
Figure 3-2 Electrical spectrum of the generated dual-band RF signal………... 43
Figure 3-3 FFT spectrum of the received down converted signal……….. 46
Figure 3-4 Measured displacement vs. real displacement………... 46
Figure 3-5 Difference between the real and the measured displacements………..….. 47
Figure 3-6 Cross-correlation function of the received echo for 0km and 3 km distant targets……….…... 48
Figure 3-7 Mean error of the displacement measurement error……….….. 49
Figure 3-8 Range resolution scheme and targets position ………...…….…..……… 52
Figure 3-9 Typical processing for extracting the phase information from the range cells………..…. 53
Figure 3-10 Displacement error estimates from simulations of the S-band radar data with 5 targets.….. 55
Figure 3-11 FFT spectrum of the SFCW signal produced by DDS………..…... 56
Figure 3-12 Digitally Controlled Motorized Linear Stage……….……….. 57
Figure 3-13 the position of the antenna and target……….……….. 58
Figure 3-14 The FFT spectrum of the SFCW signal after IF down conversion………. 59
Figure 3-15. FFT spectrum of the SFCW of If down converted received signal …………...………. 59
Figure 3-16 Phase unwrapping. Profile along a hypothetical displacement………..………... 61
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List of tables
Table 3-1 Summary of the main parameters of the photonics-based radar system and SFCW……….52 Table 3-2 Phase information before and after manually Unwrapping processing……….60
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Chapter 1
The project’s goal and its relevance in our life
Inherent risk created by landslides and other ground failures are accountable for big waste, as human suffering, massive economic loses, and environmental decay. Many strategies for decreasing losses and impact from this kind of events have been proposed, for example, to develop a real-time monitoring and prediction capability in the regions most affected by landslides and applying remote sensing technologies [1].
Several different approaches for controlling and mapping the movement of ground displacements have been extracted as extensometers, crack meters, inclinometers, laser or sonar range finders, GPS receivers, etc. However, all these solutions provide only a localized deformation information and, even for this reason, are expensive and time consuming to be installed, as well as ineffective when monitoring broad areas, such as landslides monitoring on inaccessible areas [2]. To overcome these limitations, a remote sensing approach, based on radars, is well suited as it is able to provide complete and precise deformation maps of the whole area under surveillance [3]-[4].
The merchant radar system relies on the Stepped-Frequency Continuous-Wave (SFCW) technique and differential interferometry [4]. Using the various sinusoidal signals bye the SFCW technique on a single RF band, being all coherent between them with constant frequency separation, to illuminate the scenario. The distance among the antenna and the target is calculated by measuring the phase difference between the sent and received signal, coming out from the signal round
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trip time. Target distance measurement is executed by measuring the phase difference between the several frequency components of the SFCW signal [5]. Merging a large signal bandwidth for a better range cell resolution and to
decrease the system noise for an increase differential phase estimation is allowed by the use of various sinusoidal waveforms.As well as it known the differential phase estimation precision is guaranteed by the phase coherence between the sinusoidal waveforms. As we calculated the phase of the echo signal, differential interferometry is used to evaluate the signal phase difference among two or more successive coherent reception of the same target.So this information can be expound into a very small change in range, noted a very precise displacement measures.
At the present time this technique is bounded to utilization of coherent sinusoidal waveforms in a single RF band, due to the issues in generating high frequency multi-band coherent signals. Although in the electronic domain it's
feasible to produce multi-band coherent signals [5], there is a large increment in the elaboration and the charge of the system for high frequency signals, restricting the operation to a few GHz.
Anyway, the ability to use coherent multi-bands may be favorable to growth the combined signal bandwidth, for improving the range cell resolution and the displacement measurement precision. In addition, it would be favorable to ameliorate the facility to tune the operative RF carriers for adjusting the system to the environment and to the perception range [6].This would effect on the ultimate precision in product again the metamorphosis maps chip into a trustful primary hazard tracing.
During the preceding year, photonics technologies have been checked as a means to dominate the RF electronic device restriction. As we see in the successive
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chapter, the first realized Indicative of a photonics-based coherent radar system that extracts photonics for the production and demodulation of the RF signals without the use of noisy and narrow band mixers and putting the RF clock with an extremely permanent optical clock. Using photonics has been also showed the facility of the producing and discovering several RF signals in the same time, in several RF bands insuring an inherent elevate phase coherence between them, let coherent multiband radar operations [7].
Here we accurate movement calculation through extracting the innovative aspect of the photonics based radar. Actually, general resolution based on SFCW and differential interferometry, coming out from the large bandwidth and pliability of photonics, that letting us to achieve the various bands (two in this case) with the ability of the dynamically tune operative carrier in RF .We avoid to utilization of the correction algorithms because of the inherent high phase coherence among the bands. In addition, we are capable to exploit all the RF carriers permit for a decrease of the whole system power consumption and footprint through the applying of a single photonics-based transceiver.
Mainly, this thesis reports an experiment of small displacement measures based on SFCW technique and differential interferometry using a photonics-based coherent radar system operating in S- and X-bands and synthesizing bandwidth up to 7.4 GHz, and it demonstrates the capability to reach an accuracy lower than < 200µm over a range up to 3 km without correction algorithms, and after that we arrive to a specific characteristic of the SFCW that allow us to observing and calculation of the same experiment for multiple target.
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Chapter 2
In this section, a radio architecture, based on photonic integrated multi-band coherent radar system is proposed and approved. Accuracy and flexibility of photonic technologies are operated for the production and the simultaneous detection of multiple radar signals in a very wide frequency range.
In addition, the digital approach allows software-defined of our radio model, which allows flexible use of advanced radar techniques such as a waveform diversity and frequency hopping. So the proposed architecture is promising for future radar systems that need to adapt to different scenarios for a better understanding of the situation.
A single laser unit is used for multi-band transmitter and receiver sections for the proposed system, reducing the complexity of architecture with the potential benefits of the system size, cost and reliability.
This section details the operation of the multi-band coherent radar system, and describes a concept of dual-band transceiver, operating in X-band and S- bands simultaneously and independently.
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2-1: Introduction
One of the basic characteristics that the next generation of radar must demonstrate is, the ability to provide a full representation of the scenario in question. To do this, they need to collect a variety of information, allowing the interpretation of complex scenes. For this purpose, multi-band radar has benefited with different characteristics of the environment and targets pursuant to the applied carrier frequencies (CFS). For example, the long-range target detection and monitoring are two key features (both for military and civilian radar) that normally need to select the appropriate frequency band.
S-band signal (in the frequency band from 2 to 4 GHz) shows the strong safety against weathering disorder, and is widely adopted for the early warning radar applications. On the other hand, radar operating in the X-band (in the frequency band from 8 to 12 GHz) often used to produce narrow beams to track the target, and to improve the spatial resolution for the imaging target. Furthermore, the K-band radar (in the 27-40 GHz) is limited in range due to a weakening of the atmosphere, although they allow the higher angle and spatial discrimination. It is commonly used as a radar airport surface movement [8]. Therefore, multi-band radar is able to perform the function of several radars, greatly reducing cost and increasing performance.
In fact, compared to normal radar systems, multi-band radar allows to improve the targets classification and recognition algorithms [9] - [10], and the imaging operation of a complex range of objectives [11].
Furthermore, the integration of data from different radar in different frequency bands is a promising method to improve coverage in weather monitoring
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applications [12]. And at the same time, signal processing with multiple bandwidth frequencies in one device is difficult, mainly due to the bandwidth limitations of electronic devices.
Some examples of multiband radar transceiver has been reported [5], [13],but the features are limited and fixed, so making them is appropriate only for very specific applications. To achieve a multi-functional approach, the radar systems need to be configured and have a software-defined RF signal generators,to be able and flexible for producing broadband carriers waveform for very high frequencies (EHFs), while maintaining the required stability of the phase for coherent signal processing.
In addition, a direct analog to digital conversion at radar receiver applied in RF domain is necessary to reconfigure the required ability and reliability, as well as reducing noise and costs due to its analog counterpart [14].The expansion of analog-to-digital converters (ADCs) and direct digital synthesizers (DDSs) let the execution of common-aim totally digital transceivers, which can figure out every transmission protocol through software coding, providing the software defined radio (SDR) model [15].
Nevertheless, the bandwidth of these digital components is today moderate to
a few gigahertz, and will extremely develop farther. In recent years, the research scope of microwave photonics has been offered the facility of draw out the large bandwidth and flexibility of photonics for discover RF section that cover the frequency range from a few megahertz up to several tens of gigahertz [16]–[17].The compatibility of the Photonic procedures for producing high-quality RF signals, with CF flexibility, has already been shown by the author in [18].In that design, a phase-modulated signal, at a low IF, has been up-converted alternatively to the X-band and to the Ka-band by the identical implementation, retaining great phase stability. In addition, in [19], the author has also shown that photonics is engaged for the accurate
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digitization of high-frequency RF signals. There, a photonics-based ADC has been able to sampling signals directly in X-band and Ka-band with an efficient number of bits (ENOB) greater than 7, then prevailing the restriction of the electronic ADCs. These significances have been applied to the expansion of the first photonics-based coherent radar system, come off under the ERC-funded project PHODIR [20], [21], thus proving the suitability of the photonic approach for radar systems [22].
Here we ameliorate those conclusion offer and we accredit the architecture of a multi-band and multi-waveform coherent radar system. The layout combines the frequency flexibility of the photonics-based RF generator with the enormous input band of the photonic ADC, discovering a dual-band affirmation of working together coherently, and apart in the S-band and X-band, with a decreased system complication.
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2-2: The Principles of System Design and Signal
Processing
The layout principle of the suggested architecture is drawn in Fig. 2-1 (left). To create a multi-band transceiver, the first part is a mode-locked laser (MLL), which produces a very accurate optical pulse train with a repetition rate FMLL.
The optical spectrum of the laser is compound by a chain of modes, phase locked to each other, distanced by FMLL, as shown in Fig. 2-1(A).
In the transmitter part, the laser is amplitude-modulated via a DDS by a Mach– Zehnder modulator (MZM). The DDS produce X (t), which is the sum of N several baseband signals Sn(t) at different low Ifs IFn, their spectrum is depicted in Fig.
2-1(B) as follows:
X(t) = ∑ cos(2π × IFn × t) × Sn(t)
N n=1
(𝟏)
Along of the amplitude modulation, the signals at IF are transferred as lower and upper sidebands around each optical mode of the MLL [see Fig. 2-1(C)]. In order to prevent the overlapping of these sidebands, the maximum frequency generated by the DDS must be shorter than mid of the laser frequency rate. When the optical signal thus modulated is found out with a photodiode (PD), all the spectral components are heterodyned together.
So in this way, a double of the signals from the DDS is acquired at each kFMLL±IFn by entity of a positive integer [see Fig. 2-1(D)], up to the bandwidth of
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bandpass filters (BPFs) may be applied to exploit the doubles of the signals up-converted to the favorite CFs, as shown in Fig. 2-1(E). At the end, the signals are joined together so that they can be amplified by a wideband amplifier (WBA) before being transmitted. Alternatively, the output of the BPFs can be routed to frequency-specific electrical chains in order to use separate high gain antennas.
Fig. 2-1. (left) layout of origin of the multi-band photonic transceiver. (right) (A) Optical spectrum of the MLL. (B) Electrical spectrum of the modulating signal. (C) Optical spectrum after the MZM. (D) Electrical spectrum at the
output of the PD. (E) Electrical spectrum after the filter bank. (F) Electrical spectrum of the echo signal at the receiver ADC [7].
In the receiver part, the incoming reflection signals, passed from the filter to refute false tones and noise accumulated by the antenna, modulate the MLL by use of another MZM. We carry out the optical sampling of the RF signals directly at the CF, taking advantage of the extremely low time jitter of the MLL [23]. Thus, corresponding to the sampling theory, the signals are aliased down to their basic IFs, and a narrow-bandwidth high-precision ADC can digitize the photo detected signal [see Fig. 2-1(F)]. As the RF signals are born starting from signals at IF in the zone
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of 0<IF<FMLL/2, their under sampling at FMLL folds the signals back to the original
IFs and does not produce overlapping.
As coherent radar detection need a reference signal for estimation the reflection Doppler shift, in the case of tracking, the transmitter and receiver construction are doubled to operate the reference signals as well as, sketched in Fig. 2-1 (left). In addition, in the present layout, a single reference can handle all the diverse CFs.
The required of the reference signal for coherent digital signal processing (DSP) is sampled twice to catch both the in-phase and quadrature (I/Q) samples. This can be passed resorting to the orthogonal sampling, in which the Q signal is directly derived by the I signal (or vice versa) [14]. For this, it is compulsory that all the signals at CFs CFn verify the rule:
FMLL
i =
4 × CFn
4j + 1 (𝟐)
Where i and j are integer numbers with i>0 and j≥0. Using this limitation, coherent digital signal processing can be quickly organized together on all the reflection signals from the diverse frequency bands. The block diagram of the proposed DSP is depicted in Fig. 2-2(A).
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Fig. 2-2. (A) Block diagram of the implemented DSP. (B) Spectra in the different state [7].
We propound a dual-signal formation, like a model in Fig. 2-2(B), where the two signals, after the under sampling, crease at1⁄8 FMLL and 3⁄8 FMLL, respectively. The time-domain reflection signal, accumulated by the ADC with a sampling rate FMLL, is provided by a channels separation block. It first dose the Fast Fourier Transform (FFT) to move the signal in the frequency domain, where two ideal digital BPFs, stack up from 0 to FMLL /4 and from FMLL /4 to FMLL /2, respectively, separating both of them into two channels, an Inverse Fast Fourier Transform (IFFT) returns the signals in the time domain.
In this section, the two channels convey a signal with a bandwidth of FMLL/4,
so the vectors can be decimated to halve the sampling rate, thus aliasing both the signal at a1⁄8 FMLL.
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For carrying out the perfect coherent demodulation, our signals must be proceed to the zero frequency by the use of another digital down-conversion (DDC). As both the signals are aliased at the similar frequency, and thought that in pulsed radar the reference signals is a continuous wave (CW), a single reference can be used.
Once obtained, its vector is decimated halving the sampling rate, and thanks to the orthogonal sampling the Q components can be derived by the in-phase components by simply shifting the samples. In the DDC blocks, the reflection signals are mixed with the I and Q reference signal and the samples can be filtered by using a suitable combination of cascaded integrator-comb (CIC) Finite-Impulse Response (FIR) filters. At the end, the pulse-Doppler processing is performed to extract the range and velocity information of the targets [8].
This procedure can be quickly made larger to more than two signals, scaling the frequencies and the decimation rates and avoiding the overlapping of the aliased signals.
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2-3: Developed Proof-of-Concept Photonics-Based
Transceiver
Until executed the usefulness of our structure, a dual-band indicant will be extended and distinguished reflecting its ability to produce and detect different types of signals in S-band and X-band. Also, its implementation on the phase noise, noise figure (NF), and spurious-free dynamic range (SFDR) have been considered.
The transceiver that is shown in Fig. 2-3, utilizes a semiconductor passive MLL of a repetition rate about 400 MHz, generating 1-ps-long optical pulses at a central wavelength of 1550 nm with a time jitter of about 10 fs integrated in the range [5 kHz–10 MHz].
Fig. 2-3. Photograph of the developed photonic transceiver [7].
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The 5 kHz limitation is referred to the reverse of the coherence time necessary toward the radar coherent detection, and the 10 MHz limitation show the offset frequency where the signal source analyzer gets its noise floor region. The signals and the reference are produced directly at IFs trough a two-channel 180-MHz-bandwidth 400-Msample/s DDS with 16 bits of precision (NI PXIe-5451), clocked by the laser.
Corresponding to the SDR model, by using a simple instrumentation of the block diagram in Fig.2-4(A), the waveforms are digitally produced. Two waveform construction functions create the baseband waveforms with the wanted parameters, i.e., the pulse width (PW), the pulse repetition frequency (PRF), and the type of modulation. The produced digital waveforms are moved to the respective IFs, and collected simultaneously sample by sample. So, the resulting waveform is present to be generated by the DDS.
In Fig.2-4(A), the parameters of the waveforms used to verify the transceiver helpfulness are also reported, and the spectrum of the signal generated by the DDS is shown in Fig.2-4(B). The parameters chosen for the two waveforms correspond to those typically used in radar systems, except for the PRF that is very high for simple time-domain visualizations, and they are completely independent, as requested by the SDR radar paradigm.
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Fig. 2-4. (A) Block diagram of the digital waveform generation, and values of the set parameters. (B) Electrical spectrum of the signal generated by the DDS signals' zoom (RBW = kHz) [7].
Pursuant to the rules of utilization that have been illustrated before, the compound signal from the DDS carries to a 10-GHz-bandwidth MZM modulating the laser optical pulses, and a 12-GHz-bandwidth PD (Discovery DSC40S) produces all the replicas of the two modulating signals at the IF.
The resulting electrical signal from PD is split into two paths where two 20-MHz-bandwidth five-cavity BPFs extract the favorable replicas in the S-band and the X-band.
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In fact, the chirped signal generated at IF1 = 75 MHz is up-converted to CF1=9925
MHz, i.e., using the lower sideband of the 25th harmonic, and the phase-coded signal at IF2 = 125 MHz is moved to CF2=2525 MHz, i.e., the upper sideband of the sixth
laser harmonic. The two born signals are explained by:
S1 = ∑ rect(t − nTr1 PW1) n ∙ cos [2π ∙ (CF1 −Δf 2) ∙ t + π ∙ Δf PW1 ∙ t 2](3) S2 = ∑ rect(t −nTr2 PW1) n ∙ Aj∙ cos(2π ∙ CF2∙ t)(4)
Where Tr1 and Tr2 show the pulse repetition interval PRI, i.e., the inverse of the PRF,
Δf is the frequency deviation of the linear chirp, and Aj represent the jth-order ±1
Barker sequence coefficient.
The two CFs satisfy (2) for i=4, j=99 and i=4, j=25, respectively, thus
allowing the orthogonal under sampling and the use of a single reference signal. This one is generated by the secondary channel of the DDS as a CW at 75 MHz, up-converted to CF1 by means of the series of MZM, PD, and BPF identical to the one
demonstrated above.
Our modulated radar signals after the BPFs are demonstrated in Fig.2-5, specifying the ability of the system to coherent produce radar signals with several PRF and various PW.
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Fig. 2-5.Sampling oscilloscope trace of the generated signals at the BPFs outputs, scale 2 s div [7].
The Fig.2-6 shows a zoom of the generated radar pulses, stabilizing the different CFs and their coherence. After the BPFs, the two signal are boosted and loan entirely through a wideband directorial coupling, and a 2–18-GHz WBA (Ciao Wireless CA218) raising the power of the acquire signal up to a total power of 0 dBm.
Fig.2-6.Sampling oscilloscope trace of the generated signals at the BPFs outputs, scale 200 ps/div [7].
Fig.2-7 demonstrates the spectrum at the output of WBA, which specifies the absence of spurious tones due to the WBA, as well as the zooms of the modulated
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carriers. It can be understood that the spectra produced at IFs by the DDS are correctly up-converted to the S-band and the X-band together.
Fig.2-7 Electrical spectrum of the signal at the WBA output (MHz); Insets: zooms around the two CFs (kHz) [7].
As explained in [18], with the assumption about the phase noise of the modulating signal is poorly referred to the Nth harmonics of the MLL, the time jitter of the carrier in the up-conversion procedure remains without variation like the same as the starting point of MLL, while the phase-noise “PN” curve rises as [24].
PN ∝ 20 ∙ Log10( CFn
FMLL)(5) In the case of exam, it is considered only for the X-band signal, as can be seen in Fig.2-8. The phase noise of the S-band signal is a little aggravated by the DDS. However, the two carriers show excellent phase constancy, where they have a time jitter of 16 fs for the X-band and 18 fs for the S-band.
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Fig.2-8.Phase noise of the MLL, of the DDS at its two IFs, and of the two optically generated carriers [7].
Now, our signal which obtained at the top, is able to be transmitted (or further amplified for long-range radar applications), matching to the scheme in Fig.2-1 (left), for executing the transceiver efficiency, a sample of the transmit signal has been sent immediately to the receiver stage.
The received signal is split to be filtered by two BPFs identical to the transmitter ones, and coupled again to a single path.In fact, we are thrown down the out-of-band noise of the WBA, and in case of a real radar system, clear the false signals accumulated by the receiving antenna, as any other communication signals, like mobile networks signals, satellite television... As explained in the prior part, the filtered multiband signal is optically sampled by the laser pulses with a MZM, and the resulting optical signal, opportunely amplified and dispersed by a 2.5-km-long spool of dispersion compensating fiber (DCF) [20], is sent to a 2-GHz-bandwidth PD. The necessity to barricade PD saturation is presence of the DCF, and it is also
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helpful to accredit the low phase noise of the system as it operates like a delay line of 12.5 s
Fig.2-9. Two-tone SFDR of the photonic receiver [7].
Identical techniques are exploited to recover the reference signal, separated from the bank of filter that does not necessarily exist.
In Fig.2-9, the two-tone SFDR measurements of the photonic receiver (avoiding electrical amplifiers) are reported for both the reception of S-band and X-band signals.The two bands represent equal efficiency, and an SFRD of 98 dB∙Hz23
is used, As ADCs, two channels of a 12-bit and 600-MHz-bandwidth real-time oscilloscope (LeCroy HRO66Zi) have been employed to execute the needed DSP. Its sample rate has been reduced to 400 Msample/s, so it is given an instantaneous bandwidth of 200MHz.
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As it is shown in Fig.2-10, the FFT of the received signal, illustrates the two waveforms that can be seen at their original IFs, i.e., the chirped signal at 75 MHz and the phase coded signal at 125MHz. At this point, two digital filters can isolate the two signals, and the sample vectors are decimated by four, thus decreasing the sampling rate to 100 Msamples/s and aliasing both the signals at 25MHz. So using this approach, the frequency is quite one-fourth of the sampling rate, and a 90 degree shifted replica can be arrived simply shifting the sample vector by one status, allowing the digital development of the (I/Q) components of the reference signal.
Fig.2-10. FFT of the received signal sampled at 400 MS/s [7].
Fig.2-11 shows the decimated vector from the channels separation digital block for the first channel, showing that the 1- µs-long chirped pulses have been correctly received from the X-band.
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Fig.2-11.IFFT of the signal filtered at 75 MHz and decimated to 100 MS/s [7].
Furthermore, Fig.2-12 reports the autocorrelation function of the received signal, digitally down-converted to zero frequency, showing the range resolution improvement of a chirped pulse with respect to an unmodulated pulse.In a similar way, the channel at S-band taking 2 µs-long pulses with a PRF of 200 kHz and phase modulated with a 13-bit Barker code is shown in Fig.2-13(A), as well as its autocorrelation in Fig.2-13(B).
34
Fig.2-13 (A) Autocorrelation function of the signal at 75 IF. (B) Autocorrelation function of the signal at 125 IF [7]
This demonstrates the efficiency of the offered plan of principle, which is capable in the same time and separately produce S-band radar signals, as well as the X-band, letting them also their reception and digital demodulation. So, the coordinate of two variant radar systems using in various frequency bands can be elide in a single device.
Moreover, the clashes notice show some restriction on the dynamic range, NF, and output power. The system's linearity is usually moderated by the natural of the MZM, but the two-tone SFDR of 98 dB∙Hz23, also if less than the electrical model,
is appropriate for most utilization.
The 25 dB NF of the receiver is very elevate, but it is mainly according to the real application performance where the received signals are amplified only by a 3.5-dB-NF and 30 dB-gain WBA. As the photonics frequency transformation stand a conversion loss about 35 dB, the gain of the first stage is basic for the NF of the system.
35
If a suitably designed RF front-end with about 100-dB gain and 7-dB NF [20] was employed, relevant to the Friis formula the NF would decrease lower than 10 dB, appropriate for real long-range radar applications. Also, the selection of applying fully noisy commercial WBAs instead of frequency-specific components is still motivated by practical reasons (e.g., available devices, cables, antenna, etc.). However, low-noise multi-band booster are being expanded [25], and multi-band antennas are already on the market [26].
36
2-4: Radar Demonstration
To better prove the suitability of the proposed scheme for radar applications, a proof-of-concept demonstration detecting a real moving target has been conducted. With respect to the previous section describing the capabilities of the photonic transceiver, few simplifications were needed.
First of all, the buffer of the real time oscilloscope cannot manage long-lasting acquisitions so the ADCs were replaced by a dual channel 80-Msample/s ADC (NI PXIe-5122) embedded in a PXIe controller. To satisfy (2) with the new sampling rate, the CFs have been slightly changed, moving the S-band carrier to 2530 MHz, and the X-band carrier to 9930 MHz. The available WBAs cannot provide sufficient power for long-range detections so the target needed to be placed at less than 10 m from the system. This limitation, considering also the limited bandwidth determined by the slower ADCs, made it impossible to perform range measurements so we focused on Doppler detections.
The experimental setup is depicted in Fig.2-14. The DDS is set to generate two CWs at 70 and 130 MHz from its primary channel, and a CW at 70 MHz from the secondary channel to be used as reference for the Doppler processing.
These RF signals are fed to the photonic transceiver, not modified with respect to the previous section, that up-converts IF1=70 MHz to CF1=9930 MHz and
37
Fig.2-14. Experimental setup of the moving targets' detection demonstration [7].
The WBA boosts the two carriers to 0 dBm and a wideband horn antenna with a gain of 6 dBm transmits them to illuminate the target. On the other hand, the reference consists only in CF1, and is sent directly to the photonic receiver. The
employed target is a planar metallic reflector, mounted on a trolley running on tracks. The implemented system works in a two-antenna configuration so the echo is collected by another horn antenna, placed besides the transmitting one, which feeds the receiver that down-converts back to IF the echo and the reference. The signals from the receiver are acquired at 80Msample/s, thus aliasing at 10 MHz (IF1) and at
30 MHz (IF2), and after the channels separation and the sample decimation, they are
38
Fig.2-15 shows the Doppler shifts measured detecting the target approaching the antennas, for the carriers at 2530 and 9930 MHz, respectively. The two curves present different values and an opposite sign. The different sign is explained by the different aliasing experienced by the two carriers. In fact, the first down-conversion from CFs to IFs changes the sign of the signal at CF1, which is a lower sideband of
a laser harmonic, while it maintains the sign of CF2. In the second step, instead, both
the signals change their sign. To explain the different values, we must consider that the Doppler frequency shift FD is given by
FD = 2V ∙CFn
c (6)
There, C is the speed of light and V represents the target speed with respect to the observer. This equation clearly shows that higher CFs give larger Doppler shifts. More in details, the S-band carrier presents a frequency deviation of 15 Hz, corresponding to a speed of 0.88 m/s, while the shift of the X-band carrier is 60 Hz, which gives a speed of 0.90 m/s.
39
The target movement is well presented in Fig.2-16, which shows the velocity in different instants, as measured by the two carriers. Initially the target is stationary, then it is accelerated up to 1.8 m/s, and slows down until it is stopped at the end of the track. As can be seen, the two curves present a small difference. This is due to the different velocity resolution achievable in the two frequency bands. In fact, the coherent integration time of 200 ms gives a Doppler resolution of 5 Hz, which corresponds to a precision of about 0.08 m/s at 9930 MHz, while at 2530 MHz the resolution is only 0.3 m/s.
Fig.2-16. Velocity of the target as detected by the two carriers [7].
To consider this ambiguity, error bars have been added on the graph, showing that the values at X-band are always included in the confidence interval of the signal at S-band, thus confirming once again the effectiveness of the scheme.
40
Chapter 3
As observed in the precedent chapter, the phase coherency of the Multi-Band signals generated by the photonic-based radar is the key aspect to carry out the differential phase estimation in order to improve the precise displacement calculation.
After the stepped frequency continuous wave signal modulation, and its main capabilities have been introduced, a dual-band operative setup is considered for exploits the S-band and X-band signal. This can allow us to estimate the differential phase over the synthetic band up to 7.4 GHz.
41
3-1: Operation Principle and photonics-based radar
characteristic
The displacement measurement relies on the differential phase calculation. Exploiting the SFCW module, the target is illuminated by a fixed number of N sinusoidal signals, coherent to each other, with frequency “fn = f0+ n ∙ Δf”, whit a separation of 𝛥f, where “n=0,….,N-1”. Each frequency component of the backscattered echo acquires a diverse phase shift φn, relevant to the round trip time T:
T = 2d
c (1) Where d is the traveled distance. The diverse phase shift φn could be achieved by:
φn = 2π ∙ fn ∙2d
c (2) Then, if the range is fixed, for two frequencies separated by 𝛥f, will achieve a respective phase difference Φ
Φ = 2π ∙ Δf ∙2d
c (3) Therefore every change in the distance 𝛥d causes a phase variation 𝛥Φ, and
Δd = c
42
If the phase is shifted more than 2π, we have ambiguities phase shift. Actually in this technique, it is offer an unambiguous range given by
Ru = c
2 ∙ Δf(5) By considering equation (4), we can observed that the accuracy in the phase measurement matches to a different precision in the range calculation, depending on Δf. The larger is the frequency separation “Δf”, the more accurate is the distance calculation; at the cost of reducing the unambiguous range. To reach at this purpose, the photonic transceiver presented in the second chapter [7], insures the phase coherency over different frequency bands, furnishing a very stableΔf. In addition, its low phase noise lets an accurate differential phase calculation, which is ideal for our purpose, i.e. the precise displacement measure. The scheme of photonics-based dual-band radar system used to carry out the differential phase displacement measures is shown in Fig.3-1.
43
A mode-locked laser (MLL) feeds the transceiver which produces optical pulses with a very accurate frequency of FMLL = 400MHz, and high phase stability to up convert and down convert RF signals in S-band and X-band.More precisely, in the transmitter module the laser is modulated by a Mach-Zehnder modulator (MZM) with a Direct Digital Synthesizer (DDS), where the DDS produces the compound of two SFCW waveforms at the intermediate frequencies (IFs) of 75MHz and 125MHz.
In this procedure the electrical produced IF waveforms are shifted to the lower and upper sidebands around every optical mode of the MLL, and after the photo detection they result up converted to every multiple frequency of the MLL. At the end, a set of electrical band-pass filters extract the replica of interest, at 2475MHz (S-band) and 9875MHz (X-band) as shown in Fig.3-2. A wideband amplifier (WBA) amplifies the signal [10] to be transmitted by a TX Horn antenna.
Fig.3-2. Electrical spectrum of the generated dual-band RF signal of 20 steps divided in 1MHz, Both S and X-band [27].
44
The RX Horn antenna receives the scattered echoes. The received signal is amplified by a WBA and used to modulate the laser pulses in another MZM. This carries out the optical down-conversion of the RF signal, so that after the photo detection the two waveforms are aliased back to their original IFs [7].
Finally, a 400 MSample/s analog-to-digital converter (ADC) converts the IF signal into a digital signal, as well as a copy of the one produced by the DDS. This procedure allow to get a correct phase reference.
45
3-2: One Target Displacement Measurement
To operate the system, the two antennas (one for transmission and one for reception) are installed on a digitally controlled motorized linear stage, able to provide a very precise shifts of 1µm. Actually, in order to provide a preliminary
displacement calculation we perform by moving the radar TX and RX antenna pair and using the laboratory wall like a fixed target. At first, the rail has been positioned in the central position of the motorized linear stage, and used as reference location, for achieving the both positive and negative target displacements test.
The SFCW signal, where spectrum is shown in Fig.3-2, is composed by Nstep=20 frequency steps for both S-band and X-band of BW=20MHz, equally
spaced by 𝛥f=1MHz, with a duration of Tstep =200μs each.By this representation of
the SFCW, we are able to carry out the standard radar processing [8]in order to acquire an irregular range estimate, with a resolution “ΔR = c
2∙BW” given by the
signal bandwidth BW, i.e. is 7.5m.In addition, the differential phase calculation on
our SFCW signals provides 20 separately measures with a 𝛥f of 7.4GHz, and since the unambiguous range in this case is just 2cm, other couples of tones with different 𝛥f can be finally exploited to remove the ambiguities in this second stage of the high accurate range evaluation.
The signal processing is operated on the two recorded signals, i.e. the down converted echo (Fig.3-3) and the reference (fig.3-2), in the frequency domain via a Fast Fourier Transform (FFT). The first spectrum is multiplied by the complex conjugate of the second, and the phase of each of the 40 tones is calculated.
46
Fig.3-3. FFT spectrum of the received IF down-converted signal [27].
Fig.3-4 shows the displacement measured by the radar system at the varying of the stage position, by steps as small as 0.2mm and with a 10 mm-long excursion. Therefore, received signal is acquired 50 times. The differential phase is evaluated for each of them and, at the end, the phases are manually unwrapped and stored in the right scale (hear mm). This procedure will be explained in more detail in the following sections.
47
Fig.3-4.Measured displacement vs. real displacement [27].
As it is shown in the figure, the system provides a fine accordance between real position and displacement. The error between the measured and the real displacement in each point, reported in Fig.3-5, is limited to less than ±0.2 mm.
48
In a formal request for this application, like a mine or structural monitoring, the target can be placed up to a distance of a few kms. Therefore, we apply the calculation for different target separations. As a miscalculation in the phase difference estimation is also the inherent phase noise of the radar system, applying a delay between the signal generation and reception de-correlates the reflection and the reference signal, thus furnishing a higher approximation of the efficiency.
A delay is applied by adding one and two 1km-long optical fiber spools in the transceiver. Since the refractive index of fibers is about 1.5, the spools simulate a distance of 1.5km and 3km, respectively.
The target distance, has been calculated by measuring the peak of cross-correlation function between the received and the reference signals. As shown in Fig.3-6, in which the blue plot stores the 0km target, the red plot the 1.5km case and the green one the 3km case, the calculated ranges match with the nominal lengths of the fiber spools.
Fig.3-6. Cross-correlation function of the received echo for 0km (Blue), 1.5km (Red) and 3 km (Green) distant targets [27].
49
With respect to the precision of the differential phase approximation, the mean absolute error for displacement measure at different range is shown in Fig.3-7. It is clear that the error remains always below 0.2 mm, this proving the stability of the phase of our system and its suitability for phase interferometry techniques.
Fig.3-7. Mean error of the displacement measurement error [27].
In the proposed scenario, the phase sensitivity could degrade over long path lengths due to the losses of the amplifiers and filters in the receiving, alternatively augmenting the total noise of the system. However, with respect to available transceivers, the dual band ability makes the system robust versus atmospheric propagation and weather conditions. In this case, ad hoc of post-processing could be required to increase the Signal-to-Noise Ratio “SNR”, and to grant the sub-mm measurement precision of the system.
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3-3: Multi-Target simulation Displacement
Measurement
As seen in section 3-2, the system exploits the combination of SFCW modulation and differential phase measurements for precisely determining the target displacement [27]. As we can observe a shift in range of the target results in a shift in phase of the radar return, relative to the wavelength of the radar signal.
A SFCW radar effectively samples the frequency response of a target at specific
frequencies within a given bandwidth. The sampled frequency response can then be transformed into the time domain using the Inverse Discrete Fourier Transform (IDFT), in order to provide the range profile of the target, for each band (we omit the index for simplicity). As for the single target case, the targets are illuminated by a sequence of N sinusoidal signals, coherent to each other, with frequency “fm=f0+[(m-1)∙Δf]”, with “m=1,…,N,” frequency separation step Δf, total signal bandwidth “BW=N∙Δf” and carrier frequency f0. Each frequency component of the echo accumulates a phase shift depending on the travelled radial distance d. Two frequencies separated by Δf will present a relative phase difference Φ.
Φ = 2π ∙ Δf ∙2d
c (1) Thus, any changes in the radial distance Δd, measured at range cell m, with m=1,…,N, will induce a phase variation ΔΦ of the mth Fourier harmonic f
m, as
follows:
Δd = c
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The range cell resolution ΔR, which is achievable by applying standard algorithms on the received signal, is equal to:
ΔR = c
2 ∙ BW(3) This means that two targets with relative distance smaller than ΔR cannot be distinguished as separate targets. If more than one target is present in scene under analysis, and they belong to distinct range cells (i.e. their distance is larger than ΔR), proper decomposition algorithms can be applied to the received signal, thus allowing to determine the separate displacement of each target [28].
First, we have conducted numerical simulations in order to verify the feasibility of the experiments to be carried out. The description of the simulated system goes as follows.We simulate two SFCW signals (one for S-band and one for X-band), which originally would be generated by the direct digital synthesizer (DDS), each composed by “Nstep=66” frequency steps, spaced by “Δf=300 kHz”, for
a total bandwidth of “BW=20 MHz”, and with step duration of “Tstep=50 ns”. This
setup allows to reach a maximum unambiguous range of Ru=500 m, divided into range cells of extension ΔR of about 7.5 m.Signal parameters are chosen according to the actual system constraints on the front-end filters, whose bandwidth is limited to 20 MHz. The signal intermediate frequencies (IFs) are “IF1=75 MHz” and
“IF2=125 MHz”.The waveforms are then up-converted by the photonics-based radar
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All the parameters used in the numerical simulations are summarized in Table.3-1.
Table.3-1 - Summary of the main parameters of the photonics-based radar system and transmitted SFCW signal [33]
Our aim was to simulate a number of ideal targets along the monitored range Ru to verify the influence of a multiple scatterer scenario in an ideal environment. Five targets are simulated at 25, 145, 220, 300 and 400 m, each of them with different displacement trends, the scene scheme reported in the fig. 3-8.
Fig.3-8 Range resolution scheme and targets position
Parameter S-band X-band
Intermediate Freq. IF1=75 MHz IF2=125 MHz
Carrier Freq. f0 6FMLL-IF1
2.475 GHz
25FMLL-IF2
9.875 GHz
Sampling Freq. fc 400 MHz
No. of steps Nstep 66
Frequency separation ∆f 300 kHz
Bandwidth BW ≈ 20 MHz
Step duration Tstep 50 ns
No. of SFCW periods per data record
1000
53
Fig.3-9 Typical processing for extracting the phase information from the range cells [33]
Fig.3-9 shows the processing applied to the data received. Let us respectively call r(t) and s(t) the reference and received signals, at the varying of time for each received band, down-converted at IF1 and IF2 frequencies, as shown in Fig.2-1(right)
F, without lacking generality, the S- and X-band subscripts will be omitted.
First of all, aiming at selecting the range cells of the targets of interest, we apply a cross-correlation between r(t) and s(t), which corresponds to the beat frequencies of the signal.The graph on the left in Fig.3-9 shows the cross-correlation and the five targets in their respective range distances. Parallel to this processing, for each band, r(t) and s(t) are mixed then low-pass filtered to obtain the mixed signal m(t).
54
A Discrete Fourier Transform (DFT) is applied to the data in order to obtain the frequency domain components of the signal, as depicted in the right graph in Fig.3-9. Finally, the local maxima, i.e., the peaks of the spectrum representing the five targets simulated in different range cells (colored crosses in Fig.3-9(left)) are automatically selected from the spectrum M(f) of m(t), as summarized in Fig.3-9. Considering that our goal was only to verify how the phase estimate for one target is influenced by the presence of the others, we have not considered the amplitude decrement due to the distance.
It is important to note that if the frequency sweep of the modulating waveform is positive, negative frequencies will be selected, and vice versa. Differential phase measurements are then carried out by evaluating the phase difference of each of the selected frequencies (peaks or local maxima) in both bands.
The presented algorithm is then applied to calculate ΔΦ between two consecutive measurements. The results for the numerical simulations are depicted in Fig.3-10, which shows the error curves between the true and the estimated displacements, for the five targets we have considered. In the simulations, fifteen records (sets of measurements) were generated, each composed of 1000 repetitions of the SFCW signal. The first record is taken as reference for phase calculations. These records represent the x-axis in Fig.3-10.
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As we can observe the typical precision is lower than 0.2mm for simulated distances up to 400m, but depending on the relative positions of the targets. The farthest targets show a higher displacement error. However, it is important to observe how the error curves show a certain degree of correlation (e.g. the green and cyan curves of targets 4 and 5). A possible solution could be to use a different windowing function to evaluate the DFT of the data, at the cost of worsening the range resolution.
Fig.3-10 Displacement error estimates from computer simulations of the S-band radar data with five targets [33]
We also have run real experiments in order to validate the simulations done so far. In these experiments the same radar system parameters reported in Tab. 1 have been used. We report the signal produced by the DDS for IF1=75 MHz and
56
Fig.3-11. FFT spectrum of the SFCW signal produced by DDS.
Moreover, the system displacement measurement capability has been tested with metal disk targets placed at given distances from the radar system. Both the displacements computed from the interferometric phase and the measurement errors are being evaluated and preliminary results show good agreement with the performed simulations.
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3-4: Manual phase unwrapping process
In the previous sections we described the techniques used to achieve the displacement measurement. In this section, we report the experiment setup with the signal characteristic that can be found procedure in the previous chapter, in order to show the manual phase unwrapping.
The experiment is carried out on the roof of the laboratory by installing a rectangular metal plate target on a digitally controlled motorized linear stage, as shown in fig.3-12. This is controlled with a computer and it is able to a precision of 1µm. The target is illuminated with two horn antenna, one for TX part and another for the RX part. We installed them as depicted in fig.3-13. The target is positioned in the first range cell.
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At first, the rail has been posted in the zero position of the motorized linear stage, used as reference location, for achieving the target displacements test from 0 to 9 mm by the steps of 0.9 mm and from 9 to 27 by steps of 9 mm.
Fig.3-13.The target and antenna position.
The transmitted SFCW signal spectrum is shown in Fig.3-11, with a resolution given by the signal bandwidth of 20MHz, i.e. 7.5m. The operated signal processing contains on move the two accumulated signals, i.e. the down converted echo (Fig.3-14) and the reference (fig.3-11), to the frequency domain via a Fast Fourier Transform (FFT), multiplying together the first one and the complex conjugate of the second. Then, with the pick picking techniques, we choose the pick of each 132
59
tones, for both positive and negative frequency (showed in the fig.3-15), to extract their phase, and after this taking the average from them to achieve the high accuracy differential phase for each of the frequency bands.
Fig.3-14. the FFT spectrum of the SFCW signal after IF down conversion
60
As shown in Table.3-2, the phase of a given number of test is reported. This implies, that the phase values are known modulo2π, ranging in the interval [-π, π], that is returned by the arctangent function. These are the so-called wrapped phases. Fig3-16 illustrates the concept of wrapped and unwrapped phases.
Table.3-2. Phase information before and after manually Unwrapping processing.
Since the full displacement phases are wrapped around the 2π interval, the full phase values need to be reconstructed by estimating the integer number of cycles to beadded to their wrapped version:
ΔΦunwrapped = ΔΦwrapped+ 2π ∙ k(1)
Where k is an integer number. This operation is named phase unwrapping. Test number ref mm wrapped phase Rad K unwrapped phase Rad 1 0 -2.8190000 0 -2.819000000 2 0.9 -2.8180000 1 3.4651853071 3 1.8 -2.8190000 2 9.7473706143 4 2.7 -2.8190000 3 16.030555921 5 3.6 -2.8170000 4 22.315741228 6 4.5 -2.8200000 5 28.595926535 7 5.4 -2.8190000 6 34.880111843 8 6.3 -2.8180000 7 41.164297150 9 7.2 -2.8190000 8 47.446482457 10 8.1 -2.8190000 9 53.729667764 11 9 -2.8210000 10 60.010853071 12 18 -2.8220000 11 66.293038378 13 27 -2.8260000 12 72.572223686
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The better clarify the concept of the wrapped and unwrapped phases, results of a MATLAB simulation are reported in fig.3-16.
Fig.3-16. Phase unwrapping. Profile along a hypothetical displacement with full displacement phases, ranging approximately between 35 and 60 radians (left). This profile represents the unwrapped version of the profile shown
on the right side, which contains wrapped phase values that range in the interval [-π, π]
Here we report the graphical illustration of the table.3-2 in fig.3-17, as it is show, the linearity of the chart after the unwrapping process confirms our result.
