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Abstract
In this work, we investigate the performance of an ambiguity function-shaped waveform (AFSW) using a millimeter-wave photonics-based radar system at 100 GHz. An AFSW is a radar waveform whose ambiguity function can be shaped to increase the peak-to-sidelobe ratio (PSR) for better detectability of targets in a desired range/velocity region. To the best of the authors’ knowledge, this paper is the first in the literature that investigates the performance of such a waveform in a photonics-based radar system. We experimentally compare the AFSW performance to the conventional frequency-modulated continuous wave (FMCW). The experimental results show the ability of the AFSW to achieve a PSR of 38.35 dB compared to the PSR of 14.5 dB obtained using the conventional FMCW. Moreover, we investigate the effects of some optical system impairments on the AFSW, such as: (i) optical modulator nonlinearity, (ii) optical modulator bias drift, and (iii) sampling offset error between the transmitter and receiver.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In the last decade, the usage of photonic components has been extensively investigated to address the limitations imposed by radio frequency (RF) components in radar systems [1–3]. An example is utilizing photonics for generating waveforms at high frequencies, taking advantage of their excellent performance in having large bandwidth, flat response, and immunity against electromagnetic interference [4]. Signal generation at high frequency using photonics is done by heterodyning two optical carriers in a photodiode, which produces an RF signal with a carrier frequency equal to the difference between the two optical carriers’ frequencies [5]. This is unlike the signal up-conversion using electronics which is realized through multiple stages, where each stage adds unwanted phase noise and amplitude noise due to the nonlinearities of the electronic mixers and the local oscillators. The level of such noises increases as the radio frequency gets higher [6]. In this regard, photonics can be exploited to generate radar waveforms at high frequencies, such as 100 GHz, in one stage rather than multiple stages, as is typically the case with electronics [7].
The choice of the waveform in radar systems plays a major role in enabling the system to distinguish between two closely spaced targets in either range or velocity. The most used transmit waveforms in photonics-based radars are frequency-modulated continuous wave (FMCW) [8–12] and binary-phase waveforms [13,14]. Although these waveforms may work well for certain applications, they suffer from important limitations in other ones. For example, although the FMCW waveform is one of the most popular and widely-used radar waveforms for its high performance and low complexity, the FMCW waveform suffers from high sidelobes in the range-velocity response. These high sidelobes may mask important small targets near a larger target [15].
Radar waveforms can be designed to optimize a desired performance criterion. For example, the waveform can be designed to achieve a suitable autocorrelation function (ACF) [16,17] or a suitable ambiguity function (AF) [18,19]. The AF of a waveform is a two-dimensional function of range and velocity [20]. Hence, an AFSW is a radar waveform with a shaped AF to increase the peak-to-sidelobe ratio (PSR) by suppressing the sidelobes to enhance the detectability of targets in a desired range/velocity region [21–24]. Using the AFSWs, the targets can be detected directly from the AF (the range/velocity response) of the reflected signal.
Several theoretical methods can be used for the AF waveform design. For example, in [19], the authors proposed a cognitive approach to design waveforms under the constant modulus constraint (CMC) to minimize the average value of the AF of the transmit waveform over some range-velocity bins. They devised a polynomial-time waveform optimization procedure based on the maximum block improvement (MBI) method. In [21], an iterative sequential quartic optimization algorithm was proposed to deal with the problem of shaping the AF in the presence of interferences. In [22], an adaptive sequential refinement (ASR) method was developed to increase the PSR and suppress the sidelobes in the desired region(s) in the range-velocity response. Numerical results showed that the ASR method produces a radar waveform with a high PSR and superior AF shaping. All these studies used numerical simulations to show the ability of the designed waveforms to increase the PSR and prohibit the sidelobes in the region of interest. However, to the best of the authors’ knowledge, none of these methods has been attempted in an experiment. Thus, an extremely important advance in this research field would be to experimentally evaluate the performance of such a designed AFSW in a photonics-based radar system working at high frequencies (such as 100 GHz) and investigate the effects of various photonic system impairments on the designed waveform.
In this paper, we build a 100 GHz photonics-based radar system and experimentally compare the performance of the AFSW to the conventional FMCW. We design the AFSW to increase the PSR in a region of interest. The experimental results show the ability of the designed AFSW to achieve a PSR of $38.35$ dB, which is better than conventional FMCW which can only provide a PSR of $14.5$ dB using the same photonics-based radar system. Additionally, we investigate the effects of various critical photonic system parameters on the AFSW. The investigation includes the effects of: (i) optical modulator nonlinearity, (ii) optical modulator bias drift, and (iii) sampling time-offset (mismatch) between the transmitter (Tx) and receiver (Rx).
2. Concept
2.1 Concept of the photonics-based radar
The concept of the photonics-based radar system using AFSW in our study is illustrated in Fig. 1. Here, we have three main components: (i) the optoelectronic transmitter, (ii) the targets, and (iii) the receiver. The optoelectronic transmitter comprises of an optical frequency comb generator, a waveform generator, an in-phase/quadrature (IQ) Mach-Zehnder modulator, and an optical/electrical converter. Two coherent optical carriers are selected from the optical frequency comb. The optical IQ modulator modulates one of the optical carriers. The waveform generator drives the optical modulator with either a complex AFSW or complex FMCW signals. The optical signal is then mixed with the other unmodulated carrier from the optical frequency comb for optical/electrical conversion using heterodyne beating in the photodiode to produce an RF signal at 100 GHz. The Tx antenna radiates the RF signal toward the targets. Here, we consider two scenarios for the targets: (i) the first scenario considers only one stationary target (Target 1), and (ii) the second scenario considers two targets (Target 1 and Target 2), where Target 1 is stationary while Target 2 is moving. The Rx antenna collects the reflections from the targets. The receiver downconverts the received signal and processes it to generate the range/velocity response. We analyze the system performance by measuring the PSR of the designed AFSW and compare it to that of the FMCW. Additionally, in our system, we investigate the PSR performance under the conditions of: (i) optical modulator nonlinearity; where we investigate the effect of limited spurious-free dynamic range (SFDR) in an optical MZM on the PSR, (ii) optical modulator bias drift; where we investigate the PSR penalty for each waveform under the presence of modulator bias drift which may happen due to MZM modulator thermal instability or poor bias control, and (iii) sampling time-offset; where we measure the PSR penalty induced by the sampling time-offset between the transmitter and receiver.
2.2 Design of AFSW
In this work, we design the radar waveform to increase the PSR in the region of interest in the range/velocity response by shaping its AF (i.e., range-velocity response).
Let the column vector $\mathbf {x}=[x_1,x_2,\ldots,x_N]^T\in \mathbb {C}^N$ denote the $N$ samples of the radar waveform. The samples of the range cell under test received waveform, after down-conversion to baseband, is denoted by $\mathbf {y}=[y_1,y_2,\ldots,y_N]^T\in \mathbb {C}^N$, which can be expressed as [19]:
where $\alpha _t$ denotes the complex amplitude of the target, which accounts for the channel and backscattering effects from the target within the range bin of interest, $\odot$ denotes the Hadamard product, $\nu _{t}$ is the normalized target Doppler frequency, and $\mathbf {p}(\nu _{t}) = [1, e^{j2\pi \nu _{t}},\ldots, e^{i2\pi (N-1)\nu _{t}} ]^T$. The $N$-dimensional column vectors $\mathbf {x}_{c}$ and $\mathbf {n}$ denote the interfering return samples, possibly from clutter or threatening/non-threatening targets, and noise samples, respectively. The interfering samples $\mathbf {x}_{c}$ can be expressed as:The output of filtering the received vector $\mathbf {y}$ with a filter matched to the target signature, which is $\mathbf {x}\odot \mathbf {p}(\nu _{t})$, is given by:
3. Numerical analysis
3.1 Generation of the AFSW
The following parameters are used to design the waveform: the waveform duration $T=2~\mu s$, the sampling rate $f_s=512$ MSample/s corresponding to a bandwidth of 256 MHz, the carrier frequency $f_c=100$ GHz, and the number of samples per pulse $N=1024$. For demonstration, we chose the region of interest with range and velocity intervals of $\left [-20,20\right ]$ m and $\left [-3,3\right ]$ m/s, respectively. It is worth mentioning that other regions of different intervals could also be selected. However, increasing the region of interest of the range-response will produce degradation in the velocity-response and vice versa. This is due to the fact that the total volume of the ambiguity surface of the range-velocity response of a radar waveform must remain constant [20].
3.2 Numerical analysis of the AFSW
In this work, we consider a monostatic radar system which transmits $M$ ($=N$, as in [19]) consecutive waveforms during a single coherence processing interval (CPI). This is required to estimate the velocity of a moving target [25]. At the receiver, a vector is first formed from the $M$ received waveforms such that its $m$th element is the $m$th sample of the $m$th received echo waveform ($m=1,2,\ldots,M$). Then, a bank of filters where each is matched to a specific velocity is applied to the received vector to compute the range-velocity response. Note that each waveform has $N$ samples. Therefore, the time difference between two consecutive samples of the so-formed received vector is $(N+1)/f_s$ seconds, which is a scaled version of the time difference between two consecutive samples of a single waveform ($=1/fs$ seconds). In the case of FMCW, this new time difference between the vector’s samples causes a scaling in the estimation of the range offset from the range-velocity response by a factor of ($N+1$).
Figure 2 shows the numerically generated AFs of the AFSW and FMCW with the same used parameters (bandwidth and pulse duration). The 3-D and 2-D shapes of the AF of the AFSW are, respectively, shown in Figs. 2(a) and 2(b), while the 3-D and 2-D shapes of the AF of the FMCW waveform are, respectively, shown in Figs. 2(c) and 2(d). For convenience, the region of interest is offset to be centered at ($0,0$), which corresponds to ($V$, $R$), where $V$ and $R$ are the target velocity and range, respectively. As can be seen, the AF of the AFSW has only one peak at the exact range and velocity values of the target, and the sidelobes are suppressed in the remaining parts of the region of interest. This proves that the AFSW can accurately infer the target range and velocity from the AF. In contrast, the AF of the FMCW has a tilted ridge. This ridge scans all range and velocity values and the sidelobes are not suppressed, leading to an inaccurate inference of the target range and velocity from the AF.
Figures 3(a), 3(b), and 3(c) show the range-responses of the AFSW at velocity cuts of 0 m/s, 1 m/s, and 5 m/s, respectively. Figure 3(a) shows the range-response of the AFSW at the velocity of the target. As can be seen, the range-response has a peak at the target range while the range-sidelobes have been suppressed in the region of interest with a $147$ dB PSR. Figure 3(b) shows the range-response of the AFSW at a velocity cut greater than the target velocity by 1 m/s, which is within the velocity range of interest. Similar results have been obtained where the range-sidelobes have been considerably suppressed. Figure 3(c) shows the range-response of the AFSW at a velocity cut greater than the target velocity by 5 m/s, which is outside the velocity range of interest. This cut has relatively high range-sidelobes. Therefore, perfect detection of targets can be accomplished as long as they move with velocities within the range of interest. The range-responses of the FMCW at different velocity cuts are shown in Figs. 3(d), 3(e), and 3(f). Unlike the AF of the AFSW, the AF of FMCW has a peak in the range-response for each velocity cut, leading to an inaccurate inference of the target range. The peak is slightly shifted as the velocity value increases. The PSR is about 13.4 dB. By comparing the results of Fig. 3, we can conclude that the AFSW is superior to the FMCW in terms of the PSR.
Fig. 3. Velocity (offset) cuts for AFSW (top row) and FMCW (bottom row): (a), (d) $V$ (offset) $=0$ m/s, (b), (e) $V$ (offset) $=1$ m/s, (c), (f) $V$ (offset) $=5$ m/s.
The velocity-responses at some range cuts for the AFSW and FMCW are shown in Fig. 4. The AFSW velocity-responses at range cuts of 0 m, 11 m, and 25 m are plotted in Figs. 4(a), 4(b), and 4(c), respectively. Figure 4(a) shows the velocity-response of the AFSW at the target range. As can be seen, the velocity-response has a peak at the target velocity with suppressed velocity-sidelobes in the region of interest. Figure 4(b) shows the velocity-response of the AFSW at a range cut greater than the target range by 11 m, which is within the range of the region of interest. No velocity-sidelobes are observed. Figure 4(c) shows the velocity-response of the AFSW at a range cut greater than the target range by 25 m, which is outside the region of interest. On the other hand, the velocity-responses of the FMCW at the corresponding range cuts are shown in Figs. 4(d), 4(e), and 4(f). As can be seen, there is a peak in the velocity-response for each range cut, leading to an inaccurate inference of the target velocity from the AF. The peak is shifted as the range value increases. From Fig. 4, it becomes evident that the AFSW provides a much higher PSR compared to that of the FMCW in the region of interest.
Fig. 4. Range (offset) cuts for AFSW (top row) and FMCW (bottom row): (a), (d) $R$ (offset) $=0$ m, (b), (e) $R$ (offset) $=11$ m, (c), (f) $R$ (offset) $=25$ m.
4. Experimental demonstrations
This section demonstrates a photonics-based radar system that transmits either a complex AFSW or complex FMCW with signal parameters given in Table 1. Figure 5 shows the experimental setup. The light from a laser source (LS, NKT-Photonics Koheras ADJUSTIK) is sent into two cascaded phase modulators (PM, EOspace PM-5V5-40-PFA-PFA-UV) to generate an optical frequency comb (OFC). The spacing between the frequency lines of the OFC is controlled by tuning the frequency of an RF signal generator (SG, Keysight E8257D). In this experiment, the frequency of the RF signal is set to 12.5 GHz. The OFC is amplified using a booster optical amplifier (OA, Amonics AEDFA-BO-23-B-FA). Two optical carriers (frequency lines) spaced by 100 GHz are selected from the OFC using a wavelength selective switch (WSS, Finisar 4000S). The generated OFC and the selected optical carriers are shown in Fig. 6. The AFSW is digitally generated and loaded to the arbitrary waveform generators (AWG, Keysight M8195A). Then, an IQ Mach-Zehnder modulator (IQ-MZM, Fujitsu FTM7977HQA) is used to modulate one of the selected optical carriers by the output of the AWG while the other optical carrier remains unmodulated. Both the modulated and unmodulated optical carriers are combined by an optical coupler (OC). Figure 6(b) shows the spectrum of optical carriers after modulating one of them. The output of the OC is then amplified using an OA (Amonics AEDFA-PA-40-B-FA). A tunable optical attenuator (Agilent N7764A) is used to control the input optical power of an ultra-fast 100 GHz photodetector (FINISAR XPDV4121R), which is used for heterodyne beating. The modulated and unmodulated carriers mix and convert the optical signal to an RF electrical signal at 100 GHz. The resulting electrical signal is amplified using a low-noise amplifier (LNA, QuinStar QLW-75B05030-I2) and transmitted using a horn antenna (QuinStar QGH-WPRR00).
Fig. 5. Photonics-based radar system. OFC: optical frequency comb, LS: laser source, SG: signal generator, PM: phase modulator, OA: optical amplifier, WSS: wavelength selective switch, DSP: digital signal processing, AWG: Arbitrary waveform generator, MZM: Mach-Zehender modulator, PC: polarization controller, ATT: attenuator, PD: photodiode, LNA: low noise amplifier, MXR: mixer, OSC: oscilloscope.
Fig. 6. (a) OFC and selected optical unmodulated carriers. (b) The modulated and unmodulated carrires.
Table 1. The transmitted signal parameters.
At the receiver side, a horn antenna (QuinStar QGH-WPRR00) is used. The horn antenna output signal is mixed with an RF carrier using an electronic mixer to down convert the received signal to an intermediate frequency (IF) set to 5.6 GHz. Then, a digital oscilloscope (Keysight DSOX 932048) is used to obtain the I and Q components of the received signal. Finally, digital signal processing is performed off-line to generate the range-velocity response of the received echo signal.
We consider two scenarios in conducting the experimental work. In the first scenario, one stationary target is placed at a distance of 2 meters from the radar front-end. In the second scenario, two targets are considered, where the first target is placed 2 meters from the radar front-end, and the second target is moved towards the first target with a speed of 2 m/s. The second target is initially placed at a distance of 5 meters from the first target. Both experiment scenarios are depicted in Fig. 7. The targets are metal plates that are 30 cm by 25 cm in size.
4.1 First scenario: one stationary target
Figure 8 shows the spectrum of the return signal from the target in the first scenario after down conversion to the IF frequency using the AFSW and FMCW, while Fig. 9 shows the normalized time-frequency diagrams for the corresponding baseband complex signals. Note that the time-frequency diagram is useful for analyzing FMCW signals but not very informative for AFSW signals. This is because the AFSWs are random (noise-like) signals.
Fig. 9. Normalized time-frequency diagrams of received baseband complex signal. (a) AFSW and (b) FMCW.
Figure 10 depicts the 3-D and 2-D AF shapes. Figures 10(a) and 10(b) show the AF obtained using the AFSW. The peak shows the target at 47 meters even though the true range of the target is 2 meters. This is because the optical system adds a delay corresponding to 45 meters. For comparison, Figs. 10(c) and 10(d) show the AF obtained using the FMCW. It can be seen that the AF has a tilted ridge.
Figures 11(a)-(c) present the range-responses of the AFSW at different velocity cuts. Note that in this scenario the target is stationary, hence its true velocity is zero. As can be seen from Fig. 11(a), the range-response of the zero-velocity cut shows a clear peak with a PSR of $38.35$ dB at the exact location of the stationary target ($2+45=47$ m). However, there are also other peaks of lower values. This is intuitively not surprising because the experiment has been conducted in a busy lab where there are many reflecting objects. Further, since we are experimenting with a frequency of 100 GHz, these reflections even become more pronounced. Figure 11(b) shows the range-response of the return waveform at a velocity cut greater than the target velocity by 1 m/s, which is within the velocity range of interest. This cut indicates that there are no moving targets in the region of interest with that velocity. Figure 11(c) shows the range-response of the return waveform at a velocity cut greater than the target velocity by 5 m/s, which is outside the velocity range of interest. The sidelobes in this cut are relatively high. Any moving target with that velocity cannot be easily detected. On the other hand, the corresponding velocity cuts for the FMCW are depicted in Figs. 11(d)-(f). The range-response at the 0 m/s velocity cut has a peak with a PSR of about $14.5$ dB at 47 meters, which is the target range plus the corresponding system delay. Every other velocity cut shows a peak slightly shifted in the range response. From this figure, we can observe the ability of the AFSW to show the target only at the exact range and velocity values. Moreover, the achieved PSR is about 38.35 dB compared to the 14.5 dB of the FMCW.
Fig. 11. Range responses of the AFSW (top row) and FMCW (bottom row) in the first scenario at different velocity cuts.
Figures 12(a)-(c) present the velocity-responses of the AFSW at different range cuts. Figure 12(a) shows the velocity-response of the range cut at about 47 meters, which is the range of the target plus the corresponding system delay. This cut shows a clear peak with a PSR of about 40 dB at 0 m/s which means the target is stationary. Figure 12(b) shows the velocity-response at a range cut greater than the target range by 11 m. The sidelobes are suppressed in the velocity values of interest. Figure 12(c) shows the velocity-response of the return waveform at a range cut greater than the target range by 25 m, which is outside the range of interest. For comparison, the range cuts of the FMCW are depicted in Figs. 12(d)-(f). The velocity-response at the 47 m range cut has a peak at 0 m/s with a PSR of about 13 dB. The other range cuts have peaks but at different velocity values. From this figure, we can observe the ability of the AFSW to show the target only at the exact range and velocity values. Moreover, the achieved PSR is about 40 dB compared to the 13 dB of the FMCW.
Fig. 12. Velocity responses of the AFSW (top row) and FMCW (bottom row) in the first scenario at different range cuts.
It is relevant here to mention that the noise floor in Figs. 11 and 12 can be approximately estimated from the region of interest, in case of AFSW, since the theoretical sidelobe level there is about $-147$ dB, as shown in Fig. 3(a). For the FMCW case, outside the region of interest can be considered since the theoretical sidelobe level there is about $-60$ dB, as shown in Fig. 3(d). Note that the region of interest under consideration extends from $-20$ to $20$ m in range and from $-3$ to $3$ m/s in velocity. From Fig. 11(a), the noise floor in the region of interest is about $-45$ dB below the response’s peak value; see the red dashed line. Similarly, from Fig. 11(d), the noise floor outside the region of interest, as indicated by the red dashed line, is about $-45$ dB below the response’s peak value. Similar results can be obtained from Figs. 12(a), 12(d).
4.2 Second scenario: two targets
Now, we consider the second scenario in which the first target is stationary at a distance of 2 m from the radar front-end, while the second target is moving at 2 m/s and initially located at 5 m from the first one. Due to the limitations in our lab, it is not feasible to move the second target at that velocity. However, we emulated the movement of the second target by first getting its reflection and then adding the velocity effect to the received signal through digital signal processing [25]. Then, we added both the reflected signal from the stationary target and the reflected signal from the second target after adding the velocity effect. The result is depicted in Fig. 13. As can be seen from Figs. 13(a) and 13(b), the two peaks corresponding to the two targets are clearly seen in the range-velocity response. The two peaks are at the ranges 47 m and 52 m which are the targets’ ranges plus the corresponding system delay. Therefore, the distance between the two targets is exactly 5 m, which is the actual distance. For comparison, Figs. 13(c) and 13(d) depict the range-velocity response of the FMCW of the targets in the second scenario. The range-velocity response has two narrow tilted ridges. That means for each velocity or range cut, there will be two dominant peaks, leading to an inaccurate inference of the range/velocity of the targets from the AF.
Fig. 13. Range-velocity response of the AFSW (top row) and FMCW (bottom row) in the second scenario.
The range-responses of the AFSW at 0 m/s, 1 m/s, and 2 m/s velocity cuts are shown in Figs. 14(a)-(c). As shown from Fig. 14(a), the range-response has only one peak with a PSR of about 37.9 dB at range 47 m, which is corresponding to the stationary target plus the corresponding system delay. Note that there are no unwanted returns from the second target. Figure 14(b) shows that there are no moving targets in the region of interest with velocity of 1 m/s. Thanks to the AFSW, there are no unwanted returns from both targets at that cut. Figure 14(c) shows that there is a moving target at the range of 52 m, which is corresponding to the moving target plus the system delay, and with a velocity of 2 m/s. Again, there are no unwanted returns from the first (stationary) target. For comparison, the range-responses of the FMCW at 0 m/s, 1 m/s, and 2 m/s velocity cuts are shown in Figs. 14(d)-(f). The 0 m/s velocity cuts are depicted in Fig. 14(d). In this cut, there is a peak with a PSR of about 14.5 dB at 47 m for the stationary target and another lower peak at 50 m caused by the sidelobes of the moving target. Note that the moving target is located at 52 m. The 1 m/s velocity cuts are depicted in Fig. 14(e). In this cut, there is a peak at 48 m resulting from the stationary target sidelobes. The other lower peak at 51 m resulted from the sidelobes of the moving target. The 2 m/s velocity cuts are depicted in Fig. 14(f). In this cut, there is a peak at 48.3 m resulting from the stationary target sidelobes. The other lower peak at 51.8 m resulted from the moving target. Note that the moving target range is slightly shifted from 52m to 51.8 m due to the range-Doppler coupling in the FMCW. From Fig. 14, the superiority of the AFSW is clearly observed where the ratio of the peak to the next unwanted peak in the region of interest of the ACF in Fig. 14(a) and Fig. 14(c) is much higher (better) than that of Fig. 14(d) and Fig. 14(f), respectively. Moreover, the AFSW has the ability to show the targets only at their exact ranges and velocities.
Fig. 14. Range responses of the AFSW (top row) and FMCW (bottom row) in the second scenario at different velocity cuts.
The velocity-responses at some range cuts for the AFSW and FMCW are shown in Fig. 15. The velocity-response of the AFSW at the 47 m range cut is shown in Fig. 15(a). There is only one peak at 0 m/s, which indicates that the target is stationary. Figure 15(b) shows that there is no target at the range 49 m within the range of the velocities of interest. Finally, Fig. 15(c) shows that there is a moving target at the range of 52 m and with a velocity of 2 m/s. Similarly, the range cuts of the FMCW are shown in Figs. 15(d)-(f). There are two peaks in all cuts as expected. From Fig. 15, the superiority of the AFSW is clearly observed where the AFSW succeeded in showing the peak of the target at the corresponding true range and velocity values. Moreover, the AFSW provided higher PSR in the region of interest compared to the FMCW.
Fig. 15. Velocity responses of the AFSW (top row) and FMCW (bottom row) in the second scenario at different range cuts.
To sum up, the experiment’s results reveal that the AF cuts of the AFSW show the target only at the corresponding true range/velocity values. Furthermore, the AFSW has a relatively better performance in terms of sidelobe levels and main lobe sharpness compared to the FMCW. Although the experimental results of the AFSW are satisfactory, there is a relatively high deterioration in the performance of the AFSW when compared to the theoretical analysis. Such deterioration is expected due to the various impairments imposed on the AFSW through the radar system. In the next section, we will investigate the impact of some impairments of the photonics-based radar system on the AFSW.
5. Effect of optical system impairments
In this section, we investigate some effects of the optical system impairments on the PSR performance of the AFSW. The investigation includes the effects of: (i) optical modulator nonlinearity, (ii) optical modulator bias drift, and (iii) sampling time-offset between the Tx and Rx. To simulate these impairments, we use VPIphotonics software [26] to build the simple setup shown in Fig. 16.
5.1 IQ modulator nonlinearity effect
First, we investigate the effects of the MZM nonlinearity on the PSR of the designed AFSW. We bias the IQ modulator at the quadrature point, which is in the middle of the most linear region of the modulator power transfer function. As the peak-to-peak voltage ($V_{pp}$) of the generated waveform increases, the effect of the nonlinearity increases. First, we show the theoretical performance of the AFSW and FMCW under perfect conditions (no distortion); see Figs. 17(a) and 17(c). The superiority of the AFSW can be noted since it can provide a PSR of 147 dB compared to the FMCW which can only provide a PSR of 13.47 dB. Next, we vary the $V_{pp}$ of the input waveform in steps. Figures 17(b1)-(b3) and Figs. 17(d1)-(d3) show, respectively, the ACF of the AFSW and FMCW when $V_{pp}=0.1V_{\pi }$, $V_{pp}=0.5V_{\pi }$, and $V_{pp}=V_{\pi }$. The PSR of AFSW when $V_{pp}=0.1V_{\pi }$ is reduced to 49.2 dB, while the PSR of FMCW is reduced to 13.4 dB. When the $V_{pp}$ increases to $1V_{\pi }$, the PSR of AFSW is reduced to 32.8 dB, while the PSR of FMCW is reduced to 13.1 dB.
Fig. 17. Effect of the optical modulator nonlinearity on the PSR of AFSW (top row) and FMCW (bottom row) at different $V_{pp}$ points. (a), (c) Theoretical PSR, (b1), (d1) $V_{pp}=0.1V_{\pi }$ (b2), (d2) $V_{pp}=0.5V_{\pi }$, and (b3), (d3) $V_{pp}=V_{\pi }$.
Figure 18 summarizes the PSR of both AFSW and FMCW as the normalized $V_{pp}$ increases from $0.1$ to $1$. The theoretical PSRs are also depicted. We can notice that the AFSW suffers a higher penalty than the FMCW. Despite that, the PSR curve of the AFSW still maintains higher values compared to the PSR curve of the FMCW, which indicates the superiority of the AFSW over FMCW under this scenario.
5.2 Optical IQ-modulators bias drift
For optimum operation of the I/Q modulator, both of the I/Q MZMs should be biased at the quadrature points. However, in real applications, the bias point may change over time [27]. The effect of bias drift on the AFSW and FMCW is shown in Fig. 19. First, the ACFs of both the AFSW and FMCW are depicted in Figs. 19(a) and Fig. 19(e) when there is no bias drift. In this case, the PSR of the AFSW is 51.6 dB (for $V_{pp}=0.06V_{\pi }$) and the PSR of the FMCW is 13.4 dB. The figure also shows the ACFs of both the AFSW and FMCW when we increased the bias drift to take the values $0.15V_{\pi }$, $0.35V_{\pi }$, and $0.5V_{\pi }$. The corresponding PSRs of the AFSW are 35.2 dB, 27.2 dB, and 22.2 dB, respectively. The corresponding PSRs of the FMCW are 13.2 dB, 12.9 dB, and 12.6 dB, respectively. As can be seen, the PSR decreases as the bias drift increases.
Fig. 19. Effect of DC-bias drift on the PSR performance of AFSW (top row) and FMCW (bottom row). (a), (e) $0 V_{\pi }$ drift, (b), (f) $0.15 V_{\pi }$ drift, (c), (g) $0.35 V_{\pi }$ drift, and (d), (h) $0.5 V_{\pi }$ drift.
Figure 20 summarizes the PSR for both the AFSW and FMCW as the normalized bias drift increases from $0$ to $0.5$. As can be seen, a change in the bias does not cause any significant degradation in the PSR of the FMCW, as it has a value of 12.6 dB at minimum. On the other hand, the AFSW has superior performance at the quadrature point with about 50 dB PSR. However, it is more sensitive to the change in the bias; a $0.1V_{\pi }$ drift in the bias results in a penalty of approximately 14 dB degradation in the PSR. It should be noted that although the PSR performance of the AFSW deteriorates faster than that of the FMCW as the bias drift increases, it never goes below the PSR of FMCW even at a high bias drift, such as $0.5V_{\pi }$. At this value of the bias drift, the difference in PSR between AFSW and FMCW is $9.6$ dB.
5.3 Sampling time-offset
Usually, the waveforms are oversampled (upsampled) before the conversion from the digital domain to the analog domain using a digital-to-analog converter (DAC). This is needed to facilitate the function of the antialiasing filter in the analog-to-digital converter (ADC) at the receiver side [28]. Since the AFSW has a non-constant envelope, a sampling time-offset (mismatch) between the original waveform samples and the receiver samples can significantly degrade the performance of the AFSW. To investigate this effect, we simulated the upsampling process at the waveform generator to produce the analog signal and then performed downsampling on the receiver side with induced time-offset during the downsampling process. The ACF is determined by correlating the downsampled waveform with the original AFSW.
Figure 21 illustrates the impact of sampling time-offset on the PSR of AFSW and FMCW when the sampling time-offset values (relative to the sampling period $T_s$) are set to $0$, $0.1$, $0.2$, and $0.3$. As can be seen, the PSR of the AFSW decreases as the sampling time-offset increases, while the PSR of the FMCW barely changes. The main lobe regrowth of the ACF of the AFSW and FMCW can also be observed under this effect and is more pronounced in the case of AFSW.
Fig. 21. Effect of sampling on the PSR of AFSW (top row) and FMCW (bottom row) at different time-offset values. (a), (e) No sampling time-offset, (b), (f) sampling time-offset = $0.1T_{s}$ (c), (g) sampling time-offset = $0.2T_{s}$, and (d), (h) sampling time-offset = $0.3T_{s}$.
Figure 22 summarizes the PSR performance of the AFSW and FMCW as the normalized sampling time-offset increases from 0 to 1. Initially, the PSR of the AFSW drops rapidly due to a small increase in the sampling time-offset (mismatch). The PSR of the AFSW reaches its minimum at the normalized sampling time-offset of 0.5. At this time-offset, the PSR is about 62 dB. As the sampling time-offset increases, the sampling instant becomes closer to the instant of next original sample, which interprets the improvement in the PSR. At time-shift $1T_s$, the sampling instant coincides with the time of original signal sample. The PSR then returns to its maximum value (147 dB). On the other hand, FMCW shows immunity to the sampling time-offset with a PSR of about 13.5 dB. Despite the higher sensitivity of the AFSW to the sampling time-offset, it provides a higher PSR than that of the FMCW.
6. Conclusion
In this study, the performance of an AFSW is investigated in a photonics-based radar system at 100 GHz. First, the AFSW is designed to increase the PSR by suppressing the sidelobes/unwanted returns in a region of interest. Then, the performance of the designed waveform is compared to the conventional FMCW in a photonics-based radar system. Unlike the range/velocity response of the FMCW, the range/velocity response of the AFSW shows the targets at the true range and velocity values. Furthermore, the experimental results show the ability of the designed AFSW to achieve PSR of $38.35$ dB using the photonics-based radar system compared to the PSR of $14.5$ dB obtained using the conventional FMCW. The impacts of optical system impairments on the AFSW have also been investigated, such as the IQ modulator nonlinearity, bias drift, and sampling time-offset (mismatch). Although the performance of the AFSW degrades due to the optical system impairments, it still provides a much higher PSR compared to the conventional FMCW in the region of interest. The performance of the AFSW could be improved if the effects of these impairments and other system impairments are taken into consideration during the waveform design process.
A. Appendix: The energy-constrained adaptive sequential refinement algorithm (EC-ASR)
Here, we first formulate the optimization problem for designing an AFSW and then provide its iterative solution using the EC-ASR algorithm. Without loss of generality, the Doppler frequencies are assumed to be centered to the target Doppler frequency $\nu _{t}$. The normalized Doppler interval $[-\frac {1}{2}, \frac {1}{2}]$ is divided into $N_\nu$ bins. Therefore, the normalized frequency can be represented as: $\nu _h = -\frac {1}{2} + \frac {h}{N_\nu },\ h = 0 ,\ldots, N_\nu -1$. Let $\mathcal {B}_k$ denote the set containing the Doppler bin indices of interest for the $k^{th}$ scatterer. Then, since our design criterion is to minimize the interference power at the output of the matched filter, it can be shown that the objective function can be written as [22]:
Algorithm 1. EC-ASR algorithm.
The problem of waveform design usually undergoes the constant modulus constraint. That is,
The solution for this optimization problem starts by first simplifying the form of $f(\mathbf {x})$. It has been shown in [22] that $f(\mathbf {x})$ defined in (5) can be rewritten to take the form:
The solution for (10) is outlined in Algorithm 1, and is termed the energy-constraint ASR (EC-ASR) algorithm. This algorithm is developed in a similar manner to the development of ASR algorithm based on CMC, which is detailed in [22]. Note that the development of the EC-ASR algorithm is based on converting the quadratic constraint to a linear constraint as in step 18 of Algorithm 1, which can be shown to iteratively converge to a single solution.
Funding
National Plan for Science, Technology and Innovation (MAARIFAH), King Abdulaziz City for Science and Technology, Kingdom of Saudi Arabia, Award Number (3-17-09-001-0012).
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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