Compressive spectrum sensing of radar pulses based on photonic techniques Qiang Guo, Yunhua Liang, Minghua Chen,* Hongwei Chen, and Shizhong Xie Tsinghua National Laboratory for Information Science and Technology (TNList), Department of Electronic Engineering, Tsinghua University, Beijing, 100084, China * [email protected]
Abstract: We present a photonic-assisted compressive sampling (CS) system which can acquire about 106 radar pulses per second spanning from 500 MHz to 5 GHz with a 520-MHz analog-to-digital converter (ADC). A rectangular pulse, a linear frequency modulated (LFM) pulse and a pulse stream is respectively reconstructed faithfully through this system with a sliding window-based recovery algorithm, demonstrating the feasibility of the proposed photonic-assisted CS system in spectral estimation for radar pulses. ©2015 Optical Society of America OCIS codes: (060.2360) Fiber optics links and subsystems; (230.0250) Optoelectronics; (280.5600) Radar; (060.5625) Radio frequency photonics.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006). E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006). M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008). J. H. G. Ender, “On compressive sensing applied to radar,” Signal Process. 90(5), 1402–1414 (2010). J. L. Paredes, G. R. Arce, and Z. Wang, “Ultra-wideband compressed sensing: Channel estimation,” IEEE J. Sel. Top. Signal Proc. 1(3), 383–395 (2007). J. Haupt, W. U. Bajwa, M. Rabbat, and R. Nowak, “Compressed sensing for networked data,” IEEE Signal Process. Mag. 25(2), 92–101 (2008). M. Mishali and Y. C. Eldar, “From theory to practice: Sub-nyquist sampling of sparse wideband analog signals,” IEEE J. Sel. Top. Signal Proc. 4(2), 375–391 (2010). J. Yoo, S. Becker, M. Loh, M. Monge, E. Candès, and A. Emami-Neyestanak, “A 100MHz-2GHz 12.5x subNyquist rate receiver in 90 nm CMOS,” in Proc. IEEE Radio Freq. Integr. Circ. Conf, 31–34 (2012). J. Yoo, C. Turnes, E. Nakamura, C. Le, S. Becker, E. Sovero, M. Wakin, M. Grant, J. Romberg, A. EmamiNeyestanak, and E. Candès, “A compressed sensing parameter extraction platform for radar pulse signal acquisition,” IEEE J. Emerg. Sel. Top. Circ. Syst. 2(3), 626–638 (2012). J. M. Nichols and F. Bucholtz, “Beating Nyquist with light: A compressively sampled photonic link,” Opt. Express 19(8), 7339–7348 (2011). T. P. McKenna, M. D. Sharp, D. G. Lucarelli, J. A. Nanzer, M. L. Dennis, and T. R. Clark, Jr., “Wideband photonic compressive sampling analog-to-digital converter for RF spectrum estimation,” in Proceedings of Optical Fiber Communication Conference, OFC/NFOEC (Anaheim, Calif., 2013), paper OTh3D.1. B. T. Bosworth and M. A. Foster, “High-speed ultrawideband photonically enabled compressed sensing of sparse radio frequency signals,” Opt. Lett. 38(22), 4892–4895 (2013). Y. Liang, M. Chen, H. Chen, C. Lei, P. Li, and S. Xie, “Photonic-assisted multi-channel compressive sampling based on effective time delay pattern,” Opt. Express 21(22), 25700–25707 (2013). P. D. Welch, “The use of fast Fourier transform for the estimation of power spectra: a method based on timeaveraging over short, modified periodograms,” IEEE Trans. Acoust. Speech 15, 70–73 (1967). J. Tropp and A. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53(12), 4655–4666 (2007). V. M. Patel, G. R. Easley, D. M. Healy, Jr., and R. Chellappa, “Compressed synthetic aperture radar,” IEEE J. Sel. Topics Signal Proc. 4(2), 244–254 (2010).
1. Introduction Compressive sampling (CS) is a novel sensing paradigm that samples sparse signals at their information rate which is orders of magnitude lower than the Nyquist rate [1,2]. As natural
#228565 - $15.00 USD (C) 2015 OSA
Received 27 Nov 2014; revised 27 Jan 2015; accepted 30 Jan 2015; published 12 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.004517 | OPTICS EXPRESS 4517
signals are sparse or compressible in a certain domain, CS can be successfully applied in diverse fields such as imaging, radar, communication and networks [3–6]. In radar applications, the received radar signals need to be down-converted by their carrier frequency for the bandwidth limit of the analog-to-digital converters (ADCs). On the other hand, modern radar systems operate with large bandwidths and an increasing number of channels produce a huge amount of data which is quite difficult to acquire, store and process [4]. CS provides a promising solution that the received radio frequency (RF) signals can be reconstructed from a reduced number of samples without down-conversion [7]. In [8], a fully integrated wideband receiver called the random modulation pre-integrator (RMPI) with an effective instantaneous bandwidth (EIBW) of 2GHz and a sub-Nyquist rate of 320 MSa/s is reported. In [9], a RMPIbased compressed sensing parameter extraction platform with an instantaneous bandwidth of 2.5 GHz is proposed for radar pulse acquisition. However, high frequency signals could not be sensed by these CS systems due to the bandwidth limit of the electronic devices. For this reason, the photonic-assisted CS system which can offer a very large instantaneous bandwidth is an attractive scheme for spectral estimation. A compressively sampled photonic link is first demonstrated to successfully reconstruct 1 GHz harmonic signals while sampling at only 500 MS/s in [10]. In [11], a photonic CS ADC is proposed to sample multi-band RF signals. A new architecture for high-speed compressed sensing using chirp processing with ultrafast laser pulses is reported in [12], exactly reconstructing multi-tone signals spanning from 900 MHz to 14.76 GHz. In [13], a photonic-assisted multi-channel CS scheme is proposed for spectral estimation. The aforementioned photonic-assisted CS systems have been demonstrated to successfully sense multi-tone or multi-band RF signals. Motivated by the recent work [10–13], the photonic-assisted CS scheme could be developed to sense the spectrum of radar pulse streams which is very prevalent in electronic reconnaissance or electronic warfare. In this paper, a photonic-assisted CS system is proposed to acquire radar pulse streams for the first time. An IQ modulator and a photodetector are utilized to perform the spectral compression of the radar pulses. As the received radar pulses may not be spectrally sparse during a long acquisition time, a sliding window-based recovery algorithm is proposed to guarantee the signal sparsity within a time window and realize a continuous signal reconstruction. In the experiment, about 100 radar pulses ranging from 500 MHz to 5 GHz are acquired within 100 μs and exactly recovered with a 520-MHz ADC, experimentally demonstrating the feasibility of the proposed system in spectral estimation for radar pulses. 2. Principle and experimental details The architecture of the proposed photonic-assisted CS system is shown in Fig. 1. A continuous-wave (CW) laser (100-KHz linewidth) emits an output power of 15 dBm at 1550 nm. A 127-bit pseudo-random binary sequence (PRBS) is generated by a pulse pattern generator (PPG) whose external clock is set to 10.16 GHz corresponding to a repetition rate of 80 MHz denoted as fp. The input radar pulse stream is generated by an arbitrary waveform generator (AWG) with a 50-GS/s sampling rate and 10-bit vertical resolution. The pulses may be emitted from radars in diverse places, so the time of arrival (TOA) of these radar pulses are different at the receiver as shown in Fig. 1. As the bit rate of the PRBS signal should be at least twice the Nyquist rate of the input radar signal, the bandwidth of the radar pulse stream is limited to about 5 GHz. In the experiment, a radar pulse stream (about 100 pulses) spanning from 500 MHz to 5 GHz is acquired by this system within 100 μs. Then an IQ modulator with a 3-dB bandwidth of 12.5 GHz and a half-wave voltage (Vpi) of 3.5 V is used to convert the radar pulse stream and the PRBS signal from the electrical domain to the optical form, respectively. Following the IQ modulator, a 10-GHz photodetector (8-mA photocurrent output) performs spectrum convolution between the input radar pulse stream and the PRBS signal. The spectrum of the mixing signal is a linear combination of fp-shifted copies of the input RF spectrum. All the frequency components of the acquired radar pulses are transferred to the low-frequency band and then a low pass filter can be used to extract the information of the spectrally compressed signal. The output signal is then sampled by an off-the-shelf
#228565 - $15.00 USD (C) 2015 OSA
Received 27 Nov 2014; revised 27 Jan 2015; accepted 30 Jan 2015; published 12 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.004517 | OPTICS EXPRESS 4518
electrical ADC with a 520-MHz bandwidth and an effective number of bits (ENOB) of 10 bits corresponding to a compression factor of 0.12. Finally, the obtained measurements are post processed offline by a digital signal processor (DSP).
Fig. 1. The Experimental setup of the proposed photonic-assisted compressive sampling system. CW: continuous wave, PC: polarization controller, PPG: pulse pattern generator, AWG: arbitrary waveform generator, PRBS: pseudorandom binary sequence, PD: photodetector, LPF: low pass filter, LNA: low noise amplifier, ADC: analog-to-digital converter, DSP: digital signal processor.
However, the acquired radar pulse stream may not be frequency-sparse within 100 μs, so we cannot reconstruct these radar pulses from the measurements by compressive sampling directly. To solve this problem, the concept of local Fourier sparsity is introduced. In Fig. 2, the temporal waveform and spectrogram of a chirp signal are presented and it is obvious that the chirp signal is neither time-sparse nor frequency-sparse. If the chirp signal is segmented into several pieces, each piece would present sparsity in the frequency domain as shown in Fig. 2. Similarly, the number of the acquired radar pulses will be decreased during a short acquisition time which may lead to low frequency occupancy over a wide spectrum. Therefore, a time window should be introduced to guarantee the local Fourier sparsity of the acquired radar pulses. As shown in Fig. 1, the spectrally compressed signal is segmented into several pieces by time windows and two adjacent time windows have a considerable overlap. By properly selecting the length of the time window, the sparsity of the windowed signal can be guaranteed. In addition, the time window will cause information loss which is more serious along the window edges than in the middle. To average out the errors resulting from the reduced information, overlapping time windows are proposed and a classical windowing method [14] is introduced for the continuous signal reconstruction. Moreover, as shown in Fig. 2, if the window length is extended, frequency occupancy will be increased which makes it more difficult to realize a robust reconstruction. On the contrary, a short time window will degrade the spectral resolution. In addition, the amount of overlap between windows relates to the computational complexity of the recovery algorithm. A large overlap will badly affect the speed of the recovery while a small overlap will aggravate the information loss. Therefore, a trade-off between the speed and accuracy of the reconstruction needs to be taken into consideration. With the sliding time window, a continuous signal reconstruction is realized.
#228565 - $15.00 USD (C) 2015 OSA
Received 27 Nov 2014; revised 27 Jan 2015; accepted 30 Jan 2015; published 12 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.004517 | OPTICS EXPRESS 4519
Fig. 2. The principle of local Fourier sparsity.
In the experiment, the length of a time window is set to 5 μs and the amount of overlap between windows is 2 μs. The acquired radar pulses in a time window is recovered by adopting the orthogonal matching pursuit (OMP) algorithm [15]. To evaluate the performance of the proposed photonic-assisted CS system, a rectangular pulse, a linear frequency modulated (LFM) pulse and a radar pulse stream is respectively tested through this system. 3. Experimental Results and discussion A rectangular pulse is first tested to evaluate the performance of the photonic-assisted CS system. The carrier frequency and duration of this pulse is 3.21 GHz and 2 μs, respectively. Figure 3(a) shows the power spectral density of the rectangular pulse and the normalized power spectral density of the recovered pulse is depicted in Fig. 3(b). The carrier frequency of the recovered pulse is 3.2101 GHz with a recovery error of 100 KHz and the recovered power spectrum matches well with the original one in the frequency range from 3.206 GHz to 3.216 GHz. The temporal waveform of the recovered pulse with a rectangular envelope and a pulse width of 2 μs shown in the inset of Fig. 3(b) indicates a successful recovery.
Fig. 3. Experimental results for the recovery of a rectangular pulse and a LFM pulse. (a) Original power spectral density of the rectangular pulse. (b) Recovered power spectral density of the rectangular pulse. Inset: temporal waveform of the recovered pulse. (c) Original power spectral density of the LFM pulse. (d) Recovered power spectral density of the LFM pulse. Inset: temporal waveform of the recovered LFM pulse.
In radar systems, a long-duration radar pulse contributes to improving its energy and target detection capability while it would also degrade the range resolution [16]. Hence LFM pulses are widely used in radar systems owing to their unique properties. In the experiment, a #228565 - $15.00 USD (C) 2015 OSA
Received 27 Nov 2014; revised 27 Jan 2015; accepted 30 Jan 2015; published 12 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.004517 | OPTICS EXPRESS 4520
LFM pulse with a 3.22-GHz center frequency, 10-MHz bandwidth and 1-μs pulse width as shown in Fig. 3(c) is compressively sampled by the proposed system. The normalized power spectral density of the recovered pulse is shown in Fig. 3(d). The recovered carrier frequency is 3.22012 GHz and the reconstructed power spectrum keeps consistent with the original one in the frequency range 3.21-3.23 GHz. Mixed with a local oscillator signal at 3.21 GHz, the recovered LFM pulse is then down-converted. The corresponding temporal waveform of the down-converted LFM pulse is depicted in Fig. 3(d). As the pulse width is 1 μs and the carrier frequency of the pulse after down conversion is 10 MHz, the down-converted LFM pulse contains 10 periods which agrees with the description in Fig. 3(d). A successful recovery of a radar pulse stream is presented in Fig. 4. The waterfall diagram of this radar pulse stream is shown in Fig. 4(a) where the horizontal axis is frequency and the vertical axis is time. In this figure, it can be observed that the radar pulses within a time window are frequency-sparse. A time-frequency representation (TFR) of the recovered pulse stream is depicted in Fig. 4(b). Due to the nonlinearity existing in the photonic link, the undesired high-order harmonics will emerge and aggravate the signal to noise ratio (SNR). Comparing Fig. 4(a) with (b), we can conclude that the recovery of the acquired radar pulse stream is successful and the spurious signals induced by modulation nonlinearity are weak enough relative to the recovered radar pulses.
Fig. 4. Experimental results for recovery of a radar pulse stream. (a) A waterfall diagram of the original pulse stream. (b) A waterfall diagram from CS reconstruction with sampling rate equal to 10.4% of the Nyquist rate.
Signal reconstruction is performed with the sliding window-based recovery algorithm. Figure 5(a) shows the power spectral density of the acquired radar pulses in a time window and Fig. 5(c) presents a waterfall diagram of the windowed signal. The corresponding recovery results are shown in Fig. 5(b) and (d), respectively. Making a Comparison between
#228565 - $15.00 USD (C) 2015 OSA
Received 27 Nov 2014; revised 27 Jan 2015; accepted 30 Jan 2015; published 12 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.004517 | OPTICS EXPRESS 4521
two waterfall diagrams, we can find that all the frequency components of the windowed signal are exactly recovered and some spurious signals can be eliminated by increasing the decision threshold level. However, this increase will change the shape of the signal spectrum which has a negative effect on signal reconstruction. Therefore selecting a proper decision threshold level is quite important. In general, the results in Fig. 5 present an exact recovery.
Fig. 5. Recovery results in a time window. (a) The original power spectral density. (b) The recovered power spectral density. (c) A waterfall diagram of the original signal within a time window. (d) A waterfall diagram of the recovered signal within a time window.
4. Conclusions The proposed photonic-assisted CS system has been demonstrated its capability of exactly recovering radar pulses with a sub-Nyquist ADC in this paper. In the experiment, a random measurement matrix is generated to compress the spectrum of the input radar signal. An IQ modulator and a photodetector are utilized to perform the mixing of the radar pulses and the PRBS signal. A microsecond rectangular pulse and a LFM pulse are spectrally estimated respectively and the performance of the proposed CS system is also analyzed. By using the sliding window-based algorithm, a radar pulse stream ranging from 0.5 GHz to 5 GHz is exactly recovered with a 520-MHz ADC, indicating that the photonic-assisted CS scheme is promising in spectral estimation for radar pulses. Acknowledgments This work is supported by National Program on Key Basic Research Project (973) under Contract 2012CB315703, NSFC under Contract 61271134 and 61120106001.
#228565 - $15.00 USD (C) 2015 OSA
Received 27 Nov 2014; revised 27 Jan 2015; accepted 30 Jan 2015; published 12 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.004517 | OPTICS EXPRESS 4522
Abstract: We present a photonic-assisted compressive sampling (CS) system which can acquire about 106 radar pulses per second spanning from 500 MHz to 5 GHz with a 520-MHz analog-to-digital converter (ADC). A rectangular pulse, a linear frequency modulated (LFM) pulse and a pulse stream is respectively reconstructed faithfully through this system with a sliding window-based recovery algorithm, demonstrating the feasibility of the proposed photonic-assisted CS system in spectral estimation for radar pulses. ©2015 Optical Society of America OCIS codes: (060.2360) Fiber optics links and subsystems; (230.0250) Optoelectronics; (280.5600) Radar; (060.5625) Radio frequency photonics.
References and links 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006). E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006). M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25(2), 83–91 (2008). J. H. G. Ender, “On compressive sensing applied to radar,” Signal Process. 90(5), 1402–1414 (2010). J. L. Paredes, G. R. Arce, and Z. Wang, “Ultra-wideband compressed sensing: Channel estimation,” IEEE J. Sel. Top. Signal Proc. 1(3), 383–395 (2007). J. Haupt, W. U. Bajwa, M. Rabbat, and R. Nowak, “Compressed sensing for networked data,” IEEE Signal Process. Mag. 25(2), 92–101 (2008). M. Mishali and Y. C. Eldar, “From theory to practice: Sub-nyquist sampling of sparse wideband analog signals,” IEEE J. Sel. Top. Signal Proc. 4(2), 375–391 (2010). J. Yoo, S. Becker, M. Loh, M. Monge, E. Candès, and A. Emami-Neyestanak, “A 100MHz-2GHz 12.5x subNyquist rate receiver in 90 nm CMOS,” in Proc. IEEE Radio Freq. Integr. Circ. Conf, 31–34 (2012). J. Yoo, C. Turnes, E. Nakamura, C. Le, S. Becker, E. Sovero, M. Wakin, M. Grant, J. Romberg, A. EmamiNeyestanak, and E. Candès, “A compressed sensing parameter extraction platform for radar pulse signal acquisition,” IEEE J. Emerg. Sel. Top. Circ. Syst. 2(3), 626–638 (2012). J. M. Nichols and F. Bucholtz, “Beating Nyquist with light: A compressively sampled photonic link,” Opt. Express 19(8), 7339–7348 (2011). T. P. McKenna, M. D. Sharp, D. G. Lucarelli, J. A. Nanzer, M. L. Dennis, and T. R. Clark, Jr., “Wideband photonic compressive sampling analog-to-digital converter for RF spectrum estimation,” in Proceedings of Optical Fiber Communication Conference, OFC/NFOEC (Anaheim, Calif., 2013), paper OTh3D.1. B. T. Bosworth and M. A. Foster, “High-speed ultrawideband photonically enabled compressed sensing of sparse radio frequency signals,” Opt. Lett. 38(22), 4892–4895 (2013). Y. Liang, M. Chen, H. Chen, C. Lei, P. Li, and S. Xie, “Photonic-assisted multi-channel compressive sampling based on effective time delay pattern,” Opt. Express 21(22), 25700–25707 (2013). P. D. Welch, “The use of fast Fourier transform for the estimation of power spectra: a method based on timeaveraging over short, modified periodograms,” IEEE Trans. Acoust. Speech 15, 70–73 (1967). J. Tropp and A. Gilbert, “Signal recovery from random measurements via orthogonal matching pursuit,” IEEE Trans. Inf. Theory 53(12), 4655–4666 (2007). V. M. Patel, G. R. Easley, D. M. Healy, Jr., and R. Chellappa, “Compressed synthetic aperture radar,” IEEE J. Sel. Topics Signal Proc. 4(2), 244–254 (2010).
1. Introduction Compressive sampling (CS) is a novel sensing paradigm that samples sparse signals at their information rate which is orders of magnitude lower than the Nyquist rate [1,2]. As natural
#228565 - $15.00 USD (C) 2015 OSA
Received 27 Nov 2014; revised 27 Jan 2015; accepted 30 Jan 2015; published 12 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.004517 | OPTICS EXPRESS 4517
signals are sparse or compressible in a certain domain, CS can be successfully applied in diverse fields such as imaging, radar, communication and networks [3–6]. In radar applications, the received radar signals need to be down-converted by their carrier frequency for the bandwidth limit of the analog-to-digital converters (ADCs). On the other hand, modern radar systems operate with large bandwidths and an increasing number of channels produce a huge amount of data which is quite difficult to acquire, store and process [4]. CS provides a promising solution that the received radio frequency (RF) signals can be reconstructed from a reduced number of samples without down-conversion [7]. In [8], a fully integrated wideband receiver called the random modulation pre-integrator (RMPI) with an effective instantaneous bandwidth (EIBW) of 2GHz and a sub-Nyquist rate of 320 MSa/s is reported. In [9], a RMPIbased compressed sensing parameter extraction platform with an instantaneous bandwidth of 2.5 GHz is proposed for radar pulse acquisition. However, high frequency signals could not be sensed by these CS systems due to the bandwidth limit of the electronic devices. For this reason, the photonic-assisted CS system which can offer a very large instantaneous bandwidth is an attractive scheme for spectral estimation. A compressively sampled photonic link is first demonstrated to successfully reconstruct 1 GHz harmonic signals while sampling at only 500 MS/s in [10]. In [11], a photonic CS ADC is proposed to sample multi-band RF signals. A new architecture for high-speed compressed sensing using chirp processing with ultrafast laser pulses is reported in [12], exactly reconstructing multi-tone signals spanning from 900 MHz to 14.76 GHz. In [13], a photonic-assisted multi-channel CS scheme is proposed for spectral estimation. The aforementioned photonic-assisted CS systems have been demonstrated to successfully sense multi-tone or multi-band RF signals. Motivated by the recent work [10–13], the photonic-assisted CS scheme could be developed to sense the spectrum of radar pulse streams which is very prevalent in electronic reconnaissance or electronic warfare. In this paper, a photonic-assisted CS system is proposed to acquire radar pulse streams for the first time. An IQ modulator and a photodetector are utilized to perform the spectral compression of the radar pulses. As the received radar pulses may not be spectrally sparse during a long acquisition time, a sliding window-based recovery algorithm is proposed to guarantee the signal sparsity within a time window and realize a continuous signal reconstruction. In the experiment, about 100 radar pulses ranging from 500 MHz to 5 GHz are acquired within 100 μs and exactly recovered with a 520-MHz ADC, experimentally demonstrating the feasibility of the proposed system in spectral estimation for radar pulses. 2. Principle and experimental details The architecture of the proposed photonic-assisted CS system is shown in Fig. 1. A continuous-wave (CW) laser (100-KHz linewidth) emits an output power of 15 dBm at 1550 nm. A 127-bit pseudo-random binary sequence (PRBS) is generated by a pulse pattern generator (PPG) whose external clock is set to 10.16 GHz corresponding to a repetition rate of 80 MHz denoted as fp. The input radar pulse stream is generated by an arbitrary waveform generator (AWG) with a 50-GS/s sampling rate and 10-bit vertical resolution. The pulses may be emitted from radars in diverse places, so the time of arrival (TOA) of these radar pulses are different at the receiver as shown in Fig. 1. As the bit rate of the PRBS signal should be at least twice the Nyquist rate of the input radar signal, the bandwidth of the radar pulse stream is limited to about 5 GHz. In the experiment, a radar pulse stream (about 100 pulses) spanning from 500 MHz to 5 GHz is acquired by this system within 100 μs. Then an IQ modulator with a 3-dB bandwidth of 12.5 GHz and a half-wave voltage (Vpi) of 3.5 V is used to convert the radar pulse stream and the PRBS signal from the electrical domain to the optical form, respectively. Following the IQ modulator, a 10-GHz photodetector (8-mA photocurrent output) performs spectrum convolution between the input radar pulse stream and the PRBS signal. The spectrum of the mixing signal is a linear combination of fp-shifted copies of the input RF spectrum. All the frequency components of the acquired radar pulses are transferred to the low-frequency band and then a low pass filter can be used to extract the information of the spectrally compressed signal. The output signal is then sampled by an off-the-shelf
#228565 - $15.00 USD (C) 2015 OSA
Received 27 Nov 2014; revised 27 Jan 2015; accepted 30 Jan 2015; published 12 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.004517 | OPTICS EXPRESS 4518
electrical ADC with a 520-MHz bandwidth and an effective number of bits (ENOB) of 10 bits corresponding to a compression factor of 0.12. Finally, the obtained measurements are post processed offline by a digital signal processor (DSP).
Fig. 1. The Experimental setup of the proposed photonic-assisted compressive sampling system. CW: continuous wave, PC: polarization controller, PPG: pulse pattern generator, AWG: arbitrary waveform generator, PRBS: pseudorandom binary sequence, PD: photodetector, LPF: low pass filter, LNA: low noise amplifier, ADC: analog-to-digital converter, DSP: digital signal processor.
However, the acquired radar pulse stream may not be frequency-sparse within 100 μs, so we cannot reconstruct these radar pulses from the measurements by compressive sampling directly. To solve this problem, the concept of local Fourier sparsity is introduced. In Fig. 2, the temporal waveform and spectrogram of a chirp signal are presented and it is obvious that the chirp signal is neither time-sparse nor frequency-sparse. If the chirp signal is segmented into several pieces, each piece would present sparsity in the frequency domain as shown in Fig. 2. Similarly, the number of the acquired radar pulses will be decreased during a short acquisition time which may lead to low frequency occupancy over a wide spectrum. Therefore, a time window should be introduced to guarantee the local Fourier sparsity of the acquired radar pulses. As shown in Fig. 1, the spectrally compressed signal is segmented into several pieces by time windows and two adjacent time windows have a considerable overlap. By properly selecting the length of the time window, the sparsity of the windowed signal can be guaranteed. In addition, the time window will cause information loss which is more serious along the window edges than in the middle. To average out the errors resulting from the reduced information, overlapping time windows are proposed and a classical windowing method [14] is introduced for the continuous signal reconstruction. Moreover, as shown in Fig. 2, if the window length is extended, frequency occupancy will be increased which makes it more difficult to realize a robust reconstruction. On the contrary, a short time window will degrade the spectral resolution. In addition, the amount of overlap between windows relates to the computational complexity of the recovery algorithm. A large overlap will badly affect the speed of the recovery while a small overlap will aggravate the information loss. Therefore, a trade-off between the speed and accuracy of the reconstruction needs to be taken into consideration. With the sliding time window, a continuous signal reconstruction is realized.
#228565 - $15.00 USD (C) 2015 OSA
Received 27 Nov 2014; revised 27 Jan 2015; accepted 30 Jan 2015; published 12 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.004517 | OPTICS EXPRESS 4519
Fig. 2. The principle of local Fourier sparsity.
In the experiment, the length of a time window is set to 5 μs and the amount of overlap between windows is 2 μs. The acquired radar pulses in a time window is recovered by adopting the orthogonal matching pursuit (OMP) algorithm [15]. To evaluate the performance of the proposed photonic-assisted CS system, a rectangular pulse, a linear frequency modulated (LFM) pulse and a radar pulse stream is respectively tested through this system. 3. Experimental Results and discussion A rectangular pulse is first tested to evaluate the performance of the photonic-assisted CS system. The carrier frequency and duration of this pulse is 3.21 GHz and 2 μs, respectively. Figure 3(a) shows the power spectral density of the rectangular pulse and the normalized power spectral density of the recovered pulse is depicted in Fig. 3(b). The carrier frequency of the recovered pulse is 3.2101 GHz with a recovery error of 100 KHz and the recovered power spectrum matches well with the original one in the frequency range from 3.206 GHz to 3.216 GHz. The temporal waveform of the recovered pulse with a rectangular envelope and a pulse width of 2 μs shown in the inset of Fig. 3(b) indicates a successful recovery.
Fig. 3. Experimental results for the recovery of a rectangular pulse and a LFM pulse. (a) Original power spectral density of the rectangular pulse. (b) Recovered power spectral density of the rectangular pulse. Inset: temporal waveform of the recovered pulse. (c) Original power spectral density of the LFM pulse. (d) Recovered power spectral density of the LFM pulse. Inset: temporal waveform of the recovered LFM pulse.
In radar systems, a long-duration radar pulse contributes to improving its energy and target detection capability while it would also degrade the range resolution [16]. Hence LFM pulses are widely used in radar systems owing to their unique properties. In the experiment, a #228565 - $15.00 USD (C) 2015 OSA
Received 27 Nov 2014; revised 27 Jan 2015; accepted 30 Jan 2015; published 12 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.004517 | OPTICS EXPRESS 4520
LFM pulse with a 3.22-GHz center frequency, 10-MHz bandwidth and 1-μs pulse width as shown in Fig. 3(c) is compressively sampled by the proposed system. The normalized power spectral density of the recovered pulse is shown in Fig. 3(d). The recovered carrier frequency is 3.22012 GHz and the reconstructed power spectrum keeps consistent with the original one in the frequency range 3.21-3.23 GHz. Mixed with a local oscillator signal at 3.21 GHz, the recovered LFM pulse is then down-converted. The corresponding temporal waveform of the down-converted LFM pulse is depicted in Fig. 3(d). As the pulse width is 1 μs and the carrier frequency of the pulse after down conversion is 10 MHz, the down-converted LFM pulse contains 10 periods which agrees with the description in Fig. 3(d). A successful recovery of a radar pulse stream is presented in Fig. 4. The waterfall diagram of this radar pulse stream is shown in Fig. 4(a) where the horizontal axis is frequency and the vertical axis is time. In this figure, it can be observed that the radar pulses within a time window are frequency-sparse. A time-frequency representation (TFR) of the recovered pulse stream is depicted in Fig. 4(b). Due to the nonlinearity existing in the photonic link, the undesired high-order harmonics will emerge and aggravate the signal to noise ratio (SNR). Comparing Fig. 4(a) with (b), we can conclude that the recovery of the acquired radar pulse stream is successful and the spurious signals induced by modulation nonlinearity are weak enough relative to the recovered radar pulses.
Fig. 4. Experimental results for recovery of a radar pulse stream. (a) A waterfall diagram of the original pulse stream. (b) A waterfall diagram from CS reconstruction with sampling rate equal to 10.4% of the Nyquist rate.
Signal reconstruction is performed with the sliding window-based recovery algorithm. Figure 5(a) shows the power spectral density of the acquired radar pulses in a time window and Fig. 5(c) presents a waterfall diagram of the windowed signal. The corresponding recovery results are shown in Fig. 5(b) and (d), respectively. Making a Comparison between
#228565 - $15.00 USD (C) 2015 OSA
Received 27 Nov 2014; revised 27 Jan 2015; accepted 30 Jan 2015; published 12 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.004517 | OPTICS EXPRESS 4521
two waterfall diagrams, we can find that all the frequency components of the windowed signal are exactly recovered and some spurious signals can be eliminated by increasing the decision threshold level. However, this increase will change the shape of the signal spectrum which has a negative effect on signal reconstruction. Therefore selecting a proper decision threshold level is quite important. In general, the results in Fig. 5 present an exact recovery.
Fig. 5. Recovery results in a time window. (a) The original power spectral density. (b) The recovered power spectral density. (c) A waterfall diagram of the original signal within a time window. (d) A waterfall diagram of the recovered signal within a time window.
4. Conclusions The proposed photonic-assisted CS system has been demonstrated its capability of exactly recovering radar pulses with a sub-Nyquist ADC in this paper. In the experiment, a random measurement matrix is generated to compress the spectrum of the input radar signal. An IQ modulator and a photodetector are utilized to perform the mixing of the radar pulses and the PRBS signal. A microsecond rectangular pulse and a LFM pulse are spectrally estimated respectively and the performance of the proposed CS system is also analyzed. By using the sliding window-based algorithm, a radar pulse stream ranging from 0.5 GHz to 5 GHz is exactly recovered with a 520-MHz ADC, indicating that the photonic-assisted CS scheme is promising in spectral estimation for radar pulses. Acknowledgments This work is supported by National Program on Key Basic Research Project (973) under Contract 2012CB315703, NSFC under Contract 61271134 and 61120106001.
#228565 - $15.00 USD (C) 2015 OSA
Received 27 Nov 2014; revised 27 Jan 2015; accepted 30 Jan 2015; published 12 Feb 2015 23 Feb 2015 | Vol. 23, No. 4 | DOI:10.1364/OE.23.004517 | OPTICS EXPRESS 4522
