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Abstract
We demonstrate measurement of RF signals in the 2-19 GHz band using a photonic compressive sensing (CS) receiver. The RF is modulated onto chirped optical pulses that then propagate through a multimode fiber that produces the random projections needed for CS via optical speckle. Our system makes 16 independent measurements per optical pulse and we demonstrate several calibration techniques to obtain the CS measurement matrix from these measurements. Then a standard penalized l1 norm method recovers amplitude, phase, and frequency of single-tone and two-tone RF signals with about 100 MHz resolution in a single 4.5 ns pulse. A novel subspace method recovers the frequency to about 20 kHz resolution over 100 pulses in a 2.8 microsecond time window. These experiments use discrete fiber-coupled optical components, but all necessary functions can be realized in photonic and electronic integrated circuits.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The field of compressive sensing (CS) is now more than a decade old [1–3] and a wide range of exciting results have been reported [4,5]. A particularly attractive application for CS is sensing RF signals in the GHz band because Nyquist rate sampling with electronic ADCs generates a huge amount of data and requires more size, weight and power (SWAP) than are practical for many applications; furthermore, for many applications the GHz band is sparsely occupied. These considerations motivated a lot of research into reduced sampling rate, electronic systems for detecting GHz-band signals [6-9 and references therein]. In these electronic CS systems the GHz signal of interest must be multiplied by analog waveforms that constitute the rows of the CS measurement matrix. These analog waveforms must contain frequency content at more than twice the highest frequency of the input signal, and they must be highly stable, carefully calibrated, and capable of being generated with electronic devices of small SWAP. This issue motivated us [10–19] and other researchers [20–37] to consider photonic devices for generating and applying the CS measurement matrix.
Most optical CS systems to date have been single-channel systems that apply the rows of the measurement matrix serially in time and have used free-space or fiber-coupled optical components. Neither property is desirable for general applications. Multiple channels, such as discussed for Xampling and the modulated wideband converter [7–9], are needed to achieve small time windows, and integrated optical components are needed to achieve small SWAP. These two objectives have led us to consider using optical speckle in a planar waveguide to perform the random projections needed for compressive sensing [17–20]. Optical spectrometers using speckle in multimode fibers and waveguides have recently been demonstrated by several groups [38–44], and we have shown that if one modulates an RF signal directly onto a single-frequency laser, the speckle spectrometer technique can resolve radio frequencies to an accuracy of 100 MHz [45]. This latter work used a silicon-on-insulator (SOI) planar waveguide whose output was imaged onto an InGaAs CCD camera having a frame rate of 60-100 Hz and as such was not capable of real-time measurement of RF signals. Other related work shows that optical speckle may be useful for performing the random projections needed for randomized numerical linear algebra calculations and machine learning [46–48].
For the canonical CS problem, a sparse input signal x (dimension N) is recovered from a measurement vector y (dimension M) with M<<N. The CS measurement vector y is obtained from x after multiplication by a measurement matrix (MM) Φ as in Eq. (1),
where s = Ψ x is a sparse vector with a small number K of non-zero elements, Ψ is the transform that shows the sparsity of x, and Θ (the transformed measurement matrix [5]) is the product of Φ and the inverse of Ψ. If Φ satisfies certain properties [1–6], sparse x can be recovered by a range of algorithms provided that M is somewhat greater than K [1–6]. An important issue for CS systems is finding a practical way to perform the matrix multiplication Φx or Θs in the analog domain. Since CS recovery calculations require accurate knowledge of Φ or Θ [16], Φ or Θ must be reproducible and amenable to calibration.2. Experimental set up
The experimental setup is illustrated in Fig. 1. Chirped optical pulses are obtained by passing femtosecond pulses from a mode-locked laser (35.71 MHz repetition rate) through a spool of dispersion-compensating fiber (−166 ps/nm at 1545 nm) producing ~4.5-ns FWHM pulses as shown within Fig. 1. RF signals are intensity modulated onto the chirped optical pulses by a 1x2 Mach-Zehnder LiNbO3 modulator (EOSPACE, zero-chirp X-cut; 20 GHz 3-dB bandwidth) biased at quadrature. One of the MZM outputs is monitored on an oscilloscope and used to set the MZM bias, measure the MLL rep rate, and record the relative timing between the optical pulses and RF waveforms. The second MZM output is amplified by an EDFA and butt-coupled into the multimode fiber (5 meters in length; 0.22 NA step-index; 105-μm core) within which the RF signal is mixed with wavelength-dependent speckle. To form the channels of the system (i.e., rows of the CS measurement matrix), the output core of the MMF is projected onto a 32-fiber bundle. The bundle fibers (in this case, multimode fibers similar to the speckle-mixing fiber) function as “buckets” that spatially sample the MMF output speckle pattern and transport the collected light to an array of time-integrating photodiodes (integration time matched to the MLL period). The photodiodes are configured as differential pairs, yielding a CS measurement vector with 16 elements. Outputs of the differential detectors are recorded using oscilloscopes (4 Tektronix DPO3034 4-channel scopes). Oscilloscopes were chosen for this experimental setup as the simplest solution for a multi-channel back-end digitizer satisfying sample-rate (at a minimum the MLL rep rate of 35.71 MHz) and signal-conditioning (vertical offset, variable front-end amplification, etc.) requirements. Over-sampling by the oscilloscopes (2.5 GS/s) is exploited to avoid having to synchronize sampling with the laser pulses. Using differential detectors reduces quiescent measurement components generated from the background optical pulse energy; hence the 8 read-out bits of the oscilloscopes could be better focused onto the relevant measurement components generated by the RF modulation. For an engineered system, we expect that the oscilloscopes will be replaced by an array of ADCs synchronized to and sampling at the laser repetition rate, similar to those offered commercially by Texas Instruments [49].
Fig. 1 Experimental setup (MLL: Mode-locked laser; MZM: Mach-Zehnder modulator; PD: Photodiode; EDFA: Erbium-doped fiber amplifier; MMF: Multimode fiber; AWG: Arbitrary waveform generator).
The 5-meter MMF used for speckle mixing is spooled and set out on the lab table. Steps to environmentally stabilize the MMF are yet to be implemented. Openly exposed to the lab environment, the MMF speckle pattern varies in time with cycles synchronized to the laboratory air conditioning. To work around this environmental drift for the experiments reported here, both calibration and signal measurements are performed within single oscilloscope acquisitions (400 μs duration), which are much shorter than the time scale of the drifting. The calibration procedure is described in Section 3. RF waveforms, consisting of calibration signals followed by “unknown” signals to be recovered, are generated using a high-speed arbitrary waveform generator (50 GS/s; Tektronix AWG70001A). The AWG is manually triggered and in turn triggers the oscilloscopes. Acquired scope traces are downloaded for post-processing following each trigger.
3. Calibration and determination of transformed measurement matrix
Successful recovery of unknown signals requires calibration of the CS system; in particular accurate determination of the measurement matrix Φ or the transformed measurement matrix Θ is needed. In earlier work, we took the approach of using a tunable laser to make high resolution measurements of the fiber speckle as a function of wavelength [18, 19]. Coupling these measurements with the time-wavelength mapping of the system (derived from the measured dispersion of the DCF between the MLL and the MZM) yielded the MM Φ as a function of time. This approach, however, is expected to break down for a high-resolution system where the RF modulation on the laser pulse itself changes the speckle pattern [45]. In the work reported here, we calibrate the system instead by measuring the speckle response to a dictionary of RF tones on a known grid and directly obtain the transformed measurement matrix Θ needed by recovery algorithms. To perform the calibration, the AWG in the experimental setup is programmed to step through the dictionary frequencies with each single-tone signal present for 50-100 laser pulse periods.
In our system, the input RF signals have an arbitrary start time relative to the MLL pulse train and therefore a non-deterministic absolute phase. Furthermore, the MLL repetition rate and RF signals are generally unsynchronized such that each laser pulse generally makes measurements for a different phase of the RF. However, since the MLL repetition rate is known very accurately, relative RF phases can be assigned to the dictionary measurements. Thus each dictionary tone yields two columns in the sensing matrix, which are referred to as the in-phase and quadrature components of the sensing matrix. We investigated three different methods to derive these columns from J laser pulse measurements. Method 1 uses measurements with pairs of pulses to estimate the two columns and then averages over all possible pairs of pulses. Method 2 fits the data as a function of the relative RF phase to read off the two columns. Method 3 uses the singular value decomposition of the J x 16 calibration matrix and uses the two dominant eigenvectors for the two columns of the sensing matrix [50].
Method 1 can be derived by considering a sinusoidal RF calibration signal xi of the form:
where fi, ai and ϕi are the frequency, amplitude and phase of the ith frequency in the dictionary. If we choose a time window T and sampling time δt such that fi is on the frequency grid associated with this window (i.e., there are an integral number of periods or half-periods of a sinewave at fi in the window), then the discrete Fourier transform of xi is given bywhere n is the number of time points (T/δt) and all the elements of si are zero except the terms at the positive and negative frequencies whose locations are set by fi. As discussed above, the phase ϕi is the sum of two terms, the phase of the RF signal relative to the first laser pulse, ϕi0, plus the phase introduced by the mismatch between the laser repetition rate fL and fi for each subsequent pulse, Δϕi,j where the subscript j indicates the pulse number. Since both fL and fi are known, Δϕi,j is known and given bywhere mod(a, m) gives a modulo m. When an arbitrary transformed measurement matrix Θ is probed by a signal at the ith frequency in the dictionary, two columns are sampled, Θi+ and Θi-, where the + and – signs indicate the columns associated with the two nonzero terms of si. If we make the definitions for every frequency in the dictionary then Eq. (1) can be transformed into y = Θ’s’. Since Δϕi,j is known, it is evident that the speckle patterns from two pulses at the ith frequency are sufficient to determine the ith columns Θi+’ and Θi-’. If these measurements are made for all frequencies in the dictionary, then Θ’ is determined and s’ can be recovered through a standard lasso or penalized l1 norm code.Method 2 is similar to Method 1 but instead of averaging the transformed measurement matrix columns over all possible pairs of pulses, Method 2 sorts all calibration measurements for each channel by Δϕi,j/(2π), fits these with a sine wave as shown in Fig. 2, and then uses two points on the fit to calculate the transformed measurement matrix Θ’ (since the fit is a sine wave, any pair of points on the fit give the same answer).
Fig. 2 Amplitude of the 16-channel measurement vector y as a function of Δϕij/(2π) for an illustrative RF frequency: data in red and fits in blue.
Method 3 uses Singular Value Decomposition (SVD) [50] to extract the 2 dominant eigenvectors from the 16 x J calibration matrix X of real numbers made for each tone in the dictionary. The SVD of X is USVT, where U is a J x J orthogonal matrix of left singular vectors, V is an M x M orthogonal matrix of right singular vectors and S is an J x M diagonal matrix of singular values (sorted from greatest to least). In our system, there are two dominant eigenvalues corresponding to the in-phase and quadrature parts of the RF calibration signal and 14 much smaller eigenvalues corresponding to system noise and distortion. The first two columns of V are the estimate of the columns of the transformed measurement matrix associated with this dictionary frequency. The calculation is repeated for each frequency in the dictionary.
Because they handle the noise in the calibration measurements differently, the measurement matrices derived from (1) the averaged two-pulse method, (2) the phase fit method, and (3) the SVD method are not identical. Figure 3 shows illustrative penalized l1 norm recovery results for a two-tone signal using the three measurement matrices calculated from the same data. In Figs. 3(a)-3(c), recovered in-phase and quadrature components are superposed for 100 single-pulse measurements. The differences between the results obtained with the different methods are analogous to the differences in estimating frequencies from the maximum of a discrete Fourier transform as opposed to a subspace method such as the Multiple Signals Classification (MUSIC) algorithms [51]. The SVD method is the most efficient computationally and is used for the remainder of the paper.
Fig. 3 Recovered in-phase and quadrature amplitudes for a two-tone signal as a function of frequency for 100 pulses. Figure 3(a) used SVD to calculate the measurement matrix; (b) used the phase-fit method and (c) used the two-pulse method. The solid vertical lines are the locations of the RF frequencies.
4. Results
4.1 Two-tone measurements
Two-tone measurements to demonstrate signal recovery and test accuracy and resolution were performed over a frequency band of 8.77 to 10.73 GHz using a dictionary grid of 57 frequencies spaced 35 MHz apart. Two-tone signals were formed by fixing one tone at 9.75 GHz and stepping the second tone through the remaining dictionary frequencies. For calibrating the system, we first collected measurements for J = 100 laser pulses for each of the 57 dictionary frequencies. Then, within the same oscilloscope acquisition, measurements were made for 100 pulses for each of the 56 two-tone signals. From the calibration measurements, the 114 x 16 transformed MM was obtained by the SVD method. Next, we use this transformed MM and penalized l1 norm codes [52] to determine the frequency, amplitude and phase of two tones distributed in this band with one tone fixed at 9.75 GHz and the other tone scanned in 35 MHz increments from 8.77 to 10.73 GHz with 100 pulses per tone. In these calculations, the transformed MM is normalized by the Frobenius Norm and the penalty parameter (usually denoted as λ, see [52] for example) is set to 3 although the results are not very sensitive to the choice of λ between 1 and 10.
Figure 4 shows illustrative results for single pulse measurements at a single pair of frequencies: 4(a) shows a pulse in which one tone has only the in-phase component while the other tone has only the quadrature component; 4(b) the in-phase component is dominant for both tones; 4(c) all 4 components are present. For this same pair of frequencies, one can superpose the results of the recovered amplitudes for 100 pulses as shown in Figs. 3(a)-3(c). As can be seen in Fig. 3, the recovered frequency often differs from the input frequency by one or more steps in the frequency grid. Figure 5 shows a 3D plot of the absolute value of the recovered amplitude as a function of frequency and pulse number for the in-phase and quadrature components of 100 pulses with the same input frequencies. We note that the difference between the recovered frequency and the known frequency for a single pulse is on the order of 100 MHz, about 3 steps in the 35 MHz grid. This is not surprising since the duration of a single pulse (~4.5 ns) corresponds to about half a period for a 100 MHz beat signal. Figure 6 shows an array plot of the average recovered amplitude (sum of the squares of the in-phase and quardrature amplitudes) for the entire set of two tone measurements. From the central region of this figure where the blue “lines” cross, one can see that two tones closer than about 3 grid frequencies (105 MHz) cannot be separated.
Fig. 4 Amplitude as a function of frequency for the two-tone signal recovered from a single 5 ns pulse: (a) shows a pulse where the in-phase of one frequency is zero and the quadrature of the other is zero; (b) the in-phase is the dominant component at both frequencies; (c) where the in-phase and quadrature components are non-zero for both frequencies. The black lines indicate the input signal frequencies.
Fig. 5 In-phase and quadrature amplitude as a function of frequency and pulse number for 100 5-ns pulses of a two-tone signal.
Fig. 6 Array plot of average recovered amplitude as a function of recovered frequency and time with one frequency held constant at 9.75 GHz and the second frequency scanned from 8.77 to 10.73 GHz in steps of 35 MHz.
We can calculate histograms of the frequencies for the two-tone test shown in Figs. 5 and 6 by finding the peaks in the sum of the squares of the in-phase and quadrature amplitudes of the recovered signal as a function of frequency. Figure 7 shows illustrative histograms for 100 pulses at 3 different pairs of frequencies. When the frequencies are well separated as in Fig. 7(a), the average frequency is within 10 MHz of the input frequency and the standard deviation is less than 2 frequency steps (70MHz). In Fig. 7(b) the two tones are separated by 175 MHz and the 2 tones can barely be discerned in the historgram. For separations less than 140 MHz, two peaks cannot be discerned as shown for example in Fig. 7(c) where the separation is 70 MHz.
Fig. 7 Histogram amplitude as a function of frequency for 100 pulses. (a) Two well separated frequencies. (b) Two frequencies separated by 175 MHz, the smallest separation for which 2 separate peaks are resolved. (c) Two frequencies separated by 70 MHz that cannot be resolved. The solid lines indicate the input frequencies.
4.2 Broadband measurements
To test the maximum frequency response and available bandwidth of our system, we programmed the RF AWG to step from 2 to 20 GHz in units of 75 MHz and from 13 to 19 GHz in units of 25 MHz. In a separate experiment, we increased the length of the DCF, shown in Fig. 1 between the MLL and the MZM, by a factor of 4 to increase the pulse length to 18 ns and stepped the RF from 2 to 10 GHz in units of 35 MHz. Given the limitations in our storage buffers, we could allow only 50 pulses per frequency for these measurements. By analogy with machine learning applications, we used the first 25 pulses at each frequency for calculating the transformed MM and the second 25 pulses to test the recovery. Figure 8 shows array plots of the recovered amplitudes as a function of frequency and time (since the input frequency from the AWG is scanned in time, time is proportional to input frequency), and Fig. 9 shows the peak amplitude as a function of frequency for the three sets of data.
Fig. 8 Array plot of amplitude as a function of recovered frequency and time. (a) for a pulse length of 4.5 ns and input frequency scanned from 2 to 20 GHz in 75 MHz steps, (b) for a pulse length of 4.5 ns and 13-19 GHz in 25 MHz steps, (c) for a pulse length of 18 ns and 2 to 10 GHz in 35 MHz steps.
Fig. 9 Recovered amplitude as a function of frequency. Red curve: single scan with 4.5 ns pulses from 2 to 20 GHz. Blue curve: 4.5 ns pulses from13 to 19 GHz. Magenta curve: 18 ns pulses from 2 to 10 GHz.
Multiple factors contribute to the roll-off in recovered amplitude observed in Figs. 8 and 9. In the scan from 2 to 19 Ghz (Fig. 8(a) and Fig. 9 red curve) the roll off is attributed to the decrease in AWG output power, MZM electro-optic response, and speckle response for given optical chirp and fiber length [18]. Figures 8(b) and Fig. 9 blue curve show that the recovered amplitude obtained between 13 and 19 GHz increases substantially when a bandpass RF amplifier is placed between the AWG and MZM. It is also true that the dictionary for the high band is limited to 240 frequencies between 13 and 19 GHz in steps of 25 MHz while the dictionary for the lower band is 240 frequencies from 2 to 20 GHz in steps of 75MHz and that the limited frequency extent of the high-band dictionary improves recovery. More channels and more pulses per frequency in the calibration process could also improve recovery at higher frequencies. To isolate the dependence on speckle response, we increased the pulse length by a factor of 4 and obtained the results shown in Fig. 8(c) and Fig. 9 magenta curve. Since the AWG output and MZM response are relatively flat from 2 to 10 GHz (as seen from the red curve in Fig. 9), the roll-off in Fig. 8(c) and Fig. 9 magenta curve can be attributed primarily to the decrease in speckle response.
In sum, Figs. 8 and 9 show that the maximum bandwidth obtained in a single scan with our system is about 15 GHz (2 to 17 GHz) (Fig. 8(a)), the maximum frequency signal recovered is 19 GHz (Fig. 8(b)) and increasing the pulse length by a factor of 4 results in a sharp roll off in response at about 8 GHz for 18 ns optical pulses, which corresponds to a limiting response of about 32 GHz for 4.5 ns pulses. This limit is applicable to a 5-m fiber and can be increased with longer fiber [18].
4.3 Fine frequency
A first estimate of the input signal frequency f is determined by solving a lasso problem as described in the previous sections. This estimate can be obtained from 1 pulse or averaged over many pulses. If the accuracy of this frequency estimate is smaller than the laser repetition rate fL (35.71 MHz), the frequency can be further refined by measuring Δf = mod(f, fL) from multiple consecutive measurement vectors.
The following method is used to derive the fine frequency. As discussed above, SVD factors a J x 16 matrix X of measurements for J pulses for an input tone into X = USVT where U is a J x J orthogonal matrix of left singular vectors, V is a 16x16 orthogonal matrix of right singular vectors and S is a J x 16 diagonal matrix of singular values (sorted from greatest to least). Define V0 and V1 to be the first two columns of V and s0 and s1 to be the two largest singular values. Let α = V0XT/s0, β = V1XT/s1 and B = α + i β (i = the square root of −1) where α, β and B are vectors of length J. For a vector B ∈ CJ (where C is the set of complex numbers) we measure the phase difference between the first J-1 components and the last J-1 components of B. Let B0 = {b0, b1, …, bJ-2} and B1 = {b1, b2, …, bJ-1}. An estimate of the change in phase Δϕ over one laser pulse repetition interval is given by
where * indicates the complex conjugate. The fine frequency fF is the given byFigure 10 shows calculations of the frequency error, the input frequency fI minus the fine frequency fF for J = 100 measurements. The large spike at 9.855 GHz is caused by the fact that mod(fI,fL) is nearly an integer and hence individual pulses do not really yield independent measurements.Fig. 10 Frequency error as a function of frequency. (a) Input frequency ranged from 8.77 to 10.73 GHz in steps of 35 MHz. (b) 17.45 to 18.55 GHz in steps of 25 MHz.
4.4 Phase
Figure 5 implies that one can estimate the relative phase between the RF and the optical pulse from the arctangent of the ratio of the in-phase to quadrature amplitude, but this estimate is not very accurate because the in-phase and quadrature terms sometimes occur at different frequencies (Fig. 4). To better estimate phase, we constructed an overcomplete transformed measurement matrix ΘOC by interpolating between the in-phase columns I and quadrature columns Q of Θ using the equation ΘOC = cosϕ I + sinϕ Q. We use a 5 degree grid in phase and perform penalized l1 norm calculations to determine the phase. Results of these calculations for single-tone signals from 8.77 to 10.695 GHz are shown in Fig. 11. The different patterns are determined by the relation between the laser repetition rate and the RF. For example, if the RF frequency were an integral multiple of the PRF of the laser, the phase between the RF and the laser pulse would constant for the 100 pulses and a horizontal line would be seen in that panel of Fig. 11. If the RF frequency divided by the PRF were an integer plus 0.5, the phase of the RF would jump back and forth between two values separted by π as a function of pulse number. Neither of these conditions exactly occurs for the grid of frequencies used to obtain Fig. 11, but one can use the frequency and PRF to estimate the slope of the phase profiles as illustrated by the red dots on the first, 5th, 6th and 7th panels in the top row of Fig. 11. Since the phase reference or RF phase relative to the first pulse is unknown, one cannot expect perfect agreement. The panels in which it is hard to discern a slope correspond to cases where the RF/PRF is far from an integer or an integer plus 0.5 or even 0.25.
Fig. 11 Phase as a function of pulse number for 56 frequencies from 8.77 to 10.695 GHz in units of 35 MHz.
5. Conclusions
We present results for a 16-channel system that uses laser speckle in a multimode fiber and time-wavelength mapping to obtain the random projections needed for compressive sensing of RF signals with frequency from 2 to 19 GHz. We develop three methods for calibrating our system and determining the CS transformed measurement matrix. We show that our system could recover up to two RF signals (a total of 6 unknowns) with accuracy on the order of 100 MHz in a single 4.5 ns pulse and an accuracy on the order of 20 kHz using 100 pulses over a time window of 2.8 μs. Also, our system recovers signals across broad bands from 2 to about 15 GHz with one setting on the AWG and from 13 to 19 GHz with a second setting on the AWG. Finally, an overcomplete dictionary method is used to determine the phase of RF relative to the laser repetition rate in a single pulse, and to demonstrate the difference in performance for signals that are nearly commensurate with (multiples of) the repetition rate compared to those that are incommensurable.
All of the components in our system can be achieved using integrated photonics components [19, 48] and future work is directed towards achieving a small form factor, integrated photonic CS receiver for GHz-band RF signals.
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