Digitally programmable arbitrary analog fir LTI filters

Abstract

Linear time-invariant (LTI) filters are a staple of modern signal processing. LTI filters perform many operations including signal estimation, bandpass / stopband selection, encoding / decoding, modulation / demodulation, etc. Most modem LTI filters operate in the digital domain due to the flexibility of programming digital processors. However, this means that the incoming signal must be digitized. This introduces the signals to the bottlenecks and latency of digital computing, where digital electronics cannot process the increasing high data-rate and high-bandwidth data in dense spectral environments in real-time. The programmable analog LTI (PA-LTI) architecture is a hardware accelerator that performs real-time LTI operations in the analog domain and is fully programmable, enabling unprecedented acceleration of front-end PHY layer communications processing. Conventional analog LTI processors, such as RLC circuits or amplifiers, can perform only one LTI function. A PA-LTI processor can be digitally programmed to implement any arbitrary finite impulse response (FIR) LTI operation in the analog domain.

Description

Attorney Docket No. MIT-25674WO01 Digitally Programmable Arbitrary Analog FIR LTI Filters CROSS-REFERENCE TO RELATED APPLICATION(S)

[0001] This application claims the priority benefit, under 35 U.S.C.119(e), of U.S. Application No.63 / 638,172, filed April 24, 2024, which is incorporated herein by reference in its entirety for all purposes. BACKGROUND

[0002] In conventional signal processing architectures, linear time-invariant (LTI) functions are used for a variety of purposes. In many cases, such as digital communications, the majority of the LTI functions are performed digitally. This is because in the digital domain the signal quality is more predictable, within the control of the system designers, and because the digital domain offers immense flexibility in the types of LTI filters that can be applied. However, for high-bandwidth signals, 5G multiple-input multiple-output (MIMO), millimeter-wave (mmWave) communications, and radar spectrum sensing, it can be challenging if not impossible for electronics to analyze a large quantity of these waveforms in real-time due to the limited speed of the analog-to-digital converters (ADCs) (typically limited to around 10 Gs / s) and the latency of data movement from the memory to the processor. SUMMARY

[0003] The present technology provides a new LTI computing paradigm for circumventing problems associated with analyzing high-bandwidth signals in real time in the digital domain. Embodiments of the present technology include a method of programmable analog linear time- invariant (PA-LTI) processing. This method includes generating an analog programmable filter signal selected to implement a finite impulse response (FIR) linear time-invariant (LTI) filtering operation, such as Wiener filtering, matched filtering, scrambling, or unscrambling. The analog programmable filter signal and an analog input signal are modulated onto an optical carrier using single-sideband, suppressed-carrier modulation to produce an analog optical filter signal and an analog optical input signal, respectively. In some cases, the optical carrier is split into first and second portions, and the analog programmable filter signal and analog input signal are modulated onto the first and second portions, respectively. The analog optical filter signal and the analog optical input signal are multiplied to produce an analog filtered signal, forAttorney Docket No. MIT-25674WO01 instance, by detecting interference of the analog optical filter signal and the analog optical input signal with a balanced photodetector. The analog filtered signal can then be converted into a digital filtered signal.

[0004] In some aspects, the techniques described herein relate to a method of programmable analog linear time-invariant (PA-LTI) processing, the method including: generating a programmable filter signal selected to implement a FIR LTI filtering operation; performing single-sideband, suppressed-carrier modulation of the programmable filter signal onto a first portion of an optical carrier to produce an analog optical filter signal; performing single- sideband, suppressed-carrier modulation of an analog input signal onto a second portion of the optical carrier to produce an analog optical input signal; and multiplying the analog optical filter signal with the analog optical input signal to produce an analog LTI-filtered signal.

[0005] In some aspects, the techniques described herein relate to a method, wherein the FIR LTI filtering operation includes at least one of Wiener filtering, matched filtering, scrambling, or unscrambling.

[0006] In some aspects, the techniques described herein relate to a method, wherein performing single-sideband, suppressed-carrier modulation of the programmable filter signal onto the first portion of the optical carrier includes modulating the programmable filter signal onto one of an upper sideband or a lower sideband of the optical carrier and performing single-sideband, suppressed-carrier modulation of the analog input signal onto the second portion of the optical carrier includes modulating the analog input signal onto the other of the upper sideband or the lower sideband of the optical carrier.

[0007] In some aspects, the techniques described herein relate to a method, further including, before performing single-sideband, suppressed-carrier modulation of the programmable filter signal or of the analog input signal: splitting the optical carrier into a first portion and a second portion, wherein performing single-sideband, suppressed-carrier modulation of the programmable filter signal onto the optical carrier is performed on the first portion and performing single-sideband, suppressed-carrier modulation of the analog input signal onto the optical carrier is performed on the second portion.

[0008] In some aspects, the techniques described herein relate to a method, wherein multiplying the analog optical filter signal with the analog optical input signal includes detecting interference of the analog optical filter signal and the analog optical input signal with a balanced photodetector.Attorney Docket No. MIT-25674WO01

[0009] In some aspects, the techniques described herein relate to a method, wherein the analog LTI-filtered signal represents a mapping of an LTI filtering operation in the frequency domain between the programmable filter signal and the analog input signal.

[0010] In some aspects, the techniques described herein relate to a method, further including: converting the analog LTI-filtered signal into a digital LTI-filtered signal.

[0011] In some aspects, the techniques described herein relate to a PA-LTI processor including: a laser to emit an optical carrier; a first single-sideband, suppressed-carrier modulator, in optical communication with the laser, to modulate a first portion of the optical carrier with an analog programmable filter signal, the analog programmable filter signal selected to implement a FIR LTI filtering operation; a second single-sideband, suppressed- carrier modulator, in optical communication with the laser, to modulate a second portion of the optical carrier with an analog input signal; and a photodetector, in optical communication with the first single-sideband, suppressed-carrier modulator and the second single-sideband, suppressed-carrier modulator, to transduce optical-domain interference of the first portion of the optical carrier modulated with the analog programmable filter signal and the second portion of the optical carrier modulated with the analog input signal into an electronic-domain analog LTI-filtered signal.

[0012] In some aspects, the techniques described herein relate to a PA-LTI processor, wherein the FIR LTI filtering operation includes at least one of Wiener filtering, matched filtering, scrambling, or unscrambling.

[0013] In some aspects, the techniques described herein relate to a PA-LTI processor, wherein the first single-sideband, suppressed-carrier modulator is configured to modulate the analog programmable filter signal onto one of an upper sideband or a lower sideband of the optical carrier and the second single-sideband, suppressed-carrier modulator is configured to modulate the analog programmable filter signal onto the other of the upper sideband or the lower sideband +of the optical carrier.

[0014] In some aspects, the techniques described herein relate to a PA-LTI processor, wherein the second single-sideband, suppressed-carrier modulator is in series with the first single- sideband, suppressed-carrier modulator.

[0015] In some aspects, the techniques described herein relate to a PA-LTI processor, wherein the photodetector is a balanced photodetector.Attorney Docket No. MIT-25674WO01

[0016] In some aspects, the techniques described herein relate to a PA-LTI processor, wherein the analog LTI-filtered signal represents a mapping of an LTI filtering operation in the frequency domain between the analog programmable filter signal and the analog input signal.

[0017] In some aspects, the techniques described herein relate to a PA-LTI processor, wherein the first single-sideband, suppressed-carrier modulator is a dual-parallel Mach-Zehnder modulator including a first sub-Mach-Zehnder modulator (MZM) and a second sub-MZM.

[0018] In some aspects, the techniques described herein relate to a PA-LTI processor, further including: a bias controller, operably coupled to the dual-parallel Mach-Zehnder modulator, to bias the first sub-MZM, the second sub-MZM, and interference between the first sub-MZM and the second sub-MZM.

[0019] In some aspects, the techniques described herein relate to a PA-LTI processor, further including: an arbitrary waveform generator or an application-specific integrated circuit, operably coupled to the first single-sideband, suppressed-carrier modulator, to generate the analog programmable filter signal.

[0020] In some aspects, the techniques described herein relate to a PA-LTI processor, further including: a beam splitter, in optical communication with the laser, the first single-sideband, suppressed-carrier modulator, and the second single-sideband, suppressed-carrier modulator, to divide the optical carrier into the first portion and the second portion, to direct the first portion to the first single-sideband, suppressed-carrier modulator, and to direct the second portion to the second single-sideband, suppressed-carrier modulator.

[0021] In some aspects, the techniques described herein relate to a PA-LTI processor, further including: an analog-to-digital converter, operably coupled to the photodetector, to convert the electronic-domain analog LTI-filtered signal in a digital LTI-filtered signal.

[0022] In some aspects, the techniques described herein relate to a PA-LTI processor including: a laser to emit an optical carrier; a first dual-parallel Mach-Zehnder modulator (DP- MZM), in optical communication with the laser, to modulate one of an upper sideband band or a lower sideband of a first portion of the optical carrier with an analog programmable filter signal to yield an analog optical filter signal, the analog programmable filter signal selected to implement a FIR LTI filtering operation; a second DP-MZM, in optical communication with the laser, to modulate the other of the upper sideband band or the lower sideband of a second portion of the optical carrier with an analog input signal to yield an analog optical input signal; a beam splitter, in optical communication with the first DP-MZM and the second DP-MZM, toAttorney Docket No. MIT-25674WO01 combine the analog optical filter signal and the analog optical input signal; a balanced photodetector, in optical communication with the beam splitter, to transduce optical-domain interference of the analog optical filter signal and the analog optical input signal into an electronic-domain analog LTI-filtered signal; and an analog-to-digital converter, operably coupled to the photodetector, to convert the electronic-domain analog LTI-filtered signal in a digital LTI-filtered signal.

[0023] In some aspects, the techniques described herein relate to a PA-LTI processor, further including: a bias controller, operably coupled to the first DP-MZM, to bias the first DP-MZM for single-sideband, suppressed-carrier modulation.

[0024] All combinations of the foregoing concepts and additional concepts discussed in greater detail below (provided such concepts are not mutually inconsistent) are contemplated as being part of the inventive subject matter disclosed herein. In particular, all combinations of claimed subject matter appearing at the end of this disclosure are contemplated as being part of the inventive subject matter disclosed herein. Terminology explicitly employed herein that also may appear in any disclosure incorporated by reference should be accorded a meaning most consistent with the particular concepts disclosed herein. BRIEF DESCRIPTIONS OF THE DRAWINGS

[0025] The skilled artisan will understand that the drawings primarily are for illustrative purposes and are not intended to limit the scope of the inventive subject matter described herein. The drawings are not necessarily to scale; in some instances, various aspects of the inventive subject matter disclosed herein may be shown exaggerated or enlarged in the drawings to facilitate an understanding of different features. In the drawings, like reference characters generally refer to like features (e.g., functionally similar and / or structurally similar elements).

[0026] FIG.1A shows a graphics processing unit (GPU) receiver.

[0027] FIG.1B shows a radio frequency (RF) system on a chip (RFSoC) receiver.

[0028] FIG.1C shows a field-programmable gate array (FPGA) receiver.

[0029] FIG.1D shows a programmable analog linear time-invariant (PA-LTI) receiver.

[0030] FIG. 2A illustrates implementation of a PA-LTI filter using optical photoelectric multiplication.Attorney Docket No. MIT-25674WO01

[0031] FIG. 2B illustrates implementation of a PA-LTI filter using cascaded electro-optic modulators or an optical frequency comb / pulse shaper for an analog programmable filter signal, ^^(^^), and a modulator for the analog input signal, ^^(^^).

[0032] FIG.2C illustrates an RF mixer suitable for narrow-band filtering applications.

[0033] FIG.2D illustrates an in-phase / quadrature (IQ) mixer / heterodyne mixer for certain RF modulation schemes.

[0034] FIG.3A illustrates creation of a distorted signal from an original signal.

[0035] FIG.3B illustrates Wiener filtering to recover an estimate of an original signal from a distorted signal.

[0036] FIG.4A illustrates implementation of an experimental linear frequency-invariant (LFI) Wiener filter.

[0037] FIG. 4B shows an overlay of outputs from theoretical and experimental LFI Wiener filters.

[0038] FIG.5 illustrates implementation of an LFI matched filter.

[0039] FIG.6A illustrates implementation of an experimental LFI matched filter.

[0040] FIG. 6B shows an overlay of outputs from theoretical and experimental LFI matched filters.

[0041] FIG.7 illustrates an LTI filter that recovers a scrambled signal.

[0042] FIG.8 illustrates experimental implementation of the scrambled signal recovery of FIG. 7.

[0043] FIG.9 illustrates mitigation of frequency crosstalk and noise with a PA-LTI receiver.

[0044] FIGS.10A–10C illustrate tracking a frequency-hopping signal with a PA-LTI receiver.

[0045] FIG. 11 illustrates encrypted wireless communications with a PA-LTI transmitter and a PA-LTI receiver.

[0046] FIG.12 illustrates noise smoothing with a PA-LTI receiver. DETAILED DESCRIPTION

[0047] Innovative aspects of a programmable analog linear time-invariant (PA-LTI) filtering architecture include its processing and hardware architecture. A PA-LTI filtering architectureAttorney Docket No. MIT-25674WO01 provides true fully analog LTI operations using analog hardware. It can implement arbitrary finite impulse response (FIR) LTI filters in any time-frequency domain. With a PA-LTI filtering architecture, data reaches a PA-LTI processor, also called a programmable FIR LTI filter, in the analog domain, before reaching an analog-to-digital converter (ADC). The PA- LTI processor multiplies two signals, either in the time domain, frequency domain, or some combination, that are selected or programmed to achieve frequency-domain LTI operations.

[0048] A frequency-domain version of PA-LTI filtering architecture, called a linear frequency- invariant (LFI) filtering architecture, is detailed below for a photoelectric multiplication setup. Suitable frequency-domain LFI operations include Wiener filtering, matched filtering, signal scrambling, and signal unscrambling. The LFI framework generalizes to other use-cases and to the more general PA-LTI framework.

[0049] FIGS.1A–1D show conventional digital receivers 100a–100c with LTI filtering and a PA-LTI receiver 100d. The digital receiver 100a in FIG.1A is graphics processing unit (GPU) receiver 100a with an analog receiver (Rx) front end or analog Rx block 110 coupled to an antenna 102 or other analog signal source. The analog Rx block 110 includes a pre-selector circuit and may include a low-noise amplifier, bandpass filter(s), and frequency up / down conversion elements, such as mixers and local oscillators. Cables 104 couple an analog signal from the analog Rx block 110 to an analog-to-digital converter (ADC) 120, which converts the analog signal into a digital signal and couples it via a serial interface 106 to a central processing unit (CPU) 130. The CPU 130 connects the final processed signal to the end user / end system that takes action based on / in response to the final processed signal, e.g., such as switching modulation schemes or displaying information about the spectrum to a user. A GPU 140 coupled to the CPU 130 via another serial interface 108 uses one or more processing cores 142 and virtual random access memory (virtual RAM, or VRAM) 144 to perform an LTI filtering operation on the digital signal.

[0050] The digital receiver 100b in FIG. 1B is a radio frequency (RF) system on a chip (RFSoC) receiver. The ADC 120 is implemented in a field-programmable gate array (FPGA) 150, which is coupled to the CPU 130 via a serial interface 108 and implements a digital signal processing (DSP) block 152, dynamic RAM (DRAM) 154, block RAM (BRAM) 156, and UltraRAM (URAM) 158. The DSP block 152 contains the computational cores of the FPGA 150 and uses them for LTI filtering. However, the number of computational cores in the DSP block 152 may be (very) limited, so the BRAM 156 and / or URAM 158 move data back and forth quickly (tiling) so that the DSP block 152 can perform the computations on chunks of RFAttorney Docket No. MIT-25674WO01 data at a time. The DRAM 154 stores much larger pieces of RF data and / or provides longer- term storage.

[0051] The digital receiver 100c in FIG.1C is an FPGA receiver similar to the RFSoC receiver 100b in FIG.1B. In this instance, the ADC 120 is implemented separately from an FPGA 160, which is coupled to the ADC 120 and CPU 130 via respective serial interfaces 106 and 108. This FPGA 160 is programmed to implement DSP block 162, DRAM 164, BRAM 166, and URAM 168. Again, the DSP block 162 contains the FPGA’s computational cores, which perform the LTI filtering. The BRAM 166 and / or URAM 168 move data back and forth quickly so that the limited number of computational cores can operate on chunks of RF data at a time, while the DRAM 154 stores larger pieces of RF data and / or provides longer-term storage.

[0052] In each of the digital receivers shown in FIGS. 1A–1C, data first travels through the ADC 120 and digital memory (e.g., VRAM 144, DRAM 154 / 164) before being LTI-filtered with a digital processor (GPU 140, DSPs 152 / 162). In the PA-LTI receiver 100d shown in FIG. 1D, the data is LTI-filtered real-time in the analog domain using a PA-LTI filter before the ADC 120. Put differently, in the PA-LTI receiver 100d, the data reaches the compute hardware before it reaches the ADC 120. This allows for real-time analog LTI processing of signals without the high cost, size, weight, and power of high-performance ADCs and thus alleviates digital processing requirements.

[0053] More specifically, the PA-LTI receiver 100d includes a PA-LTI processor 170 (e.g., as described below with respect to FIGS. 2A–2D) whose input is coupled to the output of the analog Rx front end 110 and whose output is coupled to the input of the ADC 120. The PA- LTI processor 170 is also coupled to an application-specific integrated circuit (ASIC) 172, arbitrary waveform generator (AWG), or other device suitable for generating a weight signal, which is selected based on the filtering operation performed by the PA-LTI processor 170. The weight signals can be selected so that the output of the analog Rx front end 110 is at baseband or a higher frequency; for higher efficiency, the weight signal can be programmed so that the output of the analog Rx front end 110 at baseband. The PA-LTI processor 170 multiplies the weight signal with an analog input signal from the analog Rx front end 110 in the analog domain. Multiplication in the time-domain (frequency-domain) is the Fourier equivalent of a frequency-domain (time-domain) convolution and yields an analog LTI-filtered signal that the ADC 120 converts into a digital LTI-filtered signal. Generally, the analog LTI-filtered signal is higher bandwidth than the input signal unless it is bandpass-filtered because a frequency domain convolution does not remove spectral content and can spectral content.Attorney Docket No. MIT-25674WO01 Frequency-Domain PA-LTI Hardware Setup

[0054] A PA-LTI processor can be implemented using a variety of hardware setups. The PA- LTI processor multiplies two signals in the analog domain (whether in the time domain, frequency domain, or some combination domain through wavelet transforms—e.g., spectrograms, short-time Fourier transforms (STFT), Haar waveforms, or square waveforms). One of these signals may be an input signal (e.g., a signal received by an antenna at the end of a communications link, as in FIG.1D) and the other signal is a programmable weight or filter signal, such as with an AWG or ASIC that performs a similar operation as an AWG.

[0055] An LFI architecture can perform conventional LTI operations in the frequency domain. A physical implementation of LFI performs a multiplication in the time domain (which corresponds to the LTI convolution in the frequency domain).

[0056] FIGS. 2A–2D shows various processors for implementing LFI versions of PA-LTI filtering, where an analog input signal ^^(^^)is multiplied by an analog programmable filtersignal ^^(^^) in the time domain to produce ^^(^^) = ^^(^^) ⋅ ^^(^^) (or a similar product thatmultiplies some function of ^^(^^) and ^^(^^). Thus, the time-domain multiplication yields afrequency-domain LTI convolution. The filter signal ^^(^^) can be determined by themathematics of the LTI or LFI operation being performed by the LTI / LFI processor. The signal^^(^^) received is an incoming wireless signal, a wired electrical signal, a free-spaceoptical / fiber-optic signal, or some other data-carrying signal. The signal ^^(^^) is programmed through either an AWG, ASIC, or pulse-shaping (for optical frequency combs). The exact programming depends on the use case. The hardware configurations can be categorized into wideband and narrowband use cases.

[0057] More specifically, FIG.2A shows a PA-LTI processor 200a includes an optoelectronic multiplier with a laser 210, first and second single-sideband, suppressed-carrier (SSB-SC) modulators 220a and 220b (e.g., acousto-optic or dual-parallel Mach-Zehnder modulators (DP- MZMs)), beam splitter 222, balanced photodetector 230, and optional bias controller 240. This PA-LTI processor 200a may also include another beam splitter (not shown) that splits an optical carrier (laser beam) emitted by the laser 210 into first and second portions. The first SSB-SC modulator 220a modulates a programmable filter or weight signal ^^(^^) from an AWG or ASIC (not shown) onto the upper or lower sideband of the first portion of the optical carrier to yield an analog optical filter signal. At the same time, the second SSB-SC modulator 220b modulates an analog input signal ^^(^^), e.g., from an antenna or receiver front end, onto theAttorney Docket No. MIT-25674WO01 opposite (i.e., lower or upper) sideband of the second portion of the optical carrier to yield an analog optical input signal. A second beam splitter 222 combines the analog optical filter and weight signals for detection by the balanced photodetector 230, which emits an analog LTI- filtered signal ^^(^^) equal to the product of the filter signal ^^(^^) and the analog input signal ^^(^^). This analog LTI-filtered can be digitized by an ADC 120 (FIG. 1D) for additional processing in the digital domain.

[0058] If the first and second SSB-SC modulators 220a and 220b are acousto-optic modulators, for example, the analog optical input and filter signals can be modulated onto different sidebands of the optical carrier using the +1 and –1 diffracted orders, respectively, of the first and second SSB-SC modulators 220a and 220b. The SSB-SC modulators 220a and 220b are DP-MZMs, then the bias controller 240 can be used to bias the SSB-SC modulators 220a and 220b so as to suppress the left and right sidebands, respectively (the left sideband is the sideband with frequency lower than the laser carrier and the right sideband has higher frequency than the laser carrier). A DP-MZM includes two sub-MZMs and so has three DC biases—one for each sub-MZM and a phase bias for interfering the outputs of the sub-MZMs together. The bias controller 240 biases each DP-MZM for SSB-SC modulation by biasing each sub-MZMs at its minimum point and then biasing the phase with the polarity of the quadrature setting (positive versus negative quadrature) set based on which sideband is suppressed. One copy of the modulation (i.e., the analog optical input or filter signal) is sent to one arm of the DP-MZM while a 90° phase-shifted copy is sent to the other arm.

[0059] FIG.2B shows an alternative PA-LTI processor 200b that includes a laser 210, first and second SSB-SC modulators 220a and 220b (e.g., acousto-optic or dual-parallel Mach-Zehnder modulators), optional bias controller 240 (for DP-MZMs), and photodetector 232. The first and second SSB-SC modulators 220a and 220b are arranged in series between the laser 210 and the photodetector 232. Again, the first SSB-SC modulator 220a modulates a programmable filter or weight signal ^^(^^) from an AWG or ASIC (not shown) onto one sideband of the optical carrier. The second SSB-SC modulator 220b modulates an analog input signal ^^(^^), e.g., from an antenna or receiver front end, onto the other sideband of the optical carrier. The photodetector 232 transduces the output of the second SSB-SC modulator 220b into an analog LTI-filtered signal ^^(^^) suitable digitization by an ADC for additional processing in the digital domain.

[0060] LTI filtering uses the phase information in the final readout of a fast Fourier transform (FFT) of the output waveform to distinguish between positive and negative frequencies. InsteadAttorney Docket No. MIT-25674WO01 of using a spectrum analyzer or taking the magnitude of the FFT from an oscilloscope, taking the imaginary part of the FFT retains the phase information. The correct timing of each waveform can be inferred with a time-domain matched filter and by observing the gradient and location of the phases (as a time delay deterministically introduces a phase gradient).

[0061] Electro-optic architectures like the optoelectronic multiplier shown in FIG.2A and the cascaded optical modulators shown in FIG. 2B are especially well-suited for wideband use cases, which involve processing signals that occupy a large portion of spectrum, such as for spectrum monitoring. They take advantage of the wide bandwidth of optics since in the optical domain, RF signals are usually relatively narrowband. This context focuses more on pure functionality rather than cost and form factor, though it can also perform well on those metrics. This allows for real-time LFI operations on wideband signals that are otherwise impossible to do in real-time with electronics.

[0062] Narrowband use cases usually focus on contexts with simpler hardware configurations and narrowband communications. For typical (narrowband) 5G signals, for example, each wireless channel ranges from 1–10 MHz. This can be done with conventional electronic components such as RF mixers 200c as in FIG.2C or in-phase / quadrature (IQ) mixers 200d as in FIG.2D. (Unlike the inputs to an RF mixer, the inputs to an IQ mixer are phase shifted by 90 degrees with respect to each other.) This allows for LFI operations on well-defined communications protocols to offload the work from the digital processors and speed up the system latency. Such use cases include channel estimation, modulation / demodulation, and error correction. Frequency-Domain PA-LTI / LFI Framework

[0063] Frequency-domain PA-LTI filtering, also called LFI filtering, can be physicallyachieved with an analog time-domain multiplication where the filter signal ^^(^^) isappropriately programmed. The hardware setup in FIG.2A can be used to map photoelectric multiplication to arbitrary frequency-domain FIR LTI operations.

[0064] The definitions of convolution and cross-correlation for discrete signals ^^[^^] and ^^[^^] are: Convolution: CrossAttorney Docket No. MIT-25674WO01(Continuous signals are denoted with parentheses like ^^(^^)and discrete LTI-analogous signals are denoted with brackets like ^^[^^]. Physically, ^^[^^] is the amplitude / phase of the input ^^(^^) at frequency ^^, and ^^[^^] is the amplitude / phase of the weights ^^(^^) at frequency ^^.)

[0065] Suppose that ^^(^^) and ^^(^^) first exist as two electrical or electronic-domain signals:The electrical weight signal, ^^^^(^^), can be generated (in the digital domain) by an AWG, ASIC, or other appropriately programmed device, where the amplitude, ^^^^, and the upper bound of the summation, ^^, are constrained by the LTI problem and Δ^^ is a design choice, typically constrained by hardware. These electrical signals are single sideband-suppressed carrier (SSB-SC) modulated onto a laser carrier with frequency Δ^^, where ^^(^^) and ^^(^^) are on opposite sidebands of the laser carrier (i.e., ^^(^^) and ^^(^^) are modulated onto to upper and lower sidebands, respectively, or vice versa) to produce analog optical input and weight signals, ^^^^(^^) and ^^^^(^^), respectively:Then the output of the photoelectric multiplication is: Expand the expression for the photodetector output: ^^^^^^^^(^^) ∝ Im[^^^∗^(^^)^^^^(^^)]Attorney Docket No. MIT-25674WO01∝(ℱ[^^^^(^^)] ⋆ ℱ[^^^^(^^)])(^^) − (ℱ[^^^^(^^)] ⋆ ℱ[^^^^(^^)])(^^)Theoretical result: ℱ[^^^^^^^^(^^)](^^) ∝ (ℱ[^^^^(^^)] ⋆ ℱ[^^^^(^^)])(^^) − (ℱ[^^^^(^^)] ⋆ ℱ[^^^^(^^)])(^^)

[0066] Now explicitly calculate the output of this equation:Final result:

[0067] The final equation includes only the output of the positive frequencies, as they do not alias with the negative ones. Hence, this corresponds to what is measured by a balanced photodetector after the photoelectric multiplication. A single photodetector (e.g., as in FIG. 2B) measures the intensity of the light, which is changed twice, once by ^^(^^) and once by ^^(^^).

[0068] This corresponds to a frequency-domain LTI convolution. Let the LTI signal ^^[^^]correspond to the (^^ + 1) frequency mode of ^^(^^), and respectively for ^^[^^] and ^^(^^). Thus:LTI Setup: ^^[^^] ≡ ^^^^+1 → (^^ + 1)^^^^; ^^[^^] ≡ ^^^^+1 → (^^ + 1)^^^^With this setup, ^^[0] does not start at DC but at a positive frequency, making it possible to perform any FIR LTI operation. In other words, by shifting the weights up in frequency, ^^[0] can be interpreted as the first non-zero frequency mode, rather than the DC mode: LTI Convolution setup:Attorney Docket No. MIT-25674WO01

[0069] This corresponds to the LTI convolution operator: ^^^^^^^^[^^] = ^^[^^ − 1] = ^^[^^] ∗ ^^[^^] ∗ ^^[^^ − 1],where ^^^^^^^^[^^] is the LTI interpretation of the physical signal ℱ[^^^^^^^^(^^)](^^) (again where ^^^^^^^^[^^]corresponds to the (n+1) frequency mode), ^^[^^]is the theoretical LTI convolutionoutput, and ^^[^^ − 1] is the Kronecker delta function. This photoelectric multiplication directlycorresponds to an LTI convolutional output for real-valued signals and a still useful LTI-like output for complex-valued signals, except with the offset output. The small offset should rarely be problematic and in practice can be removed by applying an analog down-conversion by Δ^^ to achieve an exact LTI convolution.

[0070] This also works if the LTI signals are interpreted as the ^^[^^], ^^[^^], ^^[^^] corresponding to the nth frequency mode, except that this involves programming information at DC (0 Hz).Attorney Docket No. MIT-25674WO01 Either way, it is the physical interpretation of these three signals that maps the physics of this system onto a conventional LTI operation, thus forming the analog LFI framework.

[0071] Therefore, by appropriately programming the weight signal ^^(^^) using an AWG,ASIC, pulse shaper, or similar tool, any frequency-domain FIR LTI signal can be implemented. Both theory and experiments indicate that these analog operations are linear and invariant to shifts (in the frequency domain for LFI). Experimental Examples of LFI Signal Processing

[0072] FIGS. 3–8 illustrate experimental implementations of LFI signal processing using photoelectric multiplication (FIG. 2A). Comparing experimental implementations of LFI signal processing to traditional LTI operations validate the mathematical framework for LFI signal processing and demonstrate various use-cases.

[0073] FIGS. 3A, 3B, 4A, and 4B illustrate Wiener filtering with LFI signal processing. A Wiener filter is an LTI operation that recovers a distorted signal. For signal estimation using a Wiener Filter, for example, ^^(^^) can be determined by: (i) a stochastic statistical analysis, e.g.,calculating the covariance matrices of the random process ^^(^^) either analytically or bymeasuring an ensemble of ^^(^^) signals (where ^^(^^) is the random process and ^^(^^) is an instance of the random process); or (ii) training a deep neural network (DNN) that minimizesthe mean squared error (MSE) of the error waveform ^^(^^) − ^̂^(^^), where ^^(^^) is the originalwaveform and ^̂^(^^) is the PA-LTI estimated waveform.

[0074] The covariance matrices for a Weiner filter can be calculated as follows. Here the analog frequency modes are re-interpreted as discrete LTI signals:where ℎ[^^]is the optimal Weiner filter; ^̂^[^^]is the estimated signal; ^^^^is the mean of the ^^[^^] (the random process of the original signal); ^^^^is the mean of the ^^[^^] (the random process of the original signal after channel distortion, assuming that the cause of the distortion is known); and ^^^^^^, ^^^^^^, and ^^^^^^are the covariance matrices between the random processes ^^[^^] and ^^[^^].Attorney Docket No. MIT-25674WO01

[0075] FIGS.3A and 3B show a theoretical implementation, where the original signal ^^[^^] isdistorted into ^^[^^] (the input signal) and then ℎ[^^] is used as the Wiener filter signal(programmable signal) to get the recovered signal ^̂^[^^]from ^^[^^]. FIG.4A shows results of a corresponding LFI experimental implementation of the Wiener filter. And FIG. 4B shows an overlay between the theoretical and experimental outputs for the Wiener filter, where the error bars show the standard deviation across ten experiments. The experiment indeed matches the theory, thus validating the LFI mathematical framework above.

[0076] FIGS. 5, 6A, and 6B illustrate matched filtering with LFI signal processing. The matched filter is used to identify certain features in a target signal ^^target[^^] that is present in ^^1[^^] and ^^2[^^] (in different locations) but not in ^^3[^^]. The LFI processing is very useful for instantaneous scanning of an entire bandwidth to look for a target frequency signature. FIG.5 shows a theoretical example, FIG.6A shows the corresponding experimental results, and FIG. 6B shows the theoretical and experimental results overlaid on each other.

[0077] FIGS. 7 and 8 illustrate theoretical and experimental signal scrambling, respectively,with an LTI filter. In this example, an LTI filter ℎ[^^] scrambles a signal ^^[^^] into thescrambled signal ^^[^^] for security purposes and then uses the inverse filter ℎ−1[^^] to recover the scrambled signal ^̂^[^^] from ^^[^^]. The experiments demonstrating both a filter and its inverse using LFI processing show that a PA-LTI architecture can implement arbitrary FIR LTI functions. Generalization

[0078] The optical and electrical hardware configurations in FIGS. 2A–2D can perform the LFI operations described above as well as other LFI operations. LFI operations also work with pulse-shaping techniques like orthogonal frequency-domain multiplexing (OFDM) that shape the pulses in each frequency mode. In general, PA-LTI pulses can be pulse-shaped as well.

[0079] The LFI process is one example of the broader PA-LTI process. The LFI process can be transformed into any other time-frequency domain or wavelet basis and perform digitally programmable, fully analog LTI operations in that domain / basis, provided that the analog multiplication between ^^(^^) and ^^(^^) can be performed in that domain / basis. This is clear from the fact that linear transforms, such as the Fourier transform, do not change the underlying mathematical framework due to the principles of linearity and superposition. Example Use CasesAttorney Docket No. MIT-25674WO01

[0080] FIG. 9 illustrates how a PA-LTI receiver 100d can be used to mitigate frequency crosstalk and noise. Consider a computer 902 and transmitter 904 that transmits or broadcast 1,000 signals, each over an independent frequency channel. When these signals are transmitted together, they incur crosstalk between adjacent frequency channels and then additive white Gaussian noise (AWGN). The PA-LTI receiver 100d receives distorted versions of the original signals. Using LTI theory with knowledge of the crosstalk and the AWGN, the PA-LTI receiver 100d constructs and implements a Weiner filter to mitigate the distortions on the received signals. The PA-LTI receiver 100d performs this Weiner filtering in the analog domain by taking a frequency-domain LTI convolution between the Weiner filter (weight signal) and the input signals in real-time with low latency and energy consumption.

[0081] FIGS. 10A–10C illustrate how the PA-LTI receiver 100d can scan for a frequency- hopping signature. Suppose that a computer 1002 and transmitter 1004 broadcast a frequency- hopping signal, shown at low frequency in FIG.10A, high frequency in FIG.10B, and off (no signal) in FIG. 10C. The PA-LTI receiver 100d can perform matching filtering to track this frequency-hopping signal in real-time with weights signals programmed in the frequency domain to implement an LTI matched filter. The optical version of PA-LTI enables the entire RF spectrum (50 GHz) to be simultaneously scanned with low latency and power, while tracking changes in the frequency-hopping of the target in real-time (e.g., within nanoseconds / picoseconds) across the whole spectrum.

[0082] FIG. 11 illustrates signal scrambling with a PA-LTI transmitter 1100 and signal unscrambling with a PA-LTI receiver 100d. A computer 1102 generates a signal, which the PA-LTI transmitter 1100 scrambles as described above to produce an encrypted signal. An antenna 1104 transmits the encrypted signal over a wireless (RF) channel, which adds AWGN to the encrypted signal, to the PA-LTI receiver 100d, which filters and decrypts / unscrambles the signal.

[0083] FIG.12 shows PA-LTI filtering used for smoothing noise in a wireless sensor—here, a bio-sensor 1202 that communicates with a wireless transmitter 1204. The bio-sensor 1202, which could be implanted or worn, acquires real-time analog data (e.g., blood oxygen level) from a subject and then relays that real-time analog data to a smartphone 1206, which uses a PA-LTI smoothing filter to filter the (noisy) real-time analog data before it gets digitized, allowing for real-time readout without the need to buffer digital processing. ConclusionAttorney Docket No. MIT-25674WO01

[0084] While various inventive embodiments have been described and illustrated herein, those of ordinary skill in the art will readily envision a variety of other means and / or structures for performing the function and / or obtaining the results and / or one or more of the advantages described herein, and each of such variations and / or modifications is deemed to be within the scope of the inventive embodiments described herein. More generally, those skilled in the art will readily appreciate that all parameters, dimensions, materials, and configurations described herein are meant to be exemplary and that the actual parameters, dimensions, materials, and / or configurations will depend upon the specific application or applications for which the inventive teachings is / are used. Those skilled in the art will recognize or be able to ascertain, using no more than routine experimentation, many equivalents to the specific inventive embodiments described herein. It is, therefore, to be understood that the foregoing embodiments are presented by way of example only and that, within the scope of the appended claims and equivalents thereto, inventive embodiments may be practiced otherwise than as specifically described and claimed. Inventive embodiments of the present disclosure are directed to each individual feature, system, article, material, kit, and / or method described herein. In addition, any combination of two or more such features, systems, articles, materials, kits, and / or methods, if such features, systems, articles, materials, kits, and / or methods are not mutually inconsistent, is included within the inventive scope of the present disclosure.

[0085] Also, various inventive concepts may be embodied as one or more methods, of which an example has been provided. The acts performed as part of the method may be ordered in any suitable way. Accordingly, embodiments may be constructed in which acts are performed in an order different than illustrated, which may include performing some acts simultaneously, even though shown as sequential acts in illustrative embodiments.

[0086] All definitions, as defined and used herein, should be understood to control over dictionary definitions, definitions in documents incorporated by reference, and / or ordinary meanings of the defined terms.

[0087] The indefinite articles “a” and “an,” as used herein in the specification and in the claims, unless clearly indicated to the contrary, should be understood to mean “at least one.”

[0088] The phrase “and / or,” as used herein in the specification and in the claims, should be understood to mean “either or both” of the elements so conjoined, i.e., elements that are conjunctively present in some cases and disjunctively present in other cases. Multiple elements listed with “and / or” should be construed in the same fashion, i.e., “one or more” of the elementsAttorney Docket No. MIT-25674WO01 so conjoined. Other elements may optionally be present other than the elements specifically identified by the “and / or” clause, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, a reference to “A and / or B”, when used in conjunction with open-ended language such as “comprising” can refer, in one embodiment, to A only (optionally including elements other than B); in another embodiment, to B only (optionally including elements other than A); in yet another embodiment, to both A and B (optionally including other elements); etc.

[0089] As used herein in the specification and in the claims, “or” should be understood to have the same meaning as “and / or” as defined above. For example, when separating items in a list, “or” or “and / or” shall be interpreted as being inclusive, i.e., the inclusion of at least one, but also including more than one, of a number or list of elements, and, optionally, additional unlisted items. Only terms clearly indicated to the contrary, such as “only one of” or “exactly one of,” or, when used in the claims, “consisting of,” will refer to the inclusion of exactly one element of a number or list of elements. In general, the term “or” as used herein shall only be interpreted as indicating exclusive alternatives (i.e., “one or the other but not both”) when preceded by terms of exclusivity, such as “either,” “one of,” “only one of,” or “exactly one of.” “Consisting essentially of,” when used in the claims, shall have its ordinary meaning as used in the field of patent law.

[0090] As used herein in the specification and in the claims, the phrase “at least one,” in reference to a list of one or more elements, should be understood to mean at least one element selected from any one or more of the elements in the list of elements, but not necessarily including at least one of each and every element specifically listed within the list of elements and not excluding any combinations of elements in the list of elements. This definition also allows that elements may optionally be present other than the elements specifically identified within the list of elements to which the phrase “at least one” refers, whether related or unrelated to those elements specifically identified. Thus, as a non-limiting example, “at least one of A and B” (or, equivalently, “at least one of A or B,” or, equivalently “at least one of A and / or B”) can refer, in one embodiment, to at least one, optionally including more than one, A, with no B present (and optionally including elements other than B); in another embodiment, to at least one, optionally including more than one, B, with no A present (and optionally including elements other than A); in yet another embodiment, to at least one, optionally including more than one, A, and at least one, optionally including more than one, B (and optionally including other elements); etc.Attorney Docket No. MIT-25674WO01

[0091] In the claims, as well as in the specification above, all transitional phrases such as “comprising,” “including,” “carrying,” “having,” “containing,” “involving,” “holding,” “composed of,” and the like are to be understood to be open-ended, i.e., to mean including but not limited to. Only the transitional phrases “consisting of” and “consisting essentially of” shall be closed or semi-closed transitional phrases, respectively, as set forth in the United States Patent Office Manual of Patent Examining Procedures, Section 2111.03.

Claims

Attorney Docket No. MIT-25674WO01 CLAIMS 1. A method of programmable analog linear time-invariant (PA-LTI) processing, the method comprising: generating a programmable filter signal selected to implement a finite impulse response (FIR) linear time-invariant (LTI) filtering operation; performing single-sideband, suppressed-carrier modulation of the programmable filter signal onto a first portion of an optical carrier to produce an analog optical filter signal; performing single-sideband, suppressed-carrier modulation of an analog input signal onto a second portion of the optical carrier to produce an analog optical input signal; and multiplying the analog optical filter signal with the analog optical input signal to produce an analog LTI-filtered signal.

2. The method of claim 1, wherein the FIR LTI filtering operation comprises at least one of Wiener filtering, matched filtering, scrambling, or unscrambling.

3. The method of claim 1, wherein performing single-sideband, suppressed-carrier modulation of the programmable filter signal onto the first portion of the optical carrier comprises modulating the programmable filter signal onto one of an upper sideband or a lower sideband of the optical carrier and performing single-sideband, suppressed-carrier modulation of the analog input signal onto the second portion of the optical carrier comprises modulating the analog input signal onto the other of the upper sideband or the lower sideband of the optical carrier.

4. The method of claim 1, further comprising, before performing single-sideband, suppressed-carrier modulation of the programmable filter signal or of the analog input signal: splitting the optical carrier into a first portion and a second portion, wherein performing single-sideband, suppressed-carrier modulation of the programmable filter signal onto the optical carrier is performed on the first portion and performing single-sideband, suppressed-carrier modulation of the analog input signal onto the optical carrier is performed on the second portion.

5. The method of claim 1, wherein multiplying the analog optical filter signal with the analog optical input signal comprises detecting interference of the analog optical filter signal and the analog optical input signal with a balanced photodetector.Attorney Docket No. MIT-25674WO01 6. The method of claim 1, wherein the analog LTI-filtered signal represents a mapping of an LTI filtering operation in the frequency domain between the programmable filter signal and the analog input signal.

7. The method of claim 1, further comprising: converting the analog LTI-filtered signal into a digital LTI-filtered signal.

8. A programmable analog linear time-invariant (PA-LTI) processor comprising: a laser to emit an optical carrier; a first single-sideband, suppressed-carrier modulator, in optical communication with the laser, to modulate a first portion of the optical carrier with an analog programmable filter signal, the analog programmable filter signal selected to implement a finite impulse response (FIR) linear time-invariant (LTI) filtering operation; a second single-sideband, suppressed-carrier modulator, in optical communication with the laser, to modulate a second portion of the optical carrier with an analog input signal; and a photodetector, in optical communication with the first single-sideband, suppressed- carrier modulator and the second single-sideband, suppressed-carrier modulator, to transduce optical-domain interference of the first portion of the optical carrier modulated with the analog programmable filter signal and the second portion of the optical carrier modulated with the analog input signal into an electronic-domain analog LTI-filtered signal.

9. The PA-LTI processor of claim 8, wherein the FIR LTI filtering operation comprises at least one of Wiener filtering, matched filtering, scrambling, or unscrambling.

10. The PA-LTI processor of claim 8, wherein the first single-sideband, suppressed- carrier modulator is configured to modulate the analog programmable filter signal onto one of an upper sideband or a lower sideband of the optical carrier and the second single-sideband, suppressed-carrier modulator is configured to modulate the analog programmable filter signal onto the other of the upper sideband or the lower sideband +of the optical carrier.

11. The PA-LTI processor of claim 8, wherein the second single-sideband, suppressed- carrier modulator is in series with the first single-sideband, suppressed-carrier modulator.

12. The PA-LTI processor of claim 8, wherein the photodetector is a balanced photodetector.Attorney Docket No. MIT-25674WO01 13. The PA-LTI processor of claim 8, wherein the analog LTI-filtered signal represents a mapping of an LTI filtering operation in the frequency domain between the analog programmable filter signal and the analog input signal.

14. The PA-LTI processor of claim 8, wherein the first single-sideband, suppressed- carrier modulator is a dual-parallel Mach-Zehnder modulator comprising a first sub-Mach- Zehnder modulator (MZM) and a second sub-MZM.

15. The PA-LTI processor of claim 14, further comprising: a bias controller, operably coupled to the dual-parallel Mach-Zehnder modulator, to bias the first sub-MZM, the second sub-MZM, and interference between the first sub-MZM and the second sub-MZM.

16. The PA-LTI processor of claim 8, further comprising: an arbitrary waveform generator or an application-specific integrated circuit, operably coupled to the first single-sideband, suppressed-carrier modulator, to generate the analog programmable filter signal.

17. The PA-LTI processor of claim 8, further comprising: a beam splitter, in optical communication with the laser, the first single-sideband, suppressed-carrier modulator, and the second single-sideband, suppressed-carrier modulator, to divide the optical carrier into the first portion and the second portion, to direct the first portion to the first single-sideband, suppressed-carrier modulator, and to direct the second portion to the second single-sideband, suppressed-carrier modulator.

18. The PA-LTI processor of claim 8, further comprising: an analog-to-digital converter, operably coupled to the photodetector, to convert the electronic-domain analog LTI-filtered signal in a digital LTI-filtered signal.

19. A programmable analog linear time-invariant (PA-LTI) processor comprising: a laser to emit an optical carrier; a first dual-parallel Mach-Zehnder modulator (DP-MZM), in optical communication with the laser, to modulate one of an upper sideband band or a lower sideband of a first portion of the optical carrier with an analog programmable filter signal to yield an analog optical filter signal, the analog programmable filter signal selected to implement a finite impulse response (FIR) linear time-invariant (LTI) filtering operation;Attorney Docket No. MIT-25674WO01 a second DP-MZM, in optical communication with the laser, to modulate the other of the upper sideband band or the lower sideband of a second portion of the optical carrier with an analog input signal to yield an analog optical input signal; a beam splitter, in optical communication with the first DP-MZM and the second DP- MZM, to combine the analog optical filter signal and the analog optical input signal; a balanced photodetector, in optical communication with the beam splitter, to transduce optical-domain interference of the analog optical filter signal and the analog optical input signal into an electronic-domain analog LTI-filtered signal; and an analog-to-digital converter, operably coupled to the photodetector, to convert the electronic-domain analog LTI-filtered signal in a digital LTI-filtered signal.

20. The PA-LTI processor of claim 19, further comprising: a bias controller, operably coupled to the first DP-MZM, to bias the first DP-MZM for single-sideband, suppressed-carrier modulation.

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