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Nature Communications volume 15, Article number: 8942 (2024 ) Cite this article ellipse
Controlling the temporal evolution of an electromagnetic (EM) wave’s frequency components, the so-called time-frequency (TF) distribution, in a versatile and real-time fashion remains very challenging, especially at the high speeds (> GHz regime) required in contemporary communication, imaging, and sensing applications. We propose a general framework for manipulating the TF properties of high-speed EM waves. Specifically, the TF distribution is continuously mapped along the time domain through phase-only processing, enabling its user-defined manipulation via widely-available temporal modulation techniques. The time-mapping operations can then be inverted to reconstruct the TF-processed signal. Using off-the-shelf telecommunication components, we demonstrate arbitrary control of the TF distribution of EM waves over a full bandwidth approaching 100 GHz with nanosecond-scale programmability and MHz-level frequency resolution. We further demonstrate applications for mitigating rapidly changing frequency interference terms and the direct synthesis of fast waveforms with customized TF distributions. The reported method represents a significant advancement in TF processing of EM waves and it fulfills the stringent requirements for many modern and emerging applications.
User-defined manipulation of the temporal properties of electromagnetic (EM) waves is key to many important applications in such diverse fields as telecommunications, sensing, metrology, biomedical imaging, and quantum processing1,2,3. Within these fields, the regime of EM waves from the microwave (radio frequency, RF) to the optical domain is of particular interest because these waves enable information encoding and manipulation at high speeds. The most straightforward approach to manipulate the temporal properties of such waves is through “temporal modulation” methods4,5,6,7,8. On the other hand, a temporal wave is often described using its frequency-domain (or Fourier transform) representation, which describes the relative complex weights between the frequency components of a wave9. Many important tasks (multiplexing, impairment mitigation etc2,10,11,12.) require the wave to be manipulated along this domain through linear “frequency filtering” rather than along its temporal representation1,13,14. However, in practice, the frequency spectrum of an incoming signal is rarely stationary but changes over time. As a result, the most general manipulation of a wave requires processing its joint time-frequency (TF) distribution, which describes the temporal evolution of the wave frequency content.
a–c We target to eliminate a single frequency tone around a prescribed time \({t}_{1}\) of an electromagnetic (EM) wave, which is composed of two different single-frequency components (\({f}_{1}\) and \({f}_{2}\) , respectively). This operation cannot be implemented using temporal modulation (a) or frequency filtering (b). c We show how the target manipulation requires modulating (or filtering) the joint time-frequency (TF) energy distribution of the wave. d Principle of the proposed concept for user-defined joint TF filtering of an EM wave. An input nonstationary microwave signal consists of three different frequency components. We consider that only two components are present in the first temporal segment of the signal to illustrate the case of an input non-stationary waveform. We show here how the proposed TF filtering scheme can be designed to preserve/eliminate a prescribed set of frequency components in different signal temporal segments. For this purpose, the input microwave signal is first modulated on an optical carrier (electro-optic, E-O, conversion). The optical wave then undergoes two suitable phase transformations, implemented through temporal phase modulation with a discretized quadratic phase pattern followed by a quadratic spectral phase filtering with a group-velocity dispersive medium (e.g., a reflective chirped fibre Bragg grating). These phase transformations induce a continuous mapping of the time-varying frequency spectrum of the signal, or its two-dimensional (2D) joint TF distribution, along the time domain. This distribution can then be easily manipulated at will using temporal modulation techniques (temporal filtering pattern indicated by the red-dashed traces). For the considered microwave photonics filtering scheme, recovering the processed wave simply involves compensating the input group-velocity dispersion and a photodetection step to transfer the optical processed wave back into the electrical domain. E-O: Electrical to Optical conversion, O-E: Optical to Electrical conversion, PM: electro-optic Phase Modulator.
The realisation of this wave processing paradigm requires the implementation of a dynamic or time-varying filtering (TVF) process, in contrast to the (quasi-)static time-invariant filtering that is implemented by conventional frequency filters, as illustrated in Fig. 1a–c. Furthermore, towards a full manipulation of the wave joint TF distribution, one should be able to programme and reconfigure the filter’s spectral response in a user-defined, arbitrary manner and at a speed as fast as the frequency resolution offered by the filter15, see “Methods”. Arbitrary joint TF filtering can be readily implemented using digital signal processing (DSP). However, this requires detection and digitisation of the complex-field wave profile followed by conversion of the processed digital profile back into an analogue wave. Moreover, for real-time wave manipulation, this procedure is very challenging when the wave variations are faster than just a few hundred MHz. This requires processing the acquired signal with a rate of about a few million FTs per second and a corresponding time resolution in the microsecond range, specifications that are within the operation limits of present real-time DSP engines31. Alternatively, radio frequency (RF) filters32,33 have been demonstrated to enable discrete tuning of some of the filter’s spectral response features (e.g., centre frequency and bandwidth), though over a limited operation bandwidth, up to a few GHz. Optical filtering offers much broader operation bandwidths, and technologies are available that allow for versatile reconfigurability of the filter’s spectral response1,34. However, the reconfigurability speed of these schemes (typically in the kHz range) remains orders of magnitude slower than the frequency resolution they can offer (typically in the GHz range), thus being unsuited for general TF signal manipulation tasks. Photonic processing has also been used for filtering high-speed microwave signals, so-called microwave photonic filters (MPFs)14. Some of these technologies offer an important degree of reconfigurability though still largely insufficient to enable a full arbitrary control of the wave joint TF distribution, either because of their intrinsically slow reconfiguration speed35,36,37, because fast tuning is restricted to some of the filter’s main specifications only (e.g., central frequency, or phase shift)25,38,39,40,41, and/or because they are limited to implementing a very specific TF filtering operation42.
In this work, we propose a concept for user-defined real-time manipulation of the joint TF distribution of EM waves directly in the analogue domain, ideally suited for operation on high-speed waves. Using this concept, we conceive and demonstrate a photonics scheme for TF processing of microwave and optical signals. The proposed approach combines the versatility of the DSP approach with the performance (e.g., processing speed and bandwidth) of a photonic solution. As illustrated in Fig. 1d, our strategy involves mapping the TF distribution (the STFT) of the incoming wave along the time domain in a continuous and gapless manner41,43,44,45,46, which in turn enables a user-defined manipulation of the wave’s TF distribution through the many available temporal modulation techniques. Since the time-mapped STFT is achieved using two consecutive phase transformations (along the temporal and spectral domains, respectively), the processed wave can be recovered by simply applying the opposite phase manipulations. This platform allows us to achieve an arbitrary manipulation of the joint TF distribution of the input microwave signal over a full bandwidth up to 92 GHz, with rapid tuning speeds in the nanosecond range and with a fine frequency resolution, down to a few hundreds of MHz. Through the implementation of the temporal filtering step using electro-optic modulation, the unit can be programmed electronically to provide any desired dynamic spectral response, with a reconfigurability speed inherently determined by the filter’s frequency resolution, as needed for full manipulation of the wave TF distribution. We demonstrate the use of this concept for the realisation of important functionalities beyond the potential of present technologies, including the mitigation of nonstationary interference terms in broadband signals and the direct synthesis of high-speed waves with user-defined sophisticated TF distributions.
The STFT or spectrogram (squared magnitude of the STFT) of a given input signal provides the Fourier transform, or frequency spectrum, of consecutively truncated short segments of this input signal15, effectively providing a faithful representation of the signal’s joint TF representation. Figure 1d shows a schematic of a specific implementation of the proposed concept for real-time joint TF filtering aimed at processing microwave signals using a photonic platform. The microwave signal under test (SUT) is first upconverted to the optical domain, and the STFT of the optical wave is then continuously mapped along the time domain using a configuration referred to as a Talbot array illuminator (TAI) spectrogram46 (see “Methods”). In particular, this method enables capturing changes in the signal frequency spectrum every prescribed analysis period \({T}_{r}\) , in such a way that each of the analysed spectra is time mapped along adjacent time slots, each with a duration \({T}_{r}\) . As expected for an STFT analysis, each truncated signal spectrum exhibits a frequency resolution (minimum spacing between two frequency components that are resolved by the Fourier representation) of the order of the inverse of \({T}_{r}\) 46. Once the STFT is mapped along the time domain, the two-dimensional (2D) joint TF distribution of the SUT can be modified at will using temporal modulation methods. Specifically, the frequency spectrum information of the SUT can be modified in amplitude and/or in phase, every analysis period \({T}_{r}\) . Moreover, each spectrum can be manipulated with a resolution that is ultimately limited by the frequency resolution of the performed spectrogram. Notice that this requires the temporal filtering function to be sufficiently fast, of the order of the analysis bandwidth \(B\) of the implemented STFT analysis (see “Methods”).
To recover the processed waveform, one just needs to invert the time and frequency-domain phase transformations used for the calculation of the STFT. This means propagating the processed waveform through a dispersive medium providing the exact opposite spectral phase response of the first one (i.e., opposite dispersion\(-\ddot{\phi }\) ), followed by a similar temporal phase modulation compensation process. Since our scheme is aimed at TF processing of amplitude encoded optical signals (upconverted microwave signals), the latest phase modulation compensation is not necessary, see Fig. 1d. Indeed, the processed microwave signal is down-converted to the microwave domain by photodetection, irrespective of the residual optical phase.
The experimental setup is shown at the centre of the figure. a Temporal trace of the SUT, involving two linearly-chirped sinusoidal waveforms (‘S1’ and ‘S2’) and the zoomed-in traces corresponding to three different sections of the SUT. b Numerical spectrogram of the generated SUT with pre-compensation, depicting the temporal evolution of the frequency spectrum and the uneven intensity. c Measured TM-SP with several zooms around three different relevant time analysis periods. We note that the TM-SP function exhibits a significant component at the centre location of each analysis window, corresponding to the optical carrier frequency; however, this term is strongly attenuated in the representation of the TM-SP traces shown here to facilitate observation and interpretation of the obtained TF distributions of the analysed optical signals. d 2D representation of the signal joint TF distribution that is numerically rescaled from the output measured temporal trace of (c) showing a close-up of the obtained distribution near the cross point of the two chirped signals. e Three zooms of the temporal filtering pattern with each pulse width of ∼ \({t}_{s}\) , reconfigured every \({T}_{r}\) = 1.5 ns. The varying time location of the filtering mask along each time slot aligns with the corresponding frequency terms of the increasing frequency-chirp component (S1) in order to filter in this component from the SUT. f The measured output after temporal modulation with the designed filtering pattern shows that the temporal pulses representing the frequency components of S2 are strongly attenuated. g The 2D representation of the TM-SP that is obtained after the temporal modulation. h Temporal trace of the processed microwave signal recovered after dispersion compensation and photo-detection, with the same zoomed-in regions as the input in (a). i Numerical spectrogram of the processed measured signal, confirming that S2 is nearly suppressed and a pure linear chirp with increasing frequency (S1) is recovered. SUT: Signal Under Test, TM-SP: Time-Mapped Spectrogram, STFT: Short-time Fourier Transform, 2D: two-dimensional, CW: Continuous-Wave laser, MZM: electro-optic Mach-Zehnder Modulator, PM: electro-optic Phase Modulator, LCFBG: Linearly Chirped Fibre Bragg Grating, PD: Photo-Diode.
To provide further evidence on the reconfigurability of the spectral response that is provided by the demonstrated time-varying filtering scheme, in terms of the passband shape, bandwidth, and tuning frequency, we have performed a standard characterisation of the RF spectral response of the microwave photonics filtering scheme in Fig. 2. This characterisation has been carried out under different temporal filtering specifications, i.e., by programming the corresponding modulation pattern, and the obtained results are presented in the Supplementary Fig. 5.
In dynamic practical environments, involving 5G and automotive radar systems29,47,48, interference or jamming can significantly impact detection or sensing performance. In automotive radar systems, mutual interference from other radars can degrade sensitivity and detection capabilities, potentially leading to hazardous situations47. These radar systems typically operate in the millimetre-wave range (with potential instantaneous bandwidth exceeding 30 GHz)49 and necessitate real-time mitigation of interferences and noise. In our subsequent results, we demonstrate the implementation of our proposal for user-defined interference mitigation over tens-of-GHz of instantaneous bandwidth through a suitable design of the time-frequency filtering response.
We use an input SUT consisting of a set of frequency-changing interferences along a broadband double-chirped signal. This spectrogram scheme is redesigned to offer a narrower frequency resolution (i.e., 110 MHz) while maintaining the full analysis bandwidth to \(1/{t}_{s} \sim\) 92 GHz, corresponding to ∼ 836 analysis points per spectrum. We achieved this by increasing the phase modulation period (time resolution of the implemented spectrogram) to \({T}_{r} \sim 9\,{{{\rm{ns}}}}\) (see “Methods”), as shown in Fig. 3. This scheme is then used to remove the undesired interferences from the SUT. This requires the design of a multi-passband filtering mask consisting of four square-like pulses along every analysis period (two pulses for each of the chirped components to be filtered in) in which the inter-pulse spacing is changed every analysis period (\({T}_{r}\) ) to select the four pulses corresponding to the two chirped signals. Figure 3e, f shows the measured TM-SP trace after the temporal filtering process, as well as the corresponding 2D representation. As expected, the system performs a sufficiently high selectivity to efficiently remove interference \({i}_{4}\) which is spaced by 1 GHz with respect to the double-chirped waveform. A sharper selectivity of 400 MHz has also been experimentally demonstrated (see Fig. 4 in the Supplementary document).
a Measured temporal waveform of the SUT, which is composed of two linear-chirped signals (‘S1’ and ‘S2’) and several interferences (‘\({i}_{1}\) ’ – ‘\({i}_{4}\) ’) with varying frequency locations and temporal durations. b Intensity of the numerical STFT, or spectrogram, of the SUT. c Measured TM-SP trace at the output of the STFT scheme, with zooms-in around the location of the interferences, each zoomed waveform extending over on analysis period (\({T}_{r} \sim 9\,{{{\rm{ns}}}}\) ). The top axis in each zoomed plot corresponds to the equivalent relative frequency axis. d A 2D representation of the measured TM-SP and the close-ups of the interference sections. The different frequency interference components are well discriminated even when extending over a duration as short as the analysis period (9 ns for ‘\({i}_{4}\) ’). e The measured output temporal trace (TM-SP) after the temporal filtering, with the same zoom-in regions as in (c), and (f), The corresponding 2D representation of the measured TM-SP of the processed signal, confirming that the unwanted interference components are strongly attenuated. SUT: Signal Under Test, STFT: Short-time Fourier Transform, TM-SP: Time-Mapped Spectrogram, 2D: two-dimensional.
The experimental setup is shown on the left and the results are captured from the real-time oscilloscope. a Measured temporal waveform of the SUT, which is composed of periodic sinc-like pulses, forming a nearly uniform spectrogram distribution, over a frequency range up to 23 GHz (positive side) along a duration of ∼ 300 ns. The frequency roll-off shown in the numerical STFT shows a good agreement with the measured frequency spectrum of the input pulse train (see Fig. 7 in the Supplementary file). b 1D temporal filtering mask mapped from the 2D image of the Mona Lisa painting used to manipulate the TF distribution of the SUT, and a zoom of the user-defined temporal filtering pattern of arbitrary shape. c The 2D representation of the TM-SP trace that is measured after the temporal filtering shows that the synthesised spectrogram closely follows the contour of the target image. d Using the same SUT as in (a), the filtering mask is now designed to craft a joint TF distribution resembling the Chinese character shown at the top left, which is achieved by changing the passband width and centre location of the implemented temporal filtering. The short pulses at the edge of each time window are purposedly introduced to facilitate the synchronisation between the filtering mask and TM-SP. e The 2D representation of the TM-SP trace that is measured after the temporal filtering confirms that the synthesised spectrogram closely resembles the target character. The frequency roll-off in the synthesised spectrogram is consistent with that of the input sinc-like pulses (see Fig. 8 in the Supplementary file). SUT: Signal Under Test, STFT: Short-time Fourier Transform, TM-SP: Time-Mapped Spectrogram, 2D: two-dimensional, CW: Continuous-Wave laser, MZM: electro-optic Mach-Zehnder Modulator, PM: electro-optic Phase Modulator, LCFBG: Linearly Chirped Fibre Bragg Grating.
In a final experiment, we showcase the capability of the proposed method to reshape the joint TF distribution of an input stationary broadband wave (Fig. 4). In this case, instead of using simple rectangular pulses for the time-frequency filtering mask, we design more complex filtering masks following arbitrary prescribed patterns. The STFT scheme is now reconfigured to achieve a time resolution (and analysis period) of \({T}_{r} \sim 4.5\,{{{\rm{ns}}}}\) , corresponding to a frequency resolution of ∼ 220 MHz, and a full analysis bandwidth of \(1/{t}_{s} \sim\) 46 GHz (see “Methods”). The input SUT is a train of sinc-like pulses, periodically spaced by \({T}_{r}/3\) , purposely designed to exhibit a nearly uniform joint TF energy distribution along its entire duration and full bandwidth (∼ 46 GHz), see Fig. 4a. An arbitrarily prescribed 2D image can then be inscribed along the joint TF energy distribution of the wave within the resolution specifications of the performed TF filtering scheme. In the first example, we successfully re-shape the wave TF distribution into a 2D pattern that resembles an image of the Mona Lisa painting, highlighting the ability to apply different filtering patterns across consecutive windows, see Fig. 4b and “Methods”. The measured TM-SP trace after modulation, Fig. 4c, confirms that the joint TF distribution of the processed wave has a contour proportional to that of the Mona Lisa image, though with the expected degraded resolution mainly due to the down-sampling implemented in the filtering mask generation process, and the limited bandwidth of the 28 GHz scope used for capturing the TM-SP traces. In a second instance, we use the same SUT and design a temporal filtering pattern, Fig. 4d aimed at synthesising a TF distribution following a 2D image of the Chinese character “中”, Fig. 4e. For this purpose, the temporal filtering pattern consists of a sequence of square-like passband pulses with suitable varying widths and centre locations, according to the desired frequency spectrum profile along each of the consecutive analysis periods.
We have demonstrated a technique to manipulate in real-time the TF distribution of a wave over broad bandwidths, with high temporal and frequency resolutions and an unprecedented degree of versatility. The proposed concept is demonstrated using off-the-shelf telecommunication components, and it is based on processes and technologies that are widely available in many other frequency regions as well. As such, we anticipate the potential of application of this same concept or similar strategies nearly across the entire EM spectrum. It would be relatively straightforward to utilise the demonstrated fibre-optics scheme for processing ultrafast light waves with frequency bandwidths well beyond those reported here, into the THz regime and above. Towards this aim, we envision the use of nonlinear optics mechanisms, such as cross-phase modulation or four-wave mixing50,51,52, in dedicated optical fibres or waveguides for the implementation of the needed temporal phase modulation process at much higher speeds. The proposed concept represents a significant advancement towards manipulation and control of the TF properties of EM waves, and we believe that it will prove particularly useful for the realisation of the cognitive and software-defined paradigms in future wireless and optical communications, intelligent remote sensing platforms and other systems based on broadband microwave, THz or optical waves.
In the proof-of-concept experiments, the filtering pattern is pre-designed accordingly to manipulate the incoming signal. In this case, the frequency components of the incoming signal need to be precisely known in advance. As such, this approach is ideally suited for applications where the signal characteristics are predetermined, such as in frequency hopping communication systems22,53 or chirped radar platforms47. Moreover, our approach also holds promise for processing unknown incoming waveforms through the generation of the filtering pattern on the fly, either in the digital domain or in an entirely analogue manner.
This work demonstrates advanced versatility based on a photonic method and with low latency, making it well-suited for real-time high-speed signal analysis and processing. Detailed latency estimates provided in the Supplementary (Latency estimation) confirm that the proposed photonic method outperforms DSP engines by orders of magnitude. Nonetheless, it is worth noting that our proposed scheme relies on high-speed electro-optic modulators and electronic AWGs, which may translate into significant cost and power consumption in practical implementations. However, the performance benefits of our proposed system, including high-resolution, real-time TF manipulation over tens of GHz bandwidth, outweigh the added complexity and cost. Equivalent performance specifications achieved through, e.g., DSP approaches, would require channelising a ∼ 40 GHz bandwidth signal into smaller few-GHz bandwidth segments54, as each DSP unit has limited real-time operation bandwidth. In turn, this would translate into a considerable cost and complexity for the resulting overall DSP system. Thus, the low latency advantage and the capability to handle such high-frequency waves in a real-time manner and with notable versatility make our proposed scheme particularly interesting for advanced applications.
The STFT of a signal is generated by applying a windowing function to successive short temporal segments of the signal, then performing a Fourier transform on each segment15. Changes in the signal spectrum are then captured over each time slot with a duration determined by the width of each analysed signal segment. This is often referred to as the time resolution of the STFT15. Moreover, the uncertainty principle of the Fourier transform implies that the frequency spectrum of each truncated signal segment exhibits a resolution (minimum spacing between two frequency components that are resolved by the Fourier representation) that is of the order of the inverse of the STFT time resolution15. As such, for a full manipulation of the TF distribution of a signal, one must be able to modify the signal spectrum with a prescribed minimum frequency resolution, at a speed (i.e., with a refresh rate) of the order of this frequency resolution.
The time-mapped STFT process is based on a temporal array illuminator (TAI) phase modulation of the input waveform (SUT), followed by a quadratic spectral phase filtering device46. In particular, the temporal phase is composed of a periodic set of multi-level temporal phase shifts that change in groups of \(q\) discrete steps, each of time width of \({t}_{s}\) , such that the \({n}^{{th}}\) phase shift can be expressed as:
where \(n={{\mathrm{1,2}}},\ldots,q\) represents the bin number of each step of the phase modulation profile and \(\sigma=\pm 1\) . This discrete phase profile then repeats along the time domain, resulting in a periodic phase modulation pattern with a period \({T}_{r}=q{t}_{s}\) . Examples of the temporal phase modulation implemented in the experimental demonstrations reported here are shown in Supplementary Fig. 1. The following spectral phase filtering process can, in practice, be implemented using a second-order dispersive medium, providing a second-order dispersion \(\ddot{\phi }\) (defined as the slope of the medium’s linear group delay as a function of the radial frequency) that satisfies the following condition:
In practice, this spectral phase filtering can be implemented using a simple group-velocity dispersive line (e.g., an appropriate length of single-mode optical fibre). This scheme calculates the Fourier transform (FT) over each consecutive section of duration \({T}_{r}\) of the input SUT, in such a way that the resulting consecutive spectra are mapped along the time domain, each extending over a time slot of duration \({T}_{r}\) . This mapping is produced according to the following frequency-to-time mapping law
where \(\Delta {\omega }_{t}\) and \(\Delta t\) are the radial frequency of the input signal and time variable at the system output, both relative to the centre of each analysis window. It can be shown that the maximum frequency extension of the input SUT, or instantaneous analysis bandwidth \(B\) of the implemented spectrogram, is just limited by the inverse of the time width \({t}_{s}\) of each single-phase step in the modulation profile46, \(B \sim 2{{{\rm{\pi }}}}/{t}_{s}\) . This latest condition ensures that the consecutive time-mapped spectra do not interfere with each other. Moreover, the signal frequency bandwidth must be within the spectrogram analysis bandwidth (\(B\) ). This implies that46:
where \(\Delta {\upsilon }_{{SUT}}\) is the full bandwidth of the SUT expressed in natural frequency units. Thus, the full frequency bandwidth of the SUT is ultimately limited by the speed or bandwidth of the phase modulation apparatus in the system.
Through the proposed method, spectral changes in the SUT are captured in every analysis period of \({T}_{r}\) . As such, the time resolution of the obtained spectrogram analysis is directly determined by the period length of the phase pattern, \({T}_{r}\) (duration of each analysed signal section) and the frequency resolution is inversely related to the time resolution, namely, \(\delta {\omega }_{t} \sim 2{{{\rm{\pi }}}}/{T}_{r}\) . The maximum number of points of the conducted spectral analysis is then defined as \(N \sim B/\delta {\omega }_{t}\) , which is determined by the number of phase steps per period in the phase modulation pattern25. As discussed above, at the system output, consecutive time-mapped spectra are spaced by the period \({T}_{r}=q{t}_{s}\) , in such a way that the processing speed (number of FTs calculated per unit of time) is determined by the inverse of this period, namely, \(1/{T}_{r}=1/\left(q{t}_{s}\right)\) . A higher processing speed can then be achieved by fixing a shorter period \({T}_{r}\) in the design (either by using a shorter \({t}_{s}\) or a smaller q factor), but this would unavoidably worsen the achieved frequency resolution. A fine frequency resolution requires a longer analysis window \({T}_{r}=q{t}_{s}\) . The limiting factor for the maximum width of \({T}_{r}\) is the required dispersion, as indicated by Eq. 2. For instance, as demonstrated in the results, a 660 MHz frequency resolution was achieved with a dispersion of \(\ddot{\phi } \sim {{\mathrm{2,600}}}{{{{\rm{ps}}}}}^{2}/{{{\rm{rad}}}}\) and a finer frequency resolution of 110 MHz required the use of a larger dispersion, i.e., \(\ddot{\phi } \sim {{\mathrm{15,494}}}{{{{\rm{ps}}}}}^{2}/{{{\rm{rad}}}}\) . Thus, an even finer frequency resolution would require an even larger amount of dispersion. However, a finer frequency resolution would also lead to a larger time resolution and slower tuning speed, as discussed below. On the other hand, the operation bandwidth of the system is related to the time width of each step of the TAI phase profile, \({B} \sim 2{{{\rm{\pi }}}}/{t}_{s}\) (See supplementary Table 1). This latest specification is mainly limited by the sampling rate of the arbitrary waveform generator that is used to generate the TAI phase. For some of the results presented in this paper, the phase step width is set as a single sampling point of the AWG, i.e., \({t}_{s} \sim\) 1/92 GHz, fully exploiting the AWG operation bandwidth of 92 GHz. To overcome this bottleneck, a higher-speed AWG, such as a commercially available 256 GS/s AWG55, could be used to generate a TAI phase with a narrower step width down to \({t}_{s} \sim\) 1/256 GHz, though this would significantly increase the cost of the system. Note that to accommodate such a TAI phase with a narrower step width, a phase modulator with a larger bandwidth would be also required56,57.
To analyse the signal in Fig. 2, a temporal phase modulation pattern is designed following Eq. (1) with \(q=\) 139 phase levels, each with a width of \({t}_{s} \sim 10.8\,{{{\rm{ps}}}}\) (see Supplementary Fig. 1a). This sets the maximum full analysis bandwidth of the performed spectrogram to \(1/{t}_{s} \sim\) 92 GHz, the analysis period (and time resolution) to \({T}_{r}=q\times {t}_{s} \sim\) 1.5 ns, and the corresponding frequency resolution \(\delta \omega \sim 2{{\pi }}\times 660\,{{{\rm{MHz}}}}\) , allowing for a total of 139 analysis points per spectrum. The phase-modulated light wave is then propagated through a reflective linearly chirped fibre Bragg grating (LCFBG), implementing the required group-velocity dispersion, according to Eq. (2), with a dispersion coefficient \(\ddot{\phi } \sim {{\mathrm{2,600}}}\,{{{{\rm{ps}}}}}^{2}/{{{\rm{rad}}}}\) , well over the full bandwidth of the optical modulated SUT. The frequency resolution of the obtained spectrogram can be calculated by estimating the minimum frequency spacing between two adjacent pulses that can be clearly separated, see the zoomed-in region in Fig. 2. The predicted frequency resolution of ∼ 660 MHz would require capturing the TM-SP with a resolution of ∼ 10.8 ps (phase modulation step width). The measured frequency resolution is, however, ∼ 2 GHz rather than 660 MHz, which is mainly limited by the sampling rate of the real-time oscilloscope and the photodiode bandwidth (see Fig. 3 in the Supplementary document). The input and output temporal waveforms shown in Fig. 2 are captured by a high-speed (70 GHz) electrical sampling scope, whereas the TM-SPs are measured using a 28 GHz real-time oscilloscope.
The temporal phase modulation pattern for the results in Fig. 3 is designed with \(q=\) 836 phase levels, each with a length of \({t}_{s} \sim 10.8\,{{{\rm{ps}}}}\) and the period length is \({T}_{r}=9\,{{{\rm{ns}}}}\) (see Supplementary Fig. 1b), corresponding to a LCFBG with \(\ddot{\phi } \sim {15,494}\,{{{{\rm{ps}}}}}^{2}/{{{\rm{rad}}}}\) . To show the reconfigurability of the system, the temporal phase used in Fig. 4 is set as \(q=207\) , \({t}_{s}=20.2\,{{{\rm{ps}}}}\) with a period of 4.55 ns and with the same LCFBG as that used for the results in Fig. 3. (see Supplementary Fig. 1c). Figure 3a, b shows the measured SUT and its corresponding numerical spectrogram. The frequency of the double-chirped signal (components denoted as \({S}_{1}\) and \({S}_{2}\) ) ranges from 120 MHz to 10 GHz, and the isolated interferences (denoted as ‘\({i}_{m}\) ’, with \(m\) = 1, 2, 3, 4) have durations ranging from \({20\times T}_{r}\) to \({1\times T}_{r}\) and central frequencies ranging from of 8 GHz to 20 GHz. As shown in Fig. 3c, d, the TM-SP enables accurate identification of the individual chirps (i.e., S1 and S2) and frequency interferences of the SUT at the expected time and frequency locations.
To process the time-mapped STFT waveform, the temporal modulation patterns are predesigned according to the frequency-to-time mapping law in Eq. (3). As described, for the results in Figs. 2 and 3, a set of 5 rectangular pulses is used to compose the filtering mask per analysis period of duration \({T}_{r}\) . As such, the tuning speed is equal to the time resolution, \({T}_{r}\) , which is inversely related to the frequency resolution, as mentioned above. This implies that a faster tuning speed can be achieved by use of a shorter period \({T}_{r}\) . In turn, this would result in a higher processing speed, though at the cost of a poorer frequency resolution. In order to precisely manipulate the waveform with a resolution that is limited by the frequency resolution of the TF distribution, a modulation time resolution is needed of the order of the TAI phase modulation step duration \({t}_{s}\) . Note that the time width of the modulation pulse can be user-defined and adjusted accordingly. However, insufficient bandwidth of the time-domain filtering pattern (i.e., the use of wider filtering pulses) may result in a deterioration of the filter’s frequency resolution, preventing, for instance, being able to filter out one of the two closely spaced frequencies, while the maximum bandwidth and time resolution of the performed filtering process remain essentially unaffected (see Supplementary – analysis of the impact of the filtering mask bandwidth on system characteristics and Supplementary Fig. 10).
In general, the temporal filtering mask can be mathematically expressed as:
where t defines the time variable of the temporal pattern, \({{{\rm{rect}}}}\left(\frac{t-T}{\Delta t}\right)\) denotes a rectangular function centred at T extending over a total duration of \(\Delta t\) , and \(p{T}_{r}\) identifies the central time location of each of the analysis periods (with \(p=0,\pm 1,\pm 2,\ldots\) ).
The basic set-up to implement the proposed joint TF filtering approach is shown in Supplementary Fig. 6. The optical carrier for the results in Fig. 2 is generated from a tuneable continuous wave laser (CoBrite-DX) with a central wavelength of 1553.3 nm, and the one for Figs. 3 and 4 is generated from an NKT laser (K80-152-14) centred at 1550 nm. The input microwave SUTs and temporal filtering patterns are all generated from an electronic AWG with a sampling rate of 92 GS/s (Keysight M8196A). An RF amplifier with a bandwidth of 50 GHz (Optilab MD-50) is used to boost the power of the SUT. The input SUT is modulated on the optical carrier through a 40 GHz electro-optic MZM (EOSPACE). To ensure that the optical modulated signal exhibits a nearly flat amplitude over different frequency components, the microwave SUT is properly designed to pre-compensate for the spectral roll-off of the RF amplifier and MZM (see Supplementary Fig. 2).
For the waveform recovery experiments (Figs. 2 and 3), the optical carrier must be kept in the optical modulated SUT. As a result, the MZM is biased to operate over its linear region (∼ 4.2 V). Concerning the results in Fig. 4, the input SUT is designed as a periodic set of short pulses, each with a shape defined by the function \({sinc}[46\times {10}^{9}(t\pm {T}_{r}/3)]\) to achieve a nearly flat spectrum along the 46 GHz bandwidth. In these experiments, the MZM is biased at 6.1 V to suppress the optical carrier. In all cases, an optical polarisation controller is used before the MZM to optimise the electro-optic modulation process. The optical SUT then enters the TAI-based STFT unit, composed of a 40 GHz electro-optic phase modulator with a half-wave voltage of 3.1 V at 1 GHz (EOSPACE) driven by another channel of the same AWG boosted by a 50 GHz RF amplifier (Optilab MD-50). As described above, the temporal phase pattern is pre-designed according to Eq. (1). An Erbium-Doped Fibre Amplifier (Pritel EDFA) is used before the subsequent dispersive element to compensate for the loss along the system. Concerning the dispersive elements, they are chosen to provide the corresponding amount of group-velocity dispersion, according to Eq. (2), over a sufficiently broad spectral bandwidth (exceeding the optical modulated-SUT bandwidth). For the results presented in Fig. 2, we use an LCFBG providing a total second-order dispersion \(\ddot{\phi } \sim {2,600}\,{{{{\rm{ps}}}}}^{2}/{{{\rm{rad}}}}\) , whereas an LCFBG with \(\ddot{\phi } \sim {15,494}\,{{{{\rm{ps}}}}}^{2}/{{{\rm{rad}}}}\) is used for the results of Figs. 3 and 4. Note that other devices can be employed to provide the desired amount of dispersion, such as fibre optic cables. The designed temporal filtering pattern is generated from the third channel of the same AWG followed by a 40 GHz RF amplifier (MD-40) and modulated on the optical pulses after the first LCFBG, using a second MZM with 40 GHz bandwidth and 37 dB extinction ratio (Optilab) following an optical polarisation controller. This MZM is biased at 6.2 V to achieve a high dynamic range. Finally, the microwave TF-filtered waveform is recovered after propagation through a second LCFBG and the O-E conversion using a 50 GHz bandwidth photo-detector (Finisar XPDV2120R). Note that the dispersion of the second LCFBG is the exact opposite of that of the first LCFBG used for the TAI spectrogram. An optical tuneable delay line is placed between the first LCFBG and the second MZM to synchronise the TM-SP trace of the SUT and the temporal filtering mask. The TM-SP trace of the SUTs, the output after temporal filtering, and the microwave SUTs in Figs. 3 and 4 are captured using a 28-GHz bandwidth real-time oscilloscope (Agilent DSO-X 92804 A) without averaging. The real-time oscilloscope is triggered through a reference signal from the electronic AWG with the same length as each incoming microwave SUT. The input microwave SUT and the output recovered processed microwave signal in Fig. 2 are captured using a 70-GHz bandwidth electrical sampling oscilloscope (Tektronix CSA8200), while the time-mapped spectrogram waveforms are obtained from the 28 GHz real-time oscilloscope. Note that the sampling rate of the sampling oscilloscope is set at 2,000 GS/s, and the measured waveform is down-sampled to 92 GS/s to obtain a proper frequency and time resolution in the numerical STFT of the SUT and recovered signal in Fig. 2. Further processing of the captured temporal traces, namely resampling, retiming, and rescaling to the frequency domain, is performed numerically offline. Specifically, each of the measured TM-SP traces is sampled in the real-time scope at a rate of 80 GS/s. This is then resampled to 92 GS/s to ensure uniform results throughout the analysis. The obtained temporal waveform is rescaled to have the frequency information along every analysis period (\({T}_{r}\) ), according to the corresponding time-mapped frequency factor, aligning the data with the corresponding frequency axis. The resulting 1D vector, now containing the rescaled frequency information along every defined time window, is then reshaped into a 2D matrix. This reshaping allows for the 2D representation of the TF distribution, as depicted in Figs. 2–4. Additional details on the reshaping process are shown in Supplementary Fig. 9.
The paper and/or the supplementary information contain all the data needed to evaluate the conclusions. Correspondence and material requests should be addressed to the corresponding author.
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