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We introduce a novel technique for broadband RF disambiguation which exploits a known jitter imparted onto the sampling rate of an optical pulse source in a subsampled analog optical link. Coarse disambiguation to bandwidths equal to the sample rate is achieved using pure tones as example waveforms by comparing the amplitude of the jitter-induced sidebands relative to the measured signal within the fundamental Nyquist band (frep/2). This sampling technique allows for ultra-wideband signal recovery with a single measurement. In a first-of-its-kind photonics demonstration we show reliable disambiguation for signals with center frequencies spanning 1 MHz – 40 GHz.
© 2014 Optical Society of America
1. Motivation
For over a decade, interest has increased in using low-jitter optical sampling techniques to quantize signals at frequencies outside the reach of electronics [1]. In many photonic A/D architectures, intentional aliasing (subsampling) has been used to optically downconvert high frequency RF signals to within the bandwidth of high-fidelity, low-frequency electronic A/D converters using various techniques [2–4]. In these applications, the RF input bandwidth is limited such that only the desired frequency band of interest is downconverted. In wideband sensing applications where it is desirable to have high-sensitivity detection across 100 GHz bandwidth simultaneously, such bandwidth limitations can not be tolerated. Therefore, techniques which exploit the benefits (low-jitter, wide-bandwidth) of optical sampling architectures to downconvert [5–8] in concert with disambiguation techniques are highly desirable.
Sampled analog photonic links have shown performance that rivals that of the standard continuous wave (CW) architecture [8], in addition to providing a way to circumvent the limitations imposed by stimulated Brillouin scattering in long-haul applications, allowing a substantial increase in optical launch power [9]. These photonic sampling architectures along with advanced signal processing techniques offer a realistic pathway to performing ultra-broadband signal detection in acceptably short processing times via subsampling— sampling signals at a rate less than the sampling theorem would require given the aggregate bandwidth of interest.
Recently there was a demonstration of an all electronic technique using RF sampling with chirped phase modulation that allows for Nyquist allocation and signal recovery in conjunction with post-process compressive sensing reconstruction algorithms [10]. Here we implement a similar technique optically, where the analog description of achieving down-conversion (spectral-folding, or aliasing) of RF signals [11] is used as a basis to derive the expressions for the sidelobe-to-peak ratio (SPR) in the measured RF power spectrum as it occurs with the application of some known phase/frequency modulation. This disambiguation technique is based on this modulation which takes the form of an intentionally-introduced ‘jitter’ optical pulse train within a subsampled analog link architecture. We impart this frequency jitter on our 5 GHz optical comb pulse source via frequency modulation of the comb generation signal, and its SPR growth is shown to be proportional to the square of comb line index, n. By exploiting this predictable growth of the frequency-modulation sidebands in the aliased output of the link we achieve coarse alias band disambiguation across a 40 GHz bandwidth with just a single direct measurement that requires no additional post-processing (as in, for example, compressed sampling techniques). Here we show its effectiveness using pure sinusoidal tones, but the theory is generalized and can also be applied to more complex signals with bandwidth. To our knowledge, the technique and its optical implementation, and the rigorous demonstration of its ability to achieve broadband RF disambiguation, are the first of their kind.
2. Theory
It is well known [14] that the phase power spectral density arising from fluctuations in the repetition rate (the temporal jitter) of a pulse train grows proportionally to n2 in the RF power spectrum. Here n = fn/frep is an integer number representing the ratio of the n-th harmonic of the pulse train repetition rate divided by the fundamental repetition rate. When this pulsetrain is used to sample an incoming RF signal, the phase noise from the pulse train is transferred to the input RF signal. In a subsampling link architecture, i.e. where the input RF signal is not required to reside in the fundamental Nyquist band (0 ≤ f ≤ frep/2) but the output measurement bandwidth is limited to the fundamental Nyquist band, the phase noise sidebands may be used to coarsely discern the signal’s original center frequency. In this work we apply a well-defined jitter, or fluctuation in the repetition rate of the sampling optical pulse train, through frequency modulation of the signal used to generate the sampling pulse train in an optical comb generator (see Fig. 1).
To show how the introduction of a known frequency dither (jitter) may be used to achieve wideband signal disambiguation, we begin by analyzing the time domain expression for the photocurrent at the output of a sampled analog optical link. The photocurrent derived from one output of the Mach-Zehnder intensity modulator (MZM) may be written as [8]
where p(t) is the temporal power profile of the sampling pulse train, vin(t) is the input RF voltage applied to the MZM (not limited to pure sinusoids, to be discussed further on below), hmzm(t) is the impulse response of the MZM, hpd(t) is the impulse response of the photodiode, hlpf(t) is the impulse response of the low-pass filter used to restrict the link output to the fundamental Nyquist band, and * denotes convolution. Following the analysis in [8], for input signals in the small-signal regime the double-sided RF power spectrum may be written as (for a quadrature-biased link) Here Psp(ω) is the spectrum of the pulse intensity, Hpd(ω) is the frequency response of the photodiode normalized to its DC responsivity, Vπ(ω) is the frequency-dependent halfwave voltage of the MZM, and Ro is the load resistance seen by the photodiode. The average photocurrent at quadrature (Iavg) is given by the product of the average optical power (Po) and the DC responsivity of the photodiode.When the sampling optical pulse train consists of a series of identical pulses the time-domain intensity profile may be written as
where p̃(t) is the intensity profile of a single pulse in the train, δ() is the Dirac delta function, and T is the repetition period of the pulse train – here, the pulse train is assumed to be perfectly periodic. The spectrum of the pulse intensity (normalized to the average optical power, Po) is then given by the Fourier transform of Eq. (3) Here, we see the spectrum of the pulse intensity consists of an optical comb with a line spacing given by ωrep = 2π/T weighted by the Fourier transform of the intensity of a single pulse in the train. It should be noted that for wideband operation it is desirable to have very short sampling pulses. In our system, the use of cascaded intensity and phase modulation results in a broad optical comb, however, the time-domain intensity immediately after the phase modulator corresponds to an approximately 50% duty cycle square wave at the comb repetition rate. To exploit the comb bandwidth and achieve short sampling pulses requires phase-compensation of the optical comb as it is readily shown that – for a fixed optical bandwidth – the pulse duration is minimized when the spectral phase is uniform [12]. Given the dominant spectral phase variation in our apparatus is quadratic, the pulses are readily compressed using standard single-mode optical fiber (see Experiment section) [13].If the sampling pulse train is again assumed to consist of a series of identical pulses, however, the repetition time is allowed to vary from pulse-to-pulse the time-domain intensity of the pulse train may be written as
where ΔT represents a small deviation from the fundamental period of the pulse train. Provided the timing deviation is much smaller than the pulse period a first-order Taylor expansion of Eq. (5) readily yields Following the method of [14] we define the timing deviation to be where J(t) is a function of time representing the timing deviation relative to the fundamental period T. Note, in this work J(t) is deterministic – therefore, we may perform our analysis in terms of J(t) and its complex spectrum SJ(ω) directly. The complex spectrum of the pulse train intensity is found by taking the Fourier transform of Eq. (7) and is given by Here, we see that there are two components to the complex spectrum of the intensity of the sampling pulse train. The first consists of a periodic comb of frequencies spaced by the pulse repetition rate (ωrep/2π = 1/T) and weighted by the Fourier transform of a single intensity pulse in the train. The second component consists of modulation sidebands resulting from the timing deviation of the pulse train which are also weighted by the Fourier transform of a single pulse in the train. These modulation sidebands grow linearly (in complex amplitude) with the index n of the periodic comb as predicted for phase-noise spectral growth in pulse trains exhibiting timing jitter [14].To illustrate how the timing deviation of the sampling pulse train may be used to disambiguate signals when the link operates in a subsampling (downconverting) mode we insert the complex spectrum of the pulse train Psp(ω) [Eq. (8)] into the expression for the RF power spectrum given by Eq. (2). We now consider the RF output power from the link in two cases. First, we consider the case when only the sampling pulse train is incident on the photodiode [Vin(ω) = 0] and the low-pass filter is removed. In this case the RF power spectrum consists of a comb of RF tones separated by the fundamental pulse repetition rate and the corresponding modulation sidebands arising from the pulse train timing deviation [essentially the magnitude-squared of Eq. (8)]. If we compare the ratio of powers of one of the modulation sidebands of the n-th order combline to the n-th-order combline – defined to be the sidelobe-to-peak ratio (SPR) – we find this ratio to be
As expected, this ratio grows quadratically with the combline index n [14].If we now consider the case where the RF input signal is present [Vin(ω) ≠ 0] and a low-pass filter is used to limit the output bandwidth to the fundamental Nyquist band (0 ≤ ω ≤ ωrep/2) it is clear that signals present at the link input will be aliased at the link output. Input signals within the n-th order alias band will appear at alias frequencies given by ω̃ = nωrep − ωin. Here, we define the alias band to be a frequency range with a bandwidth equal to the fundamental pulse repetition rate centered about the n-th-order RF combline. The peak power comparison of the central component and either sideband results in the same SPR given in Eq. (9) for an input signal vin(ω) with bandwidth BW, provided the FM frequency (ωj/2π) is chosen such that the spectral components of Eq. (2) centered at nωrep − ωin and nωrep − ωin ± ωj are clearly resolvable. From Eq. (2), if we compare the peak power of the input signal measured within the fundamental Nyquist band (i.e., the aliased signal sampled with a perfectly periodic optical pulse train) to the peak power of one of the modulation sidebands which appears about the input signal peak as a result of the timing deviation of the pulse train, we find it is also given by Eq. (9)
For a sinusoidal frequency modulation applied to the signal generating the optical comb, the timing deviation may be written as where κ is the FM sensitivity (kHz/V) of the synthesizer driving the comb source, Vj is the amplitude of the FM control voltage, and ωj/2π = fj is the FM frequency. This yields a sidelobe-to-peak ratio given by Therefore, we may directly determine the alias band from which the signal originated by measuring the sidelobe-to-peak ratio, SPR. We note, a second ambiguity remains in the measured signal, that is, from which half of the alias band did the signal arise (ωin < nωrep or ωin > nωrep). In cases where the spectral components are not clearly resolved, a second sampled reference signal (without FM, or with quadrature FM) would be required. For applications where further accuracy is required, a second sampling frequency may be used. This will be discussed in a separate publication.The RF gain of the subsampling link for signals in the n-th alias band may be written as (assuming there is no matching network internal to the photodiode)
Here, ωin is the original input signal frequency, the alias frequency is given by nωrep − ωin, and Ri is the input resistance of the MZM. The RF gain is seen to take a form similar to that of a conventional intensity modulation direct detection (IMDD) analog link, with additional frequency-filtering terms arising from the sampling optical pulse (intensity) shape and the low-pass filter. As noted earlier, the RF gain uniformity between alias bands improves as the sampling pulse duration decreases. For decreasing pulsewidth |Psp(nωrep)|2 varies less from band-to-band. From the Wiener-Khintchine theorem, |Psp(nωrep)|2 is readily determined from the intensity autocorrelation of the optical sampling pulse. It is important to note that the photodiode bandwidth need only cover the fundamental Nyquist band since the aliasing (downconversion) operation is the result of an optical heterodyne process. The optical modulator, however, must show high-efficiency across the RF frequency range of interest.3. Experimental results
3.1. Optical pulse source and subsampling link
A convenient method for generating tunable repetition-rate optical pulse trains is through cascaded eletrooptic amplitude and phase modulation schemes that produce wide-bandwidth optical frequency combs [13, 15]. Figure 1 depicts the setup used in this research for optical comb and short pulse generation, as well as the IMDD subsampling link architecture. We cascade a Mach-Zehnder intensity modulator (MZM) with four phase modulators which are driven with large ampltidue RF signals (relative to the modulator halfwave voltage). The large phase modulation index enables us to obtain wide optical combs from our CW laser. For this work we choose an input modulation frequency of RFin = 5 GHz, which translates into the repetition rate of the generated pulse signal and gives a Nyquist band edge of 2.5 GHz. All modulation was true time-delay matched which allows the repetition-rate to be continuously tuned over a multi-GHz range. Each phase modulator is driven with 30 dBm (1 W), and the intensity modulator is quadrature-biased and driven at roughly one-half its 5 GHz half-wave voltage (Vπ ≈ 6 V). The output pulses from the comb generator are then compressed with the proper amount of standard single-mode fiber, which was determined assuming a purely quadratic phase to be 1.57 km for this experiment.
Figure 2(a) shows the optical spectrum from the comb generator. The full root-mean-square (rms) bandwidth of the comb envelope is calculated [9] to be Δfrms ∼ 225 GHz from which the number of comb lines is determined from N = 1 + Δfrms/frep, where frep is 5 GHz. Within the RMS bandwidth the comb exhibits N ≈ 46 comblines which show about a 1 dB power variation (at full-width-at-half-maximum bandwidth, ΔfFWHM, ∼93 features are obtained). Figure 2(b) shows the autocorrelation measurement of the compressed optical pulse from which the RMS duration of the intensity pulse is determined to be approximately 6 ps. Ideal pulse compression is not achieved because of the deviation from a purely quadratic phase in our apparatus as evidenced by the bat ears in our optical spectra as well as the sidelobes visible in the intensity autocorrelation trace. A more uniform comb and moderately shorter pulse durations could be achieved by tailoring the drive waveform to obtain a more pure quadratic phase [13].
Fig. 2 (a) Optical comb spectra from output of cascaded modulators, showing comb lines within ∼1dB variation falling within Δfrms ≈ 225 GHz. (b) Measured intensity autocorrelation of the compressed optical pulse, indicatinga6ps pulse (intensity) duration.
3.2. RF gain of the subsampled analog link
For validation of the new gain expression presented in section 2, we start here with the RF gain performance of the subsampled analog link shown in Fig. 1. For this measurement, 16 different continuous-wave (CW) tones spanning the 300 MHz – 40 GHz range are individually applied to the RF input of the link at a power level of 10 dBm. The frequencies of these tones were chosen such that all signals are aliased to 300 MHz at the link output and so that there is one frequency per 2.5 GHz Nyquist bin. The peak signal power at 300 MHz is then measured with an electrical spectrum analyzer. The measured link gain versus frequency is shown in Fig. 3 (circles). For comparison, the link gain calculated from Eq. (13) using the measured frequency-dependent halfwave voltage of the modulator and an average photocurrent of Iavg = 2.5 mA is shown by the gray curve. In this calculation |Psp(ω)|2 is given by the Fourier transform of the measured intensity autocorrelation [Fig. 2(b)] and the frequency-dependent cable loss at the link input has been included. Across the 40 GHz bandwidth of the measurement the magnitude of the error is below 1 dB and is limited by the system measurement accuracy.
Fig. 3 RF gain as a function of input frequency for the subsampled analog link. The measured gain data are shown by the circles and the gain calculated from Eq. (13) is shown by the gray line.
3.3. Sidelobe-to-peak ratio
From Eq. (12), is it evident that the SPR grows as the square of the folding band → n2. Therefore, once the SPR for the n = 1 band is known, the alias band may be determined from the SPR assuming this quadratic growth. In our experiment (see Fig. 1) κ = 100 kHz/V, Vj = 50 mV, fj = 100 kHz which yields [Eq. (12)] SPRn=1 = 6.25 × 10−4, or approximately −32 dB. In Fig. 4 we illustrate the predicted increase in SPR by comparing the SPR for pure tone input signals at 300 MHz (n = 0 alias band), 4.7 GHz (n = 1 alias band), and 34.7 GHz (n = 7 alias band). Note, the measurement is taken in the fundamental Nyquist band (0 ≤ f ≤ 2.5 GHz) where all of the above signals alias to a center frequency of 300 MHz. Signals that inherently fall within the fundamental Nyquist band do not exhibit the 100 kHz modulation sidebands, and have SPR = 0 as illustrated when the input signal is 300 MHz (bottom curve). When the input signal frequency is such that aliasing occurs, the frequency-modulation sidebands grow as illustrated for input signals at fin = 4.7 GHz (middle curve) and fin = 34.7 GHz (top curve). Here, the measured SPR values for 4.7 GHz and 34.7 GHz are, respectively, SPRn=1 ≈ −32 dB and SPRn=7 ≈ −15.1 dB in nearly perfect agreement with Eq. (12). This measurement is repeated for input signals with center frequencies up to 40 GHz (corresponding to the frequencies at which the RF link gain was measured, Fig. 3) and the results are shown in Fig. 5. Here, the SPR for each input signal (circles), as well as each harmonic of the pulse train repetition rate (triangles) are normalized to the value found with Eq. (12) evaluated at n = 1. For reference, the scale below the plot shows the definition of the alias bands. It is very apparent in Fig. 5 that the SPR growth is n2 as expected illustrating that this quantity may readily be used to determine the alias band from which a given signal originated.
Fig. 4 Depiction of the change in SPR as a function of frequency, 300 MHz (n = 0), 4.7 GHz (n = 1), and 34.7 GHz (n = 7). The growth is n2 as predicted, where the ratio grows from line 1 to 7 by 49-times (16.9 dB).
Fig. 5 SPR growth as a function of frequency and alias band normalized to the SPR at frep = 5 GHz. A plot of n2 is overlayed showing agreement with a quadratic growth profile. The metered bar below the plot label the alias (n) bands.
3.4. Disambiguation
In order to show that this technique is truly capable of determining from which alias band an ambiguous signal originated, we perform an automated experiment where the input frequency to the link was randomized. This experiment utilizes a random uniform sample of 1000 different pure tone frequencies within the range of 1 MHz to 40 GHz. The aliased baseband replicas are measured for each random input and control code determines the SPR normalized to the known SPR at 5 GHz. The corresponding alias band is then determined from the square root of the normalized SPR. As is clearly illustrated in Fig. 6, the technique may be used to accurately disambiguate signals across a 40 GHz bandwidth. For every input signal the correct alias band was determined proving the technique reliable for coarse broadband RF disambiguation.
Fig. 6 The alias band as determined over a set of 1000 random input frequencies. The correspondance between input frequency and frequency disambiguation is excellent to 40 GHz.
4. Summary and future work
This paper presents a novel subsampling technique for coarse broadband RF disambiguation where we demonstrate that an ambiguous signal can be reliably designated to its originating frequency alias band, n, with just a single baseband measurement. This is achieved by comparing the amplitude of jitter-induced sidebands relative to the measured signal peak within the fundamental Nyquist band (f < frep/2) where this ratio grows predictably with the alias band index. This technique has application for realtime signal identification for signals at frequencies upwards of 100 GHz, only limited by the frequency response of the optical modulator and the duration of the optical sampling pulse. Sampled analog photonic links and disambiguation techniques offer potential to use a small number of high-fidelity receivers to cover 100 GHz simultaneously, providing a viable pathway for performing ultra-broadband signal detection in acceptably short processing times. Increasing the folding ratio (lower sampling rates) and increasing RF bandwidth and link gain are clear areas for future work. Evaluating the technique’s performance with more complex waveforms and in a multi-signal environment will also be of key importance.
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