Log-periodic temporal apertures for grating lobe suppression in k-space tomography

Conor J. Ryan; William L. Beardell; Janusz Murakowski; Dylan D. Ross; Garrett J. Schneider; Dennis W. Prather

doi:10.1364/OE.392118

1. Introduction

Massive multiple-input multiple-output (MIMO) systems promise increased data capacity for the fifth generation (5G) wireless communication through improvements in spectrum efficiency and utilization [1–4]. More specifically, spatial modulation (SM) schemes in large MIMO antenna array systems allow for multiple wireless data links to be established at the same time and frequency [5], greatly increasing the overall system data throughput. However, the key benefits of these systems are largely constrained by the ability to obtain accurate channel state information (CSI) [6].

In order to obtain precise CSI, training signal sequences, or pilots, are typically sent on the uplink transmission and retrieved at the base station (BS). These pilot signals are also used for the downlink by virtue of reciprocity between the uplink and downlink transmission channels. Challenges to obtaining CSI using this approach include the need for a transceiver calibration to realize channel reciprocity, timely acquisition of CSI in a highly dynamic radio frequency (RF) environment, difficulty scaling up to millimeter waves (mmW), and pilot contamination [7]. Consequently, there has been great interest in blind channel estimation techniques to obtain channel state information free of pilot training sequences, and to passively formulate a spatial-spectral map of the dynamic user-space.

Acquisition of channel state information via a microwave photonic approach holds several advantages over its’ purely RF counterpart [8–10], one of which being the ability to passively determine the spatial location of signals [11–15]. In order to obtain complete channel state information, as well as successfully implement spatial modulation schemes, it is necessary for spectral usage to be known in conjunction with spatial location of all users. To this end, the k-space tomographic imaging system has been developed for passive spatial-spectral localization of RF signals [16–20]. Spectral sensing is incorporated into the system by intentional manipulation of the optical fiber lengths within an mmW-imaging receiver described in [11–15]. By introducing arbitrary optical fiber lengths, new time-based signal correlations are formed for each signal impinging upon a phased antenna array. This time delay-based array is referred to as the temporal aperture and comprises an array of optical fibers of varying physical path lengths, which apply a phase shift to the signal captured by each receive element. While this technique disrupts the spatial distribution of the up-converted signals, and results in a scrambled interference pattern at a charge-coupled device (CCD) array, the camera response contains the spatial-spectral information of the present RF sources. Conveniently, the spatial and spectral sampling process is linear in nature, allowing one to obtain a solution with classical computed tomography algorithms. This tomographic reconstruction yields an amplitude distribution of the received RF power in k-space, providing both the angle of arrival and frequency of incoming waves. Thus, the k-space imaging system can ideally be utilized in conjunction with a receiver at a base station, acting as a cueing detector, which provides direction of arrival and frequency localization.

When developing such a cueing detector, it is desirable to maximize the resolution, spatial or spectral, of the system while minimizing the element count. Traditionally, phased array technology has been implemented with a regular element spacing where the element pitch is set at less than or equal to λ/2 ensuring suppression of unwanted grating-lobe artifacts within the radiation pattern [21]. Sampling a phased array in this manner makes it a necessity to increase the element count as the phased array diameters become larger, concomitantly improving the spatial resolution. As such, the synthesis of highly robust sparse phased arrays is commonly sought after for improvements in cost, size, weight and power (CSWAP), while simultaneously improving the system’s resolution. However, the presence of aliasing within the array radiation pattern creates difficulty in the design of sparse uniform phased arrays. Methods for sparse phased array synthesis for obtaining maximally sparse, random arrays while preserving desired radiation pattern characteristics have been well researched in the domain of phased antenna arrays and include simulated annealing, genetic algorithms and compressive sensing [22–28]. However, such techniques have not been applied to arrayed waveguide gratings (AWG) for the purpose of maximizing the spectral resolution of the temporal phased array. In this paper, an alternative approach to sparse array synthesis is introduced, in which a scale-invariant, log-periodic sampling is applied to suppress the grating lobes and minimize aliasing within the optical fiber-based time delay aperture utilized in k-space tomographic imaging system.

2. k-space tomography

From Fourier analysis, it is known that any RF signal may be represented by a superposition of plane waves, and each plane wave can be represented wholly by a complex-amplitude weighted four vector (ω,k), where ω is the angular frequency and k is the wave vector. Assuming propagation in free-space, the relation ω = c|k| can be used to further reduce the information required to fully characterize an incident RF signal to the three-vector k. In order to recover k, each incident wave front upon an antenna array must be adequately sampled, both spatially and spectrally. Phased antenna array technology is typically used to sense the angle of arrival of an incident RF wave through relative phase shifts corresponding to the path length differences of the adjacent antenna elements [21]. In the case of mmW-imaging receivers [11–15], plane waves are spatially sampled in this way at the antenna-array front end, and up-converted into the optical domain by electro-optic phase modulators fed by a common laser source. This field is then coupled into an array of equal-length optical fibers, where the temporal fluctuations induced by thermal and mechanical stress are monitored and compensated for via a feedback loop to maintain coherent optical phases [15]. From here, the signal passes through a lensed-fiber array, which is arranged to match a scaled version of the antenna array’s geometric configuration. Then, the optical signals are projected into a free-space optical-lens-based system where the lower-sideband is selectively filtered while the optical carrier and upper-sideband are suppressed. Projection of the filtered optical beams onto a CCD array yields an interference pattern that is a replica of the spatial distribution of RF signals in the optical domain. In order to understand how the spatial information of the RF signal is preserved and processed in this arrangement, the electric field, assuming no filtering has occurred, incident upon the CCD array is described as:

(1)$${E_n} = \frac{1}{{\sqrt N }}\sum\limits_{m = 0}^{M - 1} {{B_m}{e^{j(\omega t + {\theta _{nm}} + {\varphi _{RF}})}} + c.c.,} $$

where E_n is the electric field in the optical domain at the n^th pixel in the CCD array without filtering, B_m is the field amplitude at the output of the m^th optical fiber, ω is the optical frequency, θ_nm is the phase obtained as the optical beam propagates from the m^th optical fiber to the n^th pixel, φ_RF is the RF-modulated phase term at the m^th optical fiber, and c.c. denotes the complex conjugate term of the complex field term. The definition assumes there are M receiving elements and optical fibers, along with N elements in the CCD array.

Assuming the presence of K point sources emitting incoherently, the RF modulated phase terms becomes

(2)$${\varphi _{RF}} = \sum\limits_{k = 0}^{K - 1} {{S_k}\cos ({\Omega _k}t + {\phi _{km}}),} $$

where S_k are the amplitudes of the signals scaled by the modulation efficiency of the modulators, antenna collection efficiency, and the distance of the emitter from the aperture, the term Ω_k is the angular frequency of the k^th RF source, and ϕ_km is the phase obtained during the free space propagation of the wave from the k^th emitter to the m^th antenna. For this example, the modulation efficiency is assumed to be identical for all elements of the array.

Through substitution of the RF-modulated phase term of Eq. (2) into the expression for the electric field at the n^th pixel in the CCD array of Eq. (1), the electric field may be represented as

(3)$${E_n} = \frac{1}{{\sqrt N }}\sum\limits_{m = 0}^{M - 1} {{B_m}{e^{j(\omega t + {\theta _{nm}} + \sum\limits_k {{S_k}\cos (\Omega t + {\phi _{km}})} }}} + c.c.$$

In order to properly form an image of the RF waves on the detector array, a single sideband is selectively filtered from the response. Optical spectral filtering, presumably by a dielectric stack having a suitable pass-band response, can be utilized in the free-space optical system in order to suppress the carrier and upper-sideband. Following this filter, in a linear approximation, the electric field comprising a single lower-sideband at the CCD array becomes

(4)$$E_n^{LSB} = \frac{{j{e^{(\omega - \Omega )t}}}}{{2\sqrt N }}\sum\limits_{mk} {{B_m}{S_k}{e^{j({\theta _{nm}} + {\phi _{km}})}}} + c.c.$$

Considering the physics of the CCD array, which allows for the detection of the intensity of the incident field over a period of specified integration time, the detected signal takes the form

(5)$$\left\langle {{{|{E_n^{LSB}} |}^2}} \right\rangle = \frac{1}{{2N}}{\sum\limits_k {\left|{\sum\limits_m {{B_m}{S_k}{e^{j({\theta_{nm}} - {\phi_{km}})}}} } \right|} ^2},$$

where angle brackets $\left\langle \cdot \right\rangle $ signify averaging. Since this encoding process preserves both the amplitude and phase of the incoming RF waves, the up-converted signal is coherently imaged with a Fourier lens onto a CCD array. Accordingly, the spatial information of an incident RF wave is recovered through the Fourier transform of the signals captured by a phased antenna array, or spatial aperture. However, this process of optically imaging RF waves only recovers the spatial information of the incident wave front.

For spectral sensing, a temporal aperture must be integrated with the spatial aperture for time-based phase correlations that enable detection of the incident wave’s frequency. Functionally similar to the spatial aperture (i.e. phased antenna array), a temporal aperture is constructed by adjusting optical fiber lengths within the mmW-imaging system following the EO modulators, as illustrated in Fig. 1, to form a set of time-sampling baselines.

Fig. 1. Introduction of an arbitrary optical fiber length distribution behind an RF front end creates a different projection of k-space at each pixel in the CCD camera at the image plane.

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Creating these optical path length differences produces the following phase shifts, each proportional to the frequency of the respective emitter

(6)$${\varphi _{km}} = {\Omega _k}{t_m}$$

where t_m is the time delay generated by the delay line associated with the m^th receiving element. This delay-line configuration is functionally similar to the arrayed waveguide gratings used in wavelength division multiplexing (WDM) networks [29,30], in which an array of varying-length waveguides introduces phase shifts producing a frequency-steered diffraction pattern. The introduction of these delay lines naturally affects the electric field of light propagating in free-space, which now yields the detected power as

(7)$$\left\langle {{{|{E_n^{LSB}} |}^2}} \right\rangle = \frac{1}{{2N}}{\sum\limits_k {\left|{\sum\limits_m {{B_m}{S_k}{e^{j({\theta_{nm}} - {\phi_{km}} - {\Omega _k}{t_m})}}} } \right|} ^2}.$$

Assuming the emitters are sufficiently distant from the receiving aperture, the wave-fronts incident upon the array may be assumed to be plane waves allowing for a substitution of the phase term, ϕ_km = K_k·r_m, where K_k is the wave-vector of the k^th emitter, and r_m is the position of the m^th receiving antenna. As a result, the incident optical power density upon the CCD array may be updated as follows:

(8)$$\left\langle {{{|{E_n^{LSB}} |}^2}} \right\rangle = \frac{1}{{2N}}{\sum\limits_k {\left|{\sum\limits_m {{B_m}{S_k}{a_m}(u){e^{ - j({\textrm{K}_k} \cdot {\textrm{r}_m} + {\Omega _k}{t_m})}}} } \right|} ^2}.$$

which properly accounts for the updated phase term through the complex amplitude density a_m(u) of the optical beam propagating out of the m^th optical fiber to the n^th photodetector on the CCD array at a position u. Given the power density Eq. (8), the optical power detected by the n^th photodetector, located at spatial position u_n, is obtained through the following integration of the incident power over the surface area of the pixel, α_pixel

(9)$$\begin{array}{l} {P_n} = \int\limits_{{\alpha _{pixel}}} {\left\langle {{{|{E_n^{LSB}} |}^2}} \right\rangle {d^2}x = } \left\langle {{{|{E_n^{LSB}} |}^2}} \right\rangle {\alpha _{pixel}}\\ = \frac{{{\alpha _{pixel}}}}{{2N}}{\sum\limits_k {\left|{\sum\limits_m {{B_m}{S_k}{a_m}({u_n}){e^{ - j({\textrm{K}_k} \cdot {\textrm{r}_m} + {\Omega _k}{t_m})}}} } \right|} ^2}. \end{array}$$

This expression, which relates the detected optical power distribution to the received RF power distribution in k-space, can be concisely represented as

(10)$${P_n} = \sum\limits_k {{a_{nk}}{S_k}} $$

where a_nk is a weighting term representative of the contribution of the k^th emitter to the power detected by the n^th pixel of the CCD array, whereas S_k is the power of the k^th emitter in the RF scene. Retrieving the spatial-spectral information of any received signals from this interference pattern requires the inversion of the following equivalent linear matrix relationship:

(11)$${P_n} = {\textrm{a}_n} \cdot \textrm{S}$$

where P_n is the power detected by the n^th pixel in the CCD array, S is the RF scene in the form of a k-space power distribution vector, and a_n is a weight-map corresponding to the n^th pixel [16]. The weight-maps are obtained by placing a known emitter in the far-field relative to the receiving antenna array. The point spread function generated by the received signal, which is both spatially and spectrally dependent, is recorded by the CCD array for each k-point detectable by the array by maneuvering the receiving antenna array via a rotation stage and controlling the emitter frequency. Throughout the process of capturing each weight-map, any non-uniformity in the RF components of the receiving front-end (antenna, modulators, low-noise amplifiers, coaxial cables), and the corresponding spatial and spectral characteristics, are fully accounted for in the intensity distribution. Applying conventional computed tomography algorithms allows for an algebraic reconstruction of the RF scene provided the weight maps, a_n, are sufficiently diverse. For our purposes, the Kaczmarz method, or algebraic reconstruction technique (ART) [31] is used to invert the relation of Eq. (11).

Reconstructions of an RF scene are obtained in the form of a grid of points in k-space, referred to as k-points, where each point corresponds to a specific discrete angle and frequency combination. The sampling density of these grid points used to populate the weight-maps, a_n, is defined by the spatial resolution determined by the RF antenna spatial aperture, and the spectral resolution determined by the optical fiber-based temporal aperture. For an antenna array, the spatial resolution is set by the diffraction-limited spot size set by the geometry and size of the array, as well as the operational wavelengths. This relation is described mathematically, for the case of a one-dimensional antenna array, by the Rayleigh diffraction limit:

(12)$$\theta = \frac{\lambda }{D}$$

where θ is the minimum resolvable angle, λ is the wavelength of electro-magnetic radiation incident upon the aperture, and D is the diameter of the aperture. Similarly, the temporal aperture has a spectral resolution corresponding to the longest fiber-length difference in the array. The minimum resolvable frequency difference, Δf, can be found as:

(13)$$\Delta f = \frac{c}{{n\Delta {l_{\max }}}}$$

where c is the speed of light, n is the refractive index of the optical fiber and Δl_max is the largest difference between optical-fibers lengths. Accordingly, the k-space weight-maps in a are sampled at twice the spatial and spectral resolutions of Eqs. (12) and (13), per the Shannon-Nyquist sampling theorem. Sampling in this manner enables reconstructions of the RF source distributions in k-space such that the detected sources collapse to the nearest discrete grid points. Accurate reconstruction of the emitter distribution in k-space is not only dependent upon sufficient sampling of the weight-maps, but also upon the sampling performed at the various arrays comprising the system’s architecture, both spatial and temporal. In order to properly reconstruct the received optical power distribution, additional steps must be taken to mitigate any ambiguities which may arise from aliasing due to the geometric distribution of the elements within these arrays.

3. Temporal aperture

As previously stated, identifying and separating the spectral components of electromagnetic waves through the use of optical waveguides of varied lengths is a familiar concept, see e.g., AWG. In fiber-optics-based telecommunication, such spectral separation is typically performed at a relatively coarse resolution, with the optical path-length difference short enough to be easily fabricated on photonic integrated circuits [32–35]. In contrast, for the purpose of identifying signals operating in the commercial RF spectrum where channel bandwidth typically spans less than few tens of megahertz [36], the optical path-length differences required grow on the order of several meters. To achieve such an extent of path-length differences in the optical domain, the arrays may be implemented using inexpensive, low-loss optical fiber. Although optical fiber itself is relatively inexpensive, packaging optical fiber-based components is not. As a result, minimizing the number of separate optical fiber elements to achieve stated performance goals is essential in practical deployable systems. However, improving the spectral resolution of the array while limiting the number of elements necessarily increases the inter-element spacing in the array and, as a result, generates spectral grating lobes within the array response. In this section, the frequency-domain response of the temporal aperture is analyzed in detail to lay the groundwork for a rational design of temporal-array geometry and mitigate detrimental effects of reducing the number of elements.

3.1 Temporal aperture model

Consider an array of antenna elements coupled to electro-optic phase modulators, where an incident RF signal is up-converted to optical frequency as sidebands upon a carrier tone defined by a laser. The outputs of the modulators are fed into M optical fibers of the temporal aperture, which is outlined in Fig. 1. It can be noted in such a configuration, following the development of Eq. (8), that the spatial and spectral array factors may be treated as independent phase shifts. Thus the cumulative array factor may be treated as the individual temporal array factor given boresight illumination of the front-end (uniform spatial phases):

(14)$$\begin{array}{ll} AF &= \sum\limits_m {{B_m}{S_k}{a_m}({u_n}){e^{ - j({\textrm{K}_k} \cdot {\textrm{r}_m} + {\Omega _k}{t_m})}}} = \sum\limits_m {{S_k}{B_m}{a_m}(u){e^{ - j{\textrm{K}_k} \cdot {\textrm{r}_m}}}{e^{ - j{\Omega _k}{t_m}}}} \\ &= \sum\limits_m {{S_k}{B_m}{a_m}(u){e^{ - j{\Omega _k}{t_m}}}} \end{array}$$

Based upon this relationship, the temporal aperture will be discussed independently of the spatial array factor for the following section, where it is assumed there is no spatial phase associated with the incident signal (boresight illumination). For the following analysis, a uniform linear temporal array is assumed in which each optical fiber element has a progressive path length increment of Δd per channel, which for the eight channel case depicted results in a maximum length difference of 7Δd.

In this configuration, each Δd increment between successive fibers results in an optical phase difference equivalent to that of Eq. (6).

(15)$$\varphi = \frac{{2\pi }}{\lambda }n\Delta d,$$

where λ is the wavelength of the incident plane wave, and n is the refractive index of the optical fiber. Taking into account the progressive phase shift from length increments of the M optical fiber elements results in an array factor, assuming equal amplitude weights per each element, of the following form:

(16)$$\begin{array}{ll} AF &= 1 + {e^{j\left( {\frac{{2\pi }}{\lambda }n\Delta d} \right)}} + {e^{j2\left( {\frac{{2\pi }}{\lambda }n\Delta d} \right)}} + \ldots + {e^{j(M - 1)\left( {\frac{{2\pi }}{\lambda }n\Delta d} \right)}}\\ &= \sum\limits_{m = 1}^M {{e^{j(m - 1)\left( {\frac{{2\pi }}{\lambda }n\Delta d} \right)}}} . \end{array}$$

It is noted that this definition of the temporal array factor is equivalent to the phase term produced by the temporal array factor term introduced in Eq. (7), given the array is linearly incremented. Accordingly, observations made while examining the isolated temporal array allows to draw conclusions regarding the PSF generated by the complete k-space tomographic system. In this formulation, it becomes apparent that, unlike the phased antenna array, there is no fiber length difference, Δd, for which the higher order harmonics become imaginary [21]:

(17)$$2\pi = \frac{{2\pi }}{\lambda }n\Delta d.$$

This condition places a fundamental limit on the sampling of the operational bandwidth of the regularly spaced fiber grating, otherwise known as the free spectral range (FSR), which is given by:

(18)$$FSR = \frac{c}{{n\Delta d}}.$$

Free spectral range represents the spacing between adjacent modes within the array response, and the FSR provides the range in which signals can be properly analyzed without aliasing. The limitation imposed by the FSR upon the linear temporal array response is illustrated in Fig. 2. As shown in Fig. 2, the response of a thirty-element linear array with a maximum length difference, Δl_max, of 40 cm and unit spacing, Δd, of 2 cm (in black) compared to a similar array with maximum length difference of 80 cm and unit spacing of 4 cm (in red). In addition to the main lobe, additional harmonic maxima (grating lobes) are observed for these arrays that manifest as artifacts within the power distribution of the array response. It is noted, quite intuitively, that the harmonic spacing produced by the array having a maximum length difference of 40 cm, occurs at twice the interval of the 80-cm array with identical element count. Certainly, the uniform linear array is limited in terms of maximizing the FSR with a minimal channel count as the periodicity of the aliased harmonics within the desired bandwidth is inversely proportional to the periodicity of the element spacing. The impact of this aliasing upon the iterative k-space tomographic reconstruction will be discussed in more detail in Section 4.

Fig. 2. Array factor computed over 30 GHz bandwidth of a linear fiber array consisting of 30 equally spaced elements with maximum length difference, Δl_max, of 40 cm (red) and 80 cm (black). Adjacent element spacing equivalent to 2 cm in free space (red) and 4 cm in free space (black), corresponds to FSRs of 15 and 7.5 GHz, respectively (spacing between m and m ± 1 modes).

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3.2 Non-linear temporal aperture sampling

Ideally, incremental spacing of the optical fibers, Δd, corresponds to an FSR that is greater than the operational bandwidth of the receiving antenna to avoid aliasing. At the same time, to operate as an effective k-space cueing detector for wireless communication, the adequate spectral resolution must be better than a few tens of MHz. As a result, covering the 13.5-GHz-wide Ka band of 26.5-40 GHz, to account for the FCC’s allocated bands around 28 and 39 GHz, requires a linear fiber-grating array with hundreds of optical-fiber elements. This example illustrates that linear sampling of the temporal aperture is ill suited for achieving the spectral resolution and band coverage adequate for the nascent 5G market. Given the relationship between the periodicity of these phased arrays and the presence of harmonic modes, it is clear that an alternative time-sampling distribution is necessary for increasing the FSR with a minimum element count.

One appealing option is a logarithmic-periodic (LP) sampling configuration, which contains no regular spacing between fiber lengths and as such is expected to suppress the grating lobes present in linear fiber gratings. Similar configurations have been studied more in depth in the spatial domain where they take the form of log-periodic dipole antennas, in which the dipole elements are scaled in size and spacing according to a logarithmic periodicity and yield a scale-invariant geometry with increased bandwidth response [37,38]. Similar sampling configurations have been studied with investigations of exponentially spaced antenna elements within the array [39]. Aperiodic and non-uniform arrays have been explored nearly exclusively with respect to antenna arrays. Although the application of non-uniform arrays to AWG devices have been explored [40,41], these techniques have not been applied for the purpose of increasing the spectral resolution while maintaining a desired FSR in the same manner as antenna arrays. In order to generate such an array, the elements are chosen to be spaced between c^a and c^b, such that the longest time delay between the bounds defined by a and b corresponds to the desired frequency resolution. In this manner, the array spacing, when viewed upon a log_c scale appears to be spaced periodically, as illustrated by the comparison with the linear array as in Fig. 3.

Fig. 3. Comparison of optical fiber length distribution between linear (blue) and logarithmic periodic (red) array distributions viewed on logarithmic scale. This configuration of the logarithmic distribution of fibers is based upon a log₁₀ scale; this scale may vary in the design of the temporal aperture for an additional degree of freedom.

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Irregular spacing between the elements of the array re-distributes the energy from the grating lobes into the side-lobes of the array response as illustrated by Fig. 4. For some applications, this trade-off between maximizing the FSR versus minimizing the side-lobe energy may be undesirable. However, in the case of k-space tomography, the redistribution of the side-lobe energy may reduce ambiguities present in the sampled weight-maps a_n, which could increase the reliability of the tomographic reconstruction. This effect will be discussed in greater-detail in the section below.

Fig. 4. Array factor computed over 30 GHz bandwidth of log-periodic fiber array consisting of 30 fiber elements with a maximum length difference, Δl_max, of 80 cm. Comparing the spectral response to that of Fig. 2 for the linear array, of the same 80 cm Δl_max, shows the exact harmonics spaced at 7.5 GHz have disappeared and the optical power is spread across the side lobes of the frequency response.

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In addition to analyzing the spectral information obtained through the array of length differences, it is of key importance to analyze the aforementioned array pattern to determine the exact effect of the new distribution on the side-lobe distribution. Making slight modifications to Eq. (16) to account for the non-uniform Δd increments allows for the calculation of the LP-array factor. Numerical analysis is performed with a 30-element LP fiber array with Δl_max of 80 cm for direct comparison to that of the array in Section 3.1. The resulting array response over the same bandwidth, as shown in Fig. 4, illustrates the suppression of harmonics at the multiples of 7.5 GHz and concomitant distribution of the optical power across the side lobes in the array response.

4. Application to k-space tomography

In this section, the LP sampling configuration is explored for the purpose of maximizing spectral resolution of a k-space tomographic imaging system while maintaining the number of existing elements in the experimental setup. Following the analysis of the individual temporal array factor within Section 3, an analysis of the system response including the full array factor, including both spatial and spectral arrays, is used as the input for the tomographic reconstruction algorithm. Reconstructions performed utilizing the tomographic algorithm provide clarity whether the elevated side-lobe levels in the array response affect the viability of the technique.

4.1 Numerical analysis

Experimental data have previously been collected using a k-space tomographic imaging system consisting of thirty spatial and spectral elements illustrated in Fig. 5 [16–19].

Fig. 5. Three-dimensional render of experimental system. The distributed aperture consists of thirty tapered-slot antennas coupled to commercial EO phase modulators, which couple to the temporal aperture consisting of thirty optical-fibers of varied lengths. The optical signal passes through a Fourier-transform lens that generates an interferogram at the image plane [17,19].

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This system has a microwave front-end consisting of thirty tapered slot antennas with an operational bandwidth of 26.5–40 GHz configured as a two-dimensional distributed aperture. Each of the receive antenna elements is coupled to a commercial EO phase modulator. From the modulators, the coherently up-converted single-sideband signal passes through a temporal aperture of regular periodicity that spans a maximum path length difference of 40 centimeters to produce a minimum spectral resolution of roughly 500 MHz. This maximum spacing corresponds to an adjacent element spacing of Δd = 1.33 cm in a regular linear array, or about 2 cm in free-space path-length difference of 2.7λ at the highest frequency. This element spacing correlates to a free spectral range of 15 GHz, such that the first-order harmonics fall outside of the 13.5 GHz bandwidth of the receive antennas. Regular spacing of the fibers thus places a fundamental limit on the spectral resolution that can be achieved at 450 MHz without increasing the element count. As previously discussed, to substantially improve the system’s spectral resolution and localize wireless communications signals, it would be necessary to increase the number of elements in the linear array. The element-count increase becomes impractical as the total length of necessary optical fiber increases considerably within dense sampling schemes. For example, in order to improve the spectral resolution by a factor of 10, a maximum length difference Δl_max of 4 meters must be utilized. This Δl_max corresponds to an adjacent element path length difference (Δd) of 20 cm in free space, resulting in an FSR of 1.5 GHz for the 30-element system. Successfully implementing a linear array to account for the desired bandwidth would therefore require an identical ten-fold increase of the number of elements in the array for a total of a hundred-fold increase in the amount of optical fiber used, per Nyquist sampling of the elements. Thus, in an effort to minimize the total length of optical fibers and total number of elements within the temporal aperture, while maximizing the systems’ spectral resolution, the log-periodic temporal aperture sampling scheme is explored and the effects on the tomographic reconstruction are analyzed.

To this end, simulations were performed using the two aforementioned fiber-length configurations, with the k-space imaging receiver acting as a spectrometer. To simplify analysis, no spatial information was included within the weight-map matrix a, and instead the matrix contains only spectral information at a singular spatial location. This arrangement of weight-maps a_n, allows for the detection of 270 distinct time correlations separated spectrally by ∼50 MHz increments. To test the resolution limits of each temporal array configuration, two continuous-wave (CW) sources were simulated as operating from the same spatial position, but separated spectrally at the Rayleigh diffraction limit of the temporal aperture, 50 MHz apart. As illustrated by Fig. 6, the two sources, which are operating at 37.00 and 37.05 GHz respectively are reconstructed without harmonics for the log-periodic arrangement. In contrast, when the sparse regular linear array is used for the reconstructions, there is a severe degradation of the reconstruction quality. Notably, there is a strong presence of higher order harmonics spaced at the expected interval of 1.5 GHz over the 13.5 GHz of operational bandwidth. However, for the case of the log-periodic temporal aperture, the reconstructed spectrum lacks the presence of grating lobes and reconstruct exclusively the RF sources to positions accurate within the spectral resolution and at a much greater magnitude than the linear array. Additionally, the side-lobe artifacts observed in the temporal-array response of Fig. 4 are absent from the numerical reconstruction of the RF scene.

Fig. 6. Comparison of numerical tomographic k-space reconstruction of two CW sources operating within the Ka-band (37.00 and 37.05 GHz respectively) at the expected spectral resolution limit for linear (red) and log-periodic (blue) fiber arrays with 4 meters of maximum length difference. Harmonics (grating lobes) appear at 1.5 GHz increments in the reconstruction using the linear array corresponding to the inter-element spacing of 2 cm.

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4.2 Experimental verification

In order to verify the efficacy of the array geometry, a log-periodic temporal aperture with a maximum length difference of roughly 4 meters was fabricated. This log-periodic aperture contains a random variation of no more than +/- 1 cm per element introduced due to finite precision of the fabrication process. After installing the optical fiber array into the system, a calibration routine was performed by sampling the camera responses for all detectable k-vectors within the system’s spectral resolution limit, approximately 280 points. The calibration was performed by placing a single RF emitter at the receiving antenna’s boresight and sweeping through the 26.5–40 GHz bandwidth in 25 MHz increments with a CW source. After propagating through the temporal aperture, the interference patterns from these signals are captured on a 256 × 320 pixel SWIR CCD camera. These collected camera responses corresponding to each resolvable k-vector are then used to form the a matrix for the tomography reconstruction algorithm. Experimental scenes were collected by placing a single transmitting antenna at 3 meters’ distance, and boresight relative to the receiving array as depicted in Fig. 7, with two distinct CW RF tones radiating at an RF power of 10 dBm, separated by the 50 MHz minimum resolution limit, coupled into the same antenna.

Fig. 7. 30-Element distributed aperture antenna array (left). Experimental setup using a Ka-band horn antenna as the active emitter to be reconstructed by the k-space tomographic imaging modality.

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These sources were configured to operate within the 5G bands of 27–28.35 GHz and 37–40 GHz to illustrate the capability of the system as a 5G cueing detector. Reconstruction of the spectrum using the log-periodic temporal aperture is compared with the numerical analysis in Figs. 8 and 9. The resulting RF scene from the first case depicted in Fig. 8 shows faithful reconstruction of the RF tones separated by 50 MHz at 37.00 and 37.05 GHz respectively.

Fig. 8. Experimental tomographic k-space reconstruction of two CW sources separated by the spectral resolution limit within the Ka-band for log-periodic fiber array, 37.00 and 37.05 GHz respectively, compared to the numerical evaluation. The reconstructed sources are recovered at the expected position with greater than 20 dB suppression of the noise floor. Magnification of the recovered sources is also shown to demonstrate the spectral resolution.

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Fig. 9. Secondary analysis of experimental tomographic k-space reconstruction of two CW sources separated by the spectral resolution limit within the Ka-band for log-periodic fiber array, sources operating at 28.15 and 28.20 GHz respectively, compared to the numerical evaluation. The reconstruction confirms that the log-periodic array is capable of reconstructing the RF spectrum without the presence of harmonics or grating lobes. Magnification of the recovered sources is also shown to demonstrate the spectral resolution.

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The secondary evaluation case of Fig. 9, in which two CW tones radiating at 28.15 and 28.20 GHz demonstrate again that the log-periodic temporal aperture yields 50 MHz resolution without the presence of higher order harmonics, or grating lobes, within the array. Comparing these results with the numerical evaluation yields an agreement within roughly 10 dB between the measured and simulated signal to noise ratios.

5. Summary

We presented an application of log-periodic sampling scheme to the fiber-based temporal aperture within the k-space imaging modality for frequency detection of RF sources. Validation of this array geometry has been performed through experiments using a thirty element distributed-aperture RF-photonic k-space tomograph. Continuous wave tones emitted from an RF source provided an interferogram, which was used in computational reconstruction to identify spectral usage to within 50 MHz resolution. These experiments demonstrated the suppression of grating lobes within the array response due to the lack of periodicity within the logarithmic sampling. This approach allows the utilization of extremely sparse temporal arrays at decreased size, weight, power and cost when compared to densely populated systems operating at the same spectral resolution. As a result, the ability to monitor 5G spectral channels through the use of sparsely populated k-space imaging receivers is greatly enhanced.

Disclosures

The authors declare no conflicts of interest.

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Log-periodic temporal apertures for grating lobe suppression in k-space tomography

Abstract

1. Introduction

2. k-space tomography

3. Temporal aperture

3.1 Temporal aperture model

3.2 Non-linear temporal aperture sampling

4. Application to k-space tomography

4.1 Numerical analysis

4.2 Experimental verification

5. Summary

Disclosures

References

Cited By

Figures (9)

Equations (18)

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