# Spatio-Temporal 2D RTSA

Since the 1D RTSA spectrally decomposes a signal along one dimension of space, the maximum angular extent of the detection array is fixed to half-space, *i.e.,* 180°. For a fixed distance *a* between the observation location and the LWA, the total scanning path length is (яа). If the minimum spacing between the detectors is *dd* due to practical detector size constraints, the total number of detectors and thus the measurement of unique frequency samples is given by *яа/Sd.*

While the 1D RTSA systems have been extensively used for frequency discrimination along a single dimension, dominantly in optics, there has always been a great demand to increase their frequency sensitivity and resolution by exploiting a second dimension of space. For this reason, various 2D real-time frequency analysis systems have been recently reported in the literature, primarily at optical frequency ranges. The first system of this type was introduced in a patent filed by Dragone and Forde filed in 1999, and used an arrayed-waveguide grating, a diffraction grating, and focusing lenses [Dragone and Ford (2001)]. Several related systems have been proposed since then, all utilizing free-space propagation between two diffractive elements to achieve the same effect [Metz et al. (2014); Xiao and Weiner (2004); Yang et al. (2004)]. However, while offering enhanced resolution and sensitivity, these 2D spectrum analyzers are bulky systems. While these systems have been proposed and demonstrated at optics, their microwave counterparts have never been suggested. Based on this context, one may ask: Can the 1D microwave RTSA based on LWAs be extended to a 2D RTSA? The answer is Yes.

The general objective of a 2D spectral decomposition system is illustrated in Fig. 5.9. The temporal frequencies (w_{m}) of an input signal to analyze [u_{in}(t)] are mapped, in real-time, onto different spatial frequencies (k_{m} = *(f _{m}, в_{т}))* over a 2D region of space.

**Figure 5.9 **Illustration of a real-time temporal-to-spatial frequency decomposition of a broadband signal in 2D spatial coordinates *(в, j).* Reprinted with permission from Gupta and Caloz, 2015, Copyright 2015, IEEE.

To recall, the conventional 1D LWA of Section 5.5 decomposes the signal into its spectral components in space, where each frequency component is radiated along a specific direction of space according to the scanning law вм_{В} (ю) = sin^{-1}[b(®)/c] [Caloz et al. (2011)]. The dispersion relation *p(a>* of the radiating structure dictates the mapping between temporal and spatial frequencies, and thus a 1D

LWA acts as an analog RTSA along one dimension of space [Gupta et al. (2009)].

To extend the operation of spatial-spectral decomposition from 1D to 2D, consider the system shown in Fig. 5.10. [Gupta and Caloz (2015)]. It consists of an array of *M* LWAs, where each LWA is excited by a unique phaser [Caloz et al. (2013)], as shown in Fig. 5.10a. Let us also assume that each of the LWAs is a CRLH, covering the signal bandwidth (w_{start}, w_{stop}) [Caloz et al. (2011)]. This antenna array naturally provides spectral decomposition along the у-axis from forward to backward region, including broadside, according to b(w) *= m/m _{R} - (oJrn* [Gupta et al. (2009); Caloz et al. (2011)].

**Figure 5.10 **Proposed structure for 2D spectral decomposition, consisting of an array of LWAs fed by phasers. (a) Antenna array layout. (b) One of the phasers, which is here a cascaded C-section phaser. Reprinted with permission from Gupta et al., 2013, Copyright 2013, John Wiley and Sons and Gupta and Caloz, 2015, Copyright 2015, IEEE.

In this system, the frequency discrimination along the x-axis is achieved using a dispersive feeding network wherein the m^{th} antenna in the array is fed with a phaser, providing a frequency-dependent phase shift f_{m}(w). First assume, for the sake of understanding, that the LWAs do not scan with frequency and only radiate along broadside (в = 0° in the *f* = 90° plane). Consider the radiation at a certain frequency w_{0} in the *f* = 0° plane (or along the x-axis) in the following three cases:

(1) When all the LWAs are fed in phase, *i.e.,* |f_{m}+_{1}(®0) - *fm(®o)* | = *2kp,* where *k* is an integer: In this case, frequency w_{0} points to broadside, *i.e., в* = 0°.

- (2) When
*f*-_{m}+_{1}(w_{0})*>*0 : In this case, w_{0}points in the forward direction,*i.e.,*90° >*6*0._{MB}> - (3) When
*f*0: In this case, w_{m}+_{1}(w_{0}) - f_{m}(w) <_{0}points in the backward direction,*i.e.,*-90° <*6*0._{MB}<

Therefore, an appropriate choice of the feeding phasers enables a full-space frequency scanning along the x-axis. In addition, if the above three phase conditions are satisfied at multiple frequency points within the specified bandwidth, as will be shown shortly to be the case with cascaded C-section phasers, the backward-to-forward frequency scan will periodically repeat for each frequency sub-band. Finally, combining the x-axis scanning with the conventional y-axis scanning of the antenna provides the sought 2D frequency scanning in space.

Let us look at an example. Consider a cascaded C-section phaser as the dispersive feed element to the array, as shown in Fig. 5.10b [Gupta et al. (2013)]. This phaser consists of the series connection of C-sections, which are formed using a coupled-line coupler with a coupling coefficient k. The corresponding transmission phase

^{is} g^{iven b}y * ^{f}i {ю*) =

*tan*

^{-2}q_{0}^{_1}[V[1 +

*к,)/{! - k*

_{t})^{tan}(

^{ю}

*)*})], where

^{/[}2ю_{х}*q*is the number of C-sections in the phaser and

_{0}*ю*is the frequency at which the coupled lines are quarter-wavelength long. The radiation characteristics of the LWA array of Fig. 5.10a can then be computed taking into account the above dispersive network and CRLH LWA dispersion relation.

_{х}Figure 5.11 shows the computed array factor [Stutzman and Thiele (1997)] results for different dispersion characteristics of the feeding network [Gupta and Caloz (2015)]. First, a non-dispersive network is assumed by using *k* = 0 as shown in the first row of Fig. 5.11. In this trivial case, the LWA array exhibits conventional 1D scanning along y-axis, as all the antennas are fed in phase. Next, a small amount of dispersion in *f* (w) is introduced by using *k П* 0, as shown in the second row of Fig. 5.11. Consequently, the frequencyscanning plane is rotated to about *f* = 45° plane. In this case, the frequency point *ю _{А},* where |

*fm+*- f

_{1}(w_{A})_{m}(w

_{A})| = 0, corresponds to the broadside frequency in the

*f*= 0° plane. The most interesting case is when the dispersion characteristics of the phaser exhibit a periodic response in frequency due to its commensurate nature, within the radiating band of the LWA, as shown in the third row of Fig. 5.11. In this case, a periodic frequency scanning is achieved between the left and right half of the

*(в - f)*plane, thereby accommodating the same frequency band in a larger spatial area. Since the overall scanning path length Д/ is larger than

*pa,*a larger number of detectors can be placed in the region above the antenna, thereby achieving a larger frequency sampling of the signal spectrum.

**Figure 5.11 **Array factor computed results corresponding to the structure in Fig. 5.10 with *M x N* = 20 *x* 30 showing the dispersion profile of the phaser feeding elements at the input of each LWA and the 3 dB contour plots of the 2D radiation patterns in the *(в, f)* plane. Reprinted with permission from Gupta and Caloz, 2015, Copyright 2015, IEEE.