One-Dimensional Real-Time Spectrum Analyzer
As established in Section 5.3, an LWA may be seen as a microwave counterpart of a diffraction grating. Just like a diffraction grating is used in optics for spectrally decomposing a broadband signal in space, an LWA can also be analogously used to spectrally analyze a broadband microwave signal. This equivalence inspires us to devise an RTSA as shown in Fig. 5.7 for analog computation of STFT of a broadband signal. This system is based on the following three successive operations:
- (1) Spatial-spectral decomposition using an LWA to discriminate the frequency components of the signal.
- (2) Probing and monitoring of the time variation of each frequency component. Probing is achieved by antenna receivers, while monitoring is performed by envelope demodulation.
- (3) Post-processing, including analog/digital conversion, data processing, and display.
In the first step, an LWA first spectrally decomposes the nonstationary signal in space. While there are several choices for the antenna, metamaterial-based CRLH LWA is of particular usefulness as it offers three distinct benefits: (i) full-space radiation from backfire to endfire, including broadside in the fundamental mode, offering a simple and real-time frequency-space separation mechanism; (ii) frequency and bandwidth scalability, allowing to handle UWB signals; (iii) simple and compact design and implementation.
Figure 5.7 LWA-based real-time spectrum analyzer. Reprinted with permission from Gupta et al., 2009, Copyright 2009, IEEE and Caloz et al., 2013, Copyright 2013, IEEE.
As mentioned in Section 5.3.3, the typical dispersion curve of a CRLH transmission line always penetrates into the fast-wave region, resulting in leaky-wave radiation [Liu et al. (2002); Caloz and Itoh (2006)]. Therefore, according to the scanning law of (5.12a), if the CRLH LWA is excited by a pulse signal, the various spectral components of the signal radiate in different directions at any particular instant. Thus, the CRLH LWA performs a real-time spectral-to-spatial decomposition of the signal following the beamscanning law of the LWA, thereby discriminating the various spectral components present in the testing signal, as indicated in the left of Fig. 5.7.
Once the signal has been spatially decomposed in space by the CRLH LWA, the various frequency components need to be probed and monitored in real time. For this purpose, n antenna probes are arranged circularly in the far-field around the antenna at the positions r(a, вп = пАв), where a is the observation distance from the center of the antenna and Ав is the angular separation between two observation points. For broadband applications, the far- field distance is given by dff ~ 2/2/Amin, where Xmin is the smallest wavelength in the operating frequency range. At each time instant, the different probes, based on their angular location in space вп, receive the different frequencies, thereby achieving real-time frequency-space mapping of the signal propagating across the LWA. The voltages induced along the antenna probes are subsequently envelope-demodulated and monitored as a function of time to track the temporal evolution of the different spectral components of the input signal.
Finally, the envelope-demodulated induced voltages across the antenna probes are next converted into digital format via A/D converters, combined, and then post-processed, as shown in Fig. 5.7. After digitizing and combining the envelope-demodulated voltage waveforms from the antenna probes, an energy distribution function д(в, t) is obtained, where в is the radiation angle. If the beamscanning law was linear, the resulting spectrogram 5(ю, t) would be directly proportional to this function д(в, t) and would, therefore, be immediately available. Because the beam-scanning law of the CRLH LWA (5.12a) is a nonlinear function of frequency, a post-processing operation is required to compensate for this effect. Instead of uniform angular spacing, the antenna probes can also be placed nonuniformly corresponding to uniform increment in frequency, thus avoiding an extra operation of spectrogram linearization. However, in order to avoid spatial crowding and overlapping in the forward region where the CRLH LWA exhibits a larger fast-wave bandwidth, uniformly spaced probes were employed, which is also practically more convenient. Nonetheless, this spectrogram linearization step requires minimal computational resources and can be easily done at the end of the post-processing stage.
The typical full-wave computed spectrograms for diverse range of non-stationary signals are shown in Fig. 5.8 using the RTSA setup of Fig. 5.7. It can be seen that the instantaneous spectral features of complicated chirped signals are faithfully reproduced, thereby validating the usage of LWA for analog computation of STFT. More details about the experimental prototypes and measured results are available in [Gupta et al. (2009)].
Figure 5.8 Full-wave (CST Microwave Studio) spectrograms. (a) Multiple modulated Gaussian pulses. (b) Nonlinear cubicly chirped Gaussian pulse. (c) Doubly negative chirped Gaussian pulses. (d) Oppositely chirped Gaussian pulses. (e) Self-phase modulated pulses. (f) Dispersed pulse through a CRLH transmission line. Reprinted with permission from Gupta et al., 2009, Copyright 2009, IEEE.