LWA-Based Real-Time Spectrum Analyzers
Consider now some signal processing applications such as a general Fourier transforming operation: short-time Fourier transform (STFT) or a spectrogram, from which simpler operation of determining the overall spectral contents is implicitly obtained. This operation is useful for characterizing non-stationary signals, which are typical of most of today's ultra-wideband (UWB) systems, such as radar, security and instrumentation, and electromagnetic interference and compatibility (EMI/EMC), displaying rapid spectral variations in time [Thummler and Bednorz (2007)]. In order to effectively observe such signals, both time information and spectral information are simultaneously needed.
The STFT belongs to the general class of joint time-frequency representation. Joint time-frequency representations are 2D plots of a signal where a 1D signal is represented as an image in a time-frequency plane, with the signal energy distribution coded in the color-scale levels of the image. The joint time-frequency representation of a given signal thus provides information on the temporal location of the signal's spectral components, which depends on the temporal/spectral structure of this signal [Cohen (1989)]. Such analysis not only provides an intuitive insight into the transient behavior of the signals, but also completely characterizes their frequency, phase, and amplitude responses. Thus, the joint time-frequency representation is a highly informative tool for real-time spectrum analysis. Various numerical techniques exist to compute the joint time-frequency representations, with STFT/ Spectrogram and Wigner-Ville distributions being the most common [Cohen (1989)].
The STFT (Spectrogram) of a signal x(t) is calculated using
whereg(t) is a gate function. Numerous RTSAs are currently utilized for various applications. At microwaves, RTSAs are generally based on a digital STFT For UWB signals with ultrafast transients, small- gate duration is required in the STFT for high time resolution. Such small-gate durations inherently lead to a long acquisition time and large acquisition bandwidth. Thus, the STFT process requires heavy computational resources and large memory buffers. These restrictions severely affect the system functionalities and limit its performance to restricted acquisition bandwidth, thereby preventing its application to UWB systems.
In optics, joint time-frequency analysis is often achieved by frequency-resolved optical gating (FROG) systems, which are used to measure and characterize ultrashort optical pulses. A FROG system is an analog implementation of the STFT employing a self-gating process achieved by a nonlinear second harmonic generating crystal
[Trebino (2002)]. Another optical system capable of performing joint time-frequency analysis is the acoustic spectrum analyzer reported in [Lee and Wight (1986b)], where Bragg cell plays a similar role as the nonlinear crystal in the FROG systems. In acoustics, real-time spectrum monitoring is achieved using acoustic filters for speech signal analysis [Wood and Hewitt (1963)]. Whereas the optical and acoustic spectrum analyzers mentioned above are capable of handling and analyzing very broadband signals, the digital RTSAs available at microwaves are unfortunately restricted to bandwidths generally too narrow for practical broadband applications.
With this background in context, the objective of this section is to describe a system that achieves a spectrogram of an arbitrary nonstationary signal using analog means without resorting to digital computation, in the spirit of the R-ASP paradigm.