1 Analog Signal Processing Christophe Caloz, Fellow, IEEE, Shulabh Gupta, Member, IEEE, Qingfeng Zhang, Member, IEEE, and Babak Nikfal, Student Member, IEEE Abstract—Analog signal processing (ASP) is presented as a discrimination effects, emphasizing the concept of group delay systematic approach to address future challenges in high speed engineering, and presenting the fundamental concept of real- and high frequency microwave applications. The general concept time Fourier transformation. Next, it addresses the topic of of ASP is explained with the help of examples emphasizing basic ASP effects, such as time spreading and compression, the “phaser”, which is the core of an ASP system; it reviews chirping and frequency discrimination. Phasers, which repre- phaser technologies, explains the characteristics of the most sent the core of ASP systems, are explained to be elements promising microwave phasers, and proposes corresponding exhibiting a frequency-dependent group delay response, and enhancement techniques for higher ASP performance. Based hence a nonlinear phase response versus frequency, and various on ASP requirements, found to be phaser resolution, absolute phaser technologies are discussed and compared. Real-time Fourier transformation (RTFT) is derived as one of the most bandwidth and magnitude balance, it then describes novel fundamental ASP operations. Upon this basis, the specifications synthesis techniques for the design of all-pass transmission of a phaser – resolution, absolute bandwidth and magnitude and reflection phasers and band-pass cross-coupled phasers. balance – are established, and techniques are proposed to enhance Finally, it presents some applications using these phasers, phasers for higher ASP performance. Novel closed-form synthesis including a tunable pulse delay system for pulse position techniques, applicable to all-pass transmission-type cascaded C- section phasers, all-pass reflection-type coupled resonator phasers modulation, a real-time spectrum sniffer for cognitive radio and band-pass cross-coupled resonator phasers are described. and a real-time spectrogram analyzer for the characterization Several applications using these phasers are presented, including and processing of nonstationary signals. a tunable pulse delay system, a spectrum sniffer and a real- time spectrum analyzer (RTSA). Finally, future challenges and II. WHAT IS MICROWAVE ANALOG SIGNAL PROCESSING? opportunities are discussed. Microwave ASP might be defined as the manipulation of Index Terms—Analog signal processing (ASP), phaser, signals in their pristine analog form and in real time to realize frequency-dependent group delay, dispersion engineering, dis- specific operations enabling microwave or millimeter-wave persive delay structure, chirping, real-time Fourier transform, C-section and D-section, cross-coupled resonators, filter synthesis. and terahertz applications. The essence of ASP might be best approached by consider- ing the two basic effects described in Fig. 1, chirping with time spreading and frequency discrimination in the time domain. I. INTRODUCTION AND MOTIVATION Both effects involve a linear element with transfer function Today’s exploding demand for faster, more reliable and H(!) = ejφ(!), which is assumed to be of unity magnitude ubiquitous radio systems in communication, instrumentation, and whose phase, φ(!), is a nonlinear function of frequency, or radar and sensors poses unprecedented challenges in mi- whose group delay, τ(!) = −@φ(!)=@!, assuming the time- crowave and millimeter-wave engineering. harmonic phasor dependence ej!t, is a function of frequency. Recently, the predominant trend has been to place an Note that there is no incompatibility between the linearity of increasing emphasis on digital signal processing (DSP). How- the element and the nonlinearity of its phase: the former refers ever, while offering device compactness and processing flex- to the independence of the element’s response to the magnitude ibility, DSP suffers of fundamental drawbacks, such as high- of the input signal, whereas the latter refers to the nonlinearity cost analog-digital conversion, high power consumption and of the element’s phase versus frequency, which is an inherent poor performance at high frequencies. property of the element, independent of the input signal. Such To overcome these drawbacks, and hence address the afore- an element, with frequency-dependent group delay, is called arXiv:1307.2618v2 [physics.optics] 3 Aug 2013 mentioned challenges, one might possibly get inspiration from temporally dispersive, or simply dispersive when the context ultrafast optics [1]. In this area, ultra-short and thus huge- is unambiguous. It is to be noted that temporal dispersion bandwidth electromagnetic pulses are efficiently processed in contrasts with spatial dispersion, occurring for instance in the real time using analog and dispersive materials and compo- phenomena of radiation through an aperture or propagation nents [2], [3], [4]. across a periodic structure, where different spatial frequencies, We speculate here that the same approach could be po- corresponding to scattering in different directions, are gener- tentially applied to high-frequency and high-bandwidth mi- ated. The spectrum of H(!) is assumed to cover the entire crowave signals, leading to systematic microwave analog spectrum of the input signal. signal processing (ASP) as an alternative to DSP-based pro- In the chirping case, depicted in Fig. 1(a), a pulse (typically cessing, with particular promise for millimeter and terahertz be a gaussian pulse, whose spectrum is also a gaussian frequency applications. function [5]) modulated at an angular frequency !0 is passed The paper first explains more specifically what is meant through the element H(!), which is assumed here to ex- by microwave ASP, covering the basic chirping and frequency hibit a positive linear group delay slope over a frequency 2 τ(ω) that with the flat-step law considered here the pulses are not time-spread (Tout = T0), assuming that the pulse bandwidth fits τ0 τ3 = τ0 ω0 in the flat bands of the steps, since all the spectral components ω ω1 ω2 ω3 ω4 ω5 within each band are delayed by the same amount of time. ω0 Reference [8] provides a fundamental explanation of group t H(ω) = ejφ(ω) t delay dispersion in terms of wave interference mechanisms. III. CORE OF AN ASP SYSTEM: THE PHASER Tin = T0 (a) Tout > T0 The dispersive element H(!) in Fig. 1, which manipulates the spectral components of the input signal in the time τ(ω) ∆ω domain, is the core of an ASP system. We suggest to call τ02 such a dispersive element, in which a nonlinear phase versus ∆τ ω1,ω2 ∆τ frequency response is necessary for ASP, a phaser. Several τ01 τ02 ω terms have been used by the authors and discussed with τ01 ω01 ω02 colleagues from various academic and industrial backgrounds. ω01 ω02 t H(ω) = ejφ(ω) t After much reflection, “phaser” imposed itself as the most appropriate term. Table I lists the four main terms that were considered with their pros and cons, and shows the superiority Tout = T0 Tout = T0 of “phaser” over the other three terms. A phaser “phases” as a Tin = T0 (b) filter filters: it modifies the phase of the input signal following Fig. 1: Basic effects in ASP. (a) Chirping with time spreading. phase – or more fundamentally group delay – specifications, (b) Frequency discrimination in the time domain. with some magnitude considerations, just in the same way as a filter modifies the magnitude of the input signal following magnitude specifications, with some phase considerations. It has come to the attention of the authors that a few people band centered at the frequency ! , corresponding to a group 0 seem to dislike the terminology “phaser.” One of them asserted delay τ . Due to the dispersive nature of this element, the 0 that “this term should be reserved to Star Trek” [sic]. Well, the different spectral components of the pulse experience different fact that “phaser” refers to a directed-energy weapon in this delays and therefore emerge at different times. Here, the popular science fiction series [9] is not considered a major lower-frequency components are less delayed and therefore concern by the authors, who judge that confusion between emerge earlier than the higher-frequency components, while ASP phasers and Star Trek phasers is fairly unlikely! τ = τ the center-frequency component appears at the time 3 0. An ideal phaser is a phaser that would exhibit an arbitrary This results in an output pulse whose instantaneous frequency group delay with flat and lossless magnitude over a given is progressively increasing (would be decreasing in the case frequency band, as illustrated in Fig. 2. Obviously, such an of a phaser with a negative group delay slope), a phenomenon ideal response is practically unrealizable. Realizing a response called “chirping”, and which has experienced time spread- as close as possible to this ideal one for specific applications T > T ing ( out 0), accompanied with reduced amplitude due is the objective of phaser synthesis, which will be discussed to energy conservation. When undesired, this time spreading in Sec. VI. effect can be compensated by either using a stepped group delay phaser, as will be shown in the next paragraph, or by compression, using phasers of opposite chirps, as will be S (ω) 21 0 dB shown in the tunable pulse delay line application of Sec. VII. | | It should be noted that in many applications, time spreading is | τ 21 actually useful. For instance, it belongs to the essence of real- S time Fourier transformation, to be presented in Sec. IV. As | another example, time spreading may be exploited to increase τ(ω) group delay, the sampling rate of a signal at fixed clock [6],[7]. magnitude, In the latter case, depicted in Fig. 1(b), the input pulse is modulated by a two-tone signal, with frequencies !01 and !02, and passed through a dispersive element H(!) exhibiting a stopband passband stopband positive stepped group delay, with two steps, centered at ! 01 ω1 ω2 and !02, respectively. Based on this dispersive characteristic, frequency, ω the part of the pulse modulated at the lower frequency, !01, is delayed less than the part modulated at the higher frequency, Fig.