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## Alternatives to SAW Devices

SAW devices are by far the most employed devices for analog signal processing (ASP) due to their high TBWP (large group delay swing and relatively wide bandwidth), compact size, and low mass. However, other devices exist that can also be designed to present a linear group delay characteristic and, thus, be used in ASP applications, in particular, real-time Fourier transformation. One of these alternative devices employs magnetostatic wave (MSW) technology. These devices use ferrimagnetic films, such as liquid phase epitaxial (LPE) yttrium iron garnet (YIG) films, and present smaller group delay swing (in the order of hundreds of nanoseconds) than SAW devices. However, they have wider bandwidths and can operate at higher frequencies (several gigahertz) [27, 28]. As for drawbacks, MSW devices require magnets, and its fabrication process is intricate. In the eighties and nineties, the Analog Device Technology Group on the Lincoln Laboratory of the Massachusetts Institute of Technology (MIT) performed research on the use of superconductive devices for ASP. Several chirp filters were developed, and some compressive receivers employing such devices were reported in [1, 29–32]. Superconductive devices are four-port networks in which a pair of superconductive striplines separated by a variable distance are coupled by a cascade array of backward-wave couplers. Fig. 2.11 illustrates this device, where a portion of the energy propagating in the input line from the input port is transferred to a wave propagating to the other sense in the adjacent line, towards the output port. This mode of operation causes superconductive devices to be classified as reflection-type. The devices previously mentioned are classified as transmission-type. Compared to SAW, superconductive devices present higher bandwidths, but much smaller group delay swing (on the order of several dozen nanoseconds).

Consequently, their TBWP is also much smaller. As advantages, superconductive devices present low insertion loss and are capable of operating at high frequencies (several gigahertz), but require cryogenic temperatures and are complex to fabricate.

More recently, artificial transmission lines employing distributed structures on conventional microwave technologies have been studied and developed for ASP applications. In [33], a com2.1. Real-Time Fourier Transformers Employing SAW Devices 17 posite right/left-handed (CRLH) transmission line (TL) implemented on a Rogers Duroid 5870 substrate was employed in the realization of a compressive receiver. An integrated implementation of an active dual composite right-left handed (D-CRLH) dispersive delay line (DDL) using a distributed structure was designed in a gallium nitride (GaN) monolithic microwave integrated circuit (MMIC) technology and reported in [34]. A real-time Fourier transformer was designed using an engineered C-section all-pass network in [35], and an inverse Fourier transformer in [36]. Integrated active delay lines were first demonstrated in [37–39], implemented using a 0:13 μm complementary metal-oxide-semiconductor (CMOS) process.

Chirped microstrips (also known as chirped electromagnetic bandgaps (CEBGs), or nonuniform transmission lines (NUTLs)), a reflection-type structure, can also be designed to present a linear group delay. It seems that Laso et al. [40, 41] were the first to report the realization of a chirped microstrip where the energy spectral density of the input signal was given by the average output power reflected from the device. In [42], Schwartz et al. demonstrate a fully electronic system for the time-magnification (or time-stretching) of ultra-wideband (UWB) signals. The system employs three CEBGs fabricated on a Coorstek ADS-96R substrate and achieves a timemagnification factor of five when operating on a 0:6 ns time-windowed input signal with up to 8 GHz bandwidth. On [43], Ma et al. employ nonuniform coupled-line phasers constructed on a Rogers RO3010 substrate, and with structures resembling that of Fig. 2.11, to perform real-time Fourier transformation and inverse Fourier transformation. In [44, 45], Xiang et al. demonstrate a reconfigurable integrated DDL in 0:13 μm CMOS process capable of real-time spectrum analysis. The DDL presents a transversal filter structure, also resembling that of Fig. 2.11, but presenting tap gains.

### Radio Astronomy Spectrometers

In radio astronomy, the power spectrum is of vital importance in the analysis of physical phenomena and the study of celestial bodies. In this domain, the systems that allow the evaluation of the power spectrum are called spectrometers (or, more specifically, radio spectrometers, when they process radio signals). The CT-based systems mentioned previously constitute the core of a type of radio astronomy spectrometer, namely, a chirp transform spectrometer (CTS). Despite their specialized use, it is worth briefly discussing here the different types of radio spectrometers that exist, since the Fourier transform is the main operation performed by these instruments.

There exist five basic types of spectrometers commonly used in radio astronomy.

• Chirp transform spectrometers (CTSs)

• Filterbank spectrometers

• Autocorrelation spectrometers

• Acousto-optical spectrometers (AOSs)

• FFT spectrometers

Each type of spectrometer presents some strong and weak points when compared to the other types. The spectrometers used in radio astronomy are generally composed of discrete circuits (not ICs), even those for airborne/space applications, and have strict requirements in terms of stability, reliability, bandwidth, and spectral resolution.

Filterbank spectrometers are the simplest type of radio astronomy spectrometers. They are constituted by a bank of frequency contiguous IF filter channels. The IF channels usually include an amplifier to overcome signal splitting and filter transmission losses. Individual IF gain controls allow the overall spectrometer magnitude frequency response to be equalized [51]. Autocorrelation spectrometers (also known as digital autocorrelators or, simply, autocorrelators) have been used since the early sixties [52] and are constituted by digital autocorrelators [53]. They perform a one-bit or two-bit quantization of the time signal, being the use of more than two bits of quantization impractical, due to the high complexity of the circuits.

Acousto-optical spectrometers have been used since the early eighties [52]. They are based on a technique combining acoustic bending of a collimated coherent light beam by a Bragg cell, followed by detection by a sensitive array of photodetectors [53]. The bandwidth of an AOS is limited by the bandwidth of the Bragg cell (typically, 1:5 GHz to 2 GHz). AOSs are generally low power but commonly present a high mass due to the mechanical requiriments of their optical parts. Early AOSs often presented stability issues, that were solved after more sophisticated optical, electrical, mechanical, and thermal designs [52].

More recently, the technology scaling trend of integrated circuit (IC) technology enabled the realization of FFT spectrometers with performances surpassing those of other types of spectrometers. They are cheap, lightweight, compact, extremely flexible (reprogrammable), and present low power consumption.

#### Analog Implementation of the Fast Fourier Transform

Recently, some works have reported analog discrete-time implementations of the FFT aimed at reducing the burden of analog-to-digital converters (ADCs) in software-defined radios (SDRs), that need to operate at frequencies up to 5 GHz. The idea is to interpose a preprocessing circuit between the antenna and the ADC to precondition the radio signal. The preprocessing operations consist mainly in sampling and transforming the signal from the time to the frequency domain. In [63–65], Rivet et al. introduce what they call a sampled analog signal processor (SASP), that samples a signal and recovers its spectrum, to subsequently perform downconversion and channel presorting in the frequency domain. They developed a demonstrator implementing a 64-point radix-4 DFT (FFT butterfly) in a 65nm CMOS technology. The architecture of this system is shown in Fig. 2.13, and operates by first sampling the input signal through a track and hold and windowing the sampled signal. The sampling frequency (fs) determines the FFT processing frequency (64fs), the spectrum range (from 0 Hz to fs 2 ), and the spectral resolution ( fs 64 ). The voltage samples are then sent to the FFT butterfly circuit, which is composed of three stages. Each stage performs delay, addition, and weighting operations on the voltage samples. Delay operation is carried out by storing samples on capacitors, while addition and weighting operations are carried out in the current domain, via a voltage/current/voltage conversion. The weighting and the matrix units operate on a four by four set of samples. A digital logic circuit synchronizes and coordinates all the blocks. Their demonstrator occupies an area of 1200 μm by 1200 μm (with pads) and consumes 389mW when operating at 1:2 GHz.

Another analog implementation of the FFT was reported by Sadhu et al. in [66, 67]. Their system, named CRAFT, from charge reuse analog Fourier transform, is based on a 16-point radix-2 FFT (see Fig. 2.14a) and aims to function as an RF front-end channelizer for SDRs. It is built upon passive switched capacitors on a 65nm technology. The charge reuse analog Fourier transform (CRAFT) core (FFT butterfly) occupies an area of 300 μm by 480 μm, while the complete system, with sampling circuit, state machines, analog latches and multiplexer (cf. Fig. 2.14b), occupies an area of roughly 900 μm by 480 μm (without pads). The system can work with input rates up to 5GS=s, consuming only 3:8mW (12:2 pJ=conv:), and presenting a corrected (calibrated) signal-to-noise and distortion ratio (SNDR) of 47 dB. Due to its charge reuse scheme, the CRAFT system presents a power consumption 28 times better than the SASP system of [63–65]. However, the authors recognize the system complexity and susceptibility to circuit non-idealities, like noise, matching, and non-linearities, which calls for a careful layout. The complexity of the layout can be appraised based on the block-level scheme of the CRAFT layout floorplan shown in Fig. 2.14b. Moreover, as an additional complicating factor, circuit simulators and technology models lack support for simulating switching circuit noise, chargeinjection, and charge accumulation.

**Table of contents :**

Résumé

List of Figures

List of Tables

Acronyms

Symbols

**1 Introduction **

1.1 Motivation

1.2 Objectives

1.3 Thesis Structure

**2 Literature Review and State of the Art **

2.1 Real-Time Fourier Transformers Employing SAW Devices

2.1.1 Discrete Fourier Transform

2.1.2 Applications

2.1.3 Alternatives to SAW Devices

2.2 Radio Astronomy Spectrometers

2.3 Analog Implementation of the Fast Fourier Transform

2.4 Selected Research Direction

**3 Real-Time Analog Chirp Fourier Transformer **

3.1 Limitations of Common Implementations of the MCM Arrangement

3.2 Proposed Analog Chirp Fourier Transformer

3.3 Discussion on Analytical Results

3.4 Effects of System Imperfections

3.5 Equalizing Spectrum Amplitude and Resolution

3.6 Conclusion

**4 Filters with Engineered Group Delay or Phase – Part I: Minimum Phase Filters**

4.1 Engineered Group Delay Filters in the State of the Art

4.2 Design and Synthesis Procedures

4.2.1 Generation of Filter Polynomials

4.2.2 Synthesis of Filter Networks

4.3 Design and Synthesis Example

4.4 Conclusion

**5 Filters with Engineered Group Delay or Phase – Part II: All-Pass Filters **

5.1 Design and Synthesis Procedures

5.1.1 Transfer Function Generation

5.1.2 Network Synthesis Procedure

5.2 Design and Synthesis Examples

5.2.1 Lattice Network with Linear Group Delay of Negative Slope

5.2.2 Bridged-T Network with Linear Group Delay of Positive Slope

5.3 Active Filters

5.4 Conclusion

**6 Conclusion **

6.1 Suggestions for Future Work

6.2 Contributions

6.2.1 Articles Published in Conference Proceedings

6.2.2 Journal Articles

6.2.3 Other Articles Published in Conference Proceedings

Appendices

**A Sinusoidal Complex Signals **

A.1 Single-tone sinusoid

A.2 Sinusoidal linear chirp

A.3 Generation of a chirp signal

A.4 Fourier transform of a linear sinusoidal chirp

**B Detailed Derivation of the RTACFT System **

B.1 High Abstraction Level Model

B.1.1 Angular Frequency

B.1.2 Ordinary Frequency

B.2 Low Abstraction Level Model

B.2.1 Angular Frequency

B.2.2 Ordinary Frequency

**C Definitions of the Fourier Transform **

**D Component Models of the ST BiCMOS9MW Technology **

D.1 Capacitor Model

D.2 Inductor Model

**Bibliography**