A Digital Interpretation of Frequency-Periodic Signal-Interference Microwave Passive Filters

Abstract

This paper addresses, for the first time, the digital representation of spectrally periodic signal-interference microwave passive filters. Starting from the theoretical rational transfer function of a synthesized transmission-line signal-interference analog filter, the real-valued coefficients of its digital-domain filter counterpart are derived. This discrete-time formulation for this type of microwave filters allows to clarify some confusing concepts into them, such as filter order. Moreover, some particular phenomena observed in associated experimental proof-of-concept planar prototypes, such as the appearance of out-of-band/spurious resonance peaks, are also explained by this study as zero-pole quasi-cancellations. In addition, it is demonstrated here that the use of the adjective "transversal" for this class of microwave passive filters could be no rigorous as it is understood in a digital sense since its discrete-time modeling commonly obeys an infinite-impulse-response structure. For validation, the expounded theoretical framework is applied to three examples of signal-interference passive filters exhibiting quasi-elliptic-type single-passband, six-passband, and low-pass responses.

Behavioural digital modelling of lossy frequency‐periodic microwave passive filters

“…The periodicity in frequency for these filters comes from the fact that they are usually composed of transmission-line segments with electrical lengths related by integer numbers, whose spectral behaviour is mathematically expressed by periodic trigonometric functions. On the other hand, given the periodicity of the frequency response -even-type symmetry for the amplitude response and odd-type symmetry for the phase response -of discrete-time linear time-invariant systems described by difference equations [19], it is possible to match an associated discrete-time counterpart with the aforementioned parameter S 21 f after noting that it can be written as follows [15]:…”

“…As discussed in [15,16], the ideally synthesised power transmission response of a frequency-periodic microwave passive filter can be matched with the frequency response of an associated digital system. Thus, the coefficients a k and b k of the associated difference equation of the linear time-invariant discrete-time system can be extracted and the zero-pole diagram in the z-plane can be analysed.…”

Frequency‐periodic microwave passive circuits and their digital matching

“…As it was mentioned in the introductory section, a frequency-periodic microwave passive circuit is understood as an analogue passive device whose response can be matched to that of a digital LTI system with real-valued coefficients a k and b k . In particular, the ideal synthesised scattering-parameter-type response S ij ( f ) of this microwave passive circuit, where i and j refer to any two ports or even to the same one, can be formulated as follows [10]…”

“…One must analyse the frequency-periodic ideally-synthesised microwave structure as a function of the complex-valued variable zpreferably by making use of computer-based symbolic calculusand then write down the coefficients. The authors have recently applied this procedure to signal-interference microwave passive filters [10]. The study in [10] also revealed the existence of zero-pole cancelations (ZPCs) in the z-plane for these circuits, which enabledfor the first timethe explanation of spurious resonances observed in the measurements of some proof-of-concept prototypes of signal-interference microwave filters reported in the past [10].…”

Analysis and Design of a New Ring Bandstop Filter Using Lumped Equivalent Circuit

“…Either the concept of filter order or the principle of generation of TZs seems to be ambiguous especially in the case of BSFs based on cross coupling or signal interference techniques. Thus, many researchers have paid their attention to such matters and tried to clarify a signal-interference BSF from the viewpoint of digital filters [15].…”