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A Negative Group Delay (NGD) prototype filter design based on the reciprocal transfer function of a low-pass Butterworth filter of a given order, is presented. The out-of-band gain of the prototype transfer function is capped at a finite constant value via multiplication by a transfer function of a low-pass Butterworth filter with 3 dB bandwidth that is wider than the reciprocal function bandwidth. Such synthesized transfer function exhibits maximal magnitude characteristic flatness within the 3 dB bandwidth (Butterworth-like property), while it also exhibits NGD and satisfies Kramers-Kronig relations (causal transfer function). The prototype design achieves an NGD-bandwidth product that in the upper asymptotic limit as the design order increases, is a linear function of out-of-band gain in decibels. This is an improvement compared with previously reported cascaded first-order and second-order designs, which have NGDbandwidth functional dependency of out-of-band gain in decibels to the power of 1/2 and 3/4, respectively. It is shown that the transfer function of the corresponding design transformed to a non-zero center frequency can be exactly implemented with a Sallen-Key topology employing parallel resonators, or approximately implemented with an all-passive ladder topology. An in-band magnitude/phase distortion metric is applied to the prototype designs, evaluated for Gaussian and sinc pulse input waveforms, and compared with values obtained for a well-known commonly used medium. It is also shown that when the specified bandwidth corresponds to the entire bandwidth over which the group delay characteristic is negative, the magnitude characteristic variation approximately equals half the out-of-band gain value in decibels. Therefore, for any NGD design with large out-of-band gain (typically higher than 6 dB), using the entire bandwidth where group delay is negative can result in strong levels of distortion and should be checked for applied waveforms. [ABSTRACT FROM AUTHOR] more...

Two solutions to the polynomial filter's transfer function synthesis problem are considered for comparison in the frequency and time domain: the broad class of filters with a critical monotonic amplitude characteristic (CMAC) in the passband and filters which use Chebyshev (C) polynomials. To complete the synthesis procedure for linear phase applications, group delay correctors are considered, for which a convenient approximation procedure is proposed here. Comparisons of the original functions and the corrected ones are performed in the frequency and time domain. It is shown that when CMAC and C are compared as such, the latter is by no means preferable from the selectivity point of view, while the opposite stands when the comparison is based on passband amplitude distortions. When phase-corrected filtering functions are compared, based on circuit complexity and time domain performance, the CMAC are shown to be preferable. [ABSTRACT FROM AUTHOR] more...

Three fractional-order transfer functions are analyzed for differences in realizing ( $$1+\alpha $$ ) order lowpass filters approximating a traditional Butterworth magnitude response. These transfer functions are realized by replacing traditional capacitors with fractional-order capacitors ( $$Z=1/s^{\alpha }C$$ where $$0\le \alpha \le 1$$ ) in biquadratic filter topologies. This analysis examines the differences in least squares error, stability, $$-$$ 3 dB frequency, higher-order implementations, and parameter sensitivity to determine the most suitable ( $$1+\alpha $$ ) order transfer function for the approximated Butterworth magnitude responses. Each fractional-order transfer function for $$(1+\alpha )=1.5$$ is realized using a Tow-Thomas biquad a verified using SPICE simulations. [ABSTRACT FROM AUTHOR] more...

The subject of synthesis of critical-monotonic low-pass amplitude characteristics will be revisited. Several new contributions will be given in order to: facilitate the choice of the proper transfer function, to allow cataloguing the transfer functions, to simplify the circuit synthesis procedure, and to perform synthesis in the form of a state-variable continuous time active filter. Four main criteria for transfer function synthesis will be implemented: maximally flat at the origin, maximum slope at the band-edge, maximal asymptotic attenuation, and minimal amplitude distortion in the pass-band. For every criterion, a class of filters will be generated and the coefficients of the transfer functions will be calculated and published for the first time (with one exception). Properties of the classes so generated will be quantitatively compared for the first time. The state-variable structure will be advised as the one with the simplest synthesis procedure. The procedure will be explained and the design process will be exemplified. Statistical tolerance analysis will be performed for the example solutions in order to complete the information for comparison. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR] more...

Continuous-time ΔΣ modulators (CTDSM) used in wireless systems need to process signals in the presence of interferers. Peaking in the Signal Transfer Function of a conventional design necessitates a higher in-band dynamic range to accommodate interferers. A filter up front solves this problem at the expense of increased power dissipation and degraded linearity of the signal chain. Embedding the filter in the modulator achieves the same objective in a power efficient manner, while improving out-of-band linearity and reducing active area. However, having the filter inside a ΔΣ loop can be problematic with respect to stability. We show that such a system can be stabilized in a robust manner without extra hardware. Measurements of a CTDSM (signal BW = 2 MHz), with a built-in VGA (0 to 18 dB) and a second order Butterworth filter (4 MHz cutoff), show that the in-band/out-of-band IIP3 improves by about 3/10 dB when compared to the filter-CTDSM cascade, and achieves a similar dynamic range and consumes 25% lower power. [ABSTRACT FROM AUTHOR] more...

Abstract: A procedure for designing digital Butterworth filters is proposed. The procedure determines the denominator and the numerator of the filter transfer function based on the positions of the poles in the s-plane and zeros in the z-plane, respectively, and calculates the gain factor using a maximum point normalization method. In contrast to some conventional algorithms, the presented procedure is much simpler by directly obtaining the filter with 3-dB frequencies. This makes the presented algorithm a useful tool for determining the boundaries in electronic or communication systems’ frequency responses. Moreover, the proposed algorithm is compatible with high-order transformations which are the limitations of general pole-zero placement techniques. The proposed method is illustrated by the examples of designing the low-pass, high-pass, band-pass, and band-stop filter. [Copyright &y& Elsevier] more...