Ladder Filter

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A ladder filter is an electronic filter configuration that utilizes a series of reactive components (inductors and capacitors) arranged in a ladder-like topology to achieve specific frequency response characteristics, such as low-pass, high-pass, band-pass, or band-stop filtering. It is commonly used in signal processing and communication systems.

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Modelling Zero-Delay Feedback Transistor Ladder Filter using OTA Abstraction

by Peter Corbett

2024

The Transistor Ladder Filter's response is challenging to model using traditional DSP methods, as the response is both non-linear and implicit. Introducing a unit delay in the feedback path warps the frequency response. This can be... more

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Generalized Moog Ladder Filter: Part I–Linear Analysis and Parameterization

by Vesa Valimaki

2023, IEEE/ACM Transactions on Audio, Speech, and Language Processing

The Moog ladder filter, which consists of four cascaded first-order ladder stages in a feedback loop, falls within the class of devices that have attracted greatest interest in virtual analog research. On one hand, this work confirms that... more

A common improvement to this and other similar circuits is represented by the possibility to choose different frequency response modes based on which stages the output voltage is extracted from and fed back to the differential pair [26]. It is, then, possible to generalize the circuit to any positive number N of ladder stages, as shown in in Fig. 2. A continuous-time large-signal model of such a generalized circuit can be easily obtained by following the same reasoning as in [21], under the assumption that the current gain factor (beta) of the transistors can be considered infinite, which in turn implies that all base currents are null and that the base voltages of ladder stages are constant. The resulting model can be expressed as While the linear response of the Moog ladder filter has been already studied in previous works, e.g. [19], [22], the exact relationships between user-controlled parameters and the po- sition of filter poles have been so far only partially analyzed in [20], but otherwise mostly neglected. This, however, represents a crucial aspect in understanding the behavior of the original filter and is here regarded as a prerequisite to effectively define analogous digital structures. A similar approach in the case of another circuit was taken in [13]. The analysis contained in this section applies to the generalized circuit introduced in Section II.

Fig. 7. Proposed filter design. The dotted Gog (z) block is only present when N is odd.

Fig. 8. Magnitude responses obtained from the fourth-order (NV = 4) variant of the proposed filter design with f; = 48kHz, fe = 26.5 Hz, 70.0 Hz, 185.2 Hz, 489.9 Hz, 1296.1 Hz, 3428.9 Hz, 9071.6 Hz, 14755.3 Hz, and k = 3.

Fig. 1. (a) Differential pair and (b) ladder stage. When a null input signal Vj, = 0 is fed into the system, the derivatives in (1) and (2) are null, hence AVy = AVy_1 =

Fig. 3. Positions of poles with (a) N = 4, k = 0,1,2,3,4, and (b) N = 5, k = 0,1, 2,sec5(/5). where Ao(k) and Bo(k) are obtained from (14) and (15), respectively, by setting w = 0. More conveniently, the cut- off frequency can be expressed for any N as

Fig. 4. Relative cut-off frequency error €r,c, in percentage, w.r.t. normalized feedback gain knorm for N = 3,...,8. and knorm = 1, the error €,,, = 0, which is consistent with the reasoning on phase-shift oscillators at the beginning of this subsection.

Fig. 5. Magnitude responses of the fourth-order (N = 4) filter with k = 0,1, 2,3,4 and (a) fe = 100 Hz, (b) f. = 100Hz, (c) f. = 100Hz and cut-off-slope-gain compensation, and (d) f. = 100 Hz and relative de gain compensation.

Fig. 2. Generalized ladder filter circuit. . = AV, = —kAVy => AV; = 0 is the operating point of each stage 7. It is therefore possible to linearize tanh(x) + x around x9 = 0, leading to

Fig. 6. Magnitude responses of the eight-order (NV = 8) filter with k = 0, 1/2, 1, 3/2, sec8(a/8) and (a) fe = 100 Hz, (b) f. = 100 Hz, (c) fe = 100 Hz and cut-off-slope-gain compensation, and (d) f-¢ = 100 Hz and relative de gain compensation.

which, applying the same Laplace-to-z-domain mapping, cor- responds to A sign-inverting element completes the proposed design in order to match the output phase of the analog device.

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CFOA Based Low Pass and High Pass Ladder Filter—A New Configuration

by pranav kumar

2022, Circuits and Systems

A new technique using signal flow graph for conversion of ladder based filter into CFOA based filter has been proposed. The proposed technique converts the existing LC ladder based filter into CFOA in low pass and high pass configuration.... more

Table 1. Comparison of results of [7] and proposed realization. In addition, after implementing the conventional and CFOA based observed that the responses of the two filters are almost identical to t ow pass filter and high pass filter it was heir respective conventional circuits. The presence of inductor can be removed completely from the filter structure to make it less bulky and expensive. The shape of response can be set by varying resistors as variable ind practical. These filters imply a minimum No. of capacitors, resistance ters are suitable for fabrication of integrated circuit technique as we CFOA. The above conversion can be used to produce more active low uctance for low frequency filters are not and CFOA as compared to [7]. Such fil- replace the inductor with the equivalent pass filters and high pass filter. So, finally we can say that the CFOA based filter is a good step towards establishment of active filters.

Figure 11. CFOA based high pass filter. Figure 10. Response of ladder based high pass filter.

Figure 4. Conversion of signal flow diagram into CFOA block diagram. Conventional Ladder based low pass filter

From the graph, the cut-off frequency comes to be 430 Hz (approx.) and we can see that the response is the response for low pass filter. And both the graphs are identical to each other so the conversion is successful. The CFOA based low pass filter has been shown in Figure 7 while the response has been depicted in Figure 8.

Figure 12. Frequency response of CFOA based high pass filter.

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Stable Structures for Nonlinear Biquad Filters

by Jatin Chowdhury

2021

Biquad filters are a common tool for filter design. In this writing, we develop two structures for creating biquad filters with nonlinear elements. We provide conditions for the guaranteed stability of the nonlinear filters, and derive... more

Figure 11: Response of the nonlinear resonant lowpass filter to a 50 Hz square wave with varying input gain.

Figure 6: Instantaneous poles for a nonlinear biquad resonant lowpass filter calculated from eq. (22), with g1 = g2 = ¢ Proceedings of the 23" International Conference on Digital Audio Effects (DAFx2020), Vienna, Austria, September 2020-21

Figure 5: Nonlinear Transposed Direct Form II. The “NL” blocks refer to a generalized nonlinear element. We now propose adding nonlinear elements to the TDF-II str- cuture in the following fashion (see fig. 5). We will refer to this structure as the “Nonlinear Biquad”. The equation for the nonlin- ear biquad filter then becomes

Figure 1: Transposed Direct Form II using the quadratic equation.

Figure 10: Frequency-domain response of a lowpass filter with nonlinear feedback for a series of sine sweeps with amplitude A.

Figure 3: tanh nonlinearity, linearized at x = 1. In fig. 3, we show an example of linearizing the nonlinear function fnzx(x) = tanh(a) at zo = 1.

Figure 8: Instantaneous poles for a resonant lowpass filter with nonlinear feedback, calculated from eq. (33), with 91 = g2 = g Proceedings of the 23" International Conference on Digital Audio Effects (DAFx2020), Vienna, Austria, September 2020-21

Of particular interest to us will be saturating nonlinearities, includ- ing hard-clippers, soft-clippers, and sigmoid-like functions (see fig. 4). Saturating nonlinearities satisfy the property that. 2.3. Lyapunov Stability

Figure 9: Frequency-domain reponse of a nonlinear biquad low- pass filter for a series of sine sweeps with amplitude A.

Figure 12: Comparison between a linear resonant lowpass filter, a resonant lowpass made with a nonlinear biquad using a tanh clipper with input gain 0.283, and a SPICE simulation of a Sallen- Key lowpass. All of the lowpass filters have f. = 1 kHz, and Q = 10. The input signal in each case is a 250 Hz square wave. We then compare the outputs of the two filters for square waves at different frequencies, and use a simple staircase optimisation scheme to find the input gain for the nonlinear biquad that best matches the analog simulation. The results for the 250 Hz square wave can be seen in fig. 12. While the nonlinear biquad model is far from perfect, it does capture the damping effects of the analog filter much more accurately than the corresponding linear filter, and could be further improved with a more well-informed choice of base nonlinear functions, and a more sophisticated optimisation scheme.

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Moog Ladder Filter Generalizations Based on State Variable Filters

by Kurt J A M E S Werner and

2020, International Conference on Digital Audio Effects (DAFx-2020)

We propose a new style of continuous-time filter design composed of a cascade of 2nd-order state variable filters (SVFs) and a global feedback path. This family of filters is parameterized by the SVF cutoff frequencies and resonances, as... more

Figure 6: Low-pass SVF pole loci. Table 2: Setting parameters of the proposed generalized filter to match the Moog filter and (very nearly) the Octave CAT.

Table 1: CAT components (left) and derived quantities (right). poles (~ = +1). Since all of the poles exist in complex conjugate pairs, it is sufficient to study either ¢ = +1; we arbitrarily choose o = +1. So, satisfying (7) reduces to satisfying

Proceedings of the 23" International Conference on Digital Audio Effects (DAFx2020), Vienna, Austria, September 2020-21

Figure 2: Octave CAT filter schematic [25], including Keller’s correction [29].

Figure 4: Octave CAT integrator subcircuit.

Figure 11: Pole frequency error €¢ as a function of k; € {0, 1). Proceedings of the 23" International Conference on Digital Audio Effects (DAFx2020), Vienna, Austria, September 2020-21

Figure 5: Full Proposed Generalized filter block diagram. Proceedings of the 23" International Conference on Digital Audio Effects (DAFx2020), Vienna, Austria, September 2020-21

Figure 8: Family of hyperbolic loci. Thick lines are stable poles (0 < k < 1) and thin lines are unstable poles (1 < k). Blue lines are North-South hyperbolas (0 < r < 1), green lines are East-West hyperbolas (1 < 1) and red lines are the degenerate case (r = 1, Moog filter). Dashed lines are the path traced by the hyperbola vertices ( k =0 ), of: Fig. 6.

Figure 10: Time domain responses to a sawtooth wave of frequency w./12 for k € {0,1/8,...,6/8, 7/8}. Vertical offset added for clarity.

Figure 9: Magnitude responses of various filters for k € {0,1/8,...,7/8, 1}. Thick line represents k = 0. Proceedings of the 23" International Conference on Digital Audio Effects (DAFx2020), Vienna, Austria, September 2020-21

showing that HS (s) has 4 poles at s = Beis with ~,¢@ € +1. +1 and —1 are abbreviated in subscripts as + and — for com- pactness. Here, ¢ denotes which conjugate pole pair is being con- sidered, and 7 denotes whether its positive (S{pys} > 0) or negative Gites" < 0) member is being considered. These poles ps = Oy.6 + Wy.4) have real and imaginary parts Fig. 1 shows the Moog block diagram. Its transfer function is

This expression’ leads to four poles pee o with pode +1. Poles pee ¢ = F,¢ + Wy,¢j have real and imaginary parts This expression® leads to four poles De ¢ with yo € J Returning to the overall transfer function, HS io.4(S) can be writ- ten to show its two pairs of conjugate poles as Poles Dy. ¢ = Fb, + Wy,ej have real and imaginary parts pair of conjugate poles p§" are shown with two crucial features marked: their undamped natural frequency we (pole radius) and the trigonometric relationship between the poles’ angles from the real axis and their damping factor r. The SVF is only stable for r > 0, so we enforce that condition throughout.

The “inner” feedback loop of each SVF can be analyzed as a voltage divider feeding a non-inverting amplifier with a total gain of 27ri:2,7 € {1,2}, with Notice that the voltage divider terms are both 0.68 and Rigs = Rize. Rjj,1:2 and Rj ,2:2 are parallel combinations of op-amp in- verting inputs’ other input resistances, specifically

Figure 1: Full Moog ladder filter block diagram.

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