Lowpass Prototype Coefficients | Qucs-S S-parameter Viewer Documentation

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# Lowpass Prototype Coefficients

## Overview

All filter implementations begin with normalized lowpass prototype coefficients `gₖ`. These dimensionless values represent a normalized lowpass filter with:

- Source and load impedance = 1 Ω

- Cutoff frequency = 1 rad/s

The prototype is then scaled and transformed to produce the desired filter response.

## Coefficient Calculation

### Butterworth

Maximally flat magnitude response.

Formula:

    gₖ = 2 × sin(π(2k-1)/(2N))    for k = 1 to N
    g₀ = 1 (source)
    gₙ₊₁ = 1 (load)

Example (N=3):

    g₁ = 2 × sin(π/6) = 1.0000
    g₂ = 2 × sin(π/2) = 2.0000
    g₃ = 2 × sin(5π/6) = 1.0000

### Chebyshev

Equal-ripple in passband, steeper rolloff than Butterworth.

Formula:

    β = ln(1 / tanh(Ripple_dB / 17.37))
    γ = sinh(β / (2N))

    aₖ = sin(π(2k-1)/(2N))
    bₖ = γ² + sin²(kπ/N)

    g₁ = 2a₁ / γ
    gₖ = (4aₖ₋₁aₖ) / (bₖ₋₁gₖ₋₁)   for k = 2 to N

    g₀ = 1 (source)
    gₙ₊₁ = 1/tanh²(β/4)   if N is even
    gₙ₊₁ = 1              if N is odd

Example (N=3, Ripple=0.1 dB):

    g₁ ≈ 1.0316
    g₂ ≈ 1.1474
    g₃ ≈ 1.0316
    g₄ = 1.0000

### Bessel

Maximally flat group delay (linear phase). Coefficients are pre-calculated from Zverev tables.

Example (N=3):

    g₁ = 0.3374
    g₂ = 0.9705
    g₃ = 2.2034

### Gaussian

Gaussian-shaped impulse response. Coefficients are pre-calculated from Zverev tables.

Example (N=3):

    g₁ = 0.2624
    g₂ = 0.8167
    g₃ = 2.2262

### Legendre

Steepest monotonic rolloff (no passband ripple). Coefficients are pre-calculated from Zverev tables.

Example (N=3):

    g₁ = 1.1737
    g₂ = 1.3538
    g₃ = 2.1801

## Coefficient Tables (N = 2 to 10)

Pre-calculated coefficients for Bessel, Gaussian, and Legendre responses are stored in lookup tables for orders 2 through 10. For orders outside this range or for Butterworth/Chebyshev responses, coefficients are computed analytically.

## Transformation to Physical Filters

The normalized prototype `gₖ` values are transformed based on the desired filter type:

Filter Type | Transformation
--- | ---
Lowpass | Scale by fc and Z₀
Highpass | Swap L↔C, invert gₖ, scale by fc and Z₀
Bandpass | Replace each element with resonator, scale by fc, BW, Z₀
Bandstop | Replace with dual resonator, scale by fc, BW, Z₀

See individual filter documentation for specific transformation equations.

## Zverev Tables vs. Computed Coefficients

The tool supports two coefficient sources:

### Standard (Computed)

- Butterworth and Chebyshev computed from formulas

- Bessel, Gaussian, Legendre from pre-calculated tables

- Load impedance: g₀ = gₙ₊₁ = 1 Ω

### Zverev Tables

- All responses use pre-calculated values

- Load impedance: May differ from source (gₙ₊₁ ≠ 1)

- Historically used for hand calculations

The “Use Zverev Tables” option affects load impedance handling in the final design.

## Ripple Correction (Chebyshev Only)

For Chebyshev filters, the cutoff frequency can be adjusted so that fc corresponds to the -3 dB point rather than the ripple edge:

Lowpass/Highpass correction:

    ε = √(10^(Ripple_dB/10) - 1)
    fc_corrected = fc × cosh(acosh(1/ε) / N)

This is automatically applied when “Use Zverev Tables” is disabled.

## References

[1] Matthaei, G. L., Young, L., & Jones, E. M. T. (1964). Microwave Filters, Impedance-Matching Networks, and Coupling Structures. Artech House.

[2] Zverev, A. I. (1967). Handbook of Filter Synthesis. Wiley.