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.