Analytical Derivations and Coefficients Between 1-dB/3-dB and Ripple-Bandwidths for Chebyshev Filtering Responses

Analytical expressions between 1-dB/3-dB and ripple-bandwidths for classical Chebyshev responses are derived for the first time since ripple-bandwidths are generally required in filter synthesis while 1-dB/3-dB bandwidths are usually designated in specifications or post-circuit designs. Based on the derived expressions, coefficients for the
$n$
th-order (n ∊ [1], [10]) Chebyshev filtering responses are computed and tabulated with three critical cases synthesized for validations. Finally, a 4
^th
-order Chebyshev-type bandpass filter based on substrate integrated waveguide (SIW) technology is synthesized, designed, fabricated, and measured as a practical demonstration and verification.