General mn-type image filter

The k-type filter acts as a prototype for producing the general m_n designs. For any given desired bandform there are two classes of m_n transformation that can be applied, namely, the mid-series and mid-shunt derived sections; this terminology being more fully explained in the m-derived filter article. Another feature of m-type filters that also applies in the general case is that a half section will have the original k-type image impedance on one side only. The other port will present a new image impedance. The two transformations have equivalent transfer functions but differ in terms of their image impedances and circuit configurations.

Parts of this article or section rely on the reader's knowledge of the complex impedance representation of capacitors and inductors and on knowledge of the frequency domain representation of signals.

Mid-series multiple stopband

If Z and Y are the series impedance and shunt admittance of a constant k half section and;

${\displaystyle Z=Z_1+Z_2+Z_3+\cdots +Z_N}$

where Z₁, Z₂ etc are a cascade of antiresonators,

the transformed series impedance for a mid-series derived filter becomes;

${\displaystyle Z_{m_n}=m_1{Z}_1+m_2{Z}_2+m_3{Z}_3+\cdots +m_N{Z}_N}$

Where the m_n are arbitrary positive coefficients. For an invariant image impedance Z_iT and invariant bandform (that is, invariant cut-off frequencies ω_c) the transformed shunt admittance, expressed in terms of Z_mn, is given by;

${\displaystyle Y_{m_n}=\frac{Z_{m_n}}{k^2+Z^2-Z_{m_n}^2}}$

where ${\displaystyle k=\sqrt{\frac{Z}{Y}}}$ and is a constant by definition. When the m_n are all equal this reduces to the expression for an m-type filter and where they are all equal to one it reduces further to the k-type filter.

A result of this relationship is that the N antiresonators in Z_mn will transform into 2N resonators in Y_mn. The coefficients m_n can be adjusted by the designer to set the frequency of one of the two poles of attenuation, ω_∞, in each stopband. The second pole of attenuation is dependent and cannot be set separately.

Special cases

In the case of a filter with a stopband extending to zero frequency, one of the antiresonators in Z will reduce to a single inductor. In this case, the resonators in Y_mn are reduced by one to 2N-1. Similarly, for a filter with a stopband extending to infinity, one antiresonator will reduce to a single capacitor and the resonators will again be reduced by one. In a filter where both conditions pertain, the number of resonators will be 2N-2. For these end stopbands, there is only one pole of attenuation in each, as would be expected from the reduced number of resonators. These forms are the maximum allowable complexity while maintaining invariance of bandform and one image impedance.