# Filter synthesis

Filter synthesis is a process for the design of passive filters. From a desired filter response function, a network that realises the desired filter response can be found.

At the beginning of this process is the mathematical formulation of a filter attenuation response function. This function ‘approximates’ the ideal “brickwall” filter function for a given set of filter specifications. Specifications define passband and stopband requirements. The passband of a bandpass filter for example, usually extends from a lower cutoff frequency to a higher cutoff frequency. Stopbands exist on both sides of the passband and may have sub-ranges in which a certain rejection of unwanted signals is required. Whereas the ideal filter function is marked by the absence of transition regions between a passband and a stopband, the real filter response will always have transition regions – sometimes also called ‘guard bands’. The response function must also fulfill the requirement of realizability by a passive network. Thereby, we know from the start that the found function is also a function of a realisable passive network. The filter transfer function is also valid for narrowband non-LC filters ( B/fc < 10 % ), using coupled resonators (coaxial TEM-mode, waveguide cavity, microstrip, dielectric puck, etc.). Therefore the initial mathematical process for narrow bandpass filter synthesis can be quite independent of the physical filter realization. The LC element values of a lumped element filter network exactly producing the predetermined filter response, can be obtained via a straightforward mathematical process outlined further below.

There is no need for any experimental work in filter design – no ‘cut and try’ – we can use synthesis and go straight to the final solution within a frequency range of pure L and C behaviour. A similar synthesis process exists for coupled resonator bandpass filters with narrow bandwidth. Here we do not necessarily look for LC network values but for the coupling matrix and for resonator frequencies (see ). Synthesis starts with the formulation of a suitable filtering function that fulfills given selectivity requirements (see . From this function in the form of a polynomial, the transfer function can be obtained. The synthesis process steps are outlined below.

• filter design by classical LC filter synthesis (covers all types of filters, like lowpass, highpass, bandpass, bandstop)
• coupling matrix (LC) filter synthesis (narrow bandpass filters and narrow bandstop filters using coupled resonator topology)
• distributed element filter synthesis – especially wideband filter synthesis – using Richard’s transformation

Filter synthesis is a special case of so-called network synthesis. The start of a synthesis is usually a desired response function and the final output of the synthesis are either the LC values of a realisable filter circuit or – in the case of narrow bandpass filters using for example coaxial or dielectric resonators – the coupling matrix and resonator tuning data. The synthesis is carried out entirely in the mathematical domain. We can sub-divide the synthesis process as follows:

• Approximation (reflection function polynomials)
• Generating a filter response function for a given filter specification
• placement of transmission zeroes while maintaining a desired passband character (equal-ripple, maximally flat or a custom passband response)
• General Chebychev function
• Cameron method using recursive process for generating the reflection function polynomial
• (Orchard, Temes)
• non-recursive approach derives the reflection function polynomials directly from the finite frequency loss poles (transmission zeroes)
• separate treatment of stopband response with simplified functions allows easy optimisation of loss pole frequencies
• Synthesis
• optimization based quasi-synthesis method for narrowband coupled resonator bandpass filters (computationally simple)
• initial guess values for coupling matrix (topology matrix is sufficient)
• coupling matrix elements via fast converging optimization process (Ali Atia 1998)
• other optimisation based methods
• iterative process
• cross-couplings are inserted into a standard Chebyshev filter circuit, starting with very weak cross-couplings
• restoration of passband return loss by optimisation
• increased cross-coupling strengths & restoration until final cross-coupling value is reached
• generating transfer function polynomials from reflection function polynomials
• from mathematical relationships between filter polynomials (Feldtkeller equation)
• LC – lumped element filter synthesis (ladder networks)
• ABCD matrix can be found from filter polynomials (transfer function, characteristic function)
• Z or Y matrix polynomials can be found from ABCD matrix polynomials
• circuit elements via ‘pole removal’ from Z or Y matrix polynomials
• circuit transformations to suit preferences
• extraction in transformed variable domain for numerical accuracy
• Coupling matrix synthesis for narrow bandpass filters (LCX synthesis) using coupled resonators
• rigorous synthesis method (computationally complex)
• admittance matrix found from filter polynomials
• general coupling matrix via eigenvalues of admittance matrix
• coupling matrix for desired topology via matrix transformations
• LCX lumped element synthesis method
• classic extraction process for lumped element circuit (without cross-couplings)
• circuit transformations to include resonator triplets and/or quadruplets  (cross-couplings, non-canonical)
• coupling matrix elements from lumped element circuit elements
• Realisation of coupled resonator narrowband filters using discrete TEM-mode, waveguide or dielectric resonators
• choice of resonators, based on insertion loss & other requirements (Qu)
• 3D EM Eigenmode simulation of two-resonator models
• 3D EM driven mode simulation of full filter model or filter sub-structures
• 3D EM model refinement via co-simulation and variable space mapping

A fine summary of the LC filter design & synthesis process can be found on page 204 of Temes, La Patra, “Circuit Synthesis and Design”, McGraw Hill 1977.

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Link to a list of literature on LC filter synthesis

Link to a chronological list of IEEE MTT publications by R.J. Cameron

Link to IEEE MTT-8  ‘recommended books and papers’ on filters