Leapfrog filter
A leapfrog filter is a type of active circuit electronic filter that simulates a passive electronic ladder filter. Other names for this type of filter are active-ladder or multiple feedback filter. The arrangement of feedback loops in the signal flow-graph of the simulated ladder filter inspired the name "leapfrog filter", which was coined by Girling and Good. The leapfrog filter maintains the low component sensitivity of the passive ladder filter that it simulates. Synthesis. The definition and synthesis of leapfrog filters is described by Temes & LaPatra, Sedra & Brackett, Chen and Wait, Huelsman & Korn. Synthesis of leapfrog  filters typically includes the following steps: Examples. Generic filter. The design starts out with a known ladder filter of one of the typologies shown in the previous figure. Usually, all the elements of the ladder filter are lossless except the first and the last which are lossy. Using a four element voltage input, voltage output ladder filter as an example, the equations that relate the element voltages and currents are as follows: The signal-flow graph for these equations are shown in the second figure to the right. The arrangement of feedback loops in the signal flow-graph inspired the  name "leapfrog filter". The signal flow graph is manipulated to convert all current nodes into voltage nodes and all the impedances and admittances into dimensionless transmittances. This is equivalent to manipulating the equations either by multiplying both sides by "R" or by multiplying one side by R/R and distributing the "R" terms across the subtraction operation. This manipulation changes the equations as follows: The signal flow graph is further manipulated so that the gains into each summing node is +1. The result of all the manipulation is shown as the bottom signal-flow graph in the figure. The equations  represented by the resulting signal flow graph are as follows: The awkward annotation of -V and -V as labels of nodes in the signal flow graph indicates that these nodes present a 180° phase inversion with respect to the signals in the prototype filter. This manipulation is can be accomplished by a simple procedure: The signal-flow graph is suitable for implementation. State variable filters that are available in both inverting and non-inverting typologies are often used. Band pass filter. Passive circuit. The circuit for a band pass, passive ladder filter is first determined. The individual components in  parallel or series can be combined into general impedances or admittances. For this schematic: Signal-flow graph. The current and voltage variables can be put into cause and effect relationships as follows. A signal flow graph for these equations is shown to the right. Scaled signal-flow graph. For implementation reasons, the current variables may be multiplied by an arbitrary resistance to convert them into voltage variables which also converts all gains to dimensionless values. In this example all currents are multiplied by . This is accomplished by either multiplying both sides of an equation by R of by multiplying one  side by R/R and then distributing the R term over the currents. Manipulated signal-flow graph. It is convenient for implementation if the gains feeding the summing nodes all have the same sign. In that case, summation can be achieved with a junction of two resisters. Implementation. All the transmittances H - H, in this example, are bandpass filters. They can be implemented with the modified Tow-Thomas active biquad filter. This biquad has both positive and negative bandpass outputs so that it can realize any of the transmittances. This biquad also has summing inputs so it can also implement  the summing nodes. Tuning. A leapfrog filter can be difficult to tune because of the complicated feedback. One strategy is to open the feedback loops so that the remaining filter structure is a simple cascade design. Each section can then be tuned independently. The inner sections, H and H have infinite and may be unstable when the feedback loops are opened. These stages may be designed with a large, but finite so that they can be tuned while the feedback loops are open.