page_title: Mason's gain formula
section_title: Digital IIR biquad filter
hierarchical_section_title: Mason's gain formula / Examples / Digital IIR biquad filter
caption_attribution_description: English: The signal flow graph (SFG) for a digital infinite impulse response bi-quad filter. This SFG has three forward paths and two loops.
context_page_description: Mason's gain formula is a method for finding the transfer function of a linear signal-flow graph. The formula was derived by Samuel Jefferson Mason, whom it is also named after. MGF is an alternate method to finding the transfer function algebraically by labeling each signal, writing down the equation for how that signal depends on other signals, and then solving the multiple equations for the output signal in terms of the input signal. MGF provides a step by step method to obtain the transfer function from a SFG. Often, MGF can be determined by inspection of the SFG. The method can easily handle SFGs with many variables and loops including loops with inner loops. MGF comes up often in the context of control systems and digital filters because control systems and digital filters are often represented by SFGs.
context_section_description: Digital filters are often diagramed as signal flow graphs.
  There are two loops
   Note, the two loops touch so there is no term for their product.
  There are three forward paths
  All the forward paths touch all the loops so
caption: The signal flow graph (SFG) for a digital infinite impulse response bi-quad filter. This SFG has three forward paths and two loops.
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