Digital Signal Processing  

• FIR, or Finite-duration Impulse Response, filters have the distinctive trait that their impulse response lasts for a finite duration of time as opposed to IIR, or Infinite-duration Impulse Response, filters whose impulse response is infinite in duration. This trait is due to the fact that there are no feedback paths in the filter. All Z transform terms are in the numerator. Z transforms written in the Z^-1 notation only contain a "1" in the denominator. Z transforms written in the Z^+1 notation, only have a Z^N in the denominator, where N is the number of states in the filter.

  FIR filters offer several advantages over IIR filters:

  • (Performance) Completely constant group delay throughout the frequency spectrum.
  • (Stability) Complete stability at all frequencies regardless of the size of the filter.

  FIR filters also come with some disadvantages as well:

  • (Satisfaction of specification) The frequency response is not as easily defined as it is with IIR filters
  • (Complexity) The number of states required to meet a frequency specification may be far larger than that required for IIR filters.

  In practice (e.g., a commercial product called Filter Solutions), you can have at most 1000 taps to create your FIR filter.

  　

• FIR Window Filters

  The FIR filters Rectangular, Bartlett, Hanning, Hamming, Blackman, Kaiser, and Dolph-Chebyshev are all Window FIR filters. The name "Window" comes from the fact that these filters are created by scaling a truncated sinc (SIN(X)/X) pattern in time domain with a window to produce the desired frequency effect.  Since the ideal frequency response is in a square shape, whose time response is a sinc function and is physically unrealizable,  we truncate the sinc function in time domain so as to make it realizable.

  　

• Why Are Factor Graphs Useful?
  
  Factor graphs are useful for several reasons. First, factors graphs are very general: the variables and functions in the graph can be anything one can dream up. This allows them to visually describe a very large class of problems. 
  
  Second, factor graphs simplify problems: the original "big" function is broken down into several "small" ones, i.e., the original problem is broken down into several smaller and easier-to-deal-with pieces. It is for this reason that there is an efficient algorithm, the Sum-Product Algorithm, that can be applied to factor graphs. 
  
  Third, many efficient algorithms are special cases of the Sum-Product Algorithm applied to factor graphs, e.g., the FFT (Fast Fourier Transform), Viterbi Algorithm, Forward-Backward Algorithm, Kalman Filter and Bayesian Network Belief Propagation. This brings many good algorithms together in a common framework. 
  
  In summary, factor graphs allow one to understand many standard signal processing algorithms on a more abstract level. This deeper level of understanding is already leading to remarkable advances in VLSI technology for implementing these algorithms. 

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