FIR / IIR and Wavelets I am a self-learner of DSP by reading various books. I have accomplished basic understanding of Signals - CT and DT and a few transforms. I recently started to learn FIR / IIR Filters.

The part that I cannot understand is that they are termed as 'Filters' which for me logically means blocking / allowing from a threshold value - ex. lower values would be passed higher would be filtered or removed. So if we have a low pass filter - it would lead us to remove high values from a sequence and vice versa for high pass filters - is my understanding correct?

Like HPF (High Pass Filter):

x(n)={1,2,3,4,5,6,7,8,9}


Set threshold as >=5 so output sequence would be {5,6,7,8,9}

OK but the document states FIR / IIR about :


  Finite Impulse Response (FIR) : this type of filter gives a finite
  number of nonzero outputs (response) to an impulse function input. It
  does not use feed-back.


While


  Infinite Impulse Response (IIR) : this type of filter uses feed-back,
  so it could have an infinite number of nonzero outputs (response) to
  an impulse function input.


Now I cannot understand what FIR / IIR has to do with my concept of filters - allow / block high / low values. Where does the question of Feedback comes here?

Similarly for wavelets - 


  We call an octave a level of resolution, where each octave can be
  envisioned as a pair of FIR filters, at least for the one-dimensional
  case. One filter of the analysis (wavelet transform) pair is a lowpass
  filter (LPF), while the other is a highpass filter (HPF). Each filter
  has a down-sampler after it, to make the transform efficient.  For
  example, a simple lowpass filter may have coefficients {1/2,1/2},
  producing outputs (x[n] + x[n - 1])/2, which is clearly the average of
  two samples. A corresponding simple highpass filter would have
  coefficients {1/2,-1/2}, producing outputs (x[n] - x[n - 1])/2, half
  the difference of the samples.


I am not able to get the concept of how and why here equation : (x[n] + x[n - 1])/2 and (x[n] - x[n - 1])/2 is being referred?
A filter applies a frequency-dependent gain to be applied to a signal. 

The process you describe - removing values below a certain threshold -  is not filtering in a DSP sense. 

A typical FIR filter is shown below. It consists of delays, z^-1, and gains, the $b$ terms.



Compare with the IIR filter shown below which has part of its output fed back into the delay structure (the gains are missing on this diagram).