FIR / IIR and Wavelets - filtering

Question

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?

Answer 1

High pass filter and Low pass filter in signal processing refer to a system which respectively passes high-frequency and low-frequency contents (equivalently respectively blocks low-frequency and high-frequency contents) from the signal, not low/high values.
So for example, given a sequence which is the sum of a constant value (low-frequency component) and a more rapidly varying alternating values (high-frequency component):
x(n) = {0, 2, 0, 2, 0, 2, 0, ...}
The output of a low-pass filter computing y[n] = (x[n] + x[n-1])/2 would look like:
y(n) = {1, 1, 1, 1, 1, 1, 1, ...}
Keeping only the low-frequency constant value.
Similarly, the output of a high-pass filter computing y[n] = (x[n] - x[n-1])/2 would look like:
y(n) = {1, -1, 1, -1, 1, -1, ...}
This time keeping only the high-frequency content of the signal (the alternating values).
Now, this 'selection' of low-frequency and high-frequency component can often be achieved as the solution to difference equations (a mathematical relation which combines various input and output samples). FIR and IIR are specific categorisations of those filters, which are distinguished by the dependency on previous output samples (eg. y[n-1]) in the difference equation (hence the notion of feedback). For example, a simple IIR low-pass filter (with different characteristics as the previous FIR low-pass filter example) could be implemented as the difference equation:
y[n] = 0.9*y[n-1] + 0.1*x[n];
Where the current output sample y[n] is expressed in terms of the previous output y[n-1].