gain character

The gain character of a filter is converted into numerals by every frequency response.
A band which the gain is not reduced almost in is called 'passband',
and a band which the gain is reduced in is called 'stopband'.
In most case it is converted by half of sampling frequency for practical use.

Some gain characters are grouped by some basical characters.
*Low-pass
The character has a passband in low frequency range and stopband in high frequency range.
*High-pass
The character has a passband in high frequency range and stopband in low frequency side.
*Band-pass
The character has a passband in a spread range and stopband in out of the range.
*Band-eliminate
The character has a stopband in a spread range and passband in out of the range.
*All-pass
The character keeps 0 dB in all range. It is used for phase compensation.

cutoff frequency

It is a point of frequency which the character is changed. In case of low-pass character, it is a point from passband to stopband. The character of low-pass and high-pass have one point, band-pass and band-eliminate have two points.

phase character

The phase character of a filter is converted into numerals by every frequency response.
In case of the phase character with linear called linear phase,
and the character gives a fixed delay time to all frequency response.
A phase character without linear phase gives difference delay time to all frequency response,
so that in most cases, the signal shape are deform.
It's better to design a filter as linear phase for some signal processing must not have distorted shape.

difference equation

It is a relation of filter with input and output signal.
To design digital filters is to lead difference equations finally, either. You will be able to impliment filters as software application,
if you find the difference equations.

impuls response

You can find output signal of filter when you input impuls signal to it. Every coefficient of difference equation is same it about FIR filter. About IIR filter, although the coefficient doesn't relate it,
it's used to check the filter is stable or not.

moving average filter

The way to find new output signal from existing averaged signals. The existing signal assumes 'X(n)' and output signal assumes 'Y(n),
The difference equation is expressd in the next line.

You can take a low-pass character from it.
The 'N' is called 'tap number' and When the value is getting bigger,
the cutoff frequency is moving to low range and being steep,
but a demerit that the computional complexity is increasing comes.
All coefficients of the filter are '1/N', so that the phase character is always linear.
The character difference of input signal and signal through low-pass filter is high-pass.

In case of two taps, this filter's phase character is linear and the gain character is almost linear high-pass.
In case of three taps, this filter's phase character is not linear and you find a band over 0 dB,
so this filter is not practial.
When you are about to design a filter with three taps more, I recommend the filter as FIR.

FIR filter

FIR filter is assumed in next line.

FIR spells Finite Impulse Response and the Impulse response converges constantly.
And in spite of the kind of gain characters, you can design the filter as linear phase.
If all the coefficients are '1/N', the equation is same equation of moving average,
so moving average filter is a kind of FIR filter.

IIR filter

IIR filter is assumed in next line.

IIR spells Infinite Impulse Response and there is a case that the impulse response doesn't converge.
If the impulse response doesn't converge, it is guessed unstable.
And basically, it's difficult to design it as linear phase, you can't use it in case you must do it as linear phase.
Although In comparison with FIR ,it stands out demerit, you can take same gain character with
one devide ten taps comparing to FIR.
Instead of linear phase, It provides steeper gain character instead of linear phase.
The IIR filter that is designed in this site is stable constantly. So you don't have to consider whether stable or not.