Filter Design Notes.

Chapter 1: Intro to modern network theory 

1.1.0 Modern Network theory.

the general outline of a filter is that signals flow through it to a load, in a sense it is a black box where depending on the characteristics it can pass or attenuate signals. 

For example a Low pass filter with unity gain with a corner frequency of 1kHz will allow signals below 1kHz to pass through unaffected, for frequencies at or greater than 1kHz they will start to be attenuated. At the corner frequency [-3dB] the signal strength is ~0.707X and the magnitude 'Rolls off' further as the frequency increases.

[Attenuated: meaning having been reduced in force, effect, etc.. in this case magnitude]

1.1.1The Pole Zero Concept

The Frequency response of a filter can be expressed as the ratio of two polynomials in terms of S.

S = jw, j = -1^0.5 and w = 2*pi*frequency

from figure 1-1, the output can be expressed as T(S) = E_load / E_source = N(s) / D(s)

The roots of the Denominator [D(S)] are called Poles 

The roots of the Numerator [N(s)] are called Zeros 

The example of T(s) in the top right shows a system that contains 3 poles and no Zeros  IE, to be a pole or zero they must be a coefficient of S with an order >0

By plugging in S = jw to T(S) we can plot out the response of the filter across different frequencies 

For a filter to work and be stable it must follow some mathematical rules, else it goes all wack. 

when plotted on an axis with the X axis = real numbers [RE], Y axis = complex numbers [IM] 

1) They must occur in pairs which are conjugates of each other, except for real-axis poles and zeros

2)  Poles must also be restricted to the left plane, Zeroes may be present in either.