DSP | Z-Domain
Calculators Theory Z-Domain Filters
Z-Transform Z-Transfer Impulse Response Transfer Function
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The z-domain is a description of how to build a system out of delay blocks, using the complex value \(z\).
\[z = e^{j \omega} = cos(\omega) + j \cdot sin(\omega)\]% MATLAB
e = exp(1);
w = pi;
z = e^(1i*w)
ans =
-1.0000 + 0.0000i
cos(w) + 1i*sin(w)
ans =
-1.0000 + 0.0000i
where the irrational logarithm base constant \(e = 2.71828\) and j = \(\sqrt {-1}\).
Note that \(e^{j \omega}\) is the identity of z (real part = 1). if \(\omega = \pi/2\), then the position of the complex value is a quarter way around the circle, counter-clockwise, and is therefore;
% MATLAB
e = exp(1);
w = pi/2;
z = e^(1i*w)
ans =
-0.0000 + 1.0000i
The magnitude of \(z\) can therefore be adjusted, regardless of it’s angle, by multiplying it by a real number. Thus, \(z\) can also be written;
\[z = r \cdot e^{j \omega}\]which expands to
\[z = r \cdot cos(\omega) + r \cdot j \cdot sin(\omega)\]% MATLAB
w = pi;
r = 0.5
r*cos(w) + r*1i*sin(w)
ans =
-0.5000 + 0.0000i
since \(z\) is otherwise simply;
\[z = 1 \cdot e^{j \omega} = e^{j \omega}\]\(z\) can also be represented in rectangular form (real/imaginary part);
\[z = (x,y) = x + jy\]There is also a polar form (magnitude/angle);
\[z = (r, \theta) = r \cdot \theta = |z|e^{jarg(z)} = r \angle \theta\]The polar form can be converted to cartesian coordinates;
\[\displaylines{ x = cos(\omega) \\ y = sin(\omega) \\ r = \sqrt{x^{2}y^{2}} \\ \omega = atan2(y,x) }\]