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Introduction to DSP - IIR filters: the z transform

The equation for the filter's frequency response can be simplified by substituting a new variable, z :

Using z in the filters frequency response equation

Note that z is a complex number.

z equation

Complex numbers can be drawn using an Argand diagram. This is a plot where the horizontal axis represents the real part, and the vertical axis the imaginary part, of the number.

Argnakd diagram

The complex variable z is shown as a vector on the Argand diagram.

The z transform is defined as a sum of signal values x[n] multiplied by powers of z:

z transform equation

Which has the curious property of letting us generate an earlier signal value from a present one, because the z transform of x[n-1] is just the z transform of x[n] multiplied by(1/z):

z transform of x[n-1]

So the z transform of the last signal value can be obtained by multiplying the z transform of the current value by (1/z). This is why, in the filter diagram, the delay elements are represented formally using the 1/z notation.

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