# Z Transform exponent and sinusoid.

Cuthbert Nyack
The exponential function and its sampled version is shown below.
Applying the definition of the Z Transform gives:-
Which can be summed, resulting in
This transform has a zero at Z = 0 and a pole at Z = e-aT. Plots of the magnitude, phase(surface and contour), real and imaginary parts of this transform are shown below. Only the portion of the magnitude plot within the unit circle is shown. Pole is at 0.53.
The expression for a sine and its expansion in terms of exponentials is shown below:-
Applying the above result to transform each exponential gives:-
which simplifies to.
This transform has a zero at the origin and poles at cos(wT) ± j sin(wT).
Repeating the above for the cosine produces the following for the transform of the cosine.
This transform has zeros at the origin and at cos(wT) and poles at cos(wT) ± j sin(wT).
The expression for a damped sine and its expansion in terms of exponentials is shown below.
Using the same proceedure as above gives the z transform of the damped sine.
This transform has a zero at the origin and poles at e-aT(cos(wT) ± j sin(wT)) = e-T(a ± jw) .
The case of the damped cosine is illustrated below.-
This transfer function has 2 zeros at Z = 0 and Z = e-aTcos(wT) and 2 poles at Z = e-aTe±jwT. The plots below show the magnitude, phase(surface and contour), real and imaginary parts and the frequency spectrum for 10 samples/Period of the Z transfom of the damped cosine. Only the magnitude within the unit circle is shown.
The plots below show the magnitude, phase(surface and contour), real and imaginary parts and the frequency spectrum for 3 samples/Period of the Z transfom of the damped cosine.
eg parameters (0.1, 0.0, 0.0, 0.8168, 6.2, 1.6, 0, 60.0, 69.0, 1.5) show the magnitude of the Z transform of a damped cosine with 2 poles and 2 zeros.
eg parameters (0.03, 0.0, 0.0, 0.8168, 6.2, 1.6, 1, 100.0, 80.0, 1.5) show the phase of the Z transform of a damped cosine with 2 poles and 2 zeros.
eg parameters (0.1, 0.0, 0.0, 0.8168, 4.0, 1.5, 2, 120.0, 69.0, 1.5) show the real part of the Z transform of a damped cosine with 2 poles and 2 zeros.
eg parameters (0.1, 0.0, 0.0, 0.8168, 3.4, 1.5, 3, 60.0, 79.0, 1.5) show the imaginary part of the Z transform of a damped cosine with 2 poles and 2 zeros.
Fn = 5 to 8 show the damped sine with 1 zero and 2 poles. Fn = 9 to 12 show the decaying exponent with 1 pole and 1 zero.