The exponential function and its sampled version is shown below.

The expression for a sine and its expansion in terms of exponentials is shown below:-

Repeating the above for the cosine produces the following for the transform of the cosine.

The expression for a damped sine and its expansion in terms of exponentials is shown below.

The case of the damped cosine is illustrated below.-

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.

© 2000 Cuthbert A. Nyack.