Electrical – Is it possible to sketch a bode plot for a discrete-time transfer function

bode plotlaplace transform

For a regular s-domain transfer function, there are simple rules you can use to draw an asymptotic bode graph, basically by taking a slope of +/-20dB per decade and a phase shift of +/-90 degree for each pole/zero.

Is it possible to do the same for a z-domain transfer function?

I'm explicitly not looking to plot the exact transfer using Scipy/Matlab, but a pen-and-paper approach that gives intuition.

Basically, if I have an s-domain TF, it's fairly easy to have an intuition of the gain, phase, and corresponding stability margins. For a discrete system, I have close to no intuition about them at all.

Best Answer

In linear, time-invariant, continuous time systems, the transfer function \$H(s) \$ is a broken rational function in s. We obtain the complex frequency response through the substitution \$s=j\cdot \omega\$. In this case it is rather easy to find the asymptotic behaviour of the system and to sketch the bode diagram.

In linear, time-invariant, discrete time systems, the transfer function \$H(z) \$ is a broken rational function in z. We obtain the complex frequency response through the substitution \$s=e^{j\cdot \omega \cdot T_S}\$. Since this relationship is non-linear, there are no straight lines as asymptotes.

However, if one examines the zeros and poles of the time-discrete system, it is possible to predict the behavior of the system for frequencies between 0 and \$\frac{f_S}{2}\$ (or rather between \$-\frac{f_S}{2}\$ and \$+\frac{f_S}{2}\$) . Imagine the poles as very high mountains in the complex z-plane and the zeros as deep valleys, and then take a walk along the unit circle. This should give you a good feeling for how the system behaves.