# Why is my bode plot showing periodic behavior?

I found something very interersting when I was trying to simulate a discrete time system. So first I discretized the continuous system $G(s)$ using "c2d" function and "dbode". The resulting bode plot for the discretized system is displaying some sort of periodic behaviour after reaching about 3 hz. So is the phase plot.

Can someone explain why this maybe happening?

• looks like the sample rate is 1 Hz (which is $2\pi$ rad/s). sampled spectra is inherently periodic with period equal to the sample rate. Apr 6 '15 at 18:53

When you are discretizing you make transformation $s \to e^{s T}$. This transforms the left half plane into unit disc. But actually it only transforms the $(-j\omega_s/2, j\omega_s/2)$ band and repeats itself by circling for outside of the band, where $\omega_s = 2 \pi / T$. This is because of the periodicity of exponential function.
When you plot the frequency response, you plot $|G(j\omega)|=|\hat{G}(e^{j\omega T})|$ for all $\omega \in [0, \infty)$ where $\hat{G}$ is the discrete transfer function. Therefore the frequency response of any sampled system is periodic.
My guess is that you're looking at $\pi$ radians/second rather than 3 Hz.
Look at the $x$ axis. It is labeled as rad/s.
And $\pi$ radians/second corresponds to $\frac{f_s}{2}$, the maximum (baseband) frequency the system can deal with. Above that frequency, the spectrum / frequency response will start to repeat.