How to obtain the steady state response of the sampled periodic signal data?

Question

Below the green plot Vin is 5Hz sine voltage signal from a function generator output sampled at 12kHz sampling rate:

The blue plot Vout is the processed/filtered data by using the following code in Python:

#Transfer function
fc = 1 # roll off freq
wc = 2*numpy.pi*fc 
sys = signal.lti([1], [1/wc, 1])  # transfer function

#Output in time (Response of the filter to the input)
tout, v_out, x = signal.lsim(sys, v_in, time)  # output

And here is Vout in a more detailed view:

Basically I define a low pass filter transfer function using: sys = signal.lti([ 1 ], [1/wc, 1]).

Then the Vout is obtained from tout, v_out, x = signal.lsim(sys, v_in, time)

So Vout is the filtered response of Vin.

If you compare the Vin and Vout in above plots, Vout looks like it is processed like in Laplace domain, that is to say Vout looks like it is both the transient and steady state response of Vin.

(This problem seen more obvious for a DC input; the output Vout becomes like a transient response of an RC filter)

Having the sampled data like in green plot Vin, how can we obtain a steady state response of such filter? Or at least close to steady state.

Answer 1

The filter starts from a steady state of having no input when it is created. The sudden start of signal is seen as a step to your filter, so it takes time to settle. As the filter is 1st order, it resembles RC filter so it's time constant Tau determines that it will have settled to 98% after time of 5*Tau. Besides it looks like the sampled signal does not start from zero either, but with a negative value.

When working in time domain, you must just wait for transients to settle and then look at the response. To speed that up, you might want to take a look at the filter state when it has settled, and initialize the filter state with that.