# How to excite a transfer function with continuous white noise

I'm trying to get the output response of linear system represented by transfer function, once the white noise is given as an input. Here's the code:- For generating continuous noise, I'm getting help from the following question https://mathematica.stackexchange.com/a/158361/85034

tmax=10;
samplesPerSec=1;
\[Sigma]=1;
noise=Interpolation[Normal[RandomFunction[WhiteNoiseProcess[\[Sigma]],{0,tmax}]][]];
out=OutputResponse[tfm,noise,{t,0,tmax}]
Plot[out,{t,0,tmax}]

1. How to plot the output response?
2. How to access the output values in one variable?
3. How can I again apply ZOH (zero order hold) to sample this output?
• Are you intending to work in the frequency domain or the time domain? In the frequency domain the spectrum of the output is the product of the spectrum of the transfer function and the spectrum of the noise. The difficulties here are the length of the spectrum and associated windowing problems. If you are working in the time domain then you can solve the differential equation for your transfer function (possibly very easily if linear) with a random excitation. In the time domain the issue is step size. Overall you have the issue of spectrum shape. White noise is never really white.
– Hugh
Feb 11, 2022 at 8:56
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tmax = 10;
samplesPerSec = 0.25;
σ = 1;
tfm = TransferFunctionModel[(s + 1)/(s + 2), s];
noise = Interpolation[Normal[RandomFunction[WhiteNoiseProcess[σ], {0, tmax}]][]];
out = OutputResponse[tfm, noise[t], {t, 0, tmax}];
Show[Plot[out, {t, 0, tmax}], ListStepPlot[Table[out[], {t, 0, tmax, samplesPerSec}],
DataRange -> {0, tmax}, PlotStyle -> ColorData[97, 2]]] • Excellent @SubaThomas. Need help in passing output y(k) through G(z) to get output x(k), then to get Rxx(k). Only partial code is quoted: (Skipping the previous one due to limited space available) tfmd = TransferFunctionModel[(z^2 - 0.5 z + 0.05)/(0.5 z^2 + 0.1 z), z, SamplingPeriod -> samplesPerSec]; outputYk = Table[outputYt[], {t, 0, tmax, samplesPerSec}]; outputXk = OutputResponse[tfmd, outputYk, {k, 0, kmax}]; ListPlot[CorrelationFunction[outputXk, {lags}], Filling -> Axis, PlotRange -> All] The output xk is zero, therefore unable to get Rxx. Anyone please guide... Feb 12, 2022 at 2:35
• I assume outputYK is out in the question. And I don't know what is lags. Probably outputXk = OutputResponse[tfmd, outputYk] resolves your question. Feb 12, 2022 at 23:37
• Initial code is as under: tmax = 10; kmax = 5; lags = 5; samplesPerSec = 0.5; [Sigma] = 1; tfm = TransferFunctionModel[2/(s + 1)/(s + 2), s]; noise = Interpolation[ Normal[RandomFunction[ WhiteNoiseProcess[[Sigma]], {0, tmax}]][]]; outputYt = OutputResponse[tfm, noise[t], {t, 0, tmax}]; Show[Plot[outputYt, {t, 0, tmax}], ListStepPlot[Table[outputYt[], {t, 0, tmax, samplesPerSec}], DataRange -> {0, tmax}, PlotStyle -> ColorData[97, 2]]] outputXk now seems to resolve the issue. However, unable to plot the correlation for any arbitrary value of lags. Feb 13, 2022 at 16:20
• CorrelationFunction[outputXk[], {lags}] Feb 13, 2022 at 23:32
• Thanks a lot, @Suba Thomas. Indeed it is completely solved now. Feb 13, 2022 at 23:42