# How to fix superposition artifacts

I have a step response function defined as:

response = 1.044871614559231 +
188851.40028652488 E^(-0.0008027739947742557 t) -
481567.05488752836 E^(-0.0007618214677503482 t) +
533421.153826132 E^(-0.0006976033182554547 t) -
381881.5025204858 E^(-0.0006157908128333724 t) +
210904.71677914335 E^(-0.0005235899247183814 t) -
101302.35227639876 E^(-0.0004290764551077835 t) +
45982.83099921907 E^(-0.00034042402473086035 t) -
20314.01115736056 E^(-0.0002650553359099233 t) +
8074.680386193031 E^(-0.0002087261148901471 t) -
1985.479134371541 E^(-0.00017465012059731007 t)


and I am trying to add up the responses to certain jumps:

jump = {10, 10, 10, 10, 90, 250, 580, 1930, 1910,
120, -730, -930, -650, -420, -300, -270, -240, -230, -230, -220,
-170, -130, -110, -100}


at certain times:

jumptime = {10800, 21600, 32400, 43200, 54000, 64800, 75600, 86400,
97200, 108000, 118800, 129600, 140400, 151200, 162000, 172800,
183600, 194400, 205200, 216000, 226800, 237600, 248400, 259200}


If I define the lag-response function by:

r[j_] := jump[[j]]*
UnitStep[
t - jumptime[[j]]]*(response /. t -> (t - jumptime[[j]])) //
Simplify


and calculate the superposition of lagged jumps by:

totalresponse = Sum[r[tt], {tt, 1, 23}];


I get artifacts in the solution:

which are due to large numbers not being cancelled out. How can I fix this problem?

There is a large spike at the beginning of your response:

LogPlot[1 + response, {t, 0, 100000}, PlotRange -> Full, PlotStyle -> Red]


Remove it:

response2 = Piecewise[{{0, t < 10000}}, response];

r2[j_] := jump[[j]]*
UnitStep[t - jumptime[[j]]]*(response2 /. t -> (t - jumptime[[j]])) // Simplify

totalresponse2 = Sum[r2[tt], {tt, 23}] // PiecewiseExpand;

Plot[totalresponse2, {t, 0, 350000}, Exclusions -> None]


Use rationals rather than floating point numbers. Using your definitions of response and the jump variables:

res[t_] := Rationalize[#, 0] & /@ response;
r[j_] := jump[[j]]*UnitStep[t - jumptime[[j]]]*res[t - jumptime[[j]]];
totalresponse[t_] := Sum[r[tt], {tt, 1, 23}];
Plot[totalresponse[t], {t, 0, 250000}]
`

• Nice workaround. However, the solution should be a smooth curve. See, the curve is showing in my plot. Feb 19 '17 at 3:45