1D integration with symbolic limit

Hi all

I wish to calculate the measured response of a photodiode.

The photodiode has a gaussian instrument response function (IRF) while the input of the photodiode is a decaying exponential.

The IRF is defined as follows:

IRF = @(t) 1./(w*sqrt(2*pi)).*exp(-1/2*(t./w).^2);

where w is a numerical, known measured value.

The input decaying exponential signal is defined as follows:

decay_true = @(t) heaviside(t).*exp(-t/tau);

where tau is a numerical value.

The measured response at time t is the casual convolution of these two signals:

measured_signal = @(t) integral(@(tprime) IRF(tprime).*decay_true(t-tprime),tprime,-Inf,t)

I wish to pass this function a numerical array of "times" as follows

time = -1:0.001:1;
y = measured_signal(time);
plot(time,y);

However, MATLAB does not like variable integration limits, giving the error:

Error
A and B must be floating-point scalars.

Can anyone suggest how this integral can be computed with reasonable speed? I wish to pass this to a fitting algorithm later.

Thanks in advance!

Sincerely,

Ward Newman

Answer 1

[Walter Roberson]

If you have the Symbolic Toolbox then

syms IRF(t) decay_true(t) measured_signal(t)
syms tprime tau w Pi real
Pi = sym('pi');
IRF(t) = 1./(w*sqrt(2*Pi)).*exp(-1/2*(t./w).^2)
decay_true(t) = heaviside(t).*exp(-t/tau)
measured_signal(t) = int(IRF(tprime).*decay_true(t-tprime),tprime,-inf,t)
time = -1:0.001:1;
y = measured_signal(time);

You will find that this general solution depends upon sign(w). Also, I make the possibly unwarranted assumption that w and tau are real valued.