Symbolic expression for exponential decay

To calculate an exponential decay with a single constant τ, , why do you need to multiply the following by a heaviside function as below if you want a symbolic expression? Also, how would you do this for multiple constants, i.e., given a vector of the constants?

syms t real
tau = 0.1;
f(t) = exp(-t/tau)*heaviside(t);

I am referring to the answer here.

Answer 1

[Voss]

exp(-t/tau) is an exponential function of t, whose value is 1 at t = 0, whose value is less than 1 for t > 0, and whose value is greater than 1 for t < 0.

Here's a plot of exp(-t/tau) without the heaviside(t):

syms t real
tau = 0.1;

f(t) = exp(-t/tau);
fplot(f,[-1 1])

Notice the very large values f(t) has when t < 0.

Look at those same values on a log scale:

fplot(f,[-1 1])
set(gca(),'YScale','log')

Notice the function goes through 1 at t = 0.

In order to have the function start at t = 0 and discard the values for t < 0, you can use the heaviside() function, which has value 1 for t > 0 and value 0 for t < 0 (its value is 1/2 at t = 0):

fplot(heaviside(t),[-1 1],'LineWidth',2)

Multiplying exp(-t/tau) by heaviside(t) has the effect of zeroing-out the function for t < 0:

f(t) = exp(-t/tau)*heaviside(t);
fplot(f,[-1 1])

[L'O.G.]

@Voss Thanks, and for my related question about if there are multiple values for tau -- would that just be symsum of exp(-t/tau)*heaviside(t) where tau is now a vector ? i.e. symsum(exp(-t/tau)*heaviside(t),tau,1,N) where N is the end of the series doesn't seem quite right, since the sum is over

[Paul]

Also, we must have tau > 0 for the CTFT integral to converge.

syms t tau w real
f(t) = exp(-t/tau)*heaviside(t)

f(t) = 

fourier(f(t),t,w)

ans = 

Because it's possible that tau < 0 and yhe signal blows up for t>0, Matlab couldn't find a solution. So we need to add tau > 0 as an additional assumption

assumeAlso(tau,'positive');
fourier(f(t),t,w)

ans = 

and we get the expected result. Of course, this only matters if you keep tau as symbolic variable instead of an actual, positive number.

As for your follow on question, don't use symsum(). Just use regular sum().

syms tau1 tau2 positive
f(t) = sum(exp(-t./[tau1 tau2]))*heaviside(t)

f(t) = 

fourier(f(t),t,w)

ans = 

Can use actual values for tau1 and tau2 if you have them.

[Voss]

@L'O.G.:

"where N is the end of the series doesn't seem quite right, since the sum is over [tau_i]"

If N is the length of tau, then that seems ok to me.

Summing the signals using a vector tau:

syms t real 
tau = [1 2 0.5 0.2 0.1];
z = sum(exp(-t./tau))*heaviside(t)

z = 

[L'O.G.]

@Paul @Voss Thank you both very much.

[Voss]

You're welcome!