In the absence of a response to my questions relating to the variation about the signal, I'll explain a little about nonlinear least squares.
You can fit a model of the following form:
$y_i = c + \alpha \exp(-\alpha x_i)+\varepsilon_i$, where $E(\varepsilon_i)=0$.
If the $\varepsilon$ values are independent and of constant variance (or close to it), this should be quite a good approach (and would be my idea of a good starting point). If they're also normal it will also be maximum likelihood, and makes for simpler confidence intervals and tests (should you want those).
There's no closed form formula for the parameter estimates. They must be obtained iteratively, generally by taking a linear approximation at a current estimate to get the next estimate. Software to do this is in most stats packages.
Here's an example.
I made a tiny set of (x,y) data (here printed to 4 significant figures):
I fitted your model in R (free statistical software), as follows:
expfnfit = nls( y ~ c+a*exp(-a*x) , start=list(c=2,a=.5)) # fits the model
summary(expfnfit) # shows information about the fit
Formula: y ~ c + a * exp(-a * x)
Estimate Std. Error t value Pr(>|t|)
c 2.529316 0.008608 293.848 < 2e-16 ***
a 0.229818 0.027285 8.423 7.48e-06 ***
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error: 0.02448 on 10 degrees of freedom
Number of iterations to convergence: 6
Achieved convergence tolerance: 1.39e-06