Simplification of terms C Exp[a+b x] to C Exp[a]Exp[bx]

I have an expression of the form

C1 Exp[a1+b1 x] + C2 Exp[a2+b2 x] + C3 Exp[a3+b3 x] + ...

Where variables C, a, b are all expressions (some simpler than others) that do not depend on x. I don't know beforehand what all of these expressions look like.

I would like to first manipulate this such that the exponentials get factored, so

C1 Exp[a1]Exp[b1 x] + C2 Exp[a2]Exp[b2 x] + C3 Exp[a3]Exp[b3 x] + ...

Then I would like to combine terms of which b are equal, so for instance if we have b3=b2 then I would like to get

C1 Exp[a1]Exp[b1 x] + (C2 Exp[a2] + C3 Exp[a3]) Exp[b2 x] + ...

Now for each term (identified by unique values of b), I would like to get lists of coefficients and exponential terms, so

{C1 Exp[a1], C2 Exp[a2] + C3 Exp[a3]}

and

{b1, b2}.

For the latter, I think I can use Collect and simplify coefficients of functions (Exp[a x])

But how to get to C1 Exp[a1]Exp[b1 x] + (C2 Exp[a2] + C3 Exp[a3]) Exp[b2 x] + ...?

I looked at using Factor[] for factoring out Exp[a1+b1 x] to Exp[a1]Exp[b1 x]. I also considered using FourierTransform[] but that doesn't work either. It gives me a a sum of DiracDelta[]'s but then I still have the same problem.

Kind regards

Boudewijn Verhaar

============

EDIT:

I think I found an answer myself already. But it doesn't work flawlessly.

First, a case where it does work. I define my input expression:

blah = funcC1[x, y, z] Exp[funca1[x, y, z] + I funcb1[x, y, z] t] + funcC2[x, y, z] Exp[funca2[x, y, z] + I funcb2[x, y, z] t] + funcC3[x, y, z] Exp[funca3[x, y, z] + I funcb2[x, y, z] t]

Then I apply

ft = FourierTransform[blah, t, omg]

ft = Collect[ft, DiracDelta[__]]

divideBySqrt2Pi[x_] := Simplify[x/Sqrt[2 Pi]]

ampls = Map[divideBySqrt2Pi, Cases[ft, a_ DiracDelta[b_] -> a]]

diracArgs = Cases[ft, a_ DiracDelta[b_] -> b];

fromDiracArgToFreq[x_] := x - omg

freqs = Map[fromDiracArgToFreq, diracArgs]

This gives me the desired outcome, for ampls:

{E^funca1[x, y, z] funcC1[x, y, z], E^funca2[x, y, z] funcC2[x, y, z] + E^funca3[x, y, z] funcC3[x, y, z]}

and for freqs:

{funcb1[x, y, z], funcb2[x, y, z]}

But if I use as input only the first term

blah = funcC1[x, y, z] Exp[funca1[x, y, z] + I funcb1[x, y, z] t]

Then both statements involving Cases[ft, ...] yield empty lists. And I don't understand why, because the pattern is there.