dft vs continuous ft

[guiora]

for a continuous function the ft has the classical formula namely

. 00
Y=Σ (A n sin(ω n t) +B cos(ω n t) all are real numbers
. n=1

while for the dft the formula Found in literature invades into the complex plane
.......N-1.......-2Πink/N
F(k) = Σ f e
........n=0

and my question is : can a formula for dft be found without expanding into the complex plane
no matter how complicated it is. and if not why it exists in the countinuous case and not in the digital one

 

[elecfan]

I think this is because in DFT we work with impulses which are considered to get samples from the original signal.

it is sort of FT of each individual impulse and sum them up.