For Math experts only! find the fourier Transform (FT) of [s(t+T)-s(t)]/T. use result to find FT of ds(t)/dt?

Answer 1

Okay, I'm by no means an expert on Fourier Transforms, but here's what I did:

Let S(f) be the Fourier transform of s(t). Let X(f) be the Fourier Transform of [s(t+T)-s(t)]/T.

X(f) = Int (-inf to inf) [s(t+T)-s(t)]/T * e^(-2*pi * i * f * t) dt
= (1/T) Int (-inf to inf) s(t+T) e^(-2*pi * i * f * t) dt - (1/T) Int (-inf to inf) s(t) * e^(2*pi * i * f * t) dt
Let t' = t + T.
= (1/T) Int (-inf to inf) s(t') e^(-2*pi*i*f*(t'-T))dt - (1/T) S(f)
= (1/T)e^(2*pi*i*f*T) Int (-inf to inf) s(t') e^(-2*pi*i*f*t'-T)dt - (1/T) S(f)
= (1/T)*e^(2*pi*i*f*T)*S(f) - (1/T) S(f)
= ((e^(2*pi*i*f*T)-1)/T) * S(f)

Now to find the FT of ds(t)/dt, we only have to let T go to 0.

Edit: Made a little error here, this is now correct:
lim T->0 ((e^(2*pi*i*f*T)-1)/T) * S(f)
(Use L'Hopital's)
lim T->0 (2*pi*i*f)*e^(2*pi*i*f*T) * S(f)
= 2*pi*i*f*S(f)
The source verifies this answer.

Answer 2

2

Answer 3

Okay, I'm by no means an expert on Fourier Transforms, but here's what I did:

Let S(f) be the Fourier transform of s(t). Let X(f) be the Fourier Transform of [s(t+T)-s(t)]/T.

X(f) = Int (-inf to inf) [s(t+T)-s(t)]/T * e^(-2*pi * i * f * t) dt
= (1/T) Int (-inf to inf) s(t+T) e^(-2*pi * i * f * t) dt - (1/T) Int (-inf to inf) s(t) * e^(2*pi * i * f * t) dt
Let t' = t + T.
= (1/T) Int (-inf to inf) s(t') e^(-2*pi*i*f*(t'-T))dt - (1/T) S(f)
= (1/T)e^(2*pi*i*f*T) Int (-inf to inf) s(t') e^(-2*pi*i*f*t'-T)dt - (1/T) S(f)
= (1/T)*e^(2*pi*i*f*T)*S(f) - (1/T) S(f)
= ((e^(2*pi*i*f*T)-1)/T) * S(f)

Now to find the FT of ds(t)/dt, we only have to let T go to 0.

lim T->0 ((e^(2*pi*i*f*T)-1)/T) * S(f)
(Use L'Hopital's)
lim T->0 (e^(2*pi*i*f*T)/(2*pi*i*f)) * S(f)
= S(f)/(2*pi*i*f)