Fourier transform of complex exponential multiplied to unit step

[chemic_23]

Homework Statement

find the Fourier transform of complex exponential multiplied to a unit step.
given: v(t)=exp(-i*wo*t)*u(t)

Homework Equations

∫(v(t)*exp(-i*w*t) dt) from -∞ to +∞

The Attempt at a Solution

∫([v(t)]*exp(-i*w*t) dt) from -∞ to +∞
=∫([exp(-i*wo*t)*u(t)]*exp(-i*w*t) dt) from -∞ to +∞
=∫([exp(-i*wo*t)]*exp(-i*w*t) dt) from 0 to +∞
=1/(w0+w)

is this correct? help

 

[marcusl]

No, this is not correct. Go back to your 2nd to last equation
[tex]V=\int^\infty_0{exp[-i(\omega+\omega_0)t]dt}[/tex]
and think about what you are integrating over. The exponential oscillates wildly unless

[tex]\omega=-\omega_0[/tex]

What does that tell you?
(For a further hint, look up delta functions in your textbook.)

 

[chemic_23]

i've seen this transform: v(t)*e^(j*(wo)*t)<--->V(f-fo)

and letting u(t)=v(t)

where u(t)<--->1/(2*pi*f) +δ(f)/2

so, exp(-i*wo*t)*u(t)=1/(2*pi*(f+fo)) +δ(f+fo)/2is this correct?

 

[marcusl]

Almost! You are missing an i (or j):

[tex]\frac{1}{i2\pi (f+f_0)}+\frac{1}{2}\delta(f+f_0)[/tex]

 

[DeeNos]

Yes, your solution is correct! The Fourier transform of a complex exponential multiplied by a unit step function is 1/(w0+w), where w0 is the frequency of the exponential and w is the frequency of the Fourier transform.