negative frequency ?

hi,
how do i interpret fourier transform of a sinusoidal signal
F[cos(w0*t)] = pi*[del(w-w0)+del(w+w0)]
-- why is there a negative frequency component at -w0 ? we used only a
cos signal with positive frequency
--shouldn't the transform be just {del(w-w0)} as i have used a cos
signal with UNIT amplitude. so why does pi appear in original transform
?
--why is the FT of sin(w0*t) a complex function. shouldnt it again be
{del(w-w0)}

 

[redbelly]

brazingo@gmail.com wrote:

hi,
how do i interpret fourier transform of a sinusoidal signal
F[cos(w0*t)] = pi*[del(w-w0)+del(w+w0)]
-- why is there a negative frequency component at -w0 ? we used only a
cos signal with positive frequency
--shouldn't the transform be just {del(w-w0)} as i have used a cos
signal with UNIT amplitude. so why does pi appear in original transform
?
--why is the FT of sin(w0*t) a complex function. shouldnt it again be
{del(w-w0)}

It's all just mathematical book-keeping to keep track of the cos and
sin parts separately.

Complex numbers are used in order to distinguish between sine and
cosine signals. The convention is that the imaginary part corresponds
to sine, and real part corresponds to cosine. If BOTH sine and cosine
transformed to del(w-w0) (as you seem to think they should), there
would be no way to distinguish between sine and cosine in the frequency
domain.

As for the negative frequencies: it's more mathematical slight-of-hand.
The complex coefficients on the transform are such that, when you add
the positive and negative frequency parts, you are left with a
real-valued function. Which the original time-domain signal must be.

Mark