Hi,
why do i get a negative frequency in fourier transform of sinusoidal
wave ? arent frequencies supposed to be +ve ? i have read a few books
and they suggest that -ve frequencies are present for complex
sinusoids. so is it possible to physically generate complex sinusoids ?
-bz
Negative frequencies are simply a mathematical slight of hand
in the same way that +6 may be represented by the sum of +11 and
-5. In the usual Fourier transform of a periodic waveform, cosine
and sine terms are generated along with their coefficients. If we
use the standard exponential identities...
Cos(x) = 1/2(exp[jwt] + exp[-jwt])
Sin(x) = -j/2(exp[jwt] - exp[-jwt])
...in place of these, a series emerges with complex
coefficients and terms including both w and -w (with the real and
imaginary components being normally drawn on separate diagrams).
The negative ones however, have the same physical presence as
minus five apples has in a school arithmetic lesson. It is just
that in combination with the other terms a tangible waveform
emerges.
Having said all of that, if you regard a phasor as a
counter-clockwise rotating line at a particular frequency then a
negative one of the same frequency would be a line revolving
clockwise at the same rate. It is easy enough to think up physical
models (e.g. motor shafts) where this could be a useful insight.
The problem is that we usually think in terms of time which, as
far as we know, goes only one way. How about then substituting
some other variable that doesn't have this limitation? The maths
would be just as valid.
I would have gone on a little longer but it is Friday and I
must make the local fish 'n chip shop before they close or I'll
starve.
Cheers - Joe