Digital Signal Processing Reference
In-Depth Information
Illustration 88:
Does a sinusoidal amplitude spectrum also exist for reasons of symmetry?
An interesting question which may be regarded as a test case for the symmetry principle.
Top series: One line in the time domain (e.g. at t = 0) results in a constant progression of the amplitude
spectrum just as a single line in the amplitude spectrum at f = 0 results in a "DC voltage" in the time
domain.
Centre and bottom series: two lines (e.g. at t = -20 ms and t = +20 ms) produce a cosine shaped or
sinusoidal progression of the amplitude spectrum (assuming that negative amplitudes are admitted!), just
as two lines in the amplitude spectrum (e.g. at f = -50 Hz and f = +50 Hz) guarantee a cosine shaped or
sinusoidal progression in the time domain. The phase step from
/2, to
the zero positions of the (sinusoidal) amplitude spectrum is proof that every second "half wave" ought to
lie in the negative range.
With regard to the FOURIER transformation the time and frequency domains are largely symmetrical if, as
already mentioned, negative frequencies and amplitudes for the spectral area are admitted.
π
, or more precisely from +
π
/2 to -
π
