DFT, Effect of Time Shift on spectrum.
Cuthbert Nyack
This applet shows how the real and imaginary parts of the spectrum
and the odd and even parts of a signal change as the signal is
time shifted.
eg parameters (1.0, 1.0, 1.0, NA, NA, NA, 2, 0.0, 1.0, 0.1) show the
sequence is real. Changing Tsh to ±9 show the sequence
becoming completely odd.
It also shows the constancy of the magnitude of the spectrum and
that the 'continuous' presampled function can be obtained by summing terms
in the spectrum up to half the sampling frequency ie Nsum = 18.
Summing beyond half the sampling frequency eventually produces the sample sequence.
The phase change of the Np component of the DFT is shown by
the white text. For Np = 1 the phase changes by 10° (ie 360/36) for
a time shift of 1 sample and by 70° for Np = 1.
The applet below shows the time shift of a square wave represented by the Fourier series. The coefficients an and bn
are shown as red and orange. an and bn behave
similarly to the real and imaginary part between zero and
ws/2 of the DFT.
A time shift of ±9 samples in the above applet is equivalent to
a time shift of ±0.25 of a period in the applet below.
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COPYRIGHT © 2007 Cuthbert Nyack.