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.