DFT Phase

Discover how the phase plot of a DFT is generated.

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The magnitude plot is drawn by taking the absolute values of the complex DFT output. It might seem that the phase plot is not very useful. This is not true.

  • The magnitude plot tells us about the level of contribution in the signal from each participating sinusoid.
  • The phase plot tells us about the alignment of those sinusoids with respect to the origin.

Time shift

Let’s consider a rectangular signal in which L=3L=3 and N=16N=16. This signal has the following samples:

x1[n]={1index 0.1.1.0.0.0.0.0.0.0.0.0.0.0.0.0.}x_1[n]=\{\underbrace{1}_{\text{index}~0}. 1. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0.\}

Now recall the following two facts:

  • Due to DFT periodicity in both the input and output, the last N/2N/2 samples (from N/2N/2 to N1N-1) are the same as the samples from N/2-N/2 to 1-1. These are actually N/2N/2 samples to the left of zero.
  • In this kind of setup, the signal above is simply a right shift of the sequence below:

x0[n]={1index 0.1.0.0.0.0.0.0.0.0.0.0.0.0.0.1.}x_0[n]=\{\underbrace{1}_{\text{index}~0}. 1. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 1.\} ...