Some DFT Properties

The purpose of this article is to summarize some useful DFT properties in a table. If you feel that this particular content is not as descriptive as the other posts on this website are, you are right. As opposed to the rest of the content on the website, we do not intend to derive all the properties here. Instead, based on what we have learned, some important properties of the DFT are summarized in Table below with an expectation that the reader can derive themselves by following a similar methodology of plugging in the time domain expression in DFT definition.

Some DFT Properties

For example, in many of the figures encountered so far, we observed some kind of symmetry in DFT outputs. More specifically, I parts of the DFT had even symmetry while the Q components were odd symmetric. Similarly, magnitude plots were even symmetric and phase plots had odd symmetry. This is true only for real input signals.

Such kind of symmetry is called conjugate symmetry defined as

(1)   \begin{equation*}         S[N-k] = S^*[k]     \end{equation*}

which from the article on complex numbers implies

(2)   \begin{align*}         S_I[N-k]\: &= S_I[k] \\         S_Q[N-k] &= -S_Q[k]       \end{align*}


(3)   \begin{align*}             |S [N-k]| &= |S[k]| \\             \measuredangle S [N-k] &= - \measuredangle S[k]     \end{align*}

To see why real signals have a conjugate symmetric DFT, refer to the DFT definition. For a real signal, s_Q[n] is zero and

(4)   \begin{align*}          S_I[k]\: &= \sum \limits _{n=0} ^{N-1} s_I[n] \cos 2\pi\frac{k}{N}n \\         S_Q[k] &= \sum \limits _{n=0} ^{N-1} - s_I[n] \sin 2\pi\frac{k}{N}n       \end{align*}

Now S[N-k] is defined as

    \begin{align*}         S_I[N-k]\: &= \sum \limits _{n=0} ^{N-1} s_I[n] \cos 2\pi\frac{N-k}{N}n \\         S_Q[N-k] &= \sum \limits _{n=0} ^{N-1} - s_I[n] \sin 2\pi\frac{N-k}{N}n       \end{align*}

Using the identities \cos (A-B) = \cos A \cos B + \sin A \sin B, \sin (A-B) = \sin A \cos B - \cos A \sin B, \cos 2 \pi n = 1, \sin 2\pi n = 0, \cos (-A) = \cos A, and \sin (-A) = -\sin A, we get

(5)   \begin{align*}          S_I[N-k]\: &= \sum \limits _{n=0} ^{N-1} s_I[n] \cos 2\pi\frac{k}{N}n \\         S_Q[N-k] &= \sum \limits _{n=0} ^{N-1} s_I[n] \sin 2\pi\frac{k}{N}n       \end{align*}

Eq (5) and Eq (4) satisfy the definition of conjugate symmetry in Eq (2) and the proof is complete. Conjugate symmetry for magnitude and phase plots in Eq (3) can also be proved through their polar representation.

Observe an even symmetry in I part as well as magnitude of the DFT of a rectangular sequence in DFT examples. Similarly, an odd symmetry can be observed from the same figures for Q part and phase, respectively.

Effect of symmetry on plots

For a real input signal, due to an even symmetry in DFT I and magnitude plots while odd symmetry in DFT Q and phase plots, it is quite normal to discard the negative half of all these plots for real signals, with the understanding that the reader knows their symmetry properties.

Finally, observe that for real and even signals, the DFT is purely real as Q part is 0. This can be verified from Eq (5) because Q term is then a product of an even signal and an odd signal \sin 2\pi (k/N) n resulting in an odd signal. Half the values in an odd signal are the same as the other half but of opposite sign, and their sum is zero. This can be observed from zero Q part as well as zero phase in this Figure and many other signals in the text.

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