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Discrete Fourier transform of a real signal

by , PhD

This lecture discusses some important properties of the Discrete Fourier Transform of a real vector (signal).

Table of Contents

The transform

In what follows, the Discrete Fourier Transform (DFT) of an $N	imes 1$ vector x is another $N	imes 1$ vector X whose entries satisfy[eq1]where i is the imaginary unit.

Complex conjugation

We will repeatedly use some properties of complex conjugation, briefly reviewed here.

Remember that the complex conjugate of [eq2]is[eq3]

The complex conjugate of a sum is equal to the sum of the conjugates:[eq4]

The complex conjugate of a product is equal to the product of the conjugates:[eq5]

The complex conjugate of a complex exponential is[eq6]

Conjugate symmetry

The DFT of a real signal enjoys the following conjugate symmetry property.

Proposition Let x and X be two $N	imes 1$ vectors, such that X is the Discrete Fourier Transform of x. If all the entries of x are real numbers, then[eq7]for $k=1,ldots ,N-1$.


First of all, we apply the rules of complex conjugation:[eq8]In the last step, we have used the fact that the complex conjugate of a real number is equal to the number itself. Then, we exploit some basic properties of trigonometric functions:[eq9]Finally, we put together the previous results:[eq10]

First entry of the DFT

If the vector x is real, then the first entry of its DFT (X_1) is also real.


It suffices to set $k=0$ in the DTF formula:[eq11]The result of the latter sum is real because all the summands are real by assumption.

Nyquist frequency

Consider the so-called Nyquist frequency $N/2$.

If $N/2$ is an integer (i.e., $N$ is even) and the vector x is real, then the DFT entry $X_{N/2+1}$ is also real.


By the conjugate symmetry property, we have[eq12]But a complex number is equal to its conjugate only if it is real.


We now show a couple of numeric examples, where you can see that the properties above (conjugate symmetry, zero complex part for first and Nyquist entries) are satisfied.

Example 1 - N even

In the first example $N$ is even (equal to 8).

We mark symmetric terms with matching symbols.

Signal xDFT X (real part)DFT X (complex part)Comments and symmetry marks
1.000010.00000First entry (no complex part)
1.80000.60000Nyquist folding (no complex part)

Example 2 - N odd

In the second example $N$ is odd (equal to 9).

Signal xDFT X (real part)DFT X (complex part)Comments and symmetry marks
0.50003.90000First entry (no complex part)

Representation in terms of sines and cosines

When x is real, we can derive a frequency domain representation in terms of sines and cosines:

  1. if $N$ is even, the representation is[eq13]

  2. if $N$ is odd, the representation is[eq14]where [eq15] denotes the floor of $N/2$.


The usual frequency-domain representation is[eq16]

We will transform it, by using the following property, derived in the proofs above:[eq17]

When $N$ is even, we have

[eq18]Therefore,[eq19]When $N$ is odd, an almost identical derivation yields[eq20]

Note that in both cases (even and odd), the representation involves only the first [eq21] entries of the DFT.

The remaining entries of the DFT (those corresponding to frequencies higher than the Nyquist frequency) are not used in the representation.

In other words, when x is real, the information about x enclosed in the DFT X is somehow redundant: the values of X beyond the Nyquist frequency are not needed to reconstruct x.

Amplitude spectrum

The amplitude spectrum is an $N	imes 1$ vector A whose entries are calculated as[eq22]

As a direct consequence of the conjugate symmetry property derived previously, the amplitude spectrum of a real signal is symmetric around the Nyquist frequency:[eq23]for $k=1,ldots ,N-1$.

Here is an example.

Plot of a real signal and its symmetric amplitude spectrum.

Power spectrum

The entries of the power spectrum $P$ satisfy[eq24]

Therefore, also the power spectrum is symmetric.

How to cite

Please cite as:

Taboga, Marco (2021). "Discrete Fourier transform of a real signal", Lectures on matrix algebra.

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