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Discrete Fourier Transform - Frequencies

by , PhD

The Discrete Fourier Transform (DFT) is used to analyze the frequencies of a signal.

But what are these frequencies exactly?

Sometimes, the terminology can get a little bit confusing.

To check how the terms are used, we analyzed more than 50 different sources (books and lecture notes).

Table of Contents


Our aim is to clearly explain the meaning of the following terms:

Moreover, we will discuss angular and normalized frequencies.

Sampling of a continuous signal

We often use the Discrete Fourier Transform to analyze a continuous signal that has been sampled at discrete time intervals.

In other words, we can observe the signal $xleft( t
ight) $ at any time $t$, but we record its value every $T_{s}$ seconds:[eq1]

Sampling period

The time interval $T_{s}$ between two samples is called sampling period. It is expressed in seconds.

For example, $T_{s}=2$ means that we take a sample every two seconds.

Sampling frequency

The sampling frequency $f_{s}$ is the reciprocal of the sampling period:[eq2]

It is measured in hertz (Hz = 1 / s) or samples per second.

For example, $f_{s}=2$ means that we take two samples per second.

Number of samples

The total number of samples is denoted by $N$.

In other words, we record the samples[eq3]

Fundamental period

The fundamental period $T_{0}$ is defined as[eq4]

Therefore, the number of samples is equal to the product of the fundamental period and the sampling frequency: [eq5]

Remarks about the fundamental period

In general, the fundamental period of a periodic signal $xleft( t
ight) $ is the smallest $T_{0}$ such that[eq6]for any $t$.

In a DFT setting, the fundamental period (defined as number of samples times sampling interval) may not coincide with the fundamental period of the signal $xleft( t
ight) $.

However, when the DFT is used to analyze the discretized signal, the latter is treated as a periodic function with fundamental period $T_{0}$, even if the original signal had a different period.

Fundamental frequency

The fundamental frequency $f_{0}$ is the reciprocal of the fundamental period:[eq7]

Angular frequencies

Frequencies are sometimes expressed as angular frequencies, in radians per second (rad / s).

To get an angular frequency, we need to multiply a "regular" frequency by $2pi $.

For example, if we use angular frequencies, the fundamental frequency $omega _{0}$ is[eq8]and the sampling frequency is[eq9]

Normalized frequencies

Frequencies can also be expressed as normalized frequencies, in cycles per sample.

Given a frequency $f$, its normalized version $f^{prime }$ is obtained as:[eq10]where $f_{s}$ is the sampling rate.

For example, the normalized fundamental frequency is[eq11]

In other words, a signal at the fundamental frequency completes $1/N$ cycles per sample.

The DFT and its inverse

Let us now look at the frequencies used in the DFT.

The $N$ samples[eq12]are denoted by[eq13]

For $k=0,ldots ,N-1$, the Discrete Fourier Transform of the sampled signal is[eq14]where i is the imaginary unit.

For $j=0,ldots ,N-1$, the inverse DFT is[eq15]

In other words, the samples are linear combinations of the $N$ basis functions[eq16]for $k=0,ldots ,N-1$.

Frequencies of the basis functions

The basis function [eq17] completes one full cycle in $N$ samples. Therefore, its period is equal to the fundamental period $T_{0}$.

The basis function [eq18] completes one full cycle in $N/k$ samples.

As a consequence, the period of [eq19] is a fraction of the fundamental period: [eq20]

The frequency of [eq21] is the reciprocal of its period [eq22]where $f_{0}$ is the fundamental frequency and $f_{s}$ is the sampling frequency.

In other words, the frequency of [eq21] is k times the fundamental frequency. It is also proportional to the sampling frequency.

The normalized frequency of [eq21] (i.e., its frequency divided by the sampling frequency) is[eq25]

Harmonic frequencies

We have seen above that the frequencies of the basis functions are integer multiples of the fundamental frequency:[eq26]

These frequencies are known as harmonic frequencies.

The frequency $kf_{0}$ is the k-th harmonic.

Frequency bins

The indices k of the basis functions [eq21] are often referred to as frequency bins.

However, for some authors the k-th bin is a set of frequencies around the k-th harmonic frequency.

DC bin

The 0-th bin in called DC bin and the other bins are called non-DC.

The corresponding frequencies are called DC frequencies and non-DC frequencies.

Nyquist frequency

The Nyquist frequency is[eq28]

It has a crucial role in the sampling theorem, which states that a continuous signal can be perfectly reconstructed from its samples taken at the sampling rate $f_{s}$ provided that the signal contains only frequencies less than $f_{s}/2$.

How to cite

Please cite as:

Taboga, Marco (2021). "Discrete Fourier Transform - Frequencies", Lectures on matrix algebra.

The books

Most of the learning materials found on this website are now available in a traditional textbook format.