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:
sampling period (or interval, or time);
sampling frequency (or rate);
fundamental period (or total sampling time, or measurement period, or time gate);
fundamental frequency (or frequency resolution, or frequency increment, or bin width);
harmonic frequencies;
frequency bins.
Moreover, we will discuss angular and normalized frequencies.
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 at any time , but we record its value every seconds:
The time interval between two samples is called sampling period. It is expressed in seconds.
For example, means that we take a sample every two seconds.
Synonyms: sampling time; sampling interval.
Alternative notation: .
The sampling frequency is the reciprocal of the sampling period:
It is measured in hertz (Hz = 1 / s) or samples per second.
For example, means that we take two samples per second.
Synonyms: sampling rate.
The total number of samples is denoted by .
In other words, we record the samples
The fundamental period is defined as
Therefore, the number of samples is equal to the product of the fundamental period and the sampling frequency:
Synonyms: total sampling time, measurement period, time gate.
Alternative notation: .
In general, the fundamental period of a periodic signal is the smallest such thatfor any .
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 .
However, when the DFT is used to analyze the discretized signal, the latter is treated as a periodic function with fundamental period , even if the original signal had a different period.
The fundamental frequency is the reciprocal of the fundamental period:
Synonyms: frequency resolution; frequency increment; bin width.
Alternative notation: .
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 .
For example, if we use angular frequencies, the fundamental frequency isand the sampling frequency is
Frequencies can also be expressed as normalized frequencies, in cycles per sample.
Given a frequency , its normalized version is obtained as:where is the sampling rate.
For example, the normalized fundamental frequency is
In other words, a signal at the fundamental frequency completes cycles per sample.
Let us now look at the frequencies used in the DFT.
The samplesare denoted by
For , the Discrete Fourier Transform of the sampled signal iswhere is the imaginary unit.
For , the inverse DFT is
In other words, the samples are linear combinations of the basis functionsfor .
The basis function completes one full cycle in samples. Therefore, its period is equal to the fundamental period .
The basis function completes one full cycle in samples.
As a consequence, the period of is a fraction of the fundamental period:
The frequency of is the reciprocal of its period where is the fundamental frequency and is the sampling frequency.
In other words, the frequency of is times the fundamental frequency. It is also proportional to the sampling frequency.
The normalized frequency of (i.e., its frequency divided by the sampling frequency) is
We have seen above that the frequencies of the basis functions are integer multiples of the fundamental frequency:
These frequencies are known as harmonic frequencies.
The frequency is the -th harmonic.
Synonyms: harmonics.
The indices of the basis functions are often referred to as frequency bins.
However, for some authors the -th bin is a set of frequencies around the -th harmonic frequency.
Synonyms: bin indices.
The -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.
The Nyquist frequency is
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 provided that the signal contains only frequencies less than .
Synonyms: folding frequency.
Please cite as:
Taboga, Marco (2021). "Discrete Fourier Transform - Frequencies", Lectures on matrix algebra. https://www.statlect.com/matrix-algebra/discrete-Fourier-transform-frequencies.
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