Time and frequency representation
Discrete Fourier Transformation

Discrete Fourier transformation DFT is used to get spectrum of discrete signals. Only a finite number of sinusoids are needed to describe signal in frequency domain.

To illustrate what DFT does we use following example. MP3 player sends the speaker audio information as fluctuations in the voltage of an electrical signal. The result are moving air particles and producing sound. An audio signal’s fluctuations over time can be depicted as a graph: the x-axis is time, and the y-axis is the voltage of the electrical signal. This look like erratic wavelike squiggle which in real is sum a number of more regular squiggles, which represent different frequencies of sound. Frequency just means the rate at which air molecules go back and forth, or a voltage fluctuates

“The DFT does mathematically what the human ear does physically: decompose a signal into its component frequencies. Unlike the analog signal from, say, a record player, the digital signal from an MP3 player is just a series of numbers, representing very short samples of a real-world sound: CD-quality digital audio recording, for instance, collects 44,100 samples a second. If you extract some number of consecutive values from a digital signal –8, or 128, or 1,000 – the DFT represents them as the weighted sum of an equivalent number of frequencies. (“Weighted” just means that some of the frequencies count more than others toward the total.) “

Discrete Fourier Transformation (DFT) is mathematically defined as:

and inverse discrete Fourier transformation (IDFT) is defined as:

where N is number of samples in discrete signal and n = 0, 1, 2, ..., N–1, is very often substituted as Ω (Ω = ).

The result of the DFT of an N-point input time series is an N-point frequency spectrum, with Fourier frequencies k ranging from – (N/2 – 1), through the 0-frequency or so-called direct component, and up to the highest Fourier frequency N/2. Each bin number represents the integer number of sinusoidal periods present in the time series. The amplitudes and phases represent the amplitudes and phases of those sinusoids. In summary, each bin can be described by .

Discrete Fourier transformation is defined also for two-dimensional signals and it can be represented as the series expansion of an image function (over the 2D space domain).

Definition of forward and inverse 2D FT is following:

Spectrum is complex even if the sequence is real. Spectrum can be defined also as sum of real and imaginary part or as product of magnitude and phase.