Short-term Fourier transform

STFT of an audio signal which is represented by frequency (horizontal axis) and time (axis in the image plane). The intensity is represented by the height and colors of the individual bars

The short-time Fourier transform ( English short-time Fourier transform , short STFT ) is a method from the Fourier analysis to the time change of the frequency spectrum to represent a signal. While the Fourier transformation does not provide any information about the temporal change in the spectrum, the STFT is also suitable for signals whose frequency properties change over time. The STFT is used, among other things, in measuring devices such as spectrum analyzers .

For the transformation, the time signal is divided into individual time segments with the aid of a window function and these individual time segments are transferred into individual spectral ranges. The time sequence of the spectral ranges obtained in this way is represented by the STFT, which can be graphically represented three-dimensionally or as an area representation with different colors.

A special variant of the STFT is the Gabor transformation .

Frequency and time resolution

Graphical representation of the time-frequency resolution. With the same area of the individual rectangles, this corresponds graphically to Küpfmüller's uncertainty relation, the left diagram shows a higher time resolution, the right diagram a better frequency resolution

An essential property of the short-term Fourier transformation is the Küpfmüller uncertainty relation. This relation describes a connection between the resolution in the time domain and the resolution in the frequency domain, whereby the product of time and frequency represents a constant value. If the highest possible resolution in the time domain is desired, for example to determine the point in time when a certain signal starts or stops, then this results in a fuzzy resolution in the frequency domain. If a high resolution in the frequency domain is necessary in order to be able to determine the frequency precisely, then this results in a blurring in the time domain, i.e. the exact points in time can only be determined in a vague manner.

The following example with four different settings is intended to show the relationship between Küpfmüller's uncertainty relation and the short-term Fourier transformation. In all four cases, a harmonic test signal with a duration of 20 seconds and a sampling frequency of 400 Hz is taken and at the start time at 0 seconds, after 5 seconds, 10 seconds and at 15 seconds the frequency between initially 10 Hz, 25 Hz, 50 and finally 100 Hz changed by leaps and bounds. In each of the following four representations, with the test signal otherwise identical, the time window for the window function of the short-term Fourier transformation was changed between 25 ms, 125 ms, 375 ms and 1 s, the spectral intensity is shown in color in the diagrams:

Window width 25 ms	Window width 125 ms
Window width 375 ms	Window width 1 s

In the first display with a window function of 25 ms duration, a strong "smear" can be seen in the spectrum, the exact frequencies can hardly be determined. However, with this window width, the time resolution is very high and the switching times from one frequency to the next can be precisely determined. The intermittent and smeared display of the intensity over a wide frequency range, especially recognizable at the low frequency of 25 Hz, is due to the leakage effect .

In the last representation with a window width of 1 s, the frequency resolution is highest - the frequencies can be determined very precisely with the narrow horizontal lines. Instead, the exact time of switching between the individual frequencies is only blurred and can be recognized in the display by a light blue spot at the end of the lines.

species

In the case of the short-term Fourier transformation, a distinction is made between a time-continuous transformation and a time-discrete transformation used in digital signal processing .

Continuous time STFT

The continuous time signal is multiplied by a window function that only has values other than 0 for the selected time period. In addition to the rectangular function, the usual window functions are the Von Hann window and the Gauss window . Outside the window, the window function returns the value 0, which means that the product also disappears. The time-continuous STFT is given as: ${\ displaystyle w \!}$ ${\ displaystyle x (\ cdot) w (\ cdot)}$

{\ displaystyle X (\ tau, \ omega) = \ int _ {- \ infty} ^ {\ infty} x (t) w (t- \ tau) e ^ {- j \ omega t} \, dt}

with the angular frequency . ${\ displaystyle \ omega}$

Discrete-time STFT

The time-discrete signal is available as a signal sequence of individual sampled values , which is divided into individual sections by a discrete window function. The time axis is expressed by an index that is generally selected as an integer . The discrete STFT is given as ${\ displaystyle n}$

{\ displaystyle X (m, \ omega) = \ sum _ {n = - \ infty} ^ {\ infty} x [n] w [nm] e ^ {- j \ omega n}}

.

In the applications, the transformation is calculated using a Fast Fourier Transformation (FFT).

literature

Uwe Kiencke , Michael Schwarz, Thomas Weickert: Signal processing - time-frequency analysis and estimation methods . Oldenbourg, Munich 2008, ISBN 978-3-486-58668-8 .

Web links

Short-Time Fourier Transformation (STFT) with Matlab Implementation