Short-time Fourier transform
The short-time Fourier transform (English short -time Fourier transform, short STFT ) is a method of Fourier Analysis to the temporal change of the frequency spectrum of a signal present. While the Fourier transformation does not provide information about the variation with time of the spectrum, the STFT is also suitable for non-stationary signals, the frequency characteristics change over time. Common applications for the STFT, among others, in measuring instruments such as spectrum analyzers.
For the transformation of the time signal is subdivided into individual time sections by using a window function and transferred to each of these time segments in each individual spectral ranges. The temporal juxtaposition of the so obtained spectral represents the STFT, which can be represented in three dimensions or in surface representation graphically with different colors.
A special variant of the STFT is the Gabor transform.
Frequency and time resolution
An essential feature of the short-time Fourier transform, the Küpfmüllersche uncertainty relation dar. This relation describes a relationship between the resolution in the time domain and the resolution in the frequency domain, where the product of time and frequency is a constant value. If possible resolution desired in the time domain, for example, the time when a specific signal to determine one or suspends, then this results in a blurry resolution in the frequency domain. If a high resolution in the frequency range around the frequency needed to determine exactly then it follows a blur in the time domain, that is the exact time points can be determined only blurred.
The following example with four different settings to represent the relationship of Küpfmüllerschen uncertainty relation with respect to the short-time Fourier transform. In this case, in all four cases, a harmonic test signal is taken at 20 -second duration and a sampling frequency of 400 Hz and the starting time at 0 seconds after 5 seconds, 10 seconds and 15 seconds, the frequency is between the beginning of 10 Hz, 25 Hz, 50 and, finally, 100 Hz changed by leaps and bounds. , The time window for the window function of the short-time Fourier transform between 25 ms, 125 ms, 375 ms and 1 was in each of the following four representations with otherwise identical test signal, s changed, the spectral intensity is shown in color in the diagrams:
The first representation with a window function of length 25 ms, a strong " smearing " in the spectrum can be seen, the specific frequencies can be hardly determined. For this, the time resolution is very high in this window width and the time switch from one frequency to the next can be determined exactly. Intermittent and smeared over a wide frequency range of the intensity, especially at the low frequency of 25 Hz is seen due to the leakage effect.
In the final presentation with a window width of 1 s, the frequency resolution is highest - it can be with the narrow horizontal lines determine the frequencies very accurately. For the exact switching time between the individual frequencies is just out of focus and in the representation by a light blue spot at the end of the lines visible.
Species
In the short-time Fourier transform is distinguished between a continuous-time, and a transformation applied to digital signal processing of time- discrete transformation.
Continuous-time STFT
The continuous time signal is multiplied by a window function which only for the selected period of time values other than 0 has. Common window functions are in addition to the square function, the Hann window and the Gaussian window. Outside the window, the window function returns the value 0 which also the product disappears. The time-continuous STFT is given as:
With the angular frequency.
Discrete-time STFT
The time-discrete signal is present as a sequence of individual signal samples, which is divided by a discrete window function into sections. The time axis is expressed by an integer selected in the general index. The discrete STFT is given as
In the applications, the computation of the transformation by a fast Fourier transform (FFT) is carried out.