Leak effect

The leakage effect , window effect or leakage effect is a phenomenon of signal analysis . The term describes the fact that, due to the finite length of the observation period of a signal, in the context of spectral analyzes such as Fourier analysis there are also frequency components in the calculated frequency spectrum that would not occur in an infinitely long observation period that is only theoretically possible.

General

Spectral representation of the leak effect due to the temporal limitation of the sinusoidal oscillation in the second line

In principle, the leakage effect cannot be avoided in continuous signal analysis, since in reality every signal must have a beginning and an end and cannot be continued periodically for an indefinite period of time . The effects of the leakage effect can be minimized by suitable methods such as long observation periods or the use of special filters based on window functions . The leakage effect also occurs with time-discrete signals and their analysis, for example within the framework of the discrete Fourier transformation , but can be completely avoided in very specific situations by periodically continuing in the discrete spectrum as a special case.

Without a time limit - the theoretical case of an infinitely long sinusoidal oscillation is indicated in the figure in the upper case - the calculation of the magnitude spectrum in the context of the Fourier transformation in the range of positive frequencies results in a Dirac pulse at the angular frequency . The infinitely long sinusoidal oscillation represents a so-called power signal. If the sinusoidal oscillation is switched on at a certain point in time as shown in the figure below and then switched off - this time range is also known as the "observation interval" - the Dirac impulse is "smeared" in the Spectrum, which is called the leakage effect. The finitely long sinusoidal oscillation is transformed into an energy signal through the use of a rectangular function as a window function . The rectangular window function is shown in the figure at as a green dashed line. ${\displaystyle \omega _{0}}$ ${\displaystyle y_{w}(t)}$

Mathematically, this temporal limitation of time signals represents a multiplication by the window function, which is 0 up to the beginning of the time window, 1 during the time window duration and 0 again from the end of the time window. In the frequency domain, this corresponds to a convolution of the signal spectrum with the spectrum of the rectangular function, which is described by the si function . The amount spectrum is symmetrical with respect to the angular frequency . ${\displaystyle \omega _{0}}$

The leakage effect can be reduced, but not completely avoided, by means of window functions that differ from the rectangular function and with which the time signal is multiplied in the time domain. For this purpose, the increase or decrease of the amplitude in the window function is carried out more slowly than in the rectangular function, so that the spectrum of the window function is as concentrated as possible and as many derivatives as possible approach 0 at the edges of the window function . A common window function with a low leakage effect is the von Hann window . It should be noted that to reduce the leakage effect, the filter function is used in the time domain (and not, as is usual, in the spectral domain). ${\displaystyle \omega _{0}}$

Discrete-time systems

Leak effect in time-discrete systems

In time-discrete systems, e.g. B. in the context of digital signal processing , with the exception of a special case, with the discrete Fourier transformation (DFT) and the optimized variants of the fast Fourier transformation (FFT) built on it, a leakage effect occurs as a result of the block formation and the cyclical effects associated with it Folding on. A finite number of discrete sampled values in the time domain is used to calculate the discrete spectrum. Because of the discrete spectrum, there is a periodic continuation of the time-limited samples in the time domain.

This fact can be exploited in the special case of a harmonic oscillation, in which the observation window is an integral multiple of its period duration (i.e. the periodic continuation corresponds exactly to the signal course outside the observation interval), so that in this case there is no leakage effect. This case is shown in the figure opposite in the upper area. The observation interval includes exactly three periods of the harmonic signal frequency (shown here with dashed lines in gray), which is sampled with sampling values ​​(red dots). The absolute curve of the discrete spectrum with the circular wave number of the DFT shown on the right provides only one spectral component with a value not equal to 0. Due to the time-limited sampling - this corresponds to a multiplication with a rectangular function in the time domain - the gray dashed si function appears in the spectrum. At the zeros of this si function, all remaining spectral components are outside the signal frequency. This special case can only be achieved and maintained in a stable manner if the sampling frequency is synchronized with the signal frequency . ${\displaystyle N=8}$ ${\displaystyle k}$

If the observation window is not an integral multiple of the period - this case is shown below - a leak effect occurs: the discrete spectrum is spread over several spectral components. In this case, the maximum cannot be represented directly in the discrete spectrum, the portions are quasi “distributed” to neighboring spectral components. This case also occurs with general signal curves, which are formed, for example, from any sum of different harmonic oscillations.

To reduce the leakage effect, window functions are used in discrete-time systems , and the sampled signal sequence is first multiplied by the window function, for example a discrete von Hann window function. The discrete Fourier transform is then carried out.

literature

• Fernando Puente León, Uwe Kiencke, Holger Jäkel: Signals and Systems . 5th edition. Oldenbourg, 2011, ISBN 978-3-486-59748-6 .

• Randall, RB: Vibration Based Condition Monitoring . 1st edition. Wiley, 2011, ISBN 978-0-470-74785-8 .

Web links

• The Discrete Fourier Transform (DFT) (accessed July 16, 2018)
• Fourier Analysis and FFT (accessed July 16, 2018)
• Conception, development & evaluation of an adaptive multispectral sensor system (accessed on July 16, 2018)
• Basics of signal processing (accessed on July 16, 2018)
• Discrete Fourier Transform and Applications (accessed July 16, 2018)