Smoothing Windows for Spectral Leakage

Publish Date: Aug 22, 2011 | 39 Ratings | 4.33 out of 5 | Print

Smoothing windows are a simple means of minimizing spectral leakage associated with truncated waveforms.

The Problem--Finite Sampling Record Creates a Truncated Waveform
In practical signal-sampling applications, you can obtain only a finite record of the signal. This finite sampling record results in a truncated waveform that has different spectral characteristics from the original continuous-time signal. These discontinuities produce leakage of spectral information, resulting in a discrete-time spectrum that is a smeared version of the original continuous-time spectrum. To find out more about spectral leakage, see the Spectral Leakage topic.

The Solution--Smoothing Windows
A simple way to improve the spectral characteristics of a sampled signal is to apply smoothing windows. When performing Fourier or spectral analysis on finite-length data, you can use windows to minimize the transition edges of your truncated waveforms, thus reducing spectral leakage. When used in this manner, smoothing windows act like predefined, narrowband, lowpass filters. Using windowing also can separate a small amplitude signal from a larger amplitude signal with frequencies very close to each other.

Smoothing Windows
Leakage exists because of the finite time record of the input signal. To overcome leakage, one solution is to take an infinite time record, from -infinity to +infinity. Then the FFT would calculate one single line at the correct frequency, but this would take an infinite amount of time. Fortunately, another technique to reduce spectral leakage, known as windowing or smoothing windows, is available. The amount of spectral leakage depends on the amplitude of the discontinuity. The larger the discontinuity, the more the leakage. You can use windowing to reduce the amplitude of the discontinuities at the boundaries of each period. It consists of multiplying the time record by a finite length window whose amplitude varies smoothly and gradually towards zero at the edges. This is shown in the following figure, where the original time signal from spectral leakage figure is windowed using a Hanning window. Notice that the time waveform of the windowed signal gradually tapers to zero at the ends. Therefore, when performing Fourier or spectral analysis on finite-length data, you can use windows to minimize the transition edges of your sampled waveform. A smoothing window applied to the data before it is transformed into the frequency domain minimizes spectral leakage.

Windows available in NI-SCOPE Measurement Library

Effect of the Hanning Window on a waveform.

Using Hanning Windows to Minimize Spectral Leakage
Compare the FFT of the windowed data shown below to the FFT of the non-windowed data shown in the figure 2.

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