The Windows VIs implemented in the Analysis library in LabVIEW are designed for spectral analysis applications. In these applications, the input signal is windowed by passing it through one of the window VIs. The windowed signal is then passed to a DFT-based VI for frequency-domain display and analysis.
The window functions designed for spectral analysis must be DFT-even, a term defined by Fredric J. Harris in his paper "On the Use of Windows for Harmonic Analysis with the Discrete Fourier Transform" (Proceedings of the IEEE, Volume 66, No.1, January 1978). A window function is DFT-even if its dot product, or inner product, with integral cycles of sine sequences is identically zero. Another way to think of a DFT-even sequence is that its DFT has no imaginary component.
The following illustrations depict the Hanning window and one cycle of a sine pattern for a sample size of 8. The following illustrations show that the DFT-even Hanning window is not symmetric about its midpoint and that its last point is not equal to its first point, much like one complete cycle of a sine pattern.
Finally, the DFT considers input sequences to be periodic. That is, that the signal being analyzed is actually a concatenation of the input signal. The following illustration shows three such cycles of the previous sequences, demonstrating the smooth periodic extension of the DFT-even window and the single-cycle sine pattern.
Another type of window application is that of FIR filter design. This application requires windows that are symmetric about their midpoint. Refer to Designing FIR Filters by Windowing for more information about using windows to design FIR filters.
The following equations of the Hanning window function illustrate the difference between the DFT-even window function, spectral analysis, and the symmetrical window function, coefficient design.
Hanning window function for spectral analysis
for i = 0, 1, 2, , N 1
Hanning window function for symmetrical coefficient design
for i = 0, 1, 2, , N 1
The two equations above show that you can implement the symmetrical window functions by slightly modifying the use of the DFT-even window functions. The following illustration shows a block diagram that uses the Hanning Window VI to implement symmetrical windowing of filter coefficients.
Refer to Signal Processing Related Documentation for more information about smoothing windows.