# Windows for Spectral Analysis versus Windows for Coefficient Design

Spectral analysis and filter coefficient design place different requirements on a window. Spectral analysis requires a DFT-even window, while filter coefficient design requires a window symmetric about its midpoint.

## Spectral Analysis

The smoothing windows designed for spectral analysis must be DFT-even. A smoothing window is DFT-even if its dot product, or inner product, with integral cycles of sine sequences is identically zero. In other words, the DFT of a DFT-even sequence has no imaginary component.

The following two figures show the Hanning window for a sample size of 8 and one cycle of a sine pattern for a sample size of 8.

In the first figure above, the DFT-even Hanning window is not symmetric about its midpoint. The last point of the window is not equal to its first point, similar to one complete cycle of the sine pattern shown in the second figure above.

Smoothing windows for spectral analysis are spectral windows and include the following window types:

- Scaled time-domain window
- Hanning window
- Hamming window
- Triangle window
- Blackman window
- Exact Blackman window
- Blackman-Harris window
- Flat top window
- Kaiser window
- General cosine window
- Cosine tapered window

## Windows for FIR Filter Coefficient Design

Designing FIR filter coefficients requires a window that is symmetric about its midpoint.

Equations A and B illustrate the difference between a spectral window and a symmetrical window for filter coefficient design.

Equation A defines the Hanning window for spectral analysis.

| (A)

where N is the length of the window and w is the window value.

Equation B defines a symmetrical Hanning window for filter coefficient design.

| (B)

where N is the length of the window and w is the window value.

By modifying a spectral window, as shown in Equation B, you can define a symmetrical window for designing filter coefficients for a digital filter.