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Filter design by windowing

Module by: Hyeokho Choi. E-mail the author

Summary: (Blank Abstract)

FIR Filter Design by Windowing

Desired Ideal Filter Response

Unlike the design of IIR filters, the design of FIR filters starts from the spectrum of the desired filter in the DTFT domain. Let H d ⁢eiω H d ω be the ideal desired frequency response. Then, the impulse response h d ⁢n h d n corresponding to H d ⁢eiω H d ω is obtained by the inverse DTFT as

(1)
For general desired frequency response, h d ⁢n h d n is usually noncausal and infinitely long.

Windowing of Impulse Response

To obtain an FIR filter approximating the frequency response of the desired ideal filter, we need to find a filter impulse response a causal and finite h⁢n h n that approximates h d ⁢n h d n . The simplest way to obtain such h⁢n h n is to define a new system with impulse response h⁢n h n given by

(2)
where w⁢n w n is a finite duration window. For example, simple truncation corresponds to the boxcar window given as
(3)
Other commonly used windows include Bartlett, Hamming, Hanning, and Blackman windows.

Frequency Response of Windowed Impulse Response

Because the multiplication by the window function in time domain corresponds to a convolution in the DTFT domain, we can easily visualize the spectrum of the designed FIR filter. Different window functions have different main lobe width and peak sidelobe heights. The width of the main lobe governs the property of the frequency transition at band edges. The height of the peak sidelobes is related to the oscillations near the transition frequencies.

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Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags? tag icon

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

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