OpenStax-CNX

You are here: Home » Content » Digital Filter Structures and Quantization Error Analysis » Quantization Error in FIR Filters

Lenses

What is a lens?

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?

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

In these lenses

• Lens for Engineering

This module and collection are included inLens: Lens for Engineering
By: Sidney Burrus

Click the "Lens for Engineering" link to see all content selected in this lens.

Recently Viewed

This feature requires Javascript to be enabled.

Inside Collection (Course):

Course by: Douglas L. Jones. E-mail the author

Quantization Error in FIR Filters

Module by: Douglas L. Jones. E-mail the author

Summary: FIR filters suffer from both data and coefficient quantization; each has different effects. Double-precision accumulation inside the FIR filter structure greatly reduces the data quantization error.

In digital filters, both the data at various places in the filter, which are continually varying, and the coefficients, which are fixed, must be quantized. The effects of quantization on data and coefficients are quite different, so they are analyzed separately.

Data Quantization

Typically, the input and output in a digital filter are quantized by the analog-to-digital and digital-to-analog converters, respectively. Quantization also occurs at various points in a filter structure, usually after a multiply, since multiplies increase the number of bits.

Direct-form Structures

There are two common possibilities for quantization in a direct-form FIR filter structure: after each multiply, or only once at the end.

In the latter structure, a double-length accumulator adds all 2B1 2 B 1 bits of each product into the accumulating sum, and truncates only at the end. Obviously, this is much preferred, and should always be used wherever possible. All DSP microprocessors and most general-pupose computers support double-precision accumulation.

Transpose-form

Similarly, the transpose-form FIR filter structure presents two common options for quantization: after each multiply, or once at the end.

The transpose form is not as convenient in terms of supporting double-precision accumulation, which is a significant disadvantage of this structure.

Coefficient Quantization

Since a quantized coefficient is fixed for all time, we treat it differently than data quantization. The fundamental question is: how much does the quantization affect the frequency response of the filter?

The quantized filter frequency response is DTFT h Q =DTFT h inf. prec. +e= H inf. prec. w+ H e w DTFT h Q DTFT h inf. prec. e H inf. prec. w H e w Assuming the quantization model is correct, H e w H e w should be fairly random and white, with the error spread fairly equally over all frequencies w π π w ; however, the randomness of this error destroys any equiripple property or any infinite-precision optimality of a filter.

Exercise 1

What quantization scheme minimizes the L 2 L 2 quantization error in frequency (minimizes ππ|Hw H Q w|2d w w H w H Q w 2 )? On average, how big is this error?

Ideally, if one knows the coefficients are to be quantized to BB bits, one should incorporate this directly into the filter design problem, and find the MM BB-bit binary fractional coefficients minimizing the maximum deviation ( L L error). This can be done, but it is an integer program, which is known to be np-hard (i.e., requires almost a brute-force search). This is so expensive computationally that it's rarely done. There are some sub-optimal methods that are much more efficient and usually produce pretty good results.

Content actions

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

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?

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

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

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?

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

| External bookmarks