Skip to content Skip to navigation Skip to collection information

Connexions

You are here: Home » Content » Fundamentals of Signal Processing(thu) » Frequency Sampling Design Method for FIR filters

Navigation

Table of Contents

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? 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.

This content is ...

In these lenses

  • SigProc display tagshide tags

    This module is included inLens: Signal Processing
    By: Daniel McKennaAs a part of collection: "Fundamentals of Signal Processing"

    Click the "SigProc" link to see all content selected in this lens.

    Click the tag icon tag icon to display tags associated with this content.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.
 

Frequency Sampling Design Method for FIR filters

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

Given a desired frequency response, the frequency sampling design method designs a filter with a frequency response exactly equal to the desired response at a particular set of frequencies ω k ω k .

Procedure

∀ k ,k=o1…N−1: H d ⁢ ω k =∑ n =0M−1h⁢n⁢e−(i⁢ ω k ⁢n) k k o 1 … N 1 H d ω k n M 1 0 h n ω k n
(1)

Note:

Desired Response must incluce linear phase shift (if linear phase is desired)

Exercise 1

What is H d ⁢ω H d ω for an ideal lowpass filter, cotoff at ω c ω c ?

Solution

{e−(i⁢ω⁢M−12)  if  − ω c ≤ω≤ ω c 0  if  (−π≤ω<− ω c )∨( ω c <ω≤π) ω M 1 2 ω c ω ω c 0 ω ω c ω c ω

Note:

This set of linear equations can be written in matrix form
H d ⁢ ω k =∑ n =0M−1h⁢n⁢e−(i⁢ ω k ⁢n) H d ω k n M 1 0 h n ω k n
(2)
( H d ⁢ ω 0 H d ⁢ ω 1 ⋮ H d ⁢ ω N - 1 )=( e−(i⁢ ω 0 ⁢0)e−(i⁢ ω 0 ⁢1)…e−(i⁢ ω 0 ⁢(M−1)) e−(i⁢ ω 1 ⁢0)e−(i⁢ ω 1 ⁢1)…e−(i⁢ ω 1 ⁢(M−1)) ⋮⋮⋮⋮ e−(i⁢ ω M - 1 ⁢0)e−(i⁢ ω M - 1 ⁢1)…e−(i⁢ ω M - 1 ⁢(M−1)) )⁢( h⁢0 h⁢1 ⋮ h⁢M−1 ) H d ω 0 H d ω 1 ⋮ H d ω N - 1 ω 0 0 ω 0 1 … ω 0 M 1 ω 1 0 ω 1 1 … ω 1 M 1 ⋮ ⋮ ⋮ ⋮ ω M - 1 0 ω M - 1 1 … ω M - 1 M 1 h 0 h 1 ⋮ h M 1
(3)

or H d =W⁢h H d W h So

h=W-1⁢ H d h W H d
(4)

Note:

W W is a square matrix for N=M N M , and invertible as long as ω i ≠ ω j +2⁢π⁢l ω i ω j 2 l , i≠j i j

Important Special Case

What if the frequencies are equally spaced between 00 and 2⁢π 2 , i.e. ω k =2⁢π⁢kM+α ω k 2 k M α

Then H d ⁢ ω k =∑ n =0M−1h⁢n⁢e−(i⁢2⁢π⁢k⁢nM)⁢e−(i⁢α⁢n)=∑ n =0M−1(h⁢n⁢e−(i⁢α⁢n))⁢e−(i⁢2⁢π⁢k⁢nM)= DFT! H d ω k n M 1 0 h n 2 k n M α n n M 1 0 h n α n 2 k n M DFT! so h⁢n⁢e−(i⁢α⁢n)=1M⁢∑ k =0M−1 H d ⁢ ω k ⁢ei⁢2⁢π⁢n⁢kM h n α n 1 M k M 1 0 H d ω k 2 n k M or h⁢n=ei⁢α⁢nM⁢∑ k =0M−1 H d ⁢ ω k ⁢ei⁢2⁢π⁢n⁢kM=ei⁢α⁢n⁢IDFT⁢ H d ⁢ ω k h n α n M k M 1 0 H d ω k 2 n k M α n IDFT H d ω k

Important Special Case #2

h⁢n h n symmetric, linear phase, and has real coefficients. Since h⁢n=h⁢M−n h n h M n 1 , there are only M2 M 2 degrees of freedom, and only M2 M 2 linear equations are required.

H⁢ ω k =∑ n =0M−1h⁢n⁢e−(i⁢ ω k ⁢n)={∑ n =0M2−1h⁢n⁢(e−(i⁢ ω k ⁢n)+e−(i⁢ ω k ⁢(M−n−1)))  if   M even ∑ n =0M−32h⁢n⁢(e−(i⁢ ω k ⁢n)+e−(i⁢ ω k ⁢(M−n−1)))⁢(h⁢M−12⁢e−(i⁢ ω k ⁢M−12))  if   M odd ={e−(i⁢ ω k ⁢M−12)⁢2⁢∑ n =0M2−1h⁢n⁢cos ω k ⁢(M−12−n)  if   M even e−(i⁢ ω k ⁢M−12)⁢2⁢∑ n =0M−32h⁢n⁢cos ω k ⁢(M−12−n)+h⁢M−12  if   M odd H ω k n M 1 0 h n ω k n n M 2 1 0 h n ω k n ω k M n 1 M even n M 3 2 0 h n ω k n ω k M n 1 h M 1 2 ω k M 1 2 M odd ω k M 1 2 2 n M 2 1 0 h n ω k M 1 2 n M even ω k M 1 2 2 n M 3 2 0 h n ω k M 1 2 n h M 1 2 M odd
(5)

Removing linear phase from both sides yields A⁢ ω k ={2⁢∑ n =0M2−1h⁢n⁢cos ω k ⁢(M−12−n)  if   M even 2⁢∑ n =0M−32h⁢n⁢cos ω k ⁢(M−12−n)+h⁢M−12  if   M odd A ω k 2 n M 2 1 0 h n ω k M 1 2 n M even 2 n M 3 2 0 h n ω k M 1 2 n h M 1 2 M odd Due to symmetry of response for real coefficients, only M2 M 2 ω k ω k on ω∈ 0 π ω 0 need be specified, with the frequencies − ω k ω k thereby being implicitly defined also. Thus we have M2 M 2 real-valued simultaneous linear equations to solve for h⁢n h n .

Special Case 2a

h⁢n h n symmetric, odd length, linear phase, real coefficients, and ω k ω k equally spaced: ∀ k ,0≤k≤M−1: ω k =n⁢π⁢kM k 0 k M 1 ω k n k M

h⁢n=IDFT⁢ H d ⁢ ω k =1M⁢∑ k =0M−1A⁢ ω k ⁢e−(i⁢2⁢π⁢kM)⁢M−12⁢ei⁢2⁢π⁢n⁢kM=1M⁢∑ k =0M−1A⁢k⁢ei⁢(2⁢π⁢kM⁢(n−M−12)) h n IDFT H d ω k 1 M k M 1 0 A ω k 2 k M M 1 2 2 n k M 1 M k M 1 0 A k 2 k M n M 1 2
(6)

To yield real coefficients, A⁢ω A ω mus be symmetric (A⁢ω=A⁢−ω)⇒(A⁢k=A⁢M−k) A ω A ω A k A M k

h⁢n=1M⁢(A⁢0+∑ k =1M−12A⁢k⁢(ei⁢2⁢π⁢kM⁢(n−M−12)+e−(i⁢2⁢π⁢k⁢(n−M−12))))=1M⁢(A⁢0+2⁢∑ k =1M−12A⁢k⁢cos2⁢π⁢kM⁢(n−M−12))=1M⁢(A⁢0+2⁢∑ k =1M−12A⁢k⁢−1k⁢cos2⁢π⁢kM⁢(n+12)) h n 1 M A 0 k M 1 2 1 A k 2 k M n M 1 2 2 k n M 1 2 1 M A 0 2 k M 1 2 1 A k 2 k M n M 1 2 1 M A 0 2 k M 1 2 1 A k 1 k 2 k M n 1 2
(7)

Simlar equations exist for even lengths, anti-symmetric, and α=12 α 1 2 filter forms.

Comments on frequency-sampled design

This method is simple conceptually and very efficient for equally spaced samples, since h⁢n h n can be computed using the IDFT.

H⁢ω H ω for a frequency sampled design goes exactly through the sample points, but it may be very far off from the desired response for ω≠ ω k ω ω k . This is the main problem with frequency sampled design.

Possible solution to this problem: specify more frequency samples than degrees of freedom, and minimize the total error in the frequency response at all of these samples.

Extended frequency sample design

For the samples H⁢ ω k H ω k where 0≤k≤M−1 0 k M 1 and N>M N M , find h⁢n h n , where 0≤n≤M−1 0 n M 1 minimizing ∥ H d ⁢ ω k −H⁢ ω k ∥ H d ω k H ω k

For ∥l∥∞ l norm, this becomes a linear programming problem (standard packages availble!)

Here we will consider the ∥l∥2 2 l norm.

To minimize the ∥l∥2 2 l norm; that is, ∑ n =0N−1| H d ⁢ ω k −H⁢ ω k | n N 1 0 H d ω k H ω k , we have an overdetermined set of linear equations: ( e−(i⁢ ω 0 ⁢0)…e−(i⁢ ω 0 ⁢(M−1)) ⋮⋮⋮ e−(i⁢ ω N - 1 ⁢0)…e−(i⁢ ω N - 1 ⁢(M−1)) )⁢h=( H d ⁢ ω 0 H d ⁢ ω 1 ⋮ H d ⁢ ω N - 1 ) ω 0 0 … ω 0 M 1 ⋮ ⋮ ⋮ ω N - 1 0 … ω N - 1 M 1 h H d ω 0 H d ω 1 ⋮ H d ω N - 1 or W⁢h= H d W h H d

The minimum error norm solution is well known to be h=W¯⁢W-1⁢W¯⁢ H d h W W W H d ; W¯⁢W-1⁢W¯ W W W is well known as the pseudo-inverse matrix.

Note:

Extended frequency sampled design discourages radical behavior of the frequency response between samples for sufficiently closely spaced samples. However, the actual frequency response may no longer pass exactly through any of the H d ⁢ ω k H d ω k .

Collection Navigation

Content actions

Download:

Collection as:

PDF | EPUB (?)

What is an EPUB file?

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

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Module as:

PDF | EPUB (?)

What is an EPUB file?

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

Downloading to a reading device

For detailed instructions on how to download this content's EPUB to your specific device, click the "(?)" link.

| More downloads ...

Add:

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? 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.

| 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? 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.

| External bookmarks