# Connexions

You are here: Home » Content » Digital Signal Processing: A User's Guide » Linear Phase Filters

### Recently Viewed

This feature requires Javascript to be enabled.

Inside Collection (Course):

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

# Linear Phase Filters

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

In general, for πωπ ω Hω=|Hω|e(jθω) H ω H ω θ ω Strictly speaking, we say Hω H ω is linear phase if Hω=|Hω|e(jωK)e(j θ 0 ) H ω H ω ω K θ 0 Why is this important? A linear phase response gives the same time delay for ALL frequencies! (Remember the shift theorem.) This is very desirable in many applications, particularly when the appearance of the time-domain waveform is of interest, such as in an oscilloscope. (see Figure 1)

## Restrictions on h(n) to get linear phase

Hω=h=0M1hne(jωn)=h0+h1e(jω)+h2e(j2ω)++hM1e(jω(M1))=e(jωM12)(h0ejωM12++hM1e(jωM12))=e(jωM12)((h0+hM1)cosM12ω+(h1+hM2)cosM32ω++j(h0sinM12ω+)) H ω h 0 M 1 h n ω n h 0 h 1 ω h 2 2 ω h M 1 ω M 1 ω M 1 2 h 0 ω M 1 2 h M 1 ω M 1 2 ω M 1 2 h 0 h M 1 M 1 2 ω h 1 h M 2 M 3 2 ω h 0 h M 1 M 1 2 ω
(1)
For linear phase, we require the right side of Equation 1 to be e(j θ 0 )(real,positive function of ω) θ 0 (real,positive function of ω) . For θ 0 =0 θ 0 0 , we thus require h0+hM1=real number h 0 h M 1 real number h0hM1=pure imaginary number h 0 h M 1 pure imaginary number h1+hM2=pure real number h 1 h M 2 pure real number h1hM2=pure imaginary number h 1 h M 2 pure imaginary number Thus hk= h * M1k h k h * M 1 k is a necessary condition for the right side of Equation 1 to be real valued, for θ 0 =0 θ 0 0 .

For θ 0 =π2 θ 0 2 , or e(j θ 0 )=j θ 0 , we require h0+hM1=pure imaginary h 0 h M 1 pure imaginary h0hM1=pure real number h 0 h M 1 pure real number hk=( h * M1k) h k h * M 1 k

Usually, one is interested in filters with real-valued coefficients, or see Figure 2 and Figure 3.

Filter design techniques are usually slightly different for each of these four different filter types. We will study the most common case, symmetric-odd length, in detail, and often leave the others for homework or tests or for when one encounters them in practice. Even-symmetric filters are often used; the anti-symmetric filters are rarely used in practice, except for special classes of filters, like differentiators or Hilbert transformers, in which the desired response is anti-symmetric.

So far, we have satisfied the condition that Hω=Aωe(j θ 0 )e(jωM12) H ω A ω θ 0 ω M 1 2 where Aω A ω is real-valued. However, we have not assured that Aω A ω is non-negative. In general, this makes the design techniques much more difficult, so most FIR filter design methods actually design filters with Generalized Linear Phase: Hω=Aωe(jωM12) H ω A ω ω M 1 2 , where Aω A ω is real-valued, but possible negative. Aω A ω is called the amplitude of the frequency response.

### excuse:

Aω A ω usually goes negative only in the stopband, and the stopband phase response is generally unimportant.

### note:

|Hω|=±Aω=Aωe(jπ12(1signAω)) H ω ± A ω A ω 1 2 1 sign A ω where signx={1  if  x>0-1  if  x<0 sign x 1 x 0 -1 x 0

### Example 1

#### Lowpass Filter

Time-delay introduces generalized linear phase.
##### note:
For odd-length FIR filters, a linear-phase design procedure is equivalent to a zero-phase design procedure followed by an M12 M 1 2 -sample delay of the impulse response. For even-length filters, the delay is non-integer, and the linear phase must be incorporated directly in the desired response!

## 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.

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