Connexions

You are here: Home » Content » Linear-Phase FIR Filters
Content Actions
Lenses

What is a lens?

Lenses

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

What is in a lens?

Lens makers point to Connexions 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 Connexions member, a community, or a respected organization.

This content is ...
Affiliated with (?)
This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.

  • This module is included inLens: Rice University OpenCourseWare
    By: OpenCourseWare ConsortiumAs a part of collection:"Intro to Digital Signal Processing"

    Click the "Rice University OCW" link to see all content affiliated with them.

    Rice University OCW
Tags

(?)

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

Linear-Phase FIR Filters

Module by: Ivan Selesnick

Summary: (Blank Abstract)

THE AMPLITUDE RESPONSE

If the real and imaginary parts of H f ω H f ω are given by
H f ω=ℜω+ⅈℑω H f ω ω ω (1)
the magnitude and phase are defined as | H f ω|=ℜω2+ℑω2 H f ω ω 2 ω 2 pω=arctanℑωℜω p ω ω ω so that
H f ω=| H f ω|ⅇⅈpω H f ω H f ω p ω (2)
With this definition, | H f ω| H f ω is never negative and pω p ω is usually discontinuous, but it can be very helpful to write H f ω H f ω as
H f ω=Aωⅇⅈθω H f ω A ω θ ω (3)
where Aω A ω can be positive and negative, and θω θ ω continuous. Aω A ω is called the amplitude response. Figure 1 illustrates the difference between | H f ω| H f ω and Aω A ω .
ma.png
Figure 1
A linear-phase phase filter is one for which the continuous phase θω θ ω is linear. H f ω=Aωⅇⅈθω H f ω A ω θ ω with θω=-Mω+B θ ω M ω B We assume in the following that the impulse response hn h n is real-valued.

WHY LINEAR-PHASE?

If a discrete-time cosine signal x 1 n=cos ω 1 n+ φ 1 x 1 n ω 1 n φ 1 is processed through a discrete-time filter with frequency response H f ω=Aωⅇⅈθω H f ω A ω θ ω then the output signal is given by y 1 n=A ω 1 cos ω 1 n+ φ 1 +θ ω 1 y 1 n A ω 1 ω 1 n φ 1 θ ω 1 or y 1 n=A ω 1 cos ω 1 n+θ ω 1 ω 1 + φ 1 y 1 n A ω 1 ω 1 n θ ω 1 ω 1 φ 1 The LTI system has the effect of scaling the cosine signal and delaying it by θ ω 1 ω 1 θ ω 1 ω 1 .
Problem 1
When does the system delay cosine signals with different frequencies by the same amount?
[ Click for Solution 1 ]
Solution 1
  • θωω=constant θ ω ω constant
  • θω=Kω θ ω K ω
  • The phase is linear.
[ Hide Solution 1 ]
The function θωω θ ω ω is called the phase delay. A linear phase filter therefore has constant phase delay.

WHY LINEAR-PHASE: EXAMPLE

Consider a discrete-time filter described by the difference equation
yn=-0.1821xn+0.7865xn-1-0.6804xn-2+xn-3+0.6804yn-1-0.7865yn-2+0.1821yn-3 y n -0.1821 x n 0.7865 x n 1 0.6804 x n 2 x n 3 0.6804 y n 1 0.7865 y n 2 0.1821 y n 3 (4)
When ω 1 =0.31π ω 1 0.31 , then the delay is -θ ω 1 ω 1 =2.45 θ ω 1 ω 1 2.45 . The delay is illustrated in Figure 2:
lin1.png
Figure 2
Notice that the delay is fractional --- the discrete-time samples are not exactly reproduced in the output. The fractional delay can be interpreted in this case as a delay of the underlying continuous-time cosine signal.

WHY LINEAR-PHASE: EXAMPLE (2)

Consider the same system given on the previous slide, but let us change the frequency of the cosine signal.
When ω 2 =0.47π ω 2 0.47 , then the delay is -θ ω 2 ω 2 =0.14 θ ω 2 ω 2 0.14 .
lin2.png
Figure 3
note: For this example, the delay depends on the frequency, because this system does not have linear phase.

WHY LINEAR-PHASE: MORE

From the previous slides, we see that a filter will delay different frequency components of a signal by the same amount if the filter has linear phase (constant phase delay).
In addition, when a narrow band signal (as in AM modulation) goes through a filter, the envelop will be delayed by the group delay or envelop delay of the filter. The amount by which the envelop is delayed is independent of the carrier frequency only if the filter has linear phase.
Also, in applications like image processing, filters with non-linear phase can introduce artifacts that are visually annoying.

Comments, questions, feedback, criticisms?

Send feedback