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Linear Phase Filters

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

In general, for −π≤ω≤π ω H⁢ω=|H⁢ω|⁢e−(i⁢θ⁢ω) H ω H ω θ ω Strictly speaking, we say H⁢ω H ω is linear phase if H⁢ω=|H⁢ω|⁢e−(i⁢ω⁢K)⁢e−(i⁢ 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)

Figure 1
Figure 1 (imag001.png)

Restrictions on h(n) to get linear phase

H⁢ω=∑h=0M−1h⁢n⁢e−(i⁢ω⁢n)=h⁢0+h⁢1⁢e−(i⁢ω)+h⁢2⁢e−(i⁢2⁢ω)+…+h⁢M−1⁢e−(i⁢ω⁢(M−1))=e−(i⁢ω⁢M−12)⁢(h⁢0⁢ei⁢ω⁢M−12+…+h⁢M−1⁢e−(i⁢ω⁢M−12))=e−(i⁢ω⁢M−12)⁢((h⁢0+h⁢M−1)⁢cosM−12⁢ω+(h⁢1+h⁢M−2)⁢cosM−32⁢ω+…+i⁢(h⁢0⁢sinM−12⁢ω+…)) 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−(i⁢ θ 0 )⁢(real,positive function of ω) θ 0 (real,positive function of ω) . For θ 0 =0 θ 0 0 , we thus require h⁢0+h⁢M−1=real number h 0 h M 1 real number h⁢0−h⁢M−1=pure imaginary number h 0 h M 1 pure imaginary number h⁢1+h⁢M−2=pure real number h 1 h M 2 pure real number h⁢1−h⁢M−2=pure imaginary number h 1 h M 2 pure imaginary number ⋮ ⋮ Thus h⁢k= h * ⁢M−1−k 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−(i⁢ θ 0 )=−i θ 0 , we require h⁢0+h⁢M−1=pure imaginary h 0 h M 1 pure imaginary h⁢0−h⁢M−1=pure real number h 0 h M 1 pure real number ⋮ ⋮ ⇒ h⁢k=−( h * ⁢M−1−k) ⇒ h k h * M 1 k

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

Figure 2: θ 0 =0 θ 0 0 (Symmetric Filters). h⁢k=h⁢M−1−k hk h M1 k.
Figure 2 (imag002.png)
Figure 3: θ 0 =π2 θ 0 2 (Anti-Symmetric Filters). h⁢k=−h⁢M−1−k hk h M1 k.
Figure 3 (imag003.png)
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 H⁢ω=A⁢ω⁢e−(i⁢ θ 0 )⁢e−(i⁢ω⁢M−12) H ω A ω θ 0 ω M 1 2 where A⁢ω A ω is real-valued. However, we have not assured that A⁢ω 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: H⁢ω=A⁢ω⁢e−(i⁢ω⁢M−12) H ω A ω ω M 1 2 , where A⁢ω A ω is real-valued, but possible negative. A⁢ω A ω is called the amplitude of the frequency response.

excuse:

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

note:

|H⁢ω|=±⁢A⁢ω=A⁢ω⁢e−(i⁢π⁢12⁢(1−sign⁢A⁢ω)) H ω ± A ω A ω 1 2 1 sign A ω where sign⁢x={1  if  x>0-1  if  x<0 sign x 1 x 0 -1 x 0

Example 1

Lowpass Filter

Figure 4
Desired |H(ω)|
Desired |H(ω)| (imag004.png)
Figure 5: The slope of each line is −M−12 M1 2.
Desired ∠H(ω)
Desired ∠H(ω) (imag005.png)
Figure 6: A⁢ωAω goes negative.
Actual |H(ω)|
Actual |H(ω)| (imag006.png)
Figure 7: 2⁢π 2 phase jumps due to periodicity of phase. π phase jumps due to sign change in A⁢ω A ω .
Actual ∠H(ω)
Actual ∠H(ω) (imag007.png)
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 M−12 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!

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