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Linear Phase Filters

Module by: Douglas L. Jones

In general, for -π≤ω≤π ω Hω=|Hω|ⅇ-ⅈθω H ω H ω θ ω Strictly speaking, we say Hω H ω is linear phase if Hω=|Hω|ⅇ-ⅈωKⅇ-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)
imag001.png
Figure 1

Restrictions on h(n) to get linear phase

Hω=∑h=0M-1hnⅇ-ⅈωn=h0+h1ⅇ-ⅈω+h2ⅇ-ⅈ2ω+…+hM-1ⅇ-ⅈωM-1=ⅇ-ⅈωM-12h0ⅇⅈωM-12+…+hM-1ⅇ-ⅈωM-12=ⅇ-ⅈωM-12h0+hM-1cosM-12ω+h1+hM-2cosM-32ω+…+ⅈh0-hM-1sinM-12ω+… 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 h0+hM-1=real number h 0 h M 1 real number h0-hM-1=pure imaginary number h 0 h M 1 pure imaginary number h1+hM-2=pure real number h 1 h M 2 pure real number h1-hM-2=pure imaginary number h 1 h M 2 pure imaginary number ⋮ ⋮ Thus hk= 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 h0+hM-1=pure imaginary h 0 h M 1 pure imaginary h0-hM-1=pure real number h 0 h M 1 pure real number ⋮ ⋮ ⇒ hk=- 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.
imag002.png
Figure 2: θ 0 =0 θ 0 0 (Symmetric Filters). hk=hM-1-k hk h M1 k.
imag003.png
Figure 3: θ 0 =π2 θ 0 2 (Anti-Symmetric Filters). hk=-hM-1-k hk h M1 k.
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-ⅈ θ 0 ⅇ-ⅈωM-12 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-ⅈωM-12 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-ⅈπ121-signAω H ω ± A ω A ω 1 2 1 sign A ω where signx=1ifx>0-1ifx<0 sign x 1 x 0 -1 x 0
Example 1 
Lowpass Filter
Desired |H(ω)|
imag004.png
Figure 4
Desired ∠H(ω)
imag005.png
Figure 5: The slope of each line is -M-12 M1 2.
Actual |H(ω)|
imag006.png
Figure 6: AωAω goes negative.
Actual ∠H(ω)
imag007.png
Figure 7: 2π 2 phase jumps due to periodicity of phase. π phase jumps due to sign change in Aω A ω .
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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