Linear Phase Filters 1.1 2005/01/11 14:53:10 US/Central 2005/05/24 14:38:53 GMT-5 Douglas L. Jones dl-jones@uiuc.edu Douglas L. Jones dl-jones@uiuc.edu Charlet Reedstrom charlet@rice.edu Kyle Clarkson kclarks@rice.edu FIR filter design In general, for ω H ω H ω θ ω Strictly speaking, we say H ω is linear phase if 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 )

Restrictions on h(n) to get linear phase 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 … For linear phase, we require the right side of to be θ 0 (real,positive function of ω) . For θ 0 0 , we thus require h 0 h M 1 real number h 0 h M 1 pure imaginary number h 1 h M 2 pure real number h 1 h M 2 pure imaginary number ⋮ Thus h k h * M 1 k is a necessary condition for the right side of to be real valued, for θ 0 0 . For θ 0 2 , or θ 0 , we require h 0 h M 1 pure imaginary h 0 h M 1 pure real number ⋮ ⇒ h k h * M 1 k Usually, one is interested in filters with real-valued coefficients, or see and .

θ 0 0 (Symmetric Filters). hk h M1 k.

θ 0 2 (Anti-Symmetric Filters). 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 H ω A ω θ 0 ω M 1 2 where A ω is real-valued. However, we have not assured that 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 ω M 1 2 , where A ω is real-valued, but possible negative. A ω is called the amplitude of the frequency response. A ω usually goes negative only in the stopband, and the stopband phase response is generally unimportant. H ω ± A ω A ω 1 2 1 sign A ω where sign x 1 x 0 -1 x 0 Lowpass Filter

Desired |H(ω)|

Desired ∠H(ω) The slope of each line is M1 2.

Actual |H(ω)| Aω goes negative.

Actual ∠H(ω) 2 phase jumps due to periodicity of phase. phase jumps due to sign change in A ω .

Time-delay introduces generalized linear phase. For odd-length FIR filters, a linear-phase design procedure is equivalent to a zero-phase design procedure followed by an 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!