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Low-Pass Fitler Implementation: Filter Specification

Module by: Mark Butala. E-mail the author

Filter Specification

 f p f p 0.1 f s f s 0.12 δ p δ p 0.5 dB δ s δ s -60 dB Δ⁢ τ g Δ τ g ∀ f ,|f|≤ f p :|Δ⁢ τ g |≤9 f f f p Δ τ g 9

Please refer to Pictoralial filter specification for a pictorial description of how the various filter constraints correspond to the frequency response of a low-pass filter. Note that the Pictoralial filter specification does not imply that the system response must be equiripple. The constraints on the system response, to be discussed shortly, are on the maximum allowable peak errors. However, in general, an equiripple solution to a given filter specification will require the lowest filter order.

The full filter specification is given in filter specifications. The parameters f p f p and f s f s define the end of the pass-band and the beginning of the stop-band, respectively, on the frequency axis that has been normalized to 1. The parameters δ p δ p and δ s δ s define the maximum allowable ripple, in Decibels, in the pass-band and the stop-band, respectively. The parameter f ,|f| f p :|Δ τ g |9 f f f p Δ τ g 9 defines the maximum allowable deviation from a group delay of 9 in the pass-band. The parameter τ g τ g is the group delay of the system and is defined by the following:

τ g ω=dHωd x τ g ω x H ω
(1)

The parameter Δ τ g Δ τ g is defined as:

Δ τ g ω=max τ g ω |ω| ω p min τ g ω |ω| ω p Δ τ g ω ω ω p τ g ω ω ω p τ g ω
(2)

Thus, f ,|f| f p :|Δ τ g |9 f f f p Δ τ g 9 states that the group delay is not allowed to deviate more than 9 samples in the pass-band. A system that has constant group delay will have a phase response that is generalized linear-phase. Thus, deviation from constant group delay is a measure of the deviation of the phase response from linear. The MATLAB command grpdelay computes the group delay of a system as a function of frequency given the filter coefficients a a and b b.

Question:

For which of the three techniques discussed in filtering techniques must we verfiy explicitly that the group delay specification is met? All of them, some of them, or none of them?

Question:

Why do we only specify the filter coefficients for only the positive frequencies? What are we assuming? What does this imply about the coefficients a a and b b of the low-pass filter?

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