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Hossein Sameti Department of Computer Engineering Sharif University of Technology

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Definition of generalized linear-phase (GLP): Let’s focus on Type I FIR filter: 2 It can be shown that (L+1) unknown parameters a(n) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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3 Given determine coefficients of G(ω) (i.e. a(n)) such that L is minimized (minimum length of the filter). Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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G(ω) is a continuous function of ω and is as many times differentiable as we want. How many local extrema (min/max) does G(ω) have in the range ? In order to answer the above question, we have to write cos(ωn) as a sum of powers of cos(ω). 4 : sum of powers of cos(ω) Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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5 Find extrema Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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6 Polynomial of degree L-1 Maximum of L-1 real zeros Max. total number of real zeros: L+1 Conclusion: The maximum number of real zeros for (derivative of the frequency response of type I FIR filter) is L+1, where (N is the number of taps). Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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7 Given determine coefficients of G(ω) (i.e. a(n)) such that L is minimized (minimum length of the filter). Problem AProblem BProblem C Problem A Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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8 Given determine coefficients of G(ω) (i.e. a(n)) such that is minimized. ComputeGuess L Algorithm B Increase L by 1 Decrease L by 1 Yes Stop!

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9 Define F as a union of closed intervals in Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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10 where W is a positive weighting function Desired frequency response Find a(n) to minimize (same assumption as Problem B)

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We start by showing that 11 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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12 By definition: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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13 By definition: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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14 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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15 in Problem C in Problem B

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Conclusion: 16 Find a(n) such that is minimized. Problem B: Find a(n) such that is minimized. Problem C: Problem B= Problem C Problem A= Problem C We now try to solve Problem C. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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Assumptions: F: union of closed intervals G(x) to be a polynomial of order L: D = Desired function that is continuous in F. W= positive function 17

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The necessary and sufficient conditions for G(x) to be unique Lth order polynomial that minimizes is that E(x) exhibits at least L+2 alternations, i.e., there are at least L+2 values of x such that 18 for a polynomial of degree 4 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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19 Recall G(ω) can have at most L+1 local extrema. According to the alternation theorem, G(ω) should have at least L+2 alternations(local extrema) in F. Contradiction!?

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20 can also be alternation frequencies, although they are not local extrema. G(ω) can have at most L+3 local extrema in F. Ex: Polynomial of degree 7 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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21 According to the alternation theorem, we have at least L+2 alternations. According to our current argument, we have at most L+3 local extrema. Conclusion: we have either L+2 or L+3 alternations in F for the optimal case. Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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22 Extra-ripple case Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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23 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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For Type I low-pass filters, alternations always occur at If not, we potentially lose two alternations. 24 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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25 Equi-ripple except possibly at Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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26 For optimal type I low-pass filters, alternations always occur at If not, two alternations are lost and the filter is no longer optimal. Filter will be equi-ripple except possibly at Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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27 Given determine coefficients of G(ω) (i.e. a(n)) such that is minimized. At alternation frequencies, we have: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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28 Equating Eq.1 and Eq.2 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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30 L+2 linear equations and L+2 unknowns Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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31 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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32 It can be shown that if 's are known, then can be derived using the following formulae: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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33 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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34 Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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36 Original alternation frequency Next alternation frequency Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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App. estimate of L: App. Length of Kaiser filter: 37 Example: Optimal filter: Kaiser filter: Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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38 Does it meet the specs? Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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39 Increase the length of the filter by 1. Does it meet the specs? Hossein Sameti, Dept. of Computer Eng., Sharif University of Technology

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1 Chapter 7 Filter Design Techniques (cont.). 2 Optimum Approximation Criterion (1) We have discussed design of FIR filters by windowing, which is straightforward.

1 Chapter 7 Filter Design Techniques (cont.). 2 Optimum Approximation Criterion (1) We have discussed design of FIR filters by windowing, which is straightforward.

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