# Lecture 22: Design Of FIR / IIR Filters

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Lecture 22: Design of FIR / IIR FiltersFoundations of Digital Signal ProcessingOutline Designing FIR Filters with Windows Designing FIR Filters with Frequency Selection Designing FIR Filters with Equi-ripples Designing IIR Filters with Discrete Differentiation Designing IIR Filters with Impulse Invariance Designing IIR Filters with the Bilinear Transform Related Analog FiltersFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters1

News Homework #9 Due on Thursday Submit via canvas Coding Assignment #6 Due on next Monday Submit via canvasFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters2

News Exam #2 β Great Job! Mean: 86.3 Median: 87.5Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters3

Lecture 22: Design of FIR / IIR FiltersFoundations of Digital Signal ProcessingOutline Designing FIR Filters with Windows Designing FIR Filters with Frequency Selection Designing FIR Filters with Equi-ripples Designing IIR Filters with Discrete Differentiation Designing IIR Filters with Impulse Invariance Designing IIR Filters with the Bilinear Transform Related Analog FiltersFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters4

Causality and Linear PhaseQuestion: Consider a length-M symmetric, causal filter.What condition must be satisfied? Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters5

Causality and Linear PhaseQuestion: Consider a length-M symmetric, causal filter.What condition must be satisfied? π₯π₯ ππ π₯π₯ ππ ππ 1 Positive: Even symmetry Negative: Odd symmetry π₯π₯ ππ 1 ππFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters6

Causality and Linear PhaseQuestion: Consider a length-M symmetric, causal filter.What is the phase response? Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters7

Causality and Linear PhaseQuestion: Consider a length-M symmetric, causal filter.What is the phase response? Assume M is even. Even Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 Odd Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters8

Causality and Linear PhaseQuestion: Consider a length-M symmetric, causal filter.What is the phase response? Assume M is even. Even Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ ππ 12ππ0π§π§ππ 12 ππ1π§π§ππ 1 12 ππ1π§π§1 ππ 12 ππ0π§π§ ππ 12Odd Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ ππ 12ππ0π§π§ππ 12 ππ1π§π§ππ 12 11 ππ1π§π§Foundations of Digital Signal Processingππ 12 ππ0π§π§ Lecture 22: Designing FIR / IIR Filtersππ 129

Causality and Linear PhaseQuestion: Consider a length-M symmetric, causal filter.What is the phase response? Assume M is even. Even Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ ππ 12ππ0π§π§ππ 12 ππ1π§π§ππ 1 12 ππ1π§π§1 πΊπΊ ππ ππ ππ ππ jΞ ππππ 12 ππ0π§π§ ππ 12Odd Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ ππ 12ππ0π§π§ππ 12 ππ1π§π§ππ 12 11 ππ1π§π§ππ 12πΊπΊ ππ ππ ππ ππ jΞ ππFoundations of Digital Signal Processing ππ0π§π§ Lecture 22: Designing FIR / IIR Filtersππ 1210

Causality and Linear PhaseQuestion: Consider a length-M symmetric, causal filter.What is the phase response? Assume M is even. Even Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ ππ 12ππ0π§π§ππ 12 ππ1π§π§ππ 1 12πΊπΊ ππ ππ ππ ππ jΞ ππ ππ1π§π§1 ππ 120Ξ ππ ππ ππ0π§π§ ππ 12for πΊπΊ ππ 0for πΊπΊ ππ 0Odd Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ ππ 12ππ0π§π§ππ 12 ππ1π§π§ππ 12 1πΊπΊ ππ ππ ππ ππ jΞ ππ1 ππ1π§π§ππ 120Ξ ππ ππFoundations of Digital Signal Processing ππ0π§π§ ππ 12for πΊπΊ ππ 0for πΊπΊ ππ 0Lecture 22: Designing FIR / IIR Filters11

Causality and Linear PhaseQuestion: Consider a length-M symmetric, causal filter.What is the phase response? Assume M is even. Even Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ ππ 12ππ0π§π§ππ 12 ππ1π§π§ππ 1 12 ππ1π§π§ ππ ππ 1 /2 ππ ππ ππ ππ 1 /2 ππ1 ππ 12 ππ0π§π§for πΊπΊ ππ 0for πΊπΊ ππ 0 ππ 12Odd Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ ππ 12ππ0π§π§ππ 12 ππ1π§π§ππ 12 11 ππ1π§π§ ππ ππ 1 /2 ππ ππ ππ ππ 1 /2 ππFoundations of Digital Signal Processingππ 12 ππ0π§π§ ππ 12for πΊπΊ ππ 0for πΊπΊ ππ 0Lecture 22: Designing FIR / IIR Filters12

Causality and Linear PhaseQuestion: Consider a length-M symmetric, causal filter.What is the phase response? Assume M is even. Even Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ ππ 12ππ0π§π§ππ 12 ππ1π§π§ππ 1 12 ππ1π§π§Foundations of Digital Signal Processing1 ππ 12 ππ0π§π§Lecture 22: Designing FIR / IIR Filters ππ 1213

Causality and Linear PhaseQuestion: Consider a length-M symmetric, causal filter.What is the phase response? Assume M is even. Even Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ π§π§ ππ 12ππ 12ππ 1ππ0π§π§ 2ππ/2 1 ππ1π§π§ ππππ π§π§ππ 0ππ 1 12ππ 12 ππ π§π§ ππ1π§π§ Foundations of Digital Signal Processingππ 12 ππ1 ππ 12 ππ0π§π§Lecture 22: Designing FIR / IIR Filters ππ 1214

Causality and Linear PhaseQuestion: Consider a length-M symmetric, causal filter.What is the phase response? Assume M is even. Even Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ π§π§ ππ 12ππ 12ππ 1ππ0π§π§ 2ππ/2 1 ππ1π§π§ ππππ π§π§ππ 0ππ 1 12ππ 12 ππNotice thatππ π§π§ π§π§ ππ 1 ππ π§π§ 1 π§π§ ππ1π§π§ ππ 12 ππ1 ππ 12 ππ0π§π§ ππ 12 Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters15

Causality and Linear PhaseQuestion: Consider a length-M symmetric, causal filter.What is the phase response? Assume M is even. Even Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ π§π§ ππ 12ππ 12ππ 1ππ0π§π§ 2ππ/2 1 ππ1π§π§ ππππ π§π§ππ 0ππ 1 12ππ 12 ππ π§π§Pole-zero plot property?ππ π§π§ π§π§ ππ 1 ππ π§π§ 1 ππ1π§π§ ππ 12 ππ1 ππ 12 ππ0π§π§ ππ 12 Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters16

Causality and Linear PhaseQuestion: Consider a length-M symmetric, causal filter.What is the phase response? Assume M is even. Even Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ π§π§ ππ 12ππ 12ππ 1ππ0π§π§ 2ππ/2 1 ππ1π§π§ ππππ π§π§ππ 0ππ 1 12ππ 12 ππ π§π§Pole-zero plot property?ππ π§π§ π§π§ ππ 1 ππ π§π§ 1 ππ1π§π§ ππ 12 ππoππ 12o ππ0π§π§ ππ 12oox x(M-1)o o Foundations of Digital Signal Processing1 oLecture 22: Designing FIR / IIR Filterso17

Causality Question: How do we describe causal filter magnitude?Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters18

Causality Question: How do we describe causal filter magnitude?Often plottedin dB (decibels)Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters19

Causality Question: How do we describe causal filter magnitude?Often plottedin dB (decibels)Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters20

Lecture 21: Design of FIR FiltersFoundations of Digital Signal ProcessingOutline Review Downsampling & Upsampling Causality in Filters Designing FIR Filters with Windows Designing FIR Filters with Frequency Selection Designing FIR Filters with Equi-ripplesFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters21

Designing with Windows Question: How can I design an FIR filter from an ideal filter? β±Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters22

Designing with Windows Question: How can I design an FIR filter from an ideal filter? Non-causalInfinite Responseβ±Foundations of Digital Signal ProcessingIdeal filterLecture 22: Designing FIR / IIR Filters23

Designing with Windows Question: How can I design an FIR filter from an ideal filter? Non-causalInfinite Response β±Ideal filterAnswer: Window the response!Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters24

Designing with Windows Question: How can I design an FIR filter from an ideal filter? Non-causalInfinite Response β±Ideal filterAnswer: Window the response!Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters25

Designing with Windows Question: How can I design an FIR filter from an ideal filter? Non-causalInfinite Response β±Ideal filterAnswer: Window the response!Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters26

Designing with Windows Different FiltersFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters27

Designing with Windows Different FiltersFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters28

Designing with Windows Different FiltersFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters29

Designing with Windows Windowing the sinc impulse responseFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters30

Designing with Windows Windowing the sinc impulse responseFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters31

Designing with Windows Windowing the sinc impulse responseFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters32

Designing with Windows Windowing the sinc impulse responseFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters33

Designing with Windows Windowing the sinc impulse responseFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters34

Designing with Windows Windowing the sinc impulse responseFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters35

Designing with Windows Windowing the sinc impulse responseFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters36

Designing with Windows Windowing the sinc impulse responseFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters37

Designing with Windows Windowing the sinc impulse responseFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters38

Designing with Windows Windowing the sinc impulse responseFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters39

Designing with Windows Windowing the sinc impulse responseFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters40

Designing with Windows Windowing the sinc impulse responseFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters41

Lecture 22: Design of FIR / IIR FiltersFoundations of Digital Signal ProcessingOutline Designing FIR Filters with Windows Designing FIR Filters with Frequency Sampling Designing FIR Filters with Equi-ripples Designing IIR Filters with Discrete Differentiation Designing IIR Filters with Impulse Invariance Designing IIR Filters with the Bilinear Transform Related Analog FiltersFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters42

Design with Frequency Sampling Option 2: Work backwards with constraintsConsider the DFT:ππ 1β ππ π»π» ππ ππππ 0ππ2ππππππππsuch thatFoundations of Digital Signal Processingπ»π» ππ π»π» ππ ππLecture 22: Designing FIR / IIR Filters43

Design with Frequency Sampling Option 2: Work backwards with constraintsConsider the DFT:ππ 12ππ1ππ ππππβ ππ π»π» ππ ππ ππππππ 0such thatππ 1 /2ππ 1ππ 1ππ ππ 1 /22ππ1ππ ππ ππππβ ππ π»π» 0 π»π» ππ ππ ππFoundations of Digital Signal Processing π»π» ππ π»π» ππ πππ»π» ππ ππLecture 22: Designing FIR / IIR Filters2ππππ ππ ππππ44

Design with Frequency Sampling Option 2: Work backwards with constraintsConsider the DFT:ππ 12ππ1ππ ππππβ ππ π»π» ππ ππ ππππππ 0such thatππ 1 /2ππ 1 /2ππ 1ππ 1π»π» ππ π»π» ππ ππ2ππ2ππ1ππ ππ ππππππ ππ ππβ ππ π»π» 0 π»π» ππ ππ π»π» ππ ππππFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filtersππ ππ45

Design with Frequency Sampling Option 2: Work backwards with constraintsConsider the DFT:ππ 12ππ1ππ ππππβ ππ π»π» ππ ππ ππππππ 0such thatππ 1 /2ππ 1 /2ππ 1ππ 1π»π» ππ π»π» ππ ππ2ππ2ππ1ππ ππ ππππππ ππ ππβ ππ π»π» 0 π»π» ππ ππ π»π» ππ ππππππ 1 /22ππ2ππ1ππ ππππππ ππβ ππ π»π» 0 π»π» ππ ππ ππ π»π» ππ ππ ππππππ 1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filtersππ ππππ ππ46

Design with Frequency Sampling Option 2: Work backwards with constraintsConsider the DFT:ππ 12ππ1ππ ππππβ ππ π»π» ππ ππ ππππππ 0such thatππ 1 /2ππ 1 /2ππ 1ππ 1π»π» ππ π»π» ππ ππ2ππ2ππ1ππ ππ ππππππ ππ ππβ ππ π»π» 0 π»π» ππ ππ π»π» ππ ππππππ 1 /2ππ ππ2ππ2ππ1ππ ππππ ππ ππππ πππππππβ ππ π»π» 0 π»π» ππ ππ ππ π»π» ππ ππ ππ ππππππ 1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters47

Design with Frequency Sampling Option 2: Work backwards with constraintsConsider the DFT:ππ 12ππ1ππ ππππβ ππ π»π» ππ ππ ππππππ 0such thatππ 1 /2ππ 1 /2ππ 1ππ 1π»π» ππ π»π» ππ ππ2ππ2ππ1ππ ππ ππππππ ππ ππβ ππ π»π» 0 π»π» ππ ππ π»π» ππ ππππππ 1 /2ππ ππ2ππ2ππ1ππ ππ ππππ ππ ππ ππππβ ππ π»π» 0 π»π» ππ ππ π»π» ππ ππππππ 1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters48

Design with Frequency Sampling Option 2: Work backwards with constraintsConsider the DFT:ππ 12ππ1ππ ππππβ ππ π»π» ππ ππ ππππππ 0such thatππ 1 /2ππ 1 /2ππ 1ππ 1π»π» ππ π»π» ππ ππ2ππ2ππ1ππ ππ ππππππ ππ ππβ ππ π»π» 0 π»π» ππ ππ π»π» ππ ππππππ 1 /22ππ2ππ1ππ ππππ ππ ππππβ ππ π»π» 0 π»π» ππ ππ ππ ππ ππππππ 1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filtersππ ππ49

Design with Frequency Sampling Option 2: Work backwards with constraintsConsider the DFT:ππ 12ππ1ππ ππππβ ππ π»π» ππ ππ ππππππ 0such thatππ 1 /2ππ 1 /2ππ 1ππ 1π»π» ππ π»π» ππ ππ2ππ2ππ1ππ ππ ππππππ ππ ππβ ππ π»π» 0 π»π» ππ ππ π»π» ππ ππππππ 1 /212ππβ ππ π»π» 0 2 π»π» ππ cosππππππππππ 1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filtersππ ππ50

Design with Frequency Sampling An inverse DFT that forces time-symmetryππ 1 /212ππβ ππ π»π» 0 2 π»π» ππ cosππππππππππ 1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters51

Design with Frequency Sampling An inverse DFT that forces time-symmetryππ 1 /212ππβ ππ π»π» 0 2 π»π» ππ cosππππππππππ 1 Example: Consider the desired 9-sample frequency responsewith the first half defined by [1 1 0 0] Compute the frequency sampled filterFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters52

Design with Frequency Sampling An inverse DFT that forces time-symmetryππ 1 /212ππβ ππ π»π» 0 2 π»π» ππ cosππππππππππ 1 Example: Consider the desired 9-sample frequency responsewith the first half defined by [1 1 0 0] Compute the frequency sampled filter1β ππ 1 2 cos 2ππ/9 ππ2Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters53

Design with Frequency Sampling Example: Consider the desired 9-sample frequency responsewith the first half defined by [1 1 0 0] Compute the frequency sampled filter1β ππ 1 2 cos 2ππ/9 ππ2Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters54

Design with Frequency Sampling Example: Consider the desired 9-sample frequency responsewith the first half defined by [1 1 0 0] Compute the frequency sampled filter1β ππ 1 2 cos 2ππ/9 ππ2 In practice, this should be circularly shifted so that themaximum is centered.Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters55

Design with Frequency Sampling An inverse DFT that forces time-symmetryππ 1 /212ππππ 1β ππ π»π» 0 2 π»π» ππ cosππ ππππππ2ππ 1 Example: Consider the desired 9-sample frequency responsewith the first half defined by [1 1 0 0] Compute the frequency sampled filter1β ππ 1 2 cos 2ππ/9 ππ2Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters56

Design with Frequency Sampling Example: Consider the desired 9-sample frequency responsewith the first half defined by [1 1 0 0] Compute the frequency sampled filter1β ππ 1 2 cos 2ππ/9 ππ 8/22Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters57

Design with Frequency Sampling Example: Consider the desired 17-sample frequency responsewith the first half defined by [1 1 1 1 0 0 0 0] Compute the frequency sampled filter1β ππ 1 2 cos 2ππ/19 ππππ 2 cos 4ππ/19 ππππ 2 cos 6ππ/19 ππππ1716ππππ ππ 2Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters58

Design with Frequency Sampling Example: Consider the desired 17-sample frequency responsewith the first half defined by [1 1 1 1 0 0 0 0] Compute the frequency sampled filter1β ππ 1 2 cos 2ππ/19 ππππ 2 cos 4ππ/19 ππππ 2 cos 6ππ/19 ππππ1716ππππ ππ 2Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters59

Design with Frequency Sampling Example: Consider the desired 41-sample frequency responsewith the first 10 values defined by 1 Compute the frequency sampled filterFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters60

Design with Frequency Sampling Example: Consider the desired 401-sample frequency responsewith the first 100 values defined by 1 Compute the frequency sampled filter Note that in practice, this needs to be circularly shifted to thecenterFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters61

Design with Frequency Sampling Note: The definition can be slightly modified Our definition:ππ 1 /212ππππ 1β ππ π»π» 0 2 π»π» ππ cosππ ππππππ2ππ 1ππ 1 /212ππππ 1 π»π» 0 2 π»π» ππ cosππ ππππππ2 2ππ 1ππ 1 /212ππ1 π»π» 0 2 π»π» ππ cosππ ππ ππππππππ2ππ 1ππ 1 /21 π»π» 0 2 ππππ 1 1ππ π»π»Foundations of Digital Signal Processing2ππ1ππ cosππ ππππ2Lecture 22: Designing FIR / IIR Filters62

Design with Frequency Sampling Final Definitionππ 1 /21β ππ π»π» 0 2 ππππ 1 1ππ π»π»2ππ1ππ cosππ ππππ2Side note: This is very closely related to thediscrete cosine transformFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters63

Lecture 22: Design of FIR / IIR FiltersFoundations of Digital Signal ProcessingOutline Designing FIR Filters with Windows Designing FIR Filters with Frequency Sampling Designing FIR Filters with Equi-ripples Designing IIR Filters with Discrete Differentiation Designing IIR Filters with Impulse Invariance Designing IIR Filters with the Bilinear Transform Related Analog FiltersFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters64

Design with Equi-ripplesPreviously derived:ππ π§π§ ππ ππ ππ 1 2π§π§ππ/2 1 ππππ π§π§ππ 0ππ 1 ππππ 2ππ 2ππ ππππππ 12ππ/2 1ππ 12 ππ ππππ ππππ 0ππ/2 1ππππ π§π§ ππ 12 ππππ 12 ππ ππππ ππππππ 12 ππππ 1 ππππ cos ππ ππ2ππ 0Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters65

Design with Equi-ripples Equi-ripple designππ ππ ππ 1 ππππ 22ππππ/2 1ππ 1 ππππ cos ππ ππ2ππ 0Goal: Find the optimal ππππ s that satisfies passband / stopbandripple constraints.Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters66

Design with Equi-ripplesEqui-ripple design min ππ ππ π»π»ππ ππ 2ππππππ ππππππ 12ππ/2 1ππ 1 ππππ cos ππ ππ2ππ 0Desired frequencyresponseEquals:πΏπΏ2for ππ in pass bandπΏπΏ11 for ππ in stop bandFoundations of Digital Signal ProcessingπΏπΏ2 stopband rippleπΏπΏ1 passband rippleLecture 22: Designing FIR / IIR Filters67

Lecture 22: Design of FIR / IIR FiltersFoundations of Digital Signal ProcessingOutline Designing FIR Filters with Windows Designing FIR Filters with Frequency Sampling Designing FIR Filters with Equi-ripples Designing IIR Filters with Discrete Differentiation Designing IIR Filters with Impulse Invariance Designing IIR Filters with the Bilinear Transform Related Analog FiltersFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters68

IIR Filter Design from Derivatives Designing IIR Filters No easy ways to design digital IIR filters So let us start from analog filtersFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters69

IIR Filter Design from Derivatives Designing IIR Filters No easy ways to design digital IIR filters So let us start from analog filters Option 1: Preserve the difference equation!Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters70

IIR Filter Design from Derivatives Question: What is a derivative in discrete-time? In continuous-timeππππ π‘π‘ π π π π  π π ππππ In discrete-timeFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters71

IIR Filter Design from Derivatives Question: What is a derivative in discrete-time? In continuous-timeππππ π‘π‘ π π π π  π π ππππ In discrete-timeT 1ππππ π‘π‘π₯π₯ π‘π‘ π₯π₯ π‘π‘ Ξππ limΞππ 0ππππΞππππππ π‘π‘π₯π₯ ππππ π₯π₯ ππππ ππ π₯π₯ ππ π₯π₯ ππ 1 ππππ π‘π‘ ππππππFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters72

IIR Filter Design from Derivatives Question: What is a derivative in discrete-time? In continuous-timeππππ π‘π‘ π π π π  π π ππππ In discrete-timeππππ π‘π‘π₯π₯ π‘π‘ π₯π₯ π‘π‘ Ξππ limΞππ 0ππππΞππππππ π‘π‘π₯π₯ ππππ π₯π₯ ππππ ππ1 π₯π₯ ππ π₯π₯ ππ 1 ππππ π‘π‘ ππππππππFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters73

IIR Filter Design from Derivatives Question: What is a derivative in discrete-time? In continuous-timeππππ π‘π‘ π π π π  π π ππππ In discrete-timeππππ π‘π‘π₯π₯ π‘π‘ π₯π₯ π‘π‘ Ξππ limΞππ 0ππππΞππππππ π‘π‘π₯π₯ ππππ π₯π₯ ππππ ππ1 π₯π₯ ππ π₯π₯ ππ 1 ππππ π‘π‘ ππππππππππππ π‘π‘1 1 π§π§ 1 ππ π§π§ππππ π‘π‘ ππππ ππFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters74

IIR Filter Design from Derivatives Question: What is a second-derivative in discrete-time? In continuous-timeππ2π₯π₯ π‘π‘2 π π ππ π π 2πππ‘π‘ In discrete-timeππ2π₯π₯ π‘π‘ππππ π‘π‘ ππππ π‘π‘ οΏ½οΏ½οΏ½πππ π‘π‘π₯π₯ ππππ π₯π₯ ππππ ππ ππππ π‘π‘ ππππππFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters75

IIR Filter Design from Derivatives Question: What is a second-derivative in discrete-time? In continuous-timeππ2π₯π₯ π‘π‘2 π π ππ π π 2πππ‘π‘ In discrete-timeππ2π₯π₯ π‘π‘ππππ π‘π‘ ππππ π‘π‘ οΏ½οΏ½οΏ½π2π₯π₯ π‘π‘ 2πππ‘π‘π‘π‘ πππππ₯π₯ ππππ π₯π₯ ππππ ππ /ππ π₯π₯ ππππ ππ π₯π₯ ππππ 2ππ /ππ ππFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters76

IIR Filter Design from Derivatives Question: What is a second-derivative in discrete-time? In continuous-timeππ2π₯π₯ π‘π‘2 π π ππ π π 2πππ‘π‘ In discrete-timeππ2π₯π₯ π‘π‘ππππ π‘π‘ ππππ π‘π‘ οΏ½οΏ½οΏ½π2π₯π₯ π‘π‘ πππ‘π‘ 2π‘π‘ ππππ π₯π₯ ππππ 2π₯π₯ ππππ ππ π₯π₯ ππππ 2πππ₯π₯ ππ 2π₯π₯ ππ 1 π₯π₯ ππ 2 ππ2ππ2Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters77

IIR Filter Design from Derivatives Question: What is a second-derivative in discrete-time? In continuous-timeππ2π₯π₯ π‘π‘2 π π ππ π π 2πππ‘π‘ In discrete-timeππ2π₯π₯ π‘π‘ππππ π‘π‘ ππππ π‘π‘ οΏ½οΏ½οΏ½π2π₯π₯ π‘π‘ πππ‘π‘ 2π‘π‘ ππππ π₯π₯ ππππ 2π₯π₯ ππππ ππ π₯π₯ ππππ 2πππ₯π₯ ππ 2π₯π₯ ππ 1 π₯π₯ ππ 2 ππ2ππ2ππππ π‘π‘11 1 2 2 1 2π§π§ π§π§ ππ π§π§ 2 1 π§π§ 1 2ππ π§π§ππππ π‘π‘ ππππ ππππFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters78

IIR Filter Design from Derivatives Question: What is a derivative in discrete-time? Translate continuous-time to discrete-timeππππ π₯π₯ π‘π‘ππ ππ π π  π π πππ‘π‘ππππππ π₯π₯ π‘π‘1 1 ππ ππ π§π§ 1 π§π§ πππ‘π‘πππππ‘π‘ ππππFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters79

IIR Filter Design from Derivatives Question: What is a derivative in discrete-time? Translate continuous-time to discrete-timeππππ π₯π₯ π‘π‘ππ ππ π π  π π πππ‘π‘ππππππ π₯π₯ π‘π‘1 1 ππ ππ π§π§ 1 π§π§ πππ‘π‘πππππ‘π‘ ππππ1π π  1 π§π§ 1ππFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters80

IIR Filter Design from Derivatives Example: π π  1ππ1 π§π§ 1 Use the derivative conversion to transform the following biquadfilter into the discrete-time domain.1π»π» π π  π π  0.1 2 9Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters81

IIR Filter Design from Derivatives Example: π π  1ππ1 π§π§ 1 Use the derivative conversion to transform the following bi-quadfilter into the discrete-time domain.1π»π» π π  π π  0.1 2 9π»π» π§π§ 111 π§π§ 1 0.1ππ1 π§π§ 1ππ2 0.1ππ22 9 9ππ2ππ2 Foundations of Digital Signal Processingππ211 π§π§ 1 0.1ππ1 0.1ππππ22 1 π§π§Lecture 22: Designing FIR / IIR Filters2 9 9ππ282

IIR Filter Design from Derivatives Example: π π  1ππ1 π§π§ 1 Use the derivative conversion to transform the following bi-quadfilter into the discrete-time domain.1π»π» π π  π π  0.1 2 9π»π» π§π§ ππ2Finding poles1 0.1ππ π§π§ 11 0.1ππ1 0.1ππ2 1 π§π§2 1 π§π§2 9ππ2 9ππ2 0 9ππ21 0.1ππ π§π§ 1 3πππππ§π§ 1 1 0.1ππ 3ππππFoundations of Digital Signal Processing1π§π§ 1 0.1 3ππ ππLecture 22: Designing FIR / IIR Filters83

IIR Filter Design from Derivatives Example: π π  1ππ1 π§π§ 1 Use the derivative conversion to transform the following bi-quadfilter into the discrete-time domain.1π§π§ Poles1 0.1 3ππ ππFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters84

IIR Filter Design from Derivatives Example: π π  1ππ1 π§π§ 11π§π§ 1 0.1 3ππ ππFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters85

IIR Filter Design from Derivatives Question: What is a derivative in discrete-time? Translate continuous-time to discrete-time1π π  1 π§π§ 1ππPros: Relatively simple Stable IIRanalog filtersmap hereCons:1 Very limiting Stable continuous-time poles canonly be mapped to low frequenciesFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters86

Lecture 22: Design of FIR / IIR FiltersFoundations of Digital Signal ProcessingOutline Designing FIR Filters with Windows Designing FIR Filters with Frequency Sampling Designing FIR Filters with Equi-ripples Designing IIR Filters with Discrete Differentiation Designing IIR Filters with Impulse Invariance Designing IIR Filters with the Bilinear Transform Related Analog FiltersFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters87

IIR Filter Design by Impulse Invariance Designing IIR Filters No easy ways to design digital IIR filters So let us start from analog filters Option 2: Preserve the impulse response!Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters88

IIR Filter Design by Impulse Invariance Question: How else can I represent my transfer function?πΎπΎ1π»π» π π  π π  ππππππ 1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters89

IIR Filter Design by Impulse Invariance Question: How else can I represent my transfer function?πΎπΎπΎπΎππ 1ππ 11π»π» π π  ππππ πππππππ‘π‘π π  ππππFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters90

IIR Filter Design by Impulse Invariance Question: How else can I represent my transfer function?πΎπΎπΎπΎππ 1ππ 11π»π» π π  ππππ πππππππ‘π‘π π  πππππΎπΎβ π‘π‘ ππππ ππ πππππ‘π‘ππ 1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters91

IIR Filter Design by Impulse Invariance Question: How else can I represent my transfer function?πΎπΎπΎπΎππ 1ππ 11π»π» π π  ππππ πππππππ‘π‘π π  πππππΎπΎβ π‘π‘ ππππ ππ πππππ‘π‘ππ 1πΎπΎπΎπΎππ 1ππ 1β ππππ β ππ ππππ ππ ππππππππ ππππ ππ ππππππFoundations of Digital Signal ProcessingππLecture 22: Designing FIR / IIR Filters92

IIR Filter Design by Impulse Invariance Question: How else can I represent my transfer function?πΎπΎπΎπΎππ 1ππ 11π»π» π π  ππππ πππππππ‘π‘π π  πππππΎπΎβ π‘π‘ ππππ ππ πππππ‘π‘ππ 1πΎπΎπΎπΎππ 1ππ 1β ππππ β ππ ππππ ππ ππππππππ ππππ ππ οΏ½οΏ½π» π§π§ 1 ππ ππππππ π§π§ 1ππ 1Foundations of Digital Signal ProcessingππLecture 22: Designing FIR / IIR Filters93

IIR Filter Design from Derivatives Example: π»π» π§π§ πππππΎπΎ ππ 11 ππ πππππππ§π§ 1 Use impulse invariance to transform the following biquad filter intothe discrete-time domain.1π»π» π π  π π  0.1 2 9Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters94

IIR Filter Design from Derivatives Example: π»π» π§π§ πππππΎπΎ ππ 11 ππ πππππππ§π§ 1 Use impulse invariance to transform the following biquad filter intothe discrete-time domain.1π»π» π π  π π  0.1 2 9Poles:π π  0.12 9 0π π  3ππ 0.1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters95

IIR Filter Design from Derivatives Example: π»π» π§π§ πππππΎπΎ ππ 11 ππ πππππππ§π§ 1 Use impulse invariance to transform the following biquad filter intothe discrete-time domain.11/21/2π»π» π π  2π π  0.1 9 π π  3ππ 0.1 π π  3ππ 0.1Poles:π π  0.12 9 0π π  3ππ 0.1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters96

IIR Filter Design from Derivatives Example: π»π» π§π§ πππππΎπΎ ππ 11 ππ πππππππ§π§ 1 Use impulse invariance to transform the following biquad filter intothe discrete-time domain.11/21/2π»π» π π  2π π  0.1 9 π π  3ππ 0.1 π π  3ππ 0.11/21/2π»π» π§π§ 3ππ 0.1ππ 11 πππ§π§1 ππ 3ππ 0.1 ππ π§π§ 1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters97

IIR Filter Design from Derivatives Example: π»π» π§π§ π»π» π§π§ 1 ππ1 πππππππΎπΎ ππ 11 ππ πππππππ§π§ 11/2 3ππ 0.1 ππ π§π§ 11/23ππ 0.1 ππ π§π§ 1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters98

Lecture 22: Design of FIR / IIR FiltersFoundations of Digital Signal ProcessingOutline Designing FIR Filters with Windows Designing FIR Filters with Frequency Sampling Designing FIR Filters with Equi-ripples Designing IIR Filters with Discrete Diff

Designing FIR Filters with Frequency Selection Designing FIR Filters with Equi-ripples Designing IIR Filters with Discrete Differentiation Designing IIR Filters with Impulse Invariance Designing IIR Filters with the Bilinear Transform Related Analog Filters. Lecture 22: Design of FIR / IIR Filters. Foundations of Digital .

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