Lecture 22: Design Of FIR / IIR Filters
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₯ ππ π₯π₯ ππ ππ 1 Positive: Even symmetry Negative: Odd symmetry π₯π₯ ππ 1 ππFoundations 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 Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 Odd Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 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 Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ ππ 12ππ0π§π§ππ 12 ππ1π§π§ππ 1 12 ππ1π§π§1 ππ 12 ππ0π§π§ ππ 12Odd Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ ππ 12ππ0π§π§ππ 12 ππ1π§π§ππ 12 11 ππ1π§π§Foundations of Digital Signal Processingππ 12 ππ0π§π§ Lecture 22: Designing FIR / IIR Filtersππ 129
Causality and Linear PhaseQuestion: Consider a length-M symmetric, causal filter.What is the phase response? Assume M is even. Even Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ ππ 12ππ0π§π§ππ 12 ππ1π§π§ππ 1 12 ππ1π§π§1 πΊπΊ ππ ππ ππ ππ jΞ ππππ 12 ππ0π§π§ ππ 12Odd Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ ππ 12ππ0π§π§ππ 12 ππ1π§π§ππ 12 11 ππ1π§π§ππ 12πΊπΊ ππ ππ ππ ππ jΞ ππFoundations of Digital Signal Processing ππ0π§π§ Lecture 22: Designing FIR / IIR Filtersππ 1210
Causality and Linear PhaseQuestion: Consider a length-M symmetric, causal filter.What is the phase response? Assume M is even. Even Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ ππ 12ππ0π§π§ππ 12 ππ1π§π§ππ 1 12πΊπΊ ππ ππ ππ ππ jΞ ππ ππ1π§π§1 ππ 120Ξ ππ ππ ππ0π§π§ ππ 12for πΊπΊ ππ 0for πΊπΊ ππ 0Odd Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ ππ 12ππ0π§π§ππ 12 ππ1π§π§ππ 12 1πΊπΊ ππ ππ ππ ππ jΞ ππ1 ππ1π§π§ππ 120Ξ ππ ππFoundations of Digital Signal Processing ππ0π§π§ ππ 12for πΊπΊ ππ 0for πΊπΊ ππ 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 Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ ππ 12ππ0π§π§ππ 12 ππ1π§π§ππ 1 12 ππ1π§π§ ππ ππ 1 /2 ππ ππ ππ ππ 1 /2 ππ1 ππ 12 ππ0π§π§for πΊπΊ ππ 0for πΊπΊ ππ 0 ππ 12Odd Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ ππ 12ππ0π§π§ππ 12 ππ1π§π§ππ 12 11 ππ1π§π§ ππ ππ 1 /2 ππ ππ ππ ππ 1 /2 ππFoundations of Digital Signal Processingππ 12 ππ0π§π§ ππ 12for πΊπΊ ππ 0for πΊπΊ ππ 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 Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ ππ 12ππ0π§π§ππ 12 ππ1π§π§ππ 1 12 ππ1π§π§Foundations of Digital Signal Processing1 ππ 12 ππ0π§π§Lecture 22: Designing FIR / IIR Filters ππ 1213
Causality and Linear PhaseQuestion: Consider a length-M symmetric, causal filter.What is the phase response? Assume M is even. Even Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ π§π§ ππ 12ππ 12ππ 1ππ0π§π§ 2ππ/2 1 ππ1π§π§ ππππ π§π§ππ 0ππ 1 12ππ 12 ππ π§π§ ππ1π§π§ Foundations of Digital Signal Processingππ 12 ππ1 ππ 12 ππ0π§π§Lecture 22: Designing FIR / IIR Filters ππ 1214
Causality and Linear PhaseQuestion: Consider a length-M symmetric, causal filter.What is the phase response? Assume M is even. Even Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ π§π§ ππ 12ππ 12ππ 1ππ0π§π§ 2ππ/2 1 ππ1π§π§ ππππ π§π§ππ 0ππ 1 12ππ 12 ππNotice thatππ π§π§ π§π§ ππ 1 ππ π§π§ 1 π§π§ ππ1π§π§ ππ 12 ππ1 ππ 12 ππ0π§π§ ππ 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 Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ π§π§ ππ 12ππ 12ππ 1ππ0π§π§ 2ππ/2 1 ππ1π§π§ ππππ π§π§ππ 0ππ 1 12ππ 12 ππ π§π§Pole-zero plot property?ππ π§π§ π§π§ ππ 1 ππ π§π§ 1 ππ1π§π§ ππ 12 ππ1 ππ 12 ππ0π§π§ ππ 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 Symmetryππ π§π§ ππ0 ππ1π§π§ 1 ππ2π§π§ 2 ππ1π§π§ ππ 2 ππ0π§π§ ππ 1 π§π§ π§π§ ππ 12ππ 12ππ 1ππ0π§π§ 2ππ/2 1 ππ1π§π§ ππππ π§π§ππ 0ππ 1 12ππ 12 ππ π§π§Pole-zero plot property?ππ π§π§ π§π§ ππ 1 ππ π§π§ 1 ππ1π§π§ ππ 12 ππoππ 12o ππ0π§π§ ππ 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:ππ 1β ππ π»π» ππ ππππ 0ππ2ππππππππsuch thatFoundations of Digital Signal Processingπ»π» ππ π»π» ππ ππLecture 22: Designing FIR / IIR Filters43
Design with Frequency Sampling Option 2: Work backwards with constraintsConsider the DFT:ππ 12ππ1ππ ππππβ ππ π»π» ππ ππ ππππππ 0such thatππ 1 /2ππ 1ππ 1ππ ππ 1 /22ππ1ππ ππ ππππβ ππ π»π» 0 π»π» ππ ππ ππFoundations of Digital Signal Processing π»π» ππ π»π» ππ πππ»π» ππ ππLecture 22: Designing FIR / IIR Filters2ππππ ππ ππππ44
Design with Frequency Sampling Option 2: Work backwards with constraintsConsider the DFT:ππ 12ππ1ππ ππππβ ππ π»π» ππ ππ ππππππ 0such thatππ 1 /2ππ 1 /2ππ 1ππ 1π»π» ππ π»π» ππ ππ2ππ2ππ1ππ ππ ππππππ ππ ππβ ππ π»π» 0 π»π» ππ ππ π»π» ππ ππππFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filtersππ ππ45
Design with Frequency Sampling Option 2: Work backwards with constraintsConsider the DFT:ππ 12ππ1ππ ππππβ ππ π»π» ππ ππ ππππππ 0such thatππ 1 /2ππ 1 /2ππ 1ππ 1π»π» ππ π»π» ππ ππ2ππ2ππ1ππ ππ ππππππ ππ ππβ ππ π»π» 0 π»π» ππ ππ π»π» ππ ππππππ 1 /22ππ2ππ1ππ ππππππ ππβ ππ π»π» 0 π»π» ππ ππ ππ π»π» ππ ππ ππππππ 1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filtersππ ππππ ππ46
Design with Frequency Sampling Option 2: Work backwards with constraintsConsider the DFT:ππ 12ππ1ππ ππππβ ππ π»π» ππ ππ ππππππ 0such thatππ 1 /2ππ 1 /2ππ 1ππ 1π»π» ππ π»π» ππ ππ2ππ2ππ1ππ ππ ππππππ ππ ππβ ππ π»π» 0 π»π» ππ ππ π»π» ππ ππππππ 1 /2ππ ππ2ππ2ππ1ππ ππππ ππ ππππ πππππππβ ππ π»π» 0 π»π» ππ ππ ππ π»π» ππ ππ ππ ππππππ 1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters47
Design with Frequency Sampling Option 2: Work backwards with constraintsConsider the DFT:ππ 12ππ1ππ ππππβ ππ π»π» ππ ππ ππππππ 0such thatππ 1 /2ππ 1 /2ππ 1ππ 1π»π» ππ π»π» ππ ππ2ππ2ππ1ππ ππ ππππππ ππ ππβ ππ π»π» 0 π»π» ππ ππ π»π» ππ ππππππ 1 /2ππ ππ2ππ2ππ1ππ ππ ππππ ππ ππ ππππβ ππ π»π» 0 π»π» ππ ππ π»π» ππ ππππππ 1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters48
Design with Frequency Sampling Option 2: Work backwards with constraintsConsider the DFT:ππ 12ππ1ππ ππππβ ππ π»π» ππ ππ ππππππ 0such thatππ 1 /2ππ 1 /2ππ 1ππ 1π»π» ππ π»π» ππ ππ2ππ2ππ1ππ ππ ππππππ ππ ππβ ππ π»π» 0 π»π» ππ ππ π»π» ππ ππππππ 1 /22ππ2ππ1ππ ππππ ππ ππππβ ππ π»π» 0 π»π» ππ ππ ππ ππ ππππππ 1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filtersππ ππ49
Design with Frequency Sampling Option 2: Work backwards with constraintsConsider the DFT:ππ 12ππ1ππ ππππβ ππ π»π» ππ ππ ππππππ 0such thatππ 1 /2ππ 1 /2ππ 1ππ 1π»π» ππ π»π» ππ ππ2ππ2ππ1ππ ππ ππππππ ππ ππβ ππ π»π» 0 π»π» ππ ππ π»π» ππ ππππππ 1 /212ππβ ππ π»π» 0 2 π»π» ππ cosππππππππππ 1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filtersππ ππ50
Design with Frequency Sampling An inverse DFT that forces time-symmetryππ 1 /212ππβ ππ π»π» 0 2 π»π» ππ cosππππππππππ 1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters51
Design with Frequency Sampling An inverse DFT that forces time-symmetryππ 1 /212ππβ ππ π»π» 0 2 π»π» ππ cosππππππππππ 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-symmetryππ 1 /212ππβ ππ π»π» 0 2 π»π» ππ cosππππππππππ 1 Example: Consider the desired 9-sample frequency responsewith the first half defined by [1 1 0 0] Compute the frequency sampled filter1β ππ 1 2 cos 2ππ/9 ππ2Foundations 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 ππ 1 2 cos 2ππ/9 ππ2Foundations 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 ππ 1 2 cos 2ππ/9 ππ2 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-symmetryππ 1 /212ππππ 1β ππ π»π» 0 2 π»π» ππ cosππ ππππππ2ππ 1 Example: Consider the desired 9-sample frequency responsewith the first half defined by [1 1 0 0] Compute the frequency sampled filter1β ππ 1 2 cos 2ππ/9 ππ2Foundations 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 ππ 1 2 cos 2ππ/9 ππ 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 ππ 1 2 cos 2ππ/19 ππππ 2 cos 4ππ/19 ππππ 2 cos 6ππ/19 ππππ1716ππππ ππ 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 ππ 1 2 cos 2ππ/19 ππππ 2 cos 4ππ/19 ππππ 2 cos 6ππ/19 ππππ1716ππππ ππ 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:ππ 1 /212ππππ 1β ππ π»π» 0 2 π»π» ππ cosππ ππππππ2ππ 1ππ 1 /212ππππ 1 π»π» 0 2 π»π» ππ cosππ ππππππ2 2ππ 1ππ 1 /212ππ1 π»π» 0 2 π»π» ππ cosππ ππ ππππππππ2ππ 1ππ 1 /21 π»π» 0 2 ππππ 1 1ππ π»π»Foundations of Digital Signal Processing2ππ1ππ cosππ ππππ2Lecture 22: Designing FIR / IIR Filters62
Design with Frequency Sampling Final Definitionππ 1 /21β ππ π»π» 0 2 ππππ 1 1ππ π»π»2ππ1ππ cosππ ππππ2Side 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:ππ π§π§ ππ ππ ππ 1 2π§π§ππ/2 1 ππππ π§π§ππ 0ππ 1 ππππ 2ππ 2ππ ππππππ 12ππ/2 1ππ 12 ππ ππππ ππππ 0ππ/2 1ππππ π§π§ ππ 12 ππππ 12 ππ ππππ ππππππ 12 ππππ 1 ππππ cos ππ ππ2ππ 0Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters65
Design with Equi-ripples Equi-ripple designππ ππ ππ 1 ππππ 22ππππ/2 1ππ 1 ππππ cos ππ ππ2ππ 0Goal: Find the optimal ππππ s that satisfies passband / stopbandripple constraints.Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters66
Design with Equi-ripplesEqui-ripple design min ππ ππ π»π»ππ ππ 2ππππππ ππππππ 12ππ/2 1ππ 1 ππππ cos ππ ππ2ππ 0Desired frequencyresponseEquals:πΏπΏ2for ππ in pass bandπΏπΏ11 for ππ in stop bandFoundations of Digital Signal ProcessingπΏπΏ2 stopband rippleπΏπΏ1 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-timeππππ π‘π‘ π π π π π π ππππ 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-timeππππ π‘π‘ π π π π π π ππππ In discrete-timeT 1ππππ π‘π‘π₯π₯ π‘π‘ π₯π₯ π‘π‘ Ξππ limΞππ 0ππππΞππππππ π‘π‘π₯π₯ ππππ π₯π₯ ππππ ππ π₯π₯ ππ π₯π₯ ππ 1 ππππ π‘π‘ ππππππFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters72
IIR Filter Design from Derivatives Question: What is a derivative in discrete-time? In continuous-timeππππ π‘π‘ π π π π π π ππππ In discrete-timeππππ π‘π‘π₯π₯ π‘π‘ π₯π₯ π‘π‘ Ξππ limΞππ 0ππππΞππππππ π‘π‘π₯π₯ ππππ π₯π₯ ππππ ππ1 π₯π₯ ππ π₯π₯ ππ 1 ππππ π‘π‘ ππππππππFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters73
IIR Filter Design from Derivatives Question: What is a derivative in discrete-time? In continuous-timeππππ π‘π‘ π π π π π π ππππ In discrete-timeππππ π‘π‘π₯π₯ π‘π‘ π₯π₯ π‘π‘ Ξππ limΞππ 0ππππΞππππππ π‘π‘π₯π₯ ππππ π₯π₯ ππππ ππ1 π₯π₯ ππ π₯π₯ ππ 1 ππππ π‘π‘ ππππππππππππ π‘π‘1 1 π§π§ 1 ππ π§π§ππππ π‘π‘ ππππ ππFoundations 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-timeππ2π₯π₯ π‘π‘2 π π ππ π π 2πππ‘π‘ In discrete-timeππ2π₯π₯ π‘π‘ππππ π‘π‘ ππππ π‘π‘ οΏ½οΏ½οΏ½πππ π‘π‘π₯π₯ ππππ π₯π₯ ππππ ππ ππππ π‘π‘ ππππππFoundations 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-timeππ2π₯π₯ π‘π‘2 π π ππ π π 2πππ‘π‘ In discrete-timeππ2π₯π₯ π‘π‘ππππ π‘π‘ ππππ π‘π‘ οΏ½οΏ½οΏ½π2π₯π₯ π‘π‘ 2πππ‘π‘π‘π‘ πππππ₯π₯ ππππ π₯π₯ ππππ ππ /ππ π₯π₯ ππππ ππ π₯π₯ ππππ 2ππ /ππ ππFoundations 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-timeππ2π₯π₯ π‘π‘2 π π ππ π π 2πππ‘π‘ In discrete-timeππ2π₯π₯ π‘π‘ππππ π‘π‘ ππππ π‘π‘ οΏ½οΏ½οΏ½π2π₯π₯ π‘π‘ πππ‘π‘ 2π‘π‘ ππππ π₯π₯ ππππ 2π₯π₯ ππππ ππ π₯π₯ ππππ 2πππ₯π₯ ππ 2π₯π₯ ππ 1 π₯π₯ ππ 2 ππ2ππ2Foundations 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-timeππ2π₯π₯ π‘π‘2 π π ππ π π 2πππ‘π‘ In discrete-timeππ2π₯π₯ π‘π‘ππππ π‘π‘ ππππ π‘π‘ οΏ½οΏ½οΏ½π2π₯π₯ π‘π‘ πππ‘π‘ 2π‘π‘ ππππ π₯π₯ ππππ 2π₯π₯ ππππ ππ π₯π₯ ππππ 2πππ₯π₯ ππ 2π₯π₯ ππ 1 π₯π₯ ππ 2 ππ2ππ2ππππ π‘π‘11 1 2 2 1 2π§π§ π§π§ ππ π§π§ 2 1 π§π§ 1 2ππ π§π§ππππ π‘π‘ ππππ ππππFoundations 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-timeππππ π₯π₯ π‘π‘ππ ππ π π π π πππ‘π‘ππππππ π₯π₯ π‘π‘1 1 ππ ππ π§π§ 1 π§π§ πππ‘π‘πππππ‘π‘ ππππFoundations 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-timeππππ π₯π₯ π‘π‘ππ ππ π π π π πππ‘π‘ππππππ π₯π₯ π‘π‘1 1 ππ ππ π§π§ 1 π§π§ πππ‘π‘πππππ‘π‘ ππππ1π π 1 π§π§ 1ππFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters80
IIR Filter Design from Derivatives Example: π π 1ππ1 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 1ππ1 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 0.1ππ1 π§π§ 1ππ2 0.1ππ22 9 9ππ2ππ2 Foundations of Digital Signal Processingππ211 π§π§ 1 0.1ππ1 0.1ππππ22 1 π§π§Lecture 22: Designing FIR / IIR Filters2 9 9ππ282
IIR Filter Design from Derivatives Example: π π 1ππ1 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§ ππ2Finding poles1 0.1ππ π§π§ 11 0.1ππ1 0.1ππ2 1 π§π§2 1 π§π§2 9ππ2 9ππ2 0 9ππ21 0.1ππ π§π§ 1 3πππππ§π§ 1 1 0.1ππ 3ππππFoundations of Digital Signal Processing1π§π§ 1 0.1 3ππ ππLecture 22: Designing FIR / IIR Filters83
IIR Filter Design from Derivatives Example: π π 1ππ1 π§π§ 1 Use the derivative conversion to transform the following bi-quadfilter into the discrete-time domain.1π§π§ Poles1 0.1 3ππ ππFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters84
IIR Filter Design from Derivatives Example: π π 1ππ1 π§π§ 11π§π§ 1 0.1 3ππ ππFoundations 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§ 1ππPros: 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?πΎπΎ1π»π» π π π π ππππππ 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?πΎπΎπΎπΎππ 1ππ 11π»π» π π ππππ πππππππ‘π‘π π ππππFoundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters90
IIR Filter Design by Impulse Invariance Question: How else can I represent my transfer function?πΎπΎπΎπΎππ 1ππ 11π»π» π π ππππ πππππππ‘π‘π π πππππΎπΎβ π‘π‘ ππππ ππ πππππ‘π‘ππ 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?πΎπΎπΎπΎππ 1ππ 11π»π» π π ππππ πππππππ‘π‘π π πππππΎπΎβ π‘π‘ ππππ ππ πππππ‘π‘ππ 1πΎπΎπΎπΎππ 1ππ 1β ππππ β ππ ππππ ππ ππππππππ ππππ ππ ππππππFoundations of Digital Signal ProcessingππLecture 22: Designing FIR / IIR Filters92
IIR Filter Design by Impulse Invariance Question: How else can I represent my transfer function?πΎπΎπΎπΎππ 1ππ 11π»π» π π ππππ πππππππ‘π‘π π πππππΎπΎβ π‘π‘ ππππ ππ πππππ‘π‘ππ 1πΎπΎπΎπΎππ 1ππ 1β ππππ β ππ ππππ ππ ππππππππ ππππ ππ οΏ½οΏ½π» π§π§ 1 ππ ππππππ π§π§ 1ππ 1Foundations of Digital Signal ProcessingππLecture 22: Designing FIR / IIR Filters93
IIR Filter Design from Derivatives Example: π»π» π§π§ πππππΎπΎ ππ 11 ππ πππππππ§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§ πππππΎπΎ ππ 11 ππ πππππππ§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 3ππ 0.1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters95
IIR Filter Design from Derivatives Example: π»π» π§π§ πππππΎπΎ ππ 11 ππ πππππππ§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 3ππ 0.1 π π 3ππ 0.1Poles:π π 0.12 9 0π π 3ππ 0.1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters96
IIR Filter Design from Derivatives Example: π»π» π§π§ πππππΎπΎ ππ 11 ππ πππππππ§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 3ππ 0.1 π π 3ππ 0.11/21/2π»π» π§π§ 3ππ 0.1ππ 11 πππ§π§1 ππ 3ππ 0.1 ππ 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 ππ1 πππππππΎπΎ ππ 11 ππ πππππππ§π§ 11/2 3ππ 0.1 ππ π§π§ 11/23ππ 0.1 ππ 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 .
Introduction of Chemical Reaction Engineering Introduction about Chemical Engineering 0:31:15 0:31:09. Lecture 14 Lecture 15 Lecture 16 Lecture 17 Lecture 18 Lecture 19 Lecture 20 Lecture 21 Lecture 22 Lecture 23 Lecture 24 Lecture 25 Lecture 26 Lecture 27 Lecture 28 Lecture
Design of FIR Filters using 1 Rectangular window 2 Hamming window 3 Hanning window 4 Bartlet window 5 Kaiser window Design of FIR Filter using frequency sampling technique. Dr. Manjunatha. P (JNNCE) UNIT - 7: FIR Filter Design October 25, 2016 3 / 94. FIR Filter Design Introduction
EEL3135: Discrete-Time Signals and Systems FIR Filter Design: Part I - 3 - 3. Simple high-pass FIR ο¬lters 1 A. Introduction Previously (1/16 lecture notes), we had considered the following high-pass FIR ο¬lters: 1. You may want to review the 1/16 lecture notes to complement the materials i
response (FIR) or infinite impulse response (IIR) approaches. In this chapter we are concerned with just FIR designs. We will start with an overview of general digital filter design, but the emphasis of this chapter will be on real-time implementation of FIR filters using C and assembly. Basic FIR filter topologies will be reviewed
In contrast to IIR filters, FIR filters have a linear phase and inherent stability. This benefit makes FIR filters attr active enough to be designed into a large number of systems. However, for a given frequency response, FIR filters are a higher order than IIR filters, making FIR filters more computationally expensive. Figure 1 1.
FIR IIR Filter Design software is intended to design IIR and FIR digital filters in a simple and fast way, in a user friendly environment. The filter type can be LP, HP, BP, and BS. . (see previous chapter) FIR IIR Filter Design counts filter graphs from the user defined filter coefficients. Some applications of this function: determining .
The transfer function of a causal FIR filter is obtained by taking the z-transform of impulse response of FIR filtersh (n). There are many straightforward techniques for designing FIR digital filters to meet required frequency and phase response specifications, such as window design method or frequency sampling techniques.
Textbook of Algae , O. P. Sharma, Jan 1, 1986, Algae, 396 pages. Aimed to meet requirements of undergraduate students of botany. This book covers topics such as: evolution of sex and sexuality in algae; and, pigments in algae with their. An Introduction to Phycology , G. R. South, A. Whittick, Jul 8, 2009, Science, 352 pages. This text presents the subject using a systems approach and is .
