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 – 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‘₯ 𝑛𝑛 π‘₯π‘₯ 𝑛𝑛 𝑁𝑁 1 Positive: Even symmetry Negative: Odd symmetry π‘₯π‘₯ 𝑁𝑁 1 𝑛𝑛Foundations 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 Symmetry𝑋𝑋 𝑧𝑧 π‘Žπ‘Ž0 π‘Žπ‘Ž1𝑧𝑧 1 π‘Žπ‘Ž2𝑧𝑧 2 π‘Žπ‘Ž1𝑧𝑧 𝑀𝑀 2 π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 1 Odd Symmetry𝑋𝑋 𝑧𝑧 π‘Žπ‘Ž0 π‘Žπ‘Ž1𝑧𝑧 1 π‘Žπ‘Ž2𝑧𝑧 2 π‘Žπ‘Ž1𝑧𝑧 𝑀𝑀 2 π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 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 Symmetry𝑋𝑋 𝑧𝑧 π‘Žπ‘Ž0 π‘Žπ‘Ž1𝑧𝑧 1 π‘Žπ‘Ž2𝑧𝑧 2 π‘Žπ‘Ž1𝑧𝑧 𝑀𝑀 2 π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 1 𝑧𝑧 𝑀𝑀 12π‘Žπ‘Ž0𝑧𝑧𝑀𝑀 12 π‘Žπ‘Ž1𝑧𝑧𝑀𝑀 1 12 π‘Žπ‘Ž1𝑧𝑧1 𝑀𝑀 12 π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 12Odd Symmetry𝑋𝑋 𝑧𝑧 π‘Žπ‘Ž0 π‘Žπ‘Ž1𝑧𝑧 1 π‘Žπ‘Ž2𝑧𝑧 2 π‘Žπ‘Ž1𝑧𝑧 𝑀𝑀 2 π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 1 𝑧𝑧 𝑀𝑀 12π‘Žπ‘Ž0𝑧𝑧𝑀𝑀 12 π‘Žπ‘Ž1𝑧𝑧𝑀𝑀 12 11 π‘Žπ‘Ž1𝑧𝑧Foundations of Digital Signal Processing𝑀𝑀 12 π‘Žπ‘Ž0𝑧𝑧 Lecture 22: Designing FIR / IIR Filters𝑀𝑀 129

Causality and Linear PhaseQuestion: Consider a length-M symmetric, causal filter.What is the phase response? Assume M is even. Even Symmetry𝑋𝑋 𝑧𝑧 π‘Žπ‘Ž0 π‘Žπ‘Ž1𝑧𝑧 1 π‘Žπ‘Ž2𝑧𝑧 2 π‘Žπ‘Ž1𝑧𝑧 𝑀𝑀 2 π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 1 𝑧𝑧 𝑀𝑀 12π‘Žπ‘Ž0𝑧𝑧𝑀𝑀 12 π‘Žπ‘Ž1𝑧𝑧𝑀𝑀 1 12 π‘Žπ‘Ž1𝑧𝑧1 𝐺𝐺 πœ”πœ” 𝑋𝑋 πœ”πœ” 𝑒𝑒 jΘ πœ”πœ”π‘€π‘€ 12 π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 12Odd Symmetry𝑋𝑋 𝑧𝑧 π‘Žπ‘Ž0 π‘Žπ‘Ž1𝑧𝑧 1 π‘Žπ‘Ž2𝑧𝑧 2 π‘Žπ‘Ž1𝑧𝑧 𝑀𝑀 2 π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 1 𝑧𝑧 𝑀𝑀 12π‘Žπ‘Ž0𝑧𝑧𝑀𝑀 12 π‘Žπ‘Ž1𝑧𝑧𝑀𝑀 12 11 π‘Žπ‘Ž1𝑧𝑧𝑀𝑀 12𝐺𝐺 πœ”πœ” 𝑋𝑋 πœ”πœ” 𝑒𝑒 jΘ πœ”πœ”Foundations of Digital Signal Processing π‘Žπ‘Ž0𝑧𝑧 Lecture 22: Designing FIR / IIR Filters𝑀𝑀 1210

Causality and Linear PhaseQuestion: Consider a length-M symmetric, causal filter.What is the phase response? Assume M is even. Even Symmetry𝑋𝑋 𝑧𝑧 π‘Žπ‘Ž0 π‘Žπ‘Ž1𝑧𝑧 1 π‘Žπ‘Ž2𝑧𝑧 2 π‘Žπ‘Ž1𝑧𝑧 𝑀𝑀 2 π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 1 𝑧𝑧 𝑀𝑀 12π‘Žπ‘Ž0𝑧𝑧𝑀𝑀 12 π‘Žπ‘Ž1𝑧𝑧𝑀𝑀 1 12𝐺𝐺 πœ”πœ” 𝑋𝑋 πœ”πœ” 𝑒𝑒 jΘ πœ”πœ” π‘Žπ‘Ž1𝑧𝑧1 𝑀𝑀 120Θ πœ”πœ” πœ‹πœ‹ π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 12for 𝐺𝐺 πœ”πœ” 0for 𝐺𝐺 πœ”πœ” 0Odd Symmetry𝑋𝑋 𝑧𝑧 π‘Žπ‘Ž0 π‘Žπ‘Ž1𝑧𝑧 1 π‘Žπ‘Ž2𝑧𝑧 2 π‘Žπ‘Ž1𝑧𝑧 𝑀𝑀 2 π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 1 𝑧𝑧 𝑀𝑀 12π‘Žπ‘Ž0𝑧𝑧𝑀𝑀 12 π‘Žπ‘Ž1𝑧𝑧𝑀𝑀 12 1𝐺𝐺 πœ”πœ” 𝑋𝑋 πœ”πœ” 𝑒𝑒 jΘ πœ”πœ”1 π‘Žπ‘Ž1𝑧𝑧𝑀𝑀 120Θ πœ”πœ” πœ‹πœ‹Foundations of Digital Signal Processing π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 12for 𝐺𝐺 πœ”πœ” 0for 𝐺𝐺 πœ”πœ” 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 Symmetry𝑋𝑋 𝑧𝑧 π‘Žπ‘Ž0 π‘Žπ‘Ž1𝑧𝑧 1 π‘Žπ‘Ž2𝑧𝑧 2 π‘Žπ‘Ž1𝑧𝑧 𝑀𝑀 2 π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 1 𝑧𝑧 𝑀𝑀 12π‘Žπ‘Ž0𝑧𝑧𝑀𝑀 12 π‘Žπ‘Ž1𝑧𝑧𝑀𝑀 1 12 π‘Žπ‘Ž1𝑧𝑧 πœ”πœ” 𝑀𝑀 1 /2 𝑋𝑋 πœ”πœ” πœ”πœ” 𝑀𝑀 1 /2 πœ‹πœ‹1 𝑀𝑀 12 π‘Žπ‘Ž0𝑧𝑧for 𝐺𝐺 πœ”πœ” 0for 𝐺𝐺 πœ”πœ” 0 𝑀𝑀 12Odd Symmetry𝑋𝑋 𝑧𝑧 π‘Žπ‘Ž0 π‘Žπ‘Ž1𝑧𝑧 1 π‘Žπ‘Ž2𝑧𝑧 2 π‘Žπ‘Ž1𝑧𝑧 𝑀𝑀 2 π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 1 𝑧𝑧 𝑀𝑀 12π‘Žπ‘Ž0𝑧𝑧𝑀𝑀 12 π‘Žπ‘Ž1𝑧𝑧𝑀𝑀 12 11 π‘Žπ‘Ž1𝑧𝑧 πœ”πœ” 𝑀𝑀 1 /2 𝑋𝑋 πœ”πœ” πœ”πœ” 𝑀𝑀 1 /2 πœ‹πœ‹Foundations of Digital Signal Processing𝑀𝑀 12 π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 12for 𝐺𝐺 πœ”πœ” 0for 𝐺𝐺 πœ”πœ” 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 Symmetry𝑋𝑋 𝑧𝑧 π‘Žπ‘Ž0 π‘Žπ‘Ž1𝑧𝑧 1 π‘Žπ‘Ž2𝑧𝑧 2 π‘Žπ‘Ž1𝑧𝑧 𝑀𝑀 2 π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 1 𝑧𝑧 𝑀𝑀 12π‘Žπ‘Ž0𝑧𝑧𝑀𝑀 12 π‘Žπ‘Ž1𝑧𝑧𝑀𝑀 1 12 π‘Žπ‘Ž1𝑧𝑧Foundations of Digital Signal Processing1 𝑀𝑀 12 π‘Žπ‘Ž0𝑧𝑧Lecture 22: Designing FIR / IIR Filters 𝑀𝑀 1213

Causality and Linear PhaseQuestion: Consider a length-M symmetric, causal filter.What is the phase response? Assume M is even. Even Symmetry𝑋𝑋 𝑧𝑧 π‘Žπ‘Ž0 π‘Žπ‘Ž1𝑧𝑧 1 π‘Žπ‘Ž2𝑧𝑧 2 π‘Žπ‘Ž1𝑧𝑧 𝑀𝑀 2 π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 1 𝑧𝑧 𝑧𝑧 𝑀𝑀 12𝑀𝑀 12𝑀𝑀 1π‘Žπ‘Ž0𝑧𝑧 2𝑀𝑀/2 1 π‘Žπ‘Ž1𝑧𝑧 π‘Žπ‘Žπ‘˜π‘˜ π‘§π‘§π‘˜π‘˜ 0𝑀𝑀 1 12𝑀𝑀 12 π‘˜π‘˜ 𝑧𝑧 π‘Žπ‘Ž1𝑧𝑧 Foundations of Digital Signal Processing𝑀𝑀 12 π‘˜π‘˜1 𝑀𝑀 12 π‘Žπ‘Ž0𝑧𝑧Lecture 22: Designing FIR / IIR Filters 𝑀𝑀 1214

Causality and Linear PhaseQuestion: Consider a length-M symmetric, causal filter.What is the phase response? Assume M is even. Even Symmetry𝑋𝑋 𝑧𝑧 π‘Žπ‘Ž0 π‘Žπ‘Ž1𝑧𝑧 1 π‘Žπ‘Ž2𝑧𝑧 2 π‘Žπ‘Ž1𝑧𝑧 𝑀𝑀 2 π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 1 𝑧𝑧 𝑧𝑧 𝑀𝑀 12𝑀𝑀 12𝑀𝑀 1π‘Žπ‘Ž0𝑧𝑧 2𝑀𝑀/2 1 π‘Žπ‘Ž1𝑧𝑧 π‘Žπ‘Žπ‘˜π‘˜ π‘§π‘§π‘˜π‘˜ 0𝑀𝑀 1 12𝑀𝑀 12 π‘˜π‘˜Notice that𝑋𝑋 𝑧𝑧 𝑧𝑧 𝑀𝑀 1 𝑋𝑋 𝑧𝑧 1 𝑧𝑧 π‘Žπ‘Ž1𝑧𝑧 𝑀𝑀 12 π‘˜π‘˜1 𝑀𝑀 12 π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 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 Symmetry𝑋𝑋 𝑧𝑧 π‘Žπ‘Ž0 π‘Žπ‘Ž1𝑧𝑧 1 π‘Žπ‘Ž2𝑧𝑧 2 π‘Žπ‘Ž1𝑧𝑧 𝑀𝑀 2 π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 1 𝑧𝑧 𝑧𝑧 𝑀𝑀 12𝑀𝑀 12𝑀𝑀 1π‘Žπ‘Ž0𝑧𝑧 2𝑀𝑀/2 1 π‘Žπ‘Ž1𝑧𝑧 π‘Žπ‘Žπ‘˜π‘˜ π‘§π‘§π‘˜π‘˜ 0𝑀𝑀 1 12𝑀𝑀 12 π‘˜π‘˜ 𝑧𝑧Pole-zero plot property?𝑋𝑋 𝑧𝑧 𝑧𝑧 𝑀𝑀 1 𝑋𝑋 𝑧𝑧 1 π‘Žπ‘Ž1𝑧𝑧 𝑀𝑀 12 π‘˜π‘˜1 𝑀𝑀 12 π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 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 Symmetry𝑋𝑋 𝑧𝑧 π‘Žπ‘Ž0 π‘Žπ‘Ž1𝑧𝑧 1 π‘Žπ‘Ž2𝑧𝑧 2 π‘Žπ‘Ž1𝑧𝑧 𝑀𝑀 2 π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 1 𝑧𝑧 𝑧𝑧 𝑀𝑀 12𝑀𝑀 12𝑀𝑀 1π‘Žπ‘Ž0𝑧𝑧 2𝑀𝑀/2 1 π‘Žπ‘Ž1𝑧𝑧 π‘Žπ‘Žπ‘˜π‘˜ π‘§π‘§π‘˜π‘˜ 0𝑀𝑀 1 12𝑀𝑀 12 π‘˜π‘˜ 𝑧𝑧Pole-zero plot property?𝑋𝑋 𝑧𝑧 𝑧𝑧 𝑀𝑀 1 𝑋𝑋 𝑧𝑧 1 π‘Žπ‘Ž1𝑧𝑧 𝑀𝑀 12 π‘˜π‘˜o𝑀𝑀 12o π‘Žπ‘Ž0𝑧𝑧 𝑀𝑀 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:𝑁𝑁 1β„Ž 𝑛𝑛 𝐻𝐻 π‘˜π‘˜ π‘’π‘’π‘˜π‘˜ 0𝑗𝑗2πœ‹πœ‹π‘›π‘›π‘›π‘›π‘π‘such thatFoundations of Digital Signal Processing𝐻𝐻 π‘˜π‘˜ 𝐻𝐻 𝑁𝑁 π‘˜π‘˜Lecture 22: Designing FIR / IIR Filters43

Design with Frequency Sampling Option 2: Work backwards with constraintsConsider the DFT:𝑁𝑁 12πœ‹πœ‹1𝑗𝑗 π‘›π‘›π‘›π‘›β„Ž 𝑛𝑛 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒 π‘π‘π‘π‘π‘˜π‘˜ 0such that𝑁𝑁 1 /2𝑁𝑁 1π‘˜π‘˜ 1π‘˜π‘˜ 𝑁𝑁 1 /22πœ‹πœ‹1𝑗𝑗 𝑁𝑁 π‘›π‘›π‘›π‘›β„Ž 𝑛𝑛 𝐻𝐻 0 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒 𝑁𝑁Foundations of Digital Signal Processing 𝐻𝐻 π‘˜π‘˜ 𝐻𝐻 𝑁𝑁 π‘˜π‘˜π»π» π‘˜π‘˜ 𝑒𝑒Lecture 22: Designing FIR / IIR Filters2πœ‹πœ‹π‘—π‘— 𝑁𝑁 𝑛𝑛𝑛𝑛44

Design with Frequency Sampling Option 2: Work backwards with constraintsConsider the DFT:𝑁𝑁 12πœ‹πœ‹1𝑗𝑗 π‘›π‘›π‘›π‘›β„Ž 𝑛𝑛 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒 π‘π‘π‘π‘π‘˜π‘˜ 0such that𝑁𝑁 1 /2𝑁𝑁 1 /2π‘˜π‘˜ 1π‘˜π‘˜ 1𝐻𝐻 π‘˜π‘˜ 𝐻𝐻 𝑁𝑁 π‘˜π‘˜2πœ‹πœ‹2πœ‹πœ‹1𝑗𝑗 𝑁𝑁 𝑛𝑛𝑛𝑛𝑗𝑗 𝑁𝑁 π‘›π‘›β„Ž 𝑛𝑛 𝐻𝐻 0 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒𝑁𝑁Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters𝑁𝑁 π‘˜π‘˜45

Design with Frequency Sampling Option 2: Work backwards with constraintsConsider the DFT:𝑁𝑁 12πœ‹πœ‹1𝑗𝑗 π‘›π‘›π‘›π‘›β„Ž 𝑛𝑛 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒 π‘π‘π‘π‘π‘˜π‘˜ 0such that𝑁𝑁 1 /2𝑁𝑁 1 /2π‘˜π‘˜ 1π‘˜π‘˜ 1𝐻𝐻 π‘˜π‘˜ 𝐻𝐻 𝑁𝑁 π‘˜π‘˜2πœ‹πœ‹2πœ‹πœ‹1𝑗𝑗 𝑁𝑁 𝑛𝑛𝑛𝑛𝑗𝑗 𝑁𝑁 π‘›π‘›β„Ž 𝑛𝑛 𝐻𝐻 0 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒𝑁𝑁𝑁𝑁 1 /22πœ‹πœ‹2πœ‹πœ‹1𝑗𝑗 𝑛𝑛𝑛𝑛𝑗𝑗 π‘›π‘›β„Ž 𝑛𝑛 𝐻𝐻 0 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒 𝑁𝑁 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒 π‘π‘π‘π‘π‘˜π‘˜ 1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters𝑁𝑁 π‘˜π‘˜π‘π‘ π‘˜π‘˜46

Design with Frequency Sampling Option 2: Work backwards with constraintsConsider the DFT:𝑁𝑁 12πœ‹πœ‹1𝑗𝑗 π‘›π‘›π‘›π‘›β„Ž 𝑛𝑛 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒 π‘π‘π‘π‘π‘˜π‘˜ 0such that𝑁𝑁 1 /2𝑁𝑁 1 /2π‘˜π‘˜ 1π‘˜π‘˜ 1𝐻𝐻 π‘˜π‘˜ 𝐻𝐻 𝑁𝑁 π‘˜π‘˜2πœ‹πœ‹2πœ‹πœ‹1𝑗𝑗 𝑁𝑁 𝑛𝑛𝑛𝑛𝑗𝑗 𝑁𝑁 π‘›π‘›β„Ž 𝑛𝑛 𝐻𝐻 0 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒𝑁𝑁𝑁𝑁 1 /2𝑁𝑁 π‘˜π‘˜2πœ‹πœ‹2πœ‹πœ‹1𝑗𝑗 𝑛𝑛𝑛𝑛 𝑗𝑗 𝑛𝑛𝑛𝑛 π‘—π‘—π‘—π‘—π‘—π‘—π‘—β„Ž 𝑛𝑛 𝐻𝐻 0 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒 𝑁𝑁 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒 𝑁𝑁 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:𝑁𝑁 12πœ‹πœ‹1𝑗𝑗 π‘›π‘›π‘›π‘›β„Ž 𝑛𝑛 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒 π‘π‘π‘π‘π‘˜π‘˜ 0such that𝑁𝑁 1 /2𝑁𝑁 1 /2π‘˜π‘˜ 1π‘˜π‘˜ 1𝐻𝐻 π‘˜π‘˜ 𝐻𝐻 𝑁𝑁 π‘˜π‘˜2πœ‹πœ‹2πœ‹πœ‹1𝑗𝑗 𝑁𝑁 𝑛𝑛𝑛𝑛𝑗𝑗 𝑁𝑁 π‘›π‘›β„Ž 𝑛𝑛 𝐻𝐻 0 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒𝑁𝑁𝑁𝑁 1 /2𝑁𝑁 π‘˜π‘˜2πœ‹πœ‹2πœ‹πœ‹1𝑗𝑗 𝑁𝑁 𝑛𝑛𝑛𝑛 𝑗𝑗 𝑁𝑁 π‘›π‘›π‘›π‘›β„Ž 𝑛𝑛 𝐻𝐻 0 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒 𝐻𝐻 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:𝑁𝑁 12πœ‹πœ‹1𝑗𝑗 π‘›π‘›π‘›π‘›β„Ž 𝑛𝑛 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒 π‘π‘π‘π‘π‘˜π‘˜ 0such that𝑁𝑁 1 /2𝑁𝑁 1 /2π‘˜π‘˜ 1π‘˜π‘˜ 1𝐻𝐻 π‘˜π‘˜ 𝐻𝐻 𝑁𝑁 π‘˜π‘˜2πœ‹πœ‹2πœ‹πœ‹1𝑗𝑗 𝑁𝑁 𝑛𝑛𝑛𝑛𝑗𝑗 𝑁𝑁 π‘›π‘›β„Ž 𝑛𝑛 𝐻𝐻 0 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒𝑁𝑁𝑁𝑁 1 /22πœ‹πœ‹2πœ‹πœ‹1𝑗𝑗 𝑛𝑛𝑛𝑛 𝑗𝑗 π‘›π‘›π‘›π‘›β„Ž 𝑛𝑛 𝐻𝐻 0 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒 𝑁𝑁 𝑒𝑒 π‘π‘π‘π‘π‘˜π‘˜ 1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters𝑁𝑁 π‘˜π‘˜49

Design with Frequency Sampling Option 2: Work backwards with constraintsConsider the DFT:𝑁𝑁 12πœ‹πœ‹1𝑗𝑗 π‘›π‘›π‘›π‘›β„Ž 𝑛𝑛 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒 π‘π‘π‘π‘π‘˜π‘˜ 0such that𝑁𝑁 1 /2𝑁𝑁 1 /2π‘˜π‘˜ 1π‘˜π‘˜ 1𝐻𝐻 π‘˜π‘˜ 𝐻𝐻 𝑁𝑁 π‘˜π‘˜2πœ‹πœ‹2πœ‹πœ‹1𝑗𝑗 𝑁𝑁 𝑛𝑛𝑛𝑛𝑗𝑗 𝑁𝑁 π‘›π‘›β„Ž 𝑛𝑛 𝐻𝐻 0 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒 𝐻𝐻 π‘˜π‘˜ 𝑒𝑒𝑁𝑁𝑁𝑁 1 /212πœ‹πœ‹β„Ž 𝑛𝑛 𝐻𝐻 0 2 𝐻𝐻 π‘˜π‘˜ cosπ‘›π‘›π‘›π‘›π‘π‘π‘π‘π‘˜π‘˜ 1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters𝑁𝑁 π‘˜π‘˜50

Design with Frequency Sampling An inverse DFT that forces time-symmetry𝑁𝑁 1 /212πœ‹πœ‹β„Ž 𝑛𝑛 𝐻𝐻 0 2 𝐻𝐻 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-symmetry𝑁𝑁 1 /212πœ‹πœ‹β„Ž 𝑛𝑛 𝐻𝐻 0 2 𝐻𝐻 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-symmetry𝑁𝑁 1 /212πœ‹πœ‹β„Ž 𝑛𝑛 𝐻𝐻 0 2 𝐻𝐻 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„Ž 𝑛𝑛 1 2 cos 2πœ‹πœ‹/9 𝑛𝑛2Foundations 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„Ž 𝑛𝑛 1 2 cos 2πœ‹πœ‹/9 𝑛𝑛2Foundations 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„Ž 𝑛𝑛 1 2 cos 2πœ‹πœ‹/9 𝑛𝑛2 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-symmetry𝑁𝑁 1 /212πœ‹πœ‹π‘π‘ 1β„Ž 𝑛𝑛 𝐻𝐻 0 2 𝐻𝐻 π‘˜π‘˜ cos𝑛𝑛 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„Ž 𝑛𝑛 1 2 cos 2πœ‹πœ‹/9 𝑛𝑛2Foundations 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„Ž 𝑛𝑛 1 2 cos 2πœ‹πœ‹/9 𝑛𝑛 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„Ž 𝑛𝑛 1 2 cos 2πœ‹πœ‹/19 𝑛𝑛𝑐𝑐 2 cos 4πœ‹πœ‹/19 𝑛𝑛𝑐𝑐 2 cos 6πœ‹πœ‹/19 𝑛𝑛𝑐𝑐1716𝑛𝑛𝑐𝑐 𝑛𝑛 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„Ž 𝑛𝑛 1 2 cos 2πœ‹πœ‹/19 𝑛𝑛𝑐𝑐 2 cos 4πœ‹πœ‹/19 𝑛𝑛𝑐𝑐 2 cos 6πœ‹πœ‹/19 𝑛𝑛𝑐𝑐1716𝑛𝑛𝑐𝑐 𝑛𝑛 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:𝑁𝑁 1 /212πœ‹πœ‹π‘π‘ 1β„Ž 𝑛𝑛 𝐻𝐻 0 2 𝐻𝐻 π‘˜π‘˜ cos𝑛𝑛 π‘˜π‘˜π‘π‘π‘π‘2π‘˜π‘˜ 1𝑁𝑁 1 /212πœ‹πœ‹π‘π‘ 1 𝐻𝐻 0 2 𝐻𝐻 π‘˜π‘˜ cos𝑛𝑛 π‘˜π‘˜π‘π‘π‘π‘2 2π‘˜π‘˜ 1𝑁𝑁 1 /212πœ‹πœ‹1 𝐻𝐻 0 2 𝐻𝐻 π‘˜π‘˜ cos𝑛𝑛 π‘˜π‘˜ πœ‹πœ‹πœ‹πœ‹π‘π‘π‘π‘2π‘˜π‘˜ 1𝑁𝑁 1 /21 𝐻𝐻 0 2 π‘π‘π‘˜π‘˜ 1 1π‘˜π‘˜ 𝐻𝐻Foundations of Digital Signal Processing2πœ‹πœ‹1π‘˜π‘˜ cos𝑛𝑛 π‘˜π‘˜π‘π‘2Lecture 22: Designing FIR / IIR Filters62

Design with Frequency Sampling Final Definition𝑁𝑁 1 /21β„Ž 𝑛𝑛 𝐻𝐻 0 2 π‘π‘π‘˜π‘˜ 1 1π‘˜π‘˜ 𝐻𝐻2πœ‹πœ‹1π‘˜π‘˜ cos𝑛𝑛 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:𝑋𝑋 𝑧𝑧 𝑋𝑋 πœ”πœ” 𝑀𝑀 1 2𝑧𝑧𝑀𝑀/2 1 π‘Žπ‘Žπ‘˜π‘˜ π‘§π‘§π‘˜π‘˜ 0𝑀𝑀 1 𝑗𝑗𝑗𝑗 2𝑒𝑒 2𝑒𝑒 𝑗𝑗𝑗𝑗𝑀𝑀 12𝑀𝑀/2 1𝑀𝑀 12 π‘˜π‘˜ π‘Žπ‘Žπ‘˜π‘˜ π‘’π‘’π‘˜π‘˜ 0𝑀𝑀/2 1𝑗𝑗𝑗𝑗 𝑧𝑧 𝑀𝑀 12 π‘˜π‘˜π‘€π‘€ 12 π‘˜π‘˜ 𝑗𝑗𝑗𝑗 𝑗𝑗𝑗𝑗𝑀𝑀 12 π‘˜π‘˜π‘€π‘€ 1 π‘Žπ‘Žπ‘˜π‘˜ cos πœ”πœ” π‘˜π‘˜2π‘˜π‘˜ 0Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters65

Design with Equi-ripples Equi-ripple design𝑋𝑋 πœ”πœ” 𝑀𝑀 1 𝑗𝑗𝑗𝑗 22𝑒𝑒𝑀𝑀/2 1𝑀𝑀 1 π‘Žπ‘Žπ‘˜π‘˜ cos πœ”πœ” π‘˜π‘˜2π‘˜π‘˜ 0Goal: Find the optimal π‘Žπ‘Žπ‘˜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‘Šπ‘Š πœ”πœ” 𝐻𝐻𝑑𝑑 πœ”πœ” 2π‘’π‘’π‘Žπ‘Žπ‘˜π‘˜ 𝑗𝑗𝑗𝑗𝑀𝑀 12𝑀𝑀/2 1𝑀𝑀 1 π‘Žπ‘Žπ‘˜π‘˜ cos πœ”πœ” π‘˜π‘˜2π‘˜π‘˜ 0Desired frequencyresponseEquals:𝛿𝛿2for πœ”πœ” in pass band𝛿𝛿11 for πœ”πœ” in stop bandFoundations of Digital Signal Processing𝛿𝛿2 stopband ripple𝛿𝛿1 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-time𝑑𝑑𝑑𝑑 𝑑𝑑 𝑠𝑠𝑠𝑠 𝑠𝑠𝑑𝑑𝑑𝑑 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-time𝑑𝑑𝑑𝑑 𝑑𝑑 𝑠𝑠𝑠𝑠 𝑠𝑠𝑑𝑑𝑑𝑑 In discrete-timeT 1𝑑𝑑𝑑𝑑 𝑑𝑑π‘₯π‘₯ 𝑑𝑑 π‘₯π‘₯ 𝑑𝑑 Δ𝑇𝑇 limΔ𝑇𝑇 0𝑑𝑑𝑑𝑑Δ𝑇𝑇𝑑𝑑𝑑𝑑 𝑑𝑑π‘₯π‘₯ 𝑛𝑛𝑛𝑛 π‘₯π‘₯ 𝑛𝑛𝑛𝑛 𝑇𝑇 π‘₯π‘₯ 𝑛𝑛 π‘₯π‘₯ 𝑛𝑛 1 𝑑𝑑𝑑𝑑 𝑑𝑑 𝑛𝑛𝑛𝑛𝑇𝑇Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters72

IIR Filter Design from Derivatives Question: What is a derivative in discrete-time? In continuous-time𝑑𝑑𝑑𝑑 𝑑𝑑 𝑠𝑠𝑠𝑠 𝑠𝑠𝑑𝑑𝑑𝑑 In discrete-time𝑑𝑑𝑑𝑑 𝑑𝑑π‘₯π‘₯ 𝑑𝑑 π‘₯π‘₯ 𝑑𝑑 Δ𝑇𝑇 limΔ𝑇𝑇 0𝑑𝑑𝑑𝑑Δ𝑇𝑇𝑑𝑑𝑑𝑑 𝑑𝑑π‘₯π‘₯ 𝑛𝑛𝑛𝑛 π‘₯π‘₯ 𝑛𝑛𝑛𝑛 𝑇𝑇1 π‘₯π‘₯ 𝑛𝑛 π‘₯π‘₯ 𝑛𝑛 1 𝑑𝑑𝑑𝑑 𝑑𝑑 𝑛𝑛𝑛𝑛𝑇𝑇𝑇𝑇Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters73

IIR Filter Design from Derivatives Question: What is a derivative in discrete-time? In continuous-time𝑑𝑑𝑑𝑑 𝑑𝑑 𝑠𝑠𝑠𝑠 𝑠𝑠𝑑𝑑𝑑𝑑 In discrete-time𝑑𝑑𝑑𝑑 𝑑𝑑π‘₯π‘₯ 𝑑𝑑 π‘₯π‘₯ 𝑑𝑑 Δ𝑇𝑇 limΔ𝑇𝑇 0𝑑𝑑𝑑𝑑Δ𝑇𝑇𝑑𝑑𝑑𝑑 𝑑𝑑π‘₯π‘₯ 𝑛𝑛𝑛𝑛 π‘₯π‘₯ 𝑛𝑛𝑛𝑛 𝑇𝑇1 π‘₯π‘₯ 𝑛𝑛 π‘₯π‘₯ 𝑛𝑛 1 𝑑𝑑𝑑𝑑 𝑑𝑑 𝑛𝑛𝑛𝑛𝑇𝑇𝑇𝑇𝑑𝑑𝑑𝑑 𝑑𝑑1 1 𝑧𝑧 1 𝑋𝑋 𝑧𝑧𝑑𝑑𝑑𝑑 𝑑𝑑 𝑛𝑛𝑛𝑛 𝑇𝑇Foundations 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-time𝑑𝑑2π‘₯π‘₯ 𝑑𝑑2 𝑠𝑠𝑋𝑋 𝑠𝑠2𝑑𝑑𝑑𝑑 In discrete-time𝑑𝑑2π‘₯π‘₯ 𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑 ���𝑑𝑑𝑑 𝑑𝑑π‘₯π‘₯ 𝑛𝑛𝑛𝑛 π‘₯π‘₯ 𝑛𝑛𝑛𝑛 𝑇𝑇 𝑑𝑑𝑑𝑑 𝑑𝑑 𝑛𝑛𝑛𝑛𝑇𝑇Foundations 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-time𝑑𝑑2π‘₯π‘₯ 𝑑𝑑2 𝑠𝑠𝑋𝑋 𝑠𝑠2𝑑𝑑𝑑𝑑 In discrete-time𝑑𝑑2π‘₯π‘₯ 𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑 ���𝑑2π‘₯π‘₯ 𝑑𝑑 2𝑑𝑑𝑑𝑑𝑑𝑑 𝑛𝑛𝑛𝑛π‘₯π‘₯ 𝑛𝑛𝑛𝑛 π‘₯π‘₯ 𝑛𝑛𝑛𝑛 𝑇𝑇 /𝑇𝑇 π‘₯π‘₯ 𝑛𝑛𝑛𝑛 𝑇𝑇 π‘₯π‘₯ 𝑛𝑛𝑛𝑛 2𝑇𝑇 /𝑇𝑇 𝑇𝑇Foundations 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-time𝑑𝑑2π‘₯π‘₯ 𝑑𝑑2 𝑠𝑠𝑋𝑋 𝑠𝑠2𝑑𝑑𝑑𝑑 In discrete-time𝑑𝑑2π‘₯π‘₯ 𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑 ���𝑑2π‘₯π‘₯ 𝑑𝑑 𝑑𝑑𝑑𝑑 2𝑑𝑑 𝑛𝑛𝑛𝑛 π‘₯π‘₯ 𝑛𝑛𝑇𝑇 2π‘₯π‘₯ 𝑛𝑛𝑛𝑛 𝑇𝑇 π‘₯π‘₯ 𝑛𝑛𝑛𝑛 2𝑇𝑇π‘₯π‘₯ 𝑛𝑛 2π‘₯π‘₯ 𝑛𝑛 1 π‘₯π‘₯ 𝑛𝑛 2 𝑇𝑇2𝑇𝑇2Foundations 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-time𝑑𝑑2π‘₯π‘₯ 𝑑𝑑2 𝑠𝑠𝑋𝑋 𝑠𝑠2𝑑𝑑𝑑𝑑 In discrete-time𝑑𝑑2π‘₯π‘₯ 𝑑𝑑𝑑𝑑𝑑𝑑 𝑑𝑑 𝑑𝑑𝑑𝑑 𝑑𝑑 ���𝑑2π‘₯π‘₯ 𝑑𝑑 𝑑𝑑𝑑𝑑 2𝑑𝑑 𝑛𝑛𝑛𝑛 π‘₯π‘₯ 𝑛𝑛𝑇𝑇 2π‘₯π‘₯ 𝑛𝑛𝑛𝑛 𝑇𝑇 π‘₯π‘₯ 𝑛𝑛𝑛𝑛 2𝑇𝑇π‘₯π‘₯ 𝑛𝑛 2π‘₯π‘₯ 𝑛𝑛 1 π‘₯π‘₯ 𝑛𝑛 2 𝑇𝑇2𝑇𝑇2𝑑𝑑𝑑𝑑 𝑑𝑑11 1 2 2 1 2𝑧𝑧 𝑧𝑧 𝑋𝑋 𝑧𝑧 2 1 𝑧𝑧 1 2𝑋𝑋 𝑧𝑧𝑑𝑑𝑑𝑑 𝑑𝑑 𝑛𝑛𝑛𝑛 𝑇𝑇𝑇𝑇Foundations 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‘˜ 𝑋𝑋 𝑠𝑠 π‘ π‘ π‘‘π‘‘π‘‘π‘‘π‘˜π‘˜π‘‘π‘‘π‘˜π‘˜ π‘₯π‘₯ 𝑑𝑑1 1 π‘˜π‘˜ 𝑋𝑋 𝑧𝑧 1 𝑧𝑧 π‘‘π‘‘π‘‘π‘‘π‘˜π‘˜π‘‡π‘‡π‘‘π‘‘ 𝑛𝑛𝑛𝑛Foundations 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‘˜ 𝑋𝑋 𝑠𝑠 π‘ π‘ π‘‘π‘‘π‘‘π‘‘π‘˜π‘˜π‘‘π‘‘π‘˜π‘˜ π‘₯π‘₯ 𝑑𝑑1 1 π‘˜π‘˜ 𝑋𝑋 𝑧𝑧 1 𝑧𝑧 π‘‘π‘‘π‘‘π‘‘π‘˜π‘˜π‘‡π‘‡π‘‘π‘‘ 𝑛𝑛𝑛𝑛1𝑠𝑠 1 𝑧𝑧 1𝑇𝑇Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters80

IIR Filter Design from Derivatives Example: 𝑠𝑠 1𝑇𝑇1 𝑧𝑧 1 Use the derivative conversion to transform the following biquadfilter into the discrete-time domain.1𝐻𝐻 𝑠𝑠 𝑠𝑠 0.1 2 9Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters81

IIR Filter Design from Derivatives Example: 𝑠𝑠 1𝑇𝑇1 𝑧𝑧 1 Use the derivative conversion to transform the following bi-quadfilter into the discrete-time domain.1𝐻𝐻 𝑠𝑠 𝑠𝑠 0.1 2 9𝐻𝐻 𝑧𝑧 111 𝑧𝑧 1 0.1𝑇𝑇1 𝑧𝑧 1𝑇𝑇2 0.1𝑇𝑇22 9 9𝑇𝑇2𝑇𝑇2 Foundations of Digital Signal Processing𝑇𝑇211 𝑧𝑧 1 0.1𝑇𝑇1 0.1𝑇𝑇𝑇𝑇22 1 𝑧𝑧Lecture 22: Designing FIR / IIR Filters2 9 9𝑇𝑇282

IIR Filter Design from Derivatives Example: 𝑠𝑠 1𝑇𝑇1 𝑧𝑧 1 Use the derivative conversion to transform the following bi-quadfilter into the discrete-time domain.1𝐻𝐻 𝑠𝑠 𝑠𝑠 0.1 2 9𝐻𝐻 𝑧𝑧 𝑇𝑇2Finding poles1 0.1𝑇𝑇 𝑧𝑧 11 0.1𝑇𝑇1 0.1𝑇𝑇2 1 𝑧𝑧2 1 𝑧𝑧2 9𝑇𝑇2 9𝑇𝑇2 0 9𝑇𝑇21 0.1𝑇𝑇 𝑧𝑧 1 3𝑇𝑇𝑇𝑇𝑧𝑧 1 1 0.1𝑇𝑇 3𝑇𝑇𝑇𝑇Foundations of Digital Signal Processing1𝑧𝑧 1 0.1 3𝑗𝑗 𝑇𝑇Lecture 22: Designing FIR / IIR Filters83

IIR Filter Design from Derivatives Example: 𝑠𝑠 1𝑇𝑇1 𝑧𝑧 1 Use the derivative conversion to transform the following bi-quadfilter into the discrete-time domain.1𝑧𝑧 Poles1 0.1 3𝑗𝑗 𝑇𝑇Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters84

IIR Filter Design from Derivatives Example: 𝑠𝑠 1𝑇𝑇1 𝑧𝑧 11𝑧𝑧 1 0.1 3𝑗𝑗 𝑇𝑇Foundations 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-time1𝑠𝑠 1 𝑧𝑧 1𝑇𝑇Pros: 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?𝐾𝐾1𝐻𝐻 𝑠𝑠 𝑠𝑠 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‘˜ 11𝐻𝐻 𝑠𝑠 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‘˜ 11𝐻𝐻 𝑠𝑠 π‘π‘π‘˜π‘˜ π‘’π‘’π‘π‘π‘˜π‘˜π‘‘π‘‘π‘ π‘  π‘π‘π‘˜π‘˜πΎπΎβ„Ž 𝑑𝑑 π‘π‘π‘˜π‘˜ 𝑒𝑒 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‘˜ 11𝐻𝐻 𝑠𝑠 π‘π‘π‘˜π‘˜ π‘’π‘’π‘π‘π‘˜π‘˜π‘‘π‘‘π‘ π‘  π‘π‘π‘˜π‘˜πΎπΎβ„Ž 𝑑𝑑 π‘π‘π‘˜π‘˜ 𝑒𝑒 π‘π‘π‘˜π‘˜π‘‘π‘‘π‘˜π‘˜ 1πΎπΎπΎπΎπ‘˜π‘˜ 1π‘˜π‘˜ 1β„Ž 𝑛𝑛𝑛𝑛 β„Ž 𝑛𝑛 π‘π‘π‘˜π‘˜ 𝑒𝑒 π‘π‘π‘˜π‘˜π‘›π‘›π‘›π‘› π‘π‘π‘˜π‘˜ 𝑒𝑒 π‘π‘π‘˜π‘˜π‘‡π‘‡Foundations of Digital Signal Processing𝑛𝑛Lecture 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‘˜ 11𝐻𝐻 𝑠𝑠 π‘π‘π‘˜π‘˜ π‘’π‘’π‘π‘π‘˜π‘˜π‘‘π‘‘π‘ π‘  π‘π‘π‘˜π‘˜πΎπΎβ„Ž 𝑑𝑑 π‘π‘π‘˜π‘˜ 𝑒𝑒 π‘π‘π‘˜π‘˜π‘‘π‘‘π‘˜π‘˜ 1πΎπΎπΎπΎπ‘˜π‘˜ 1π‘˜π‘˜ 1β„Ž 𝑛𝑛𝑛𝑛 β„Ž 𝑛𝑛 π‘π‘π‘˜π‘˜ 𝑒𝑒 π‘π‘π‘˜π‘˜π‘›π‘›π‘›π‘› π‘π‘π‘˜π‘˜ 𝑒𝑒 ��𝐻 𝑧𝑧 1 𝑒𝑒 π‘π‘π‘˜π‘˜π‘‡π‘‡ 𝑧𝑧 1π‘˜π‘˜ 1Foundations of Digital Signal Processing𝑛𝑛Lecture 22: Designing FIR / IIR Filters93

IIR Filter Design from Derivatives Example: 𝐻𝐻 𝑧𝑧 π‘π‘π‘˜π‘˜πΎπΎ π‘˜π‘˜ 11 𝑒𝑒 π‘π‘π‘˜π‘˜π‘‡π‘‡π‘§π‘§ 1 Use impulse invariance to transform the following biquad filter intothe discrete-time domain.1𝐻𝐻 𝑠𝑠 𝑠𝑠 0.1 2 9Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters94

IIR Filter Design from Derivatives Example: 𝐻𝐻 𝑧𝑧 π‘π‘π‘˜π‘˜πΎπΎ π‘˜π‘˜ 11 𝑒𝑒 π‘π‘π‘˜π‘˜π‘‡π‘‡π‘§π‘§ 1 Use impulse invariance to transform the following biquad filter intothe discrete-time domain.1𝐻𝐻 𝑠𝑠 𝑠𝑠 0.1 2 9Poles:𝑠𝑠 0.12 9 0𝑠𝑠 3𝑗𝑗 0.1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters95

IIR Filter Design from Derivatives Example: 𝐻𝐻 𝑧𝑧 π‘π‘π‘˜π‘˜πΎπΎ π‘˜π‘˜ 11 𝑒𝑒 π‘π‘π‘˜π‘˜π‘‡π‘‡π‘§π‘§ 1 Use impulse invariance to transform the following biquad filter intothe discrete-time domain.11/21/2𝐻𝐻 𝑠𝑠 2𝑠𝑠 0.1 9 𝑠𝑠 3𝑗𝑗 0.1 𝑠𝑠 3𝑗𝑗 0.1Poles:𝑠𝑠 0.12 9 0𝑠𝑠 3𝑗𝑗 0.1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters96

IIR Filter Design from Derivatives Example: 𝐻𝐻 𝑧𝑧 π‘π‘π‘˜π‘˜πΎπΎ π‘˜π‘˜ 11 𝑒𝑒 π‘π‘π‘˜π‘˜π‘‡π‘‡π‘§π‘§ 1 Use impulse invariance to transform the following biquad filter intothe discrete-time domain.11/21/2𝐻𝐻 𝑠𝑠 2𝑠𝑠 0.1 9 𝑠𝑠 3𝑗𝑗 0.1 𝑠𝑠 3𝑗𝑗 0.11/21/2𝐻𝐻 𝑧𝑧 3𝑗𝑗 0.1𝑇𝑇 11 𝑒𝑒𝑧𝑧1 𝑒𝑒 3𝑗𝑗 0.1 𝑇𝑇 𝑧𝑧 1Foundations of Digital Signal ProcessingLecture 22: Designing FIR / IIR Filters97

IIR Filter Design from Derivatives Example: 𝐻𝐻 𝑧𝑧 𝐻𝐻 𝑧𝑧 1 𝑒𝑒1 π‘’π‘’π‘π‘π‘˜π‘˜πΎπΎ π‘˜π‘˜ 11 𝑒𝑒 π‘π‘π‘˜π‘˜π‘‡π‘‡π‘§π‘§ 11/2 3𝑗𝑗 0.1 𝑇𝑇 𝑧𝑧 11/23𝑗𝑗 0.1 𝑇𝑇 𝑧𝑧 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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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 filters 1 A. Introduction Previously (1/16 lecture notes), we had considered the following high-pass FIR filters: 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.

asset management for utilities and telecommunication providers to support network planning, design and analysis, maintenance and operations. OMS and DMS integration: A configurable standards-based software solution that provides utilities with an automated way of introducing the electric network to the OMS (Outage Management System) and DMS (Distribution Management System) via CIM (Common .