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Page 1 ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete Objectives: Filters and Difference Equations Signal Flow Graphs FIR and IIR Filters Bilinear Transform Digital Conversion of Filters Design of Analog Filters Resources: ISIP: Filter Transformations Wiki: Digital Filter Design JOS: Digital Filters Wiki: Bilinear Transform CNX: IIR Design DESIGN OF IIR FILTERS URL: Page 2 ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete Objectives: Filters and Difference Equations Signal Flow Graphs FIR and IIR Filters Bilinear Transform Digital Conversion of Filters Design of Analog Filters Resources: ISIP: Filter Transformations Wiki: Digital Filter Design JOS: Digital Filters Wiki: Bilinear Transform CNX: IIR Design DESIGN OF IIR FILTERS URL: EE 3512: Lecture 36, Slide 1 Recall our expression for a linear, constantcoefficient difference equation: This equation can be written succinctly using summations: We can draw a signal flow graph implementation of this equation: This is known as the Direct Form I implementation of the above difference equation. Can we implement this more efficiently? Converting Difference Equations To Signal Flow Graphs ] [ ... ] 1 [ ] [ ] [ ... ] 2 [ ] 1 [ ] [ 1 0 2 1 M n x b n x b n x b N n y a n y a n y a n y M N  + +  + =  + +  +  + å å = =  +   = M l l N k k l n x b k n y a n y 0 1 ] [ ] [ ] [ . . . . . . + 1  z 1  z 0 b 1 b 2 b 1  z 1  z 1 a  2 a  ] [n x ] [n y + + + + + Page 3 ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete Objectives: Filters and Difference Equations Signal Flow Graphs FIR and IIR Filters Bilinear Transform Digital Conversion of Filters Design of Analog Filters Resources: ISIP: Filter Transformations Wiki: Digital Filter Design JOS: Digital Filters Wiki: Bilinear Transform CNX: IIR Design DESIGN OF IIR FILTERS URL: EE 3512: Lecture 36, Slide 1 Recall our expression for a linear, constantcoefficient difference equation: This equation can be written succinctly using summations: We can draw a signal flow graph implementation of this equation: This is known as the Direct Form I implementation of the above difference equation. Can we implement this more efficiently? Converting Difference Equations To Signal Flow Graphs ] [ ... ] 1 [ ] [ ] [ ... ] 2 [ ] 1 [ ] [ 1 0 2 1 M n x b n x b n x b N n y a n y a n y a n y M N  + +  + =  + +  +  + å å = =  +   = M l l N k k l n x b k n y a n y 0 1 ] [ ] [ ] [ . . . . . . + 1  z 1  z 0 b 1 b 2 b 1  z 1  z 1 a  2 a  ] [n x ] [n y + + + + + EE 3512: Lecture 36, Slide 2 One of the more elementary aspects of the field of digital signal processing is to develop more efficient implementations of digital filters, as well as improve their ability to produce accurate results with less numerical precision. A more efficient implementation of our filter is a Direct Form II: This filter has the same transfer function, but shares the delay element between the feedforward (moving average/finite impulse response) and feedback (autoregressive/infinite impulse response) portions of the filter. Analog differential equations can be represented by similar signal flow graphs, but their implementation involves physical components (e.g., RLCs, op amps). Direct Form II: Sharing Delay Elements (Memory) . . . + 1  z 1  z 0 b 1 b 2 b 1 a  2 a  ] [n x ] [n y + + + + + Page 4 ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete Objectives: Filters and Difference Equations Signal Flow Graphs FIR and IIR Filters Bilinear Transform Digital Conversion of Filters Design of Analog Filters Resources: ISIP: Filter Transformations Wiki: Digital Filter Design JOS: Digital Filters Wiki: Bilinear Transform CNX: IIR Design DESIGN OF IIR FILTERS URL: EE 3512: Lecture 36, Slide 1 Recall our expression for a linear, constantcoefficient difference equation: This equation can be written succinctly using summations: We can draw a signal flow graph implementation of this equation: This is known as the Direct Form I implementation of the above difference equation. Can we implement this more efficiently? Converting Difference Equations To Signal Flow Graphs ] [ ... ] 1 [ ] [ ] [ ... ] 2 [ ] 1 [ ] [ 1 0 2 1 M n x b n x b n x b N n y a n y a n y a n y M N  + +  + =  + +  +  + å å = =  +   = M l l N k k l n x b k n y a n y 0 1 ] [ ] [ ] [ . . . . . . + 1  z 1  z 0 b 1 b 2 b 1  z 1  z 1 a  2 a  ] [n x ] [n y + + + + + EE 3512: Lecture 36, Slide 2 One of the more elementary aspects of the field of digital signal processing is to develop more efficient implementations of digital filters, as well as improve their ability to produce accurate results with less numerical precision. A more efficient implementation of our filter is a Direct Form II: This filter has the same transfer function, but shares the delay element between the feedforward (moving average/finite impulse response) and feedback (autoregressive/infinite impulse response) portions of the filter. Analog differential equations can be represented by similar signal flow graphs, but their implementation involves physical components (e.g., RLCs, op amps). Direct Form II: Sharing Delay Elements (Memory) . . . + 1  z 1  z 0 b 1 b 2 b 1 a  2 a  ] [n x ] [n y + + + + + EE 3512: Lecture 36, Slide 3 More About Types of Filters Consider a filter with only feedforward components: The transfer function is: Since the impulse response of this filter, h[n], has a finite number of nonzero terms, this filter is referred to as a finite impulse response (FIR) filter. Observe that this filter only has zeroes. Next, consider a filter with only feedback components: The transfer function is: This is an allpole filter with an infinite impulse response (IIR). Why? å å =  = = =  = M l l l M l l z b z X z Y z H l n x b n y 0 1 ) ( ) ( ) ( ] [ ] [ å å =  = + = +   = N k k k N k k z a b z H n x b k n y a n y 1 0 0 1 1 ) ( ] [ ] [ ] [ Page 5 ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete Objectives: Filters and Difference Equations Signal Flow Graphs FIR and IIR Filters Bilinear Transform Digital Conversion of Filters Design of Analog Filters Resources: ISIP: Filter Transformations Wiki: Digital Filter Design JOS: Digital Filters Wiki: Bilinear Transform CNX: IIR Design DESIGN OF IIR FILTERS URL: EE 3512: Lecture 36, Slide 1 Recall our expression for a linear, constantcoefficient difference equation: This equation can be written succinctly using summations: We can draw a signal flow graph implementation of this equation: This is known as the Direct Form I implementation of the above difference equation. Can we implement this more efficiently? Converting Difference Equations To Signal Flow Graphs ] [ ... ] 1 [ ] [ ] [ ... ] 2 [ ] 1 [ ] [ 1 0 2 1 M n x b n x b n x b N n y a n y a n y a n y M N  + +  + =  + +  +  + å å = =  +   = M l l N k k l n x b k n y a n y 0 1 ] [ ] [ ] [ . . . . . . + 1  z 1  z 0 b 1 b 2 b 1  z 1  z 1 a  2 a  ] [n x ] [n y + + + + + EE 3512: Lecture 36, Slide 2 One of the more elementary aspects of the field of digital signal processing is to develop more efficient implementations of digital filters, as well as improve their ability to produce accurate results with less numerical precision. A more efficient implementation of our filter is a Direct Form II: This filter has the same transfer function, but shares the delay element between the feedforward (moving average/finite impulse response) and feedback (autoregressive/infinite impulse response) portions of the filter. Analog differential equations can be represented by similar signal flow graphs, but their implementation involves physical components (e.g., RLCs, op amps). Direct Form II: Sharing Delay Elements (Memory) . . . + 1  z 1  z 0 b 1 b 2 b 1 a  2 a  ] [n x ] [n y + + + + + EE 3512: Lecture 36, Slide 3 More About Types of Filters Consider a filter with only feedforward components: The transfer function is: Since the impulse response of this filter, h[n], has a finite number of nonzero terms, this filter is referred to as a finite impulse response (FIR) filter. Observe that this filter only has zeroes. Next, consider a filter with only feedback components: The transfer function is: This is an allpole filter with an infinite impulse response (IIR). Why? å å =  = = =  = M l l l M l l z b z X z Y z H l n x b n y 0 1 ) ( ) ( ) ( ] [ ] [ å å =  = + = +   = N k k k N k k z a b z H n x b k n y a n y 1 0 0 1 1 ) ( ] [ ] [ ] [ EE 3512: Lecture 36, Slide 4 Design of Digital Filters Using Analog Prototypes Analog filter design theory was developed in the mid1900’s. As digital signal processing developed, it seemed reasonable to leverage existing knowledge in analog filter design. Our strategy will be to design the filter in the analog domain, and then transform the filter to the digital domain. We can derive this transformation by recalling the relationship between the Laplace transform and the ztransform: We can approximate the logarithm using a Taylor series: This transformation is known as the bilinear transform. It maps the lefthalf splane to the interior of the unit circle in the zplane. Unfortunately, it also “warps” the frequency axis, so the analog filter design must be prewarped so that it lands at the proper frequency in the zplane. Let s = s + j W and z = re j w : ) ln( 1 z T s e z sT = Û = ÷ ÷ ø ö ç ç è æ +  = ÷ ø ö ç è æ +  » =   1 1 1 1 2 1 1 2 ) ln( 1 z z T z z T z T s ÷ ÷ ø ö ç ç è æ +  = W +  w w s j j re re T j 1 1 2Read More
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Test: FIR Filters Windows Design  1 Test  10 ques 
Test: FIR Filters Windows Design  2 Test  10 ques 
Test: Frequency Sampling Method FOR Design Test  10 ques 
1. What is an IIR filter? 
2. How is the design of an IIR filter different from a FIR filter? 
3. What is the process of designing an IIR filter? 
4. What are the advantages of using an IIR filter? 
5. What are the limitations of IIR filters? 
Test: FIR Filters Windows Design  1 Test  10 ques 
Test: FIR Filters Windows Design  2 Test  10 ques 
Test: Frequency Sampling Method FOR Design Test  10 ques 

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