Lecture 10

Lecture 10 - Lecture 10 ECSE304 Signals and Systems II ECSE...

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1 Lecture 10 ECSE304 Signals and Systems II ECSE 304 Signals and Systems II Lecture 10: Discrete Time Filters and Filter Design Richard Rose McGill University Dept. of Electrical and Computer Engineering 2 Lecture 10 ECSE304 Signals and Systems II • IIR and FIR Filters • Minimum Phase Systems • Moving Average Filters • FIR Filter Design – Design by Truncation – Design by Windowing Outline
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3 Lecture 10 ECSE304 Signals and Systems II • IIR filters include the class of DLTI recursive filters represented by difference equations that include a delayed versions of the output y[n] • The nth order difference equation: • Represents an IIR filter as long as any of the coefficients are non-zero • An IIR filter has at least one pole : Infinite Impulse Response Filters a yn ayn a yn N bxn bxn b xn M NM 01 11 [] [ ] [ ] [ ] [ ] +− + + −= + + + "" Hz bb z b z aa z a z A zz pz Az zp M M N N k k M k k N k k M k k N () = ++ + + = = −− = = = = 1 1 1 1 1 1 1 1 1 1 " " { } 1 N i i a = 0 k p 4 Lecture 10 ECSE304 Signals and Systems II • The first and second order approximations to ideal filters designed in Lecture 9 are examples of simple IIR filters • Other IIR filter design methods can be found in [Oppenheim and Shafer] – Transforming continuous time transfer functions into the z-domain using the bilinear transformation – Iterative optimization techniques • Advantages of IIR Filters: – Generally require only low order (low M and N ) • Disadvantages: – Generally do not have linear phase characteristic Infinite Impulse Response Filters
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5 Lecture 10 ECSE304 Signals and Systems II • Consider the following non-recursive finite difference equation: • The system describes an Mth order FIR filter with impulse response for For a causal FIR filter: • Only the zeroes’ location in H(z) will determine frequency resp. Finite Impulse Response Filters a yn bxn bxn b xn M M 00 1 1 [] [ ] [ ] = +− + + " hn b n b n M M [ ] [ ] = + + + 01 1 δ " bn M n ,, , , = = R S T 0 0 otherwise Hz b bz b z bz b z A zz z M M MM M M k k M M () =+ + + = ++ + = −− = 1 1 1 " " 0 1 a = Impulse Response Transfer Function All poles are located at z=0 6 Lecture 10 ECSE304 Signals and Systems II • Demonstrate linear phase property for FIR filters with the symmetry condition: • Example: FIR filter with length M=5 • Symmetric about the point h[2] : h[0]=h[4] and h[1]=h[3] • Frequency response is given by: •S i n c e h[0]=h[4] and h[1]=h[3]: FIR Filters and Linear Phase 234 22 2 ( ) [0] [1] [2] [3] [4] [ [2] [0] [3] ] jj j j j j j j He h h e eh h e h e h e h e ωω ω + + + + + + 2 2 ( ) [ [2] 2 [0]cos 2 2 [1]cos ] r e h h h eH e + = [ 1 ] hM n =
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7 Lecture 10 ECSE304 Signals and Systems II • The magnitude and phase can be easily identified • Magnitude term: •P h a s e T e rm : • Phase is a linear function of frequency as long as does not change sign • It phase changes occur outside of the passband, we do not care FIR Filters and Linear Phase |( ) | ( ) jj r He H e ω = 2i f ( ) 0 () f ( ) 0 j j r j r ωπ −> = −+ < ) 2 2 ( ) [ [2] 2 [0]cos 2 2 [1]cos 2 ] r e h h h eH e ωω =+ + = j r 8 Lecture 10 ECSE304 Signals and Systems II
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This note was uploaded on 08/17/2011 for the course ECSE 304 taught by Professor Chenandbacsy during the Spring '11 term at McGill.

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Lecture 10 - Lecture 10 ECSE304 Signals and Systems II ECSE...

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