DSP-ch4

DSP-ch4 - DSP EEE 407/591 by Andreas Spanias, Ph.D. Chapter...

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Copyright ©Andreas Spanias Ch 4-1 DSP EEE 407/591 by Andreas Spanias, Ph.D. Chapter 4 spanias@asu.edu http://www.eas.asu.edu/~spanias Copyright ©Andreas Spanias Ch 4-2 FIR Digital Filters Advantages: Linear Phase Design Quite Efficient for designing notch filters Always Stable Disadvantages: Requires High Order for Narrowband Design Applications: Speech Processing, Telecommunications Data Processing, Noise Suppression, Radar Adaptive Signal Processing, Noise Cancellation, Echo Cancellation, Multipath channels
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Copyright ©Andreas Spanias Ch 4-3 FIR Digital Filters = = L i i i n x b n y 0 ) ( ) ( ( ) n y () n x T T 0 b L b ... . . . . . + + + + 1 b . L L z z X b z z X b z X b z Y + + + = ) ( ... ) ( ) ( ) ( 1 1 0 L L z b z b b z X z Y z H + + + = = ... ) ( ) ( ) ( 1 1 0 Copyright ©Andreas Spanias Ch 4-4 FIR Filter Frequency Response + + + = jL L j j e b e b b e H ... ) ( 1 0 s f f π 2 = o o foldover
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Copyright ©Andreas Spanias Ch 4-5 FIR Filter Design 1. LINEAR PHASE DESIGN 2. FOURIER SERIES DESIGN 3. ZERO PLACEMENT 4. FREQUENCY SAMPLING 5. LEAST SQUARES 6. IMPLEMENTATIONS Copyright ©Andreas Spanias Ch 4-6 LINEAR PHASE DESIGN Linear Phase (constant time delay) FIR filter design is important in pulse transmission applications where pulse dispersion must be avoided. The frequency response function of the FIR filter is written as: + + + + = jL L j j j e b e b e b b e H ... ) ( 2 2 1 0 where )) ( arg( ) ( , ) ( ) ( = Φ = j j e H e H M ) ( ) ( ) ( Φ = j j e M e H
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Copyright ©Andreas Spanias Ch 4-7 GROUP DELAY The time delay or group delay of a filter is defined as Φ = d d ) ( τ therefore if is a linear function of then is a constant. is given in terms of samples () Φ Copyright ©Andreas Spanias Ch 4-8 LINEAR PHASE AND IMPULSE RESPONSE SYMMETRIES It can be shown that linear phase is achieved if ) ( ) ( n L h n h = where h(n) is the impulse response of the filter. For L = odd = + = 2 1 0 ) ( ) )( ( ) ( L n n L n z z n h z H = = 2 1 0 2 2 cos ) ( 2 ) ( L n L j j n L n h e e H
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Copyright ©Andreas Spanias Ch 4-9 LINEAR PHASE DESIGN if we define the pseudomagnitude = = 2 1 0 1 2 cos ) ( 2 ) ( L n j n L n h e H Then < Π + = Φ 0 ) ( , 2 0 ) ( , 2 ) ( 1 1 j j e H L e H L hence the phase response is piecewise linear. Copyright ©Andreas Spanias Ch 4-10 SYMMETRIC AND ANTI-SYMMETRIC LINEAR PHASE FILTERS Two Anti-symmetries for L=even or L=odd for ) ( ) ( n L h n h = ) ( ) ( n L h n h = Two Symmetries for L=even or L=odd for
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Copyright ©Andreas Spanias Ch 4-11 EXAMPLES OF SYMMETRIES 3 3 0 0 3 L = 3 L = (b) Even symmetric around l =1.5 (d) Odd symmetric around l = 1.5 4 4 0 0 4 L = 4 L = (a) Even symmetric around l =2 (c) Odd symmetric around l =2 () hn EXAMPLES OF PHASE AND SYMMETRY IN h(n) -200 -150 -100 -50 0 50 Phase ( degrees ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 -200 -100 0 100 Normalized frequency (Nyquist == 1) Phase (degrees) 4 4 0 0 4 L = 4 L = (a) Even symmetric around l =2 (c) Odd symmetric around l =2 () hn
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Copyright ©Andreas Spanias Ch 4-13 J-DSP Simulation of an Even Symmetric h(n) Copyright ©Andreas Spanias Ch 4-14 J-DSP Simulation of Odd Symmetric h(n)
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DSP-ch4 - DSP EEE 407/591 by Andreas Spanias, Ph.D. Chapter...

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