# MODULE-8 - MODULE 8 DIGITAL FILTER DESIGN I • FIR Filter...

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Unformatted text preview: MODULE 8 DIGITAL FILTER DESIGN I • FIR Filter Design • Windowing • Frequency Sampling • CAD: Parks-McClellan Algorithm Page 8.1 INDEX Digital Filter Flow Diagrams Cascade and Parallel Realization Second-Order Sections Finite Impulse Response (FIR) Filter Design Advantages of FIR Filters Linear Phase Condition General Windowing Approach Rectangular (Truncation) Window Fundamental Window Trade-Off Hamming, Hanning, Blackman Window Comparison of Windows Kaiser Window Design Kaiser Low-Pass Design Method FIR Design by Frequency Sampling Discrete Fourier Transform CAD - The Parks-McClellan Algorithm Minimax Design Criteria Parks-McClellan Specifications Chebyshev Polynomials Minimax Polynomial Approximation Alternation Theorem MAIN INDEX Page 8.2 8. DIGITAL FILTER DESIGN I index READ : Sections 6.0-6.5, 7.0, 7.2-7.6 of Oppenheim & Schafer. Work as many related problems as possible. • Recall the generic LCCDE: y ( n ) = ∑ = K k 1 b ( k ) y ( n-k ) + ∑ = M m a ( m ) x ( n-m ). • Every linear discrete-time system/filter can be described by LCCDE's. • Taking the two-sided Z-Transform of both sides of the LCCDE (not interested in transients or initial conditions) yields: Y ( z ) = ∑ = K k 1 b ( k ) z-k Y ( z ) + ∑ = M m a ( m ) z-m X ( z ) or H ( z ) = filter transfer function = ) ( ) ( z X z Y = k K k m M m z k z m a b- =- = ∑ ∑- ) ( 1 ) ( 1 Page 8.3 Digital Filter Flow Diagrams index • A digital filter H ( z ) can be- expressed as a flow diagram- implemented as a digital circuit in many different ways (circuit topologies). • The following equivalent domain-to-domain substitutions can be made: X ( z ) ↔ x ( n ) Y ( z ) ↔ y ( n ) z-1 ↔ Unit Time Delay of course, not mixing up LHS notation with RHS notation. Page 8.4 Direct Form I Realization index Y ( z ) Σ b ( K ) b (2) b (1) z-1 z-1 z-1 X ( z ) a ( M ) a (2) a (0) a (1) z-1 z-1 z-1 Feed-forward Feed-back • This structure can be too sensitive to quantization errors- the errors are summed, fed back and re-amplified over and over. • Hence rarely used in actual implementation - except for second- order sections as will be explained. Page 8.5 Cascade or Series Realization index • By factoring the numerator and denominator we can write H ( z ) = Π i =1 I H i ( z ) which can be drawn as a cascade of smaller sections: H I ( z ) H ( z ) H 1 ( z ) H 2 ( z ) • Advantage : Smaller sections - less feedback error. • Disadvantage : Errors fed from section-to-section. • Commonly used. Page 8.6 Parallel Realization index • By performing a partial fraction expansion we can write H ( z ) = ∑ = I i 1 ) ( ˆ z H i which can be drawn as a parallel sum of smaller sections: H ( z ) H I ( z ) H 1 ( z ) H 2 ( z ) Σ • Advantages : Smaller sections - less feedback error....
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MODULE-8 - MODULE 8 DIGITAL FILTER DESIGN I • FIR Filter...

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