Digital Filter Structure PDF

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Dr. Kahtan Aziz

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digital filters signal processing IIR filters digital signal processing

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This document provides a comprehensive overview of digital filter structures, covering both Finite Impulse Response (FIR) and Infinite Impulse Response (IIR) filters. Various implementation methods such as direct forms, cascading, and parallel structures, with their advantages and disadvantages, are discussed.

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Digital Filter Structure Dr. Kahtan Aziz BASIC STRUCTURE FOR SYSTEM/FILTER The Transfer Function of LTI system can be connected in 2 ways : a. Parallel Connection : The overall transfer function, H(z) = H1(z) + H2(z) + … + HL(z) BASIC STRUCTURE FOR SYSTEM/FILTER b. Cascade...

Digital Filter Structure Dr. Kahtan Aziz BASIC STRUCTURE FOR SYSTEM/FILTER The Transfer Function of LTI system can be connected in 2 ways : a. Parallel Connection : The overall transfer function, H(z) = H1(z) + H2(z) + … + HL(z) BASIC STRUCTURE FOR SYSTEM/FILTER b. Cascade connection : The overall transfer function : H(z) = H1(z).H2(z). … HL(z) BASIC STRUCTURE FOR FIR SYSTEM/FILTER Direct form – An FIR filter of order N requires N + 1 multipliers, N adders and N delays. – An FIR filter of order 4 y[n] = hx[n] + hx[n-1] + hx[n-2] + hx[n-3]+ hx[n-4] BASIC STRUCTURE FOR FIR SYSTEM/FILTER Cascade form – Transfer function H(z) of a causal FIR filter of order N N H ( z ) =  h[k ]z − k k =0 – Factorized form k H ( z ) = h (1 + 1k z −1 +  2 k z − 2 ) k =1 Where k = N/2 if N is even and k = (N + 1)/2 if N is odd, with β2k = 0 BASIC STRUCTURE FOR FIR SYSTEM/FILTER Cascade form – FIR structure for a sixth-order FIR filter y(n) X(n) BASIC STRUCTURE FOR FIR SYSTEM/FILTER Example 4: The system of LTI is described by the following difference equation: y[n] = 0.9x[n] + x[n-1] + 0.5x[n-2] – 2.5x[n-3] - 0.2x[n-4] Draw a structure realization for the system described by this difference equation. BASIC STRUCTURE FOR IIR SYSTEM/FILTER IIR system/filter can be realized in several structures: 1. DIRECT FORM I 2. DIRECT FORM II (CANONIC) 3. CASCADE FORM 4. PARALLEL FORM BASIC STRUCTURE FOR IIR SYSTEM/FILTER Direct Form I – Consider a third order IIR described by transfer function P( z ) p0 + p1 z −1 + p3 z −3 H ( z) = = D( z ) 1 + d1 z −1 + d 3 z −3 – Implement as a cascade of two filter section BASIC STRUCTURE FOR IIR SYSTEM/FILTER Where W ( z) H1 ( z ) = = P ( z ) = p0 + p1 z −1 + p2 z − 2 + p3 z −3 X ( z) and Y ( z) 1 1 H 2 ( z) = = = W ( z ) D( z ) 1 + d1 z −1 + d 2 z − 2 + d 3 z −3 Resulting in realization indicated below BASIC STRUCTURE FOR IIR SYSTEM/FILTER Direct Form I BASIC STRUCTURE FOR IIR SYSTEM/FILTER Direct Form II (Canonic) – The two top delays can be shared BASIC STRUCTURE FOR IIR SYSTEM/FILTER Cascade Form  1 + 1k z −1 +  2 k z −2  H ( z ) = p0    −2  k  1 + 1k z +  2 k z −1  A third order transfer function  1 + 11 z −1  1 + 12 z −1 +  22 z −2  H ( z ) = p0   −1   −2   1 + 11 z  1 + 12 z +  22 z  −1 BASIC STRUCTURE FOR IIR SYSTEM/FILTER Cascade Form y(n) X(n) BASIC STRUCTURE FOR IIR SYSTEM/FILTER Parallel Form   +  1k z −1  H ( z ) =  0 +   0 k  −2  k  1 + 1k z +  2k z  −1 BASIC STRUCTURE FOR IIR SYSTEM/FILTER Parallel Form y(n) X(n) Example 1 + 2z −1 + z −2 18 25 H(z ) = =8+ − −1 1 − 0.75z + 0.125z −2 ( 1 − 0.5z −1 ) (1 − 0.25z −1 ) − 7 + 8z −1 H(z ) = 8 + 1 − 0.75z −1 + 0.125z −2 BASIC STRUCTURE FOR IIR SYSTEM/FILTER Example 5: Given the structure of system/filter shown below: a. What is the type of the structure? b. Determine the Transfer Function of the system? BASIC STRUCTURE FOR IIR SYSTEM/FILTER Example 6: Determine the Direct Form II, Cascade & Parallel realization of the following LTI systems: y(n) - ¼ y(n-1) – 3/8y(n-2) =x(n) + 2x(n-1) + x(n-2) Obtain the Transfer Function of the system. Draw the filter/system realization. FINITE PRECISION IMPLEMENTATION OF DISCRETE-TIME SYSTEM Digital implementation of DTS introduces quantization error to difference equation’s coefficients Non-linearity resulted from inclusion of quantization errors to DTS model Effect of quantization error differs depending on structure implemented EFFECT OF QUANTIZATION OF FILTER COEFFICIENT IIR System FIR System – Change poles-and- – Only affected the zeros position zeros position – Cascade & parallel – Less affected as form less sensitive errors is linearly compare to direct related form structure EFFECT OF ROUND-OFF NOISE Quantization errors is modeled as random white noise Round-off noise is introduced as additional signal source to system changing nonlinear to linear operation Quantization error minimize by increasing number of fixed-point word length or by adopting floating- point representation

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