Digital Filter Structure PDF

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