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Unformatted text preview: 1 Chapter 5: Structures of DiscreteTime Systems Rosli Besar, February 2005, FET, MMU ETM4096 Digital Signal Processing REPRESENTATION OF LCCDE Block Diagram Signal Flow Graph BASIC NETWORK STRUCTURE FOR IIR SYSTEMS Direct Form Cascade Form Parallel form BASIC NETWORK STRUCTURES FOR FIR SYSTEMS Direct Form Cascade Form 2 Once the specifications of the digital filter are given, a suitable impulse response/transfer function is obtained using one of the many approximation methods (e.g., window method). Then the next step is to find a suitable filter structure to realize the impulse response/transfer function. Realization can be done efficiently in the zdomain by converting a given transfer function, H ( z ), into a suitable filter structure. Block or flow diagrams are often used to depict filter structures and they show the computational procedure for implementing the digital filter. The structure used depends on whether the filter is an IIR or FIR filter. Introduction 3 The most frequently used realization structures for FIR and IIR filters are: Introduction Transversal (Direct) structure Cascade structure Linear phase structure Lattice structure Frequency sampling Fast convolution Direct Form I and II Cascade Parallel Lattice FIR Filters IIR Filters In this chapter, we consider the various realization structures in detail in order to provide a clear picture on the selection of filter structures. 4 The filter structures differ quite significantly with respect to complexity, number of elements, and other properties. The choice of filter structures depends largely on Introduction Whether it is FIR or IIR The implementation complexity : It refers to the number of arithmetic operations (multiplication, divisions, and additions) required to compute system output, and the memory requirements required to store the system parameters, past inputs, past outputs, and any intermediate computed values. Sensitive to coefficient quantization errors : In digital systems, the parameters of the system must be represented with finitelength, i.e. finite precision. The output of the digital system must be roundoff or truncated to fit within the limited precision constraints of the hardware used. This limited precision will degrade the performance if the system is not designed properly. 5 In Chapter 1, we considered the LCCDE of the form: The ztransform can also be used to study the properties and characteristics of LTI systems described by LCCDE. Applying the ztransform to both sides of the above equation and using the linearity property and timeshifting property, we obtain: ( 29 ) ( z X z b z Y z a k q k k k p k k = = = ) ( ) ( ) ( ) ( ) ( ) ( ] [ * ] [ ] [ z X z H z Y e X e H e Y n x n h n y j j j = = = function system : ) ( response, freq. : ) ( response, impulse : ] [ z H e H n h j Preliminary: System Function and LCCDE = = = q k k p k k k n x b k n y a ] [ ] [ 6...
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