Structural Representation in Digital Filters: A Comprehensive Guide

1. Chapter 5 Network Structures of Discrete-Time system

2. Content • Introduce • Network Representation with Signal Flow-Graph • Basic IIR System Structures • Basic FIR System Structures • Linear Phase Structures • Frequency Sample

3. 5.1 Introduce • Discrete-time system representation A discrete-time system can be described by the input-output relation, impulse response and system function.

4. 5.1 Introduce • Discrete-time system representation In the time domain, the input-output relations of an LTI discrete-time system is given by the convolution sum or, by the linear constant coefficient difference equation.

5. 5.1 Introduce • Discrete-time system representation • A discrete-time system can be implemented on a general-purpose digital computer in software or with special-purpose hardware. To this end, it is necessary to describe the input-output relationship by means of a computational algorithm.

6. 5.1 Introduce • A structural representation using interconnected basic building blocks is the first step in the hardware or software implementation of an LTI digital filter. • The structural representation provides the relations between some pertinent internal variables with the input and the output that, in turn, provides the keys to the implementation.

7. 5.1 Introduce • There are various forms of the structural representation of a digital filter. • There are literally an infinite number of equivalent structures realizing the same transfer function. • However , the accuracy, computing speed, complexity is different for the different structure.

8. 5.2 System Representation with signal flow-graph • The computational algorithm of an LTI digital filter can be conveniently represented in block diagram form using the basic building blocks representing the unit delay, the multiplier, the adder, and the pick-off nodes.

9. 5.2 System Representation with signal flow-graph Unit delay Multiplier Adder

10. 5.2 System Representation with signal flow-graph • In a signal flow-graph, the dependent and independent signal variables are represented by nodes, whereas the multiplier and the delay units are represented by directed branches. In the latter case, the directed branch has attached symbol denoting the branch-gain or the transmittance, which, for a multiplier branch, is the multiplier coefficient value and for a delay branch is simply z-1.

11. 5.2 System Representation with signal flow-graph • Source node • Output node • Input branch • Output branch 图5.2.2 信号流图

12. 5.2 System Representation with signal flow-graph In fig. 5.2.2, we can obtained

13. 5.2 System Representation with signal flow-graph • Digital filter structures represented in block diagram form can often be analyzed by writing down the expressions for the output signals of each adder as a sum of its input signals, thereby developing a set of equations relating the filter input and output signals in terms of all internal signals. • Eliminating the unwanted internal variables then results in the expression for the output signal as a function of the input signal and the filter parameters that are the multiplier coefficients.

14. 5.2 System Representation with signal flow-graph • Advantages of block diagram/signal flow chart representation (1) Easy to write down the computational algorithm by inspection. (2) Easy to analyze the block diagram to determine the explicit relation between the output and input.

15. 5.2 System Representation with signal flow-graph (3) Easy to manipulate a block diagram to derive other “equivalent” block diagrams yielding different computational algorithms. (4) Easy to determine the hardware requirements. (5) Easier to develop block diagram representations from the transfer function directly.

16. 5.2 System Representation with signal flow-graph • Basic signal flow-graph • Only basic branch in signal flow-graph; • Must have unit delay branch; • have finite nodes and branchs.

17. 5.2 System Representation with signal flow-graph • Basic signal flow-graph A digital filter structure is said to be canonic if the number of delays in the block diagram representation is equal to the order of the difference equation. Otherwise, it is nocanonic structure.

18. 5.2 System Representation with signal flow-graph • Equivalent Structures • Two digital filter structures are defined to be equivalent if they have the same transfer function. • We can obtain equivalent structures with a number of methods. However, a fairly simple way to generate an equivalent structure from a given realization is via the transpose operation.

19. 5.2 System Representation with signal flow-graph • Transpose Operation (1) Reverse all paths (2) Replace pick-off nodes by adders, and vice versa. (3) Interchange the input and output nodes • All other methods for developing equivalent structures are based on a specific algorithm for each structure.

20. 5.2 System Representation with signal flow-graph • It should be noted that under infinite-precision arithmetic, any given realization of a digital filter behaves identically to any other equivalent structure. • However, in practice, due to the finite wordlength limitations, a specific realization behaves differently from its other equivalent realizations.

21. 5.2 System Representation with signal flow-graph • Hence, it is important to choose a structure that has the least quantization effects when implemented using finite precision arithmetic. • One way to arrive at such a structure is to determine a large number of equivalent structures, analyze the finite wordlength effects in each case, and select the one showing the least effects

22. 5.2 System Representation with signal flow-graph • Classification of the digital filter • Finite Impulse Response (FIR) • Infinite Impulse Response (IIR)

23. 5.3 Basic IIR Digital Filter Structures • The causal IIR digital filters we are concerned with in this text are characterized by a real rational transfer function or, equivalently, by the constant coefficient difference equation

24. 5.3 Basic IIR Digital Filter Structures • From the difference equation representation, it can be seen that the realization of the causal IIR digital filters requires some form of feedback.

25. 5.3 Basic IIR Digital Filter Structures • Direct Form • Cascade Form • Parallel Form

26. 5.3 Basic IIR Digital Filter Structures • Direct Form • Direct forms -- Coefficients are directly the system function coefficients. • For a N-order difference equation, its system function can be written as

27. 5.3 Basic IIR Digital Filter Structures • Direct Form • For simplicity, let M=N=2, the system function become • The digital filter structure in direct form is

29. 5.3 Basic IIR Digital Filter Structures • The result structure in the figure (a) is commonly known as the direct form Ⅰ structure, figure (c) is called the direct form Ⅱ realization.

30. 5.3 Basic IIR Digital Filter Structures • Exp 5.3.1 Illustration of Direct Form Realization Consider the third-order IIR system function:

31. 5.3 Basic IIR Digital Filter Structures

32. 5.3 Basic IIR Digital Filter Structures • Direct form implementation with MATLAB yn=filter(B,A,xn)

33. 5.3 Basic IIR Digital Filter Structures • Cascade Realizations • By expressing the numerator and the denominator polynomials of the system function as a product of polynomials of lower degree, a digital filter can be realized as a cascade of low-order filter sections. • Consider, for example, H(z)=B(z)/A(z) expressed as

34. 5.3 Basic IIR Digital Filter Structures • Examples of cascade realizations obtained by different pole-zero pairings are shown below

35. 5.3 Basic IIR Digital Filter Structures

36. 5.3 Basic IIR Digital Filter Structures

37. 5.3 Basic IIR Digital Filter Structures • Exp 5.3.2 Illustration of the cascade Form Realization We develop the cascade realizations of the third-order IIR system function By factoring the numerator and denominator polynomials of H(z), We obtain

38. 5.3 Basic IIR Digital Filter Structures We arrive at a cascade realization of Another cascade realization can be given by using a different pole-zero pairing.

39. 5.3 Basic IIR Digital Filter Structures • Due to finite wordlength effects, each such cascade realization behaves differently from Others.

40. 5.3 Basic IIR Digital Filter Structures • Realization of cascade structures using MATLAB • The factorization of a polynomial can be out in MATLAB using the function factorize, which determines the second-order factors directly from the specified system function H(z).

41. 5.3 Basic IIR Digital Filter Structures • Parallel Realizations • An IIR system function can be realized in a parallel form by making use of the partial fraction expansion of the transfer function. Then we can obtain Where Hk(z) is a low-order system function.

42. 5.3 Basic IIR Digital Filter Structures

43. 5.3 Basic IIR Digital Filter Structures • The two basic parallel realizations of a 3rd order IIR transfer function are shown below

44. 5.3 Basic IIR Digital Filter Structures • General structure: • Easy to realize: No choices in section ordering and No choices in pole and zero pairing

45. 5.3 Basic IIR Digital Filter Structures Example 5.3.3 Illustration of the Parallel Form Realization

46. 5.3 Basic IIR Digital Filter Structures Their realizations are show below:

47. 5.3 Basic IIR Digital Filter Structures • MATLAB Implementation • Tf2sos: direct form to cascade form • sos2tf: cascade form to direct form • tfslatc: direct form to lattice form • latc2tf: lattice form to direct form

48. 5.4 Basic FIR Digital Filter Structures • A causal FIR filter of order N is characterized by a transfer function H(z) • In the time domain, the input-output relation of the above FIR filter is given by

49. 5.4 Basic FIR Digital Filter Structures • It can be seen that • Its realization doesn’t requires some form of feedback • Its impulse response is finite length. An FIR filter of order N－1is characterized by N coefficients and, in general, require N multipliers and N－1 two-input adders.

50. 5.4 Basic FIR Digital Filter Structures • Since FIR filters can be designed to provide exact linear phase over the whole frequency range and are always BIBO stable independent of the filter coefficients, such filters are often preferred in many application. • We now outline several realization methods for such filters.