Lecture 6: March 26, 2007

1. Lecture 6: March 26, 2007 Topics: 1. IIR Filter design: Impulse-Invariant Method 2. IIR Filter design: The Matched Z-Transform 3. IIR Filter design: Bilinear Transformation Method 4. Frequency Transformations

2. Lecture 6: March 26, 2007 Topic: 1. IIR Filter design: Impulse-Invariant Method • basic principle: sampling of impulse response of an analogue filter, • mapping: HA(p) -> H(z), • resulting filter implementation as a parallel bank of two-pole filter, • aliasing effect following from sampling process and its impact, • p -> z mapping and its impact, • summary on digital filter design.

3. Lecture 6: March 26, 2007 Topic: 2. IIR Filter design: The Matched Z-Transform • basic principle: direct mapping the poles and zeros of HA(p) (p-plane) into poles and zeros of H(z) (z-plane), • mapping: HA(p) -> H(z), • method limitations.

4. Lecture 6: March 26, 2007 Topic: 3. IIR Filter design: Bilinear Transformation Method • basic principle: application of the trapezoidal formula for numerical integration of differential equation, • mapping: HA(p) -> H(z), • summary on digital filter design.

5. Lecture 6: March 26, 2007 Topic: 4. Frequency Transformations • frequency transformations in analogue domain, • frequency transformations in digital domain.

6. Objective:to design an IIR filter having an impulse response h(n) asthe sampled version of the impulse response of the analogue filter : where T is the sampling interval. 5.3. Impulse-Invariant Method (Impulse Invariant Transformation) In consequence of this result, the frequency response of the digital filter is an aliased version of the frequency response of the corresponding analogue filter.

7. Let the transfer function of the analogue filter be given:

8. Let us assumed that the order M of the numerator is less that the order N of the denominator and that all poles of are simple. If the poles of are not simple, the discussion in this section can be appropriately modified. Then, we rewrite the transfer function of the analogue filter in its partial expansion, as follows where is the location of the k-th pole and

9. The impulse response of the analogue filter : The impulse response of the digital filter h(nT):

10. Transfer function of the digital filter:

11. Transfer function of the digital filter: Transfer function of the analogue filter: Comparing HA(p) and H(z) it can be seen that H(z) can be obtained from HA(p) by using the mapping relation:

12. Realization: parallel bank of two-pole filters (second-order section filters): Intention: to implement filters with real-valued coefficients instead of complex-valued coefficients

13. With the previous given expressions for the transfer function H(z), the IIR filter is easily realized as a parallel bank of single-pole filters:

14. With the previous given expressions for the transfer function H(z), the IIR filter is easily realized as a parallel bank of single-pole filters. If some of poles are complex-valued, they may be paired together and combined to form two-pole filter sections with real-valued coefficients:

15. When:

16. With the previous given expressions for the transfer function H(z), the IIR filter is easily realized as a parallel bank of single-pole filters. If some of poles are complex-valued, they may be paired together and combined to form two-pole filter sections with real-valued coefficients. In addition, two factors containing real-valued poles may be combined to form two-pole filters with real-valued coefficients.

17. With the previous given expressions for the transfer function H(z), the IIR filter is easily realized as a parallel bank of single-pole filters. If some of poles are complex-valued, they may be paired together and combined to form two-pole filter sections with real-valued coefficients. In addition, two factors containing real-valued poles may be combined to form two-pole filters with real-valued coefficients. Consequently, the resulting filter may be realized as a parallel bank of two-pole filters with real-valued coefficients.

18. When a continuous time signal with spectrum is sampled with sampling frequency , the spectrum of the sampled signal is given by the following expressions: Aliasing occurs if the sampling frequency is less then twice the highest frequency contained in . Aliasing Effect:

19. The next figure depicting the frequency responses of the low-pass analogue filter and the corresponding digital filter illustrates the aliasing effect following from sampling with the sampling frequency .

20. Magnitude response property illustration ALIASING !

21. Aliasing effect impact: A. The digital filter will possess (approximately) the frequency response characteristics of the corresponding analogue filter if the sampling interval T is selected sufficiently small to avoid completely or minimize the effects of aliasing. B. The impulse invariance method is inappropriate for designing high-pass filters or stop-band filters due to spectrum aliasing that results from the sampling process.

22. the p -> z mapping p - > z mapping: To investigate the mapping between the p-plane and the z-plane implied by the sampling process, we rely on a generalization of the expression relatingZ-transform of h(n) to the Laplace transform of hA(nT). This relationship is given by

23. Note that when, reduces to

24. The general characteristic of the p->z mapping defined as can be obtained by substitutions: With these substitutions we can get

25. b) -axis is mapped into the unit circle in z-plane as indicates above. Consequences: a) Then, the left-half of p-plane is mapped inside the unite circle in z-plane and right-half of p-plane is mapped into points that fall outside the unit circle in z-plane. This is one of the desirable properties of a good p -> z mapping.

26. a) The mapping of -axis into the unit circle is not one-to-one. b) c) Mapping of the adjacent strips - frequency interval: d) General case: Comments on -axis mapping:

27. B.It follows for the frequency responses of analogue filter and equivalent digital filter obtained by impulse invariant transformation that the analogue filter must be bandlimited to the range This generally requires that the analogue filter has to be suitably bandlimited prior to transformation. Conclusions: A. The mapping from the analogue frequency domain to the digital frequency domain is many-to-one. It reflects simply the effects of aliasing due to sampling.

28. Mapping: The mapping of strips of the width p-plane 0 1 0 unit circle z-plane

29. 1. Digital filter specification: 2. Transformation of requirements to the digital filter to the analogue filter: 3. The analogue filter design: 4. The analogue filter conversion to the digital filter: Summary on digital filter design:

30. The frequency response of the filter obtained by impulse invariant transformation is given by Under condition that we can obtain: Comment on scaling factor application (T):

31. in the form Then: If it is desired to get a digital filter with the same gain as the analogue filter possesses, it is necessary to transform the expression for H(z) originally given by

32. Basic principle: Mapping the poles and zeros of (from the p-plane) directly into poles and zeros of (in the z-plane). 5.4. The Matched Z-Transform

33. Transfer function of the digital filter: T is sampling interval. Comparing HA(p) and H(z) it can be seen that H(z) is obtained from HA(p) by using the mapping relation: The matched Z-transformation Transfer function of the analogue filter:

34. Comments: To preserve the frequency response characteristics of an analogue filter, the sampling interval in the matched Z-transformation must be selected properly to yield the pole and zero locations at the equivalent positions in the z-plane. Thus aliasing must be avoided by selecting T sufficiently small.

35. Comments: To preserve the frequency response characteristics of an analogue filter, the sampling interval in the matched Z-transformation must be selected properly to yield the pole and zero locations at the equivalent positions in the z-plane. Thus aliasing must be avoided by selecting T sufficiently small.

36. Although the matched Z-transformation is easy to apply, there are many cases when it is not a suitable mapping: A.If the analogue system has zeros with center frequencies greater that half sampling frequency, their z-plane positions will be greatly aliased. B.Another case where the matched Z-transformation is unsuitable is where the continuous transfer function is an all-pole system. Then the digital transfer function is an all-pole system that, in any cases, does not adequately represent the desired continuous system. In general, use of impulse-invariant transformation is to be preferred over the matched Z-transformation.

37. 5.5. Bilinear Transformation Method The IIR filter design techniques described in the previous sections have severe limitations in that they are appropriate only for low-pass filter design and limited class of band-pass filter design. In this section we describe a mapping from the p-plane to the z-plane, called the bilinear transformation, that overcomes the limitation of the other three design methods described previously. Basic principle:application of the trapezoidal formula for numerical integration of differential equation.

38. This system is also characterized by the differential equation or by the differential equations: Let us consider an analog linear filter with transfer function

39. Thus: The approximation of the previous integral by trapezoidal formula is Instead of substituting a finite difference for the derivative, suppose that we integrate the derivative and approximate the integral by the trapezoidal formula:

40. We use this expression to substitute for derivative and thus obtain a difference equation for the equivalent discrete-time system. Then: Now the differential equation evaluated in t=nT:

41. The Z-transform of the previous equation is Transfer function of the equivalent digital filter is The previous expression can be expressed in his form:

43. Transfer function of the analogue filter: Transfer function of the digital filter: z -> p: p -> z: Comparing HA(p) and H(z) it can be seen that H(z) is obtained from HA(p) by using the mapping relation:

44. From this equation it can be found Investigation of the properties of the bilinear transformation:

45. Between these limits the angle of z varies from -axis unit circle Conclusion 1: Conclusion 2:The entire range in is mapped only once into range , the aliasing errors are eliminated. However, the mapping is highly nonlinear. We observe a frequency compression or frequency warping, as it is usually called, due to these nonlinearity. Conclusion 2:The entire range in is mapped only once into range , the aliasing errors are eliminated.However, the mapping is highly nonlinear. We observe a frequency compression or frequency warping, as it is usually called, due to these nonlinearity.

47. If we obtain for z:

48. Conclusion 4: If (right-half p-plane), we find (outside unit circle) Conclusion 3: If (left-half p-plane), we find (inside unit circle)

49. 1 0 0 z-plane p-plane unit circle Mapping:

50. 1. Digital filter specification: 2. Transformation of requirements to the digital filter to the analogue filter: 3. The analogue filter design: 4. The analogue filter conversion to the digital filter: Summary on digital filter design: