1. Chapter 9 Advanced Topics in DSP

2. Sampling Rate Change
Behave as an LTI continuous-time system where freq. Response is
: New discrete-time response

3. Other approach ; involve only discrete-time operations
One approach
Reconstruct from
Resampling to obtain
Not desirable due to nonideal analog reconstruction filter, D/A, A/D converters
Other approach ; involve only discrete-time operations

4. Sampling Rate Reduction by an Integer Factor
DFT of x[n] = xc(nT)
Fig (p. 167)

5. DFT of
Since
; integer

6. Now

7. FT of xd[n] in terms of FT of x[n] infinite set of copies of freq
FT of xd[n] in terms of FT of x[n] infinite set of copies of freq. scaled through shifted by integer multiple of M copies of , freq. Scaled by M, shifted by integer of or

9. Fig (p. 170)

10. Here no aliasing since original sampled sequence is downsampled by M=2
If M > aliasing occurs
To avoid aliasing in downsampling

11. In Case of Possible Aliasing, use LPF (Prefilter)
Fig (p. 172)
It’s decimator! (downsampling by LPF followed by compression)

12. Upsampling(Increase the Sampling Rate)
Fig (p. 173)
It’s (sampling rate) expander !

13. In freq. domain, FT of is

14. (cf) freq. scaled version of FT of the input Down sampling Up sampling
1. sampling rate reduction
1. sampling rate increase
2. increase sample period
2. decrease sample period
3. decimation
3. interpolation

15. Digital Sampling Rate Increase (Interpolation)
Ideal C/D converter
Discrete-time system
Assume

17. The Up-Sampler L
n/L : an integer
otherwise
: for this example

18. Interpolation in the Frequency Domain

19. Proof of Interpolation Formula

21. Fig (p. 174)

22. General System for Interpolation
LPF
Gain : L
Cutoff :

23. Lowpass Filter

24. General Rate Change
Fig (p. 177)

26. Linear Interpolation
What is ?

27. Fig (p. 175)

28. All-Pass System All pass system The fact that and have
identical magnitude – squared when suggests that
is such that

29. (pf)

30. Simplest Example
Let (ideal delay)
such systems are called all-pass systems.
In general

31. For causal stable all-pass?
Fig (p. 275)

32. Phase of all-pass systems

33. Group delay of all-pass
Note
therefore

34. constant

35. Log magnitude
1-pole all-pass
Fig (a) (p. 276)

36. Phase
1-pole all-pass
Fig (b) (p. 276)

37. Group delay
1-pole all-pass
Fig (c) (p. 276)

38. Log magnitude
2-pole all-pass
Fig (a) (p. 277)

39. Phase
2-pole all-pass
Fig (b) (p. 277)

40. Group delay
2-pole all-pass
Fig (c) (p. 277)

41. All-pass multiple poles
Log magnitude
All-pass multiple poles
Fig (a) (p. 279)

42. All-pass multiple poles
Phase
All-pass multiple poles
Fig (b) (p. 279)

43. All-pass multiple poles
Group delay
All-pass multiple poles
Fig (c) (p. 279)

44. Minimum Phase Systems Causal & stable; i.e., poles of are
inside the unit circle &
The zeros of are also inside the unit circle.
Obviously uniquely determines for
a minimum phase system
Zeros on unit circle a special case.
May be included in some definitions of min. ph.

45. Inverse systems
LTI system
Inverse system

46. could be called an ideal compensator.
It undoes what does to the signal

47. Does a system always have a corresponding inverse?
No ex)
Inverse of rational system functions
ROC = ?

48. Example 1
ROC = ?

49. Example 2
ROC = ?

50. 1 2

51. Inverse system The zeros of are the poles of
If is causal, then its ROC is
If is also stable, then

52. Therefore : A minimum phase system has a unique, causal and stable inverse.
The above is also a possible definition of the class of minimum phase systems.

53. Assertion : Any can be represented as
Proof : suppose that has a zero at where , with all poles and remaining
zeros inside the unit circle.
Then

54. Where is minimum phase. Also then
min. ph.
All-pass
Since we can repeat this for all zeros outside the unit circle,
it follows that any system function can be represented

55. All the poles of and all the zeros of that are inside the unit circle
Conjugate reciprocals of the zeros of that are outside the unit circle

56. (1) All the zeros of that lie outside the unit circle
Poles to cancel conjugate reciprocal zeros in
Important point

57. Example – Assume is

58. a) Log magnitude
Fig (p. 284)

59. b) Phase
Fig (p. 284)

60. c) Group delay
Fig (p. 284)

61. a) Log magnitude
Fig (p. 285)

62. b) Phase
Fig (p. 285)

63. c) Group delay
Fig (p. 285)

64. a) Log magnitude
Fig (p. 286)

65. b) Phase
Fig (p. 286)

66. c) Group delay
Fig (p. 286)

67. Pairing and Ordering
SNR depends upon how zeros are paired with poles and on the order of the second-order (NS!)2 sections.
Many possible orderings and pairing
Optimization
Dynamic Programming
Random Search

68. Rule of Thumb
Pair highest Q pole with closest zero. This is last section.
(2) Repeat (1) until all poles and zeros have been paired.
(3) For unity gain filters, distribute gain as scaling multipliers.

69. Fig (p. 395)

70. Fig. 6.56(a), (b) (p. 404) Infinite-precision model
Linear-noise model for scaled system
Fig. 6.56(a), (b) (p. 404)

71. Linear-noise model with noise sources combined
Fig (c) (p. 404)

72. Pole-zero plot for Fig. 6.56
Fig. 6.57(p. 407)

73. Frequency-response functions for example system
Fig (a), (b), (c)
(p. 408)

74. Fig (d), (e), (f)
(p. 409)

75. Limit Cycles Example Ideal Fig. 6.61(p. 414) with quantization
before addition
with quantization
after addition

76. Assume and is quantized(rounded) to 4-bits
Assume initial rest condition and

78. Response of Fig. 6.61 system to an impulse
Fig. 6.62(p. 415)
If a=-1/2 , what about our statistical model?

79. Thus, we have demonstrated the existence of periodic oscillations (limit cycles) for first-order systems.
In these cases, the poles at z=±a appear to be at z=±1 when the zero input response falls into a “dead band”.

80. In the limit cycle or
; size of dead band

81. More complicated behavior for second-order systems
Conditions on coefficients for existence of limit cycles.
In cascade systems a limit cycle in an early system gets filtered by the later sections.
Limit cycles also can occur because of overflow.

82. FIR filters have no limit cycle problems!
Other structures are less sensitive and have no limit cycle problems
Wave filters
Lattice filters
Orthogonal filters