1. Multirate Processing of Digital Signals: Fundamentals
For NTUEE VLSI Signal Processing Course
Instructor: 吳安宇教授
Updated on 4/4/2001

2. Outline Introduction Sampling Rate Conversion
Multistage Implementation
Practice Structure
Polyphase Implementation

3. ? Motivation Definition
More than one sampling rate (clock) are used in a system
Module 1 ?
Module 2
clock 1
clock 2

4. Conversion Approach Analog approach
Digital approach (multirate DSP system)

5. Analog Approach Advantages Simple Straightforward
Arbitrary sampling rate
Disadvantages
D/A & A/D converter are needed
Ideal (near perfect) lowpass filter is needed
Introduced noise and distortion

6. Digital Approach Sampling rate conversion Interpolation Decimation
Increase the sampling rate
Decimation
Decrease the sampling rate

7. Sampling Theory
If the highest frequency component in a signal is fmax, then the signal should be sampled at the rate of at least 2fmax for the samples to describe the signal completely, i.e.,
For Fs < 2fmax, alias occurs in the sampling process.  Alias Distortion (aliasing)

8. Aliasing
fmax Fs f
-Fs
X(f)

9. Interpolation by L L
h(m)

10. Interpolation by L L
h(m)

11. Decimation by M
h(m) M

12. Decimation by M
h(m) M

13. Conversion by a Rational Factor M/L
Cascade of two process L
h1(m)
h2(m) M
Interpolation by L
Decimation by M

14. Conversion by a Rational Factor M/L
A more efficiency implementation L
h (m) M

15. Multistage Implementation
h(m)
L1
h(m)
L2
LI
L1
h1(m)
L2
h2(m)

16. Multistage Implementation
Advantages
Reduce the complexity
Reduce storage devices (registers)
Simplify (relax) filter design problem
Reduce the finite wordlength effect
Disadvantages
Increase the control circuit
Difficulty in choosing I and best Lj for 1 i  I

17. Interpolated FIR (IFIR) Approach
Nothing to do with interpolation and decimation
Conceptually similar
Suitable for narrowband FIR filter design
LPF
HPF
BPF

18. Application: Interpolated FIR (IFIR)
Desired narrowband response
Assume required filter order is N.
Stretched filter
Required filter order is reduced to N/2.
Desired
Undesired
Interpolated version of stretched filter
Required filter order is still N/2.
Image suppresser
Required filter order is M.
Order (N/2+M) is needed to implement!
(N/2+M) << N for small M

19. Interpolated FIR (IFIR)
(a) G(z)
(a) G(z2)
(b) I(z)
(a) G(z2)I(z)

20. Interpolated FIR (IFIR)
Quantity
Compared
Filter order
Number of
Multipliers
Adders
Conventional
Method
233
117
IFIR Method
131 66
G(z)
I(z)
Total 6 4
268 70
137

21. Some Useful Operations
Duality and Transposition
A dual system is that performs a complementary operation to that of an original system, and it can be constructed form the original system through the process of transposition.
The transposition operation is one in which the direction of all branches in the network are reversed, and the roles of the input and output of the network are interchanged.

22. Duality and Transposition
z-1
z-1

23. Duality and Transposition
They are not true in time-varying system, but can be treated as sampling rate reverse process.
transposition L L
transposition
 M
 M
transposition
 M
h(n)
 M
h(n)
transposition
 M
h(n) L
 M
h(n)
 L

24. Practical Structure
Decimation
 M
h(n)
z-1
 M
z-1
 M
z-1
 M

25. Practical Structure
Interpolation L
h(n)
z-1 L L
z-1
z-1 L

26. Application: Polyphase FIR Filter
z-1
E0(zM)
E1(zM)
EM-1(zM)
Polyphase decomposition
h(n)

27. Polyphase FIR Filter Noble identity Noble identity E (zM) M E (z) M

28. Polyphase FIR Filter 3 z-1 3 z-3 z-1 3 z-1 E0(z3) E1(z3) E2(z3)
H (z) 3
z-1 3 h0 h1 h2 h3 h4 h5 h0
z-3 3 h3 h1 h4 h2 h5
z-1

29. Polyphase FIR Filter z-3 z-1 3 z-1 3 E0(z3) 3 3 E0(z) z-1 z-1

30. Structure Comparison Direct implementation Polyphase implementation 3
z-1 3 h0 h1 h2 h3 h4 h5
z-1 3 h0 h3 h1 h4 h2 h5
Direct implementation
Polyphase implementation