1 Basic Multirate Operations
2 Interconnection of Building Blocks
1.1 Decimation and Interpolation
1.2 Digital Filter Banks
Basic Multi-rate Operations: Decimation and Interpolation
Building blocks for traditional single-rate digital signal
processing: mu

ENEE630 P t-1
Part
Part-
Supplement
TreeTree-based Filter Banks and
Multiresolution Analysis
ECE Department
C
Univ. of Maryland, College Park
Updated 10/2012 by Prof. Min Wu.
bb eng md ed (select ENEE630) min @eng md ed
bb.eng.umd.edu
ENEE630); [email protected]

7 M-channel Maximally Decimated Filter Bank
Appendix: Detailed Derivations
Multi-rate Signal Processing
7. M-channel Maximally Decmiated Filter Banks
Electrical & Computer Engineering
University of Maryland, College Park
Acknowledgment: ENEE630 slides wer

1 Basic Multirate Operations
2 Interconnection of Building Blocks
Multi-rate Signal Processing
Electrical & Computer Engineering
University of Maryland, College Park
Acknowledgment: ENEE630 slides were based on class notes developed by
Profs. K.J. Ray Liu

6 Quadrature Mirror Filter (QMF) Bank
Appendix: Detailed Derivations
Multi-rate Signal Processing
6. Quadrature Mirror Filter (QMF) Bank
Electrical & Computer Engineering
University of Maryland, College Park
Acknowledgment: ENEE630 slides were based on cl

3 The Polyphase Representation
Appendix: Detailed Derivations
Multi-rate Signal Processing
3. The Polyphase Representation
Electrical & Computer Engineering
University of Maryland, College Park
Acknowledgment: ENEE630 slides were based on class notes deve

8 General Alias-Free Conditions for Filter Banks
9 Tree Structured Filter Banks and Multiresolution Analysis
Appendix: Detailed Derivations
Multi-rate Signal Processing
8. General Alias-Free Conditions for Filter Banks
9. Tree Structured Filter Banks and

4 Multistage Implementations
5 Some Multirate Applications
Multi-rate Signal Processing
4. Multistage Implementations
5. Multirate Application: Subband Coding
Electrical & Computer Engineering
University of Maryland, College Park
Acknowledgment: ENEE630 s

ENEE630 ADSP
1.
RECITATION 5 w/ solution
Ver.201209
Consider the structures shown in Fig. RI.1, with input transforms and lter responses as
indicated. Sketch the quantities Y0 (ej ) and Y1 (ej ).
Figure RI.1:
Solution:
Comment: For a down-sampled signal,

ENEE630 ADSP
1.
RECITATION 4 w/ solution
Ver.201209
Consider the structures shown in Fig. RI.1, with input transforms and lter responses as
indicated. Sketch the quantities Y0 (ej ) and Y1 (ej ).
Figure RI.1:
Solution:
Comment: For a down-sampled signal,

ENEE630 ADSP
1.
RECITATION Part I
Consider the structures shown in Fig. RI.1, with input transforms and lter responses as
indicated. Sketch the quantities Y0 (ej ) and Y1 (ej ). (Decimator-expander)
Figure RI.1:
2.
For each case shown in the Fig. RI.2, pr

ENEE630 ADSP
1.
RECITATION 3 w/ solution
Ver.201209
Consider the structures shown in Fig. RI.1, with input transforms and lter responses as
indicated. Sketch the quantities Y0 (ej ) and Y1 (ej ).
Figure RI.1:
Solution:
Comment: For a down-sampled signal,

ENEE 630 Solution to Homework 3
Solution to 1 It can be found that
1
X(z) = z 1 [H0 (z)F0 (z) + H1 (z)F1 (z)]X(z) +
2
1 1
+ z [H0 (z)F0 (z) H1 (z)F1 (z)]X(z)
2
Or, call it simply
X(z) = T (z)X(z) + A(z)X(z)
Since H1 (z) = H0 (z), F0 (z) = H0 (z), F1 (z) =

ENEE630 ADSP
RECITATION 1 w/ solution
Material covered: Transforms and properties of the transform function of a
discrete-time LTI system.
1.
Dene H(z) =
+
n .
n= h(n)z
(1). If h(n) = an u(n), what is H(z)?
(2). If h(n) = an u(n 1), what is H(z)?
(3). Giv

ENEE 630 Homework 41
Material Covered: Multirate Maximally/Non-maximally Decimated System,
Alias Free, Distortion Function,
AC matrix, P matrix
Problem 1 Suppose the M-channel maximally decimated QMF bank is alias-free, and let T(z)
be the distortion func

ENEE 630 Solution to Homework 4
Solution to 1 Since the system is alias-free,
M T (z)
H(z)f (z) =
0
.
.
0
In other words,
H0 (z)F0 (z) + + HM 1 (z)FM 1 (z) = M T (z)
H0 (zW i )F0 (z) + + HM 1 (zW i )FM 1 (z) = 0,
i = 1, , M 1
If replace z with zW M i , i

ENEE 630 Fall 2012 Homework 11
Material covered: Basic multirate operators
and interconnection of building blocks.
Problem 1 For the system in Fig. P-1, nd an expression for y(n) in terms of x(n). Simplify
the expression as best as you can.
x(n)
2
3
3
y(n

ENEE 630 Homework 31
Material Covered: Uniform and Nonuniform QMF Filters, Perfect
Reconstruction
Problem 1 In a two-channel QMF bank, if the lters are related as
H1 (z) = H0 (z), F0 (z) = H0 (z), F1 (z) = H1 (z)
and if H0 (z) is chosen to have real coeci

ENEE 630 Solution to Homework 2
Problem 1 solution:
(a) Since W is a constant matrix, the output of W can be written as
Y0 (z)
1 1
1
Y0 (z)
2 Y (z)
Y1 (z) = 1 W W 1
1 W2 W
Y2 (z)
Y2 (z)
where
1 1
1
W = 1 W W2
1 W2 W
To get F0 (z) , set Y1 (z) and Y

ENEE 630 Homework 21
Material Covered: Polyphase Representation, Uniform DFT Filter Banks,
Analysis/Synthesis system
Problem 1 Consider the structure shown in Fig P-1(a), where W is the 3 3 DFT matrix.
This is a three channel synthesis bank with three lte

ENEE 630 Solution to Homework 1
Problem 1 solution:
x(n)
2
3
2
y(n)
2
3
y(n)
relatively prime
x(n)
2
3
3
doing nothing
x(n)
2
y(n)
2
y[n] =
1
[1 + (1)n ]x[n]
2
Problem 2 solution:
Z-transform of an does not converge except for a = 0, so the FT for x[n] =