multi-rate signal processing - university of utah
TRANSCRIPT
Lecture #10Multi-rate Signal Processing
2
The DTFT
The DTFT Change of Basis Oftentimes, it is easier
to process in a different basis
Hence, we may want to know the diagonalization of a Toeplitz matrix
4ECE 6534, Chapter 3
๐ฆ๐ฆ =
โฑ โฑ โฑ โฑ โฑ โฑโฑ โ 0 โ โ1 โ โ2 โ โ3 โฑโฑ โ 1 โ 0 โ โ1 โ โ2 โฑโฑ โ 2 โ 1 โ 0 โ โ1 โฑโฑ โ 3 โ 2 โ 1 โ 0 โฑโฑ โฑ โฑ โฑ โฑ โฑ
โฎ๐ฅ๐ฅ โ1๐ฅ๐ฅ 0๐ฅ๐ฅ 1๐ฅ๐ฅ 2โฎ
= ๐ป๐ป๐ฅ๐ฅ = ๐๐ฮ๐๐โ1๐ฅ๐ฅ
The DTFT Eigenvalue decomposition The eigenvalue decomposition of a Toeplitz matrix is
5ECE 6534, Chapter 3
๐ฆ๐ฆ =
โฑ โฑ โฑ โฑ โฑ โฑโฑ โ 0 โ โ1 โ โ2 โ โ3 โฑโฑ โ 1 โ 0 โ โ1 โ โ2 โฑโฑ โ 2 โ 1 โ 0 โ โ1 โฑโฑ โ 3 โ 2 โ 1 โ 0 โฑโฑ โฑ โฑ โฑ โฑ โฑ
โฎ๐ฅ๐ฅ โ1๐ฅ๐ฅ 0๐ฅ๐ฅ 1๐ฅ๐ฅ 2โฎ
= ๐ป๐ป๐ฅ๐ฅ = ๐๐ฮ๐๐โ1๐ฅ๐ฅ
๐๐ = The DTFT Operator
๐๐โ1 = ๐๐โ
The DTFT Eigenvalue decomposition The eigenvalue decomposition of a Toeplitz matrix is
๐ง๐ง = ๐ป๐ป๐ป๐ป๐ป๐ป = ๐๐ฮ๐ป๐ป๐๐โ1๐๐ฮG๐๐โ1๐๐ฮV๐๐โ1 = ๐๐ฮHฮ๐บ๐บฮ๐๐๐๐โ1
So what is ฮ?
6ECE 6534, Chapter 3
๐ฆ๐ฆ =
โฑ โฑ โฑ โฑ โฑ โฑโฑ โ 0 โ โ1 โ โ2 โ โ3 โฑโฑ โ 1 โ 0 โ โ1 โ โ2 โฑโฑ โ 2 โ 1 โ 0 โ โ1 โฑโฑ โ 3 โ 2 โ 1 โ 0 โฑโฑ โฑ โฑ โฑ โฑ โฑ
โฎ๐ฅ๐ฅ โ1๐ฅ๐ฅ 0๐ฅ๐ฅ 1๐ฅ๐ฅ 2โฎ
= ๐ป๐ป๐ฅ๐ฅ = ๐๐ฮ๐๐โ1๐ฅ๐ฅ
The DTFT Eigenvalue decomposition The eigenvalue decomposition of a Toeplitz matrix is
When does the inverse of filter ๐ฏ๐ฏ exist?
How do you compute the pseudo-inverse of ๐ฏ๐ฏ?
How do you compute the Weiner deconvolution of ๐ฏ๐ฏ?
7ECE 6534, Chapter 3
๐ฆ๐ฆ =
โฑ โฑ โฑ โฑ โฑ โฑโฑ โ 0 โ โ1 โ โ2 โ โ3 โฑโฑ โ 1 โ 0 โ โ1 โ โ2 โฑโฑ โ 2 โ 1 โ 0 โ โ1 โฑโฑ โ 3 โ 2 โ 1 โ 0 โฑโฑ โฑ โฑ โฑ โฑ โฑ
โฎ๐ฅ๐ฅ โ1๐ฅ๐ฅ 0๐ฅ๐ฅ 1๐ฅ๐ฅ 2โฎ
= ๐ป๐ป๐ฅ๐ฅ = ๐๐ฮ๐๐โ1๐ฅ๐ฅ
The DTFT Eigenvalue decomposition The eigenvalue decomposition of a Toeplitz matrix is
When does the inverse of filter ๐ฏ๐ฏ exist?โ When the frequency domain is all non-zero values
How do you compute the pseudo-inverse of ๐ฏ๐ฏ?โ Only invert non-zero values in the frequency domain
How do you compute the Weiner deconvolution of ๐ฏ๐ฏ?โ Perform a Tikhonov regularized inverse in frequency domain
8ECE 6534, Chapter 3
๐ฆ๐ฆ =
โฑ โฑ โฑ โฑ โฑ โฑโฑ โ 0 โ โ1 โ โ2 โ โ3 โฑโฑ โ 1 โ 0 โ โ1 โ โ2 โฑโฑ โ 2 โ 1 โ 0 โ โ1 โฑโฑ โ 3 โ 2 โ 1 โ 0 โฑโฑ โฑ โฑ โฑ โฑ โฑ
โฎ๐ฅ๐ฅ โ1๐ฅ๐ฅ 0๐ฅ๐ฅ 1๐ฅ๐ฅ 2โฎ
= ๐ป๐ป๐ฅ๐ฅ = ๐๐ฮ๐๐โ1๐ฅ๐ฅ
Exercise Exercise: An allpass filter satisfies
๐ป๐ป ๐๐๐๐๐๐ = 1
What property must by matrix satisfy to be an allpass filter?
9ECE 6534, Chapter 3
Exercise Exercise: An allpass filter satisfies
๐ป๐ป ๐๐๐๐๐๐ = 1
What property must by matrix satisfy to be an allpass filter?
Answer: The magnitudes of the eigenvalues must be equal to 1.
10ECE 6534, Chapter 3
The DFT
The DFT Diagonalization of a shift
12ECE 6534, Chapter 3
โฆโฆ๐ฅ๐ฅ 0๐ฅ๐ฅ โ1๐ฅ๐ฅ โ2 ๐ฅ๐ฅ 1 ๐ฅ๐ฅ 2 ๐ฅ๐ฅ 3
๐ฆ๐ฆ =
โฑ โฑ โฑ โฑ โฑ โฑโฑ 0 0 0 0 โฑโฑ 1 0 0 0 โฑโฑ 0 1 0 0 โฑโฑ 0 0 1 0 โฑโฑ โฑ โฑ โฑ โฑ โฑ
โฎ๐ฅ๐ฅ โ1๐ฅ๐ฅ 0๐ฅ๐ฅ 1๐ฅ๐ฅ 2โฎ
= ๐ป๐ป๐ฅ๐ฅ = ๐๐ฮ๐๐โ1๐ฅ๐ฅ
DTFT OperatorToeplitz Matrix
The DFT Diagonalization of a circular shift
13ECE 6534, Chapter 3
๐ฅ๐ฅ 0 ๐ฅ๐ฅ 1 ๐ฅ๐ฅ 2 ๐ฅ๐ฅ 3 ๐ฅ๐ฅ 4 ๐ฅ๐ฅ 5
๐ฆ๐ฆ =
0 0 0 0 0 11 0 0 0 0 00 1 0 0 0 00 0 1 0 0 00 0 0 1 0 00 0 0 0 1 0
๐ฅ๐ฅ 0๐ฅ๐ฅ 1๐ฅ๐ฅ 2๐ฅ๐ฅ 3๐ฅ๐ฅ 4๐ฅ๐ฅ 5
= ๐ป๐ป๐ฅ๐ฅ = ๐๐ฮ๐๐โ1๐ฅ๐ฅ
Circulant Matrix DFT Matrix
The DFT Circular convolution
๐ฅ๐ฅ โ โ ๐๐ = ๏ฟฝ๐๐โโค
๐ฅ๐ฅ๐๐โ๐๐๐๐๐๐ ๐๐โ๐๐,๐๐
14ECE 6534, Chapter 3
๐ฆ๐ฆ =
โ[0] โ[5] โ[4] โ[3] โ[2] โ[1]โ[1] โ[0] โ[5] โ[4] โ[3] โ[2]โ[2] โ[1] โ[0] โ[5] โ[4] โ[3]โ[3] โ[2] โ[1] โ[0] โ[5] โ[4]โ[4] โ[3] โ[2] โ[1] โ[0] โ[5]โ[5] โ[4] โ[3] โ[2] โ[1] โ[0]
๐ฅ๐ฅ 0๐ฅ๐ฅ 1๐ฅ๐ฅ 2๐ฅ๐ฅ 3๐ฅ๐ฅ 4๐ฅ๐ฅ 5
= ๐ป๐ป๐ฅ๐ฅ = ๐๐ฮ๐๐โ1๐ฅ๐ฅ
DFT Matrix
The DFT The DFT Matrix
15ECE 6534, Chapter 3
๐น๐น =1๐๐
1 1 1 1 โฏ 11 ๐๐ ๐๐2 ๐๐3 โฏ ๐๐๐๐โ1
1 ๐๐2 ๐๐4 ๐๐6 โฏ ๐๐2 ๐๐โ1
1 ๐๐3 ๐๐6 ๐๐9 โฏ ๐๐3 ๐๐โ1
โฎ โฎ โฎ โฎ โฑ โฎ1 ๐๐๐๐โ1 ๐๐2 ๐๐โ1 ๐๐3 ๐๐โ1 โฏ ๐๐(๐๐โ1) ๐๐โ1
๐๐ = ๐๐โ๐๐2๐๐๐๐
Makes matrix unitary (๐๐โ = ๐๐โ1)
Exercise Question: What property must matrices (filters) satisfy to have a zero group delay (i.e., zero phase)? Show this with matrices.
16ECE 6534, Chapter 3
Exercise Question: What property must matrices (filters) satisfy to have a zero group delay (i.e., zero phase)? Show this with matrices.
Answer: The matrix must be symmetric
This is because โ ๐ป๐ป = ๐๐ฮ๐๐โ
17ECE 6534, Chapter 3
Real if symmetric
The Graph Fourier Transform
Graph Spectrum For a given graph, there exists a shift matrix
19ECE 6534, Chapter 3
๐ฅ๐ฅ1
๐ฅ๐ฅ2๐ฅ๐ฅ3
๐ฅ๐ฅ4
๐ฅ๐ฅ5
๐ฆ๐ฆ =
0 0 1 0 0 01 0 0 0 0 10 1 0 0 0 00 1 0 0 0 00 0 0 0 1 0
๐ฅ๐ฅ 1๐ฅ๐ฅ 2๐ฅ๐ฅ 3๐ฅ๐ฅ 4๐ฅ๐ฅ 5
= ๐๐ฮ๐๐โ1๐ฅ๐ฅ
Graph Fourier Transform
Graph Spectrum Question: What are graph frequency components?
20ECE 6534, Chapter 3
๐ฅ๐ฅ1
๐ฅ๐ฅ2๐ฅ๐ฅ3
๐ฅ๐ฅ4
๐ฅ๐ฅ5
๐ฆ๐ฆ =
0 0 1 0 0 01 0 0 0 0 10 1 0 0 0 00 1 0 0 0 00 0 0 0 1 0
๐ฅ๐ฅ 1๐ฅ๐ฅ 2๐ฅ๐ฅ 3๐ฅ๐ฅ 4๐ฅ๐ฅ 5
= ๐๐ฮ๐๐โ1๐ฅ๐ฅ
Graph Fourier Transform
Multi-rate Signal ProcessingDownsampling and Upsampling
Multirate signal processing Question: What is multirate signal processing?
22ECE 6534, Chapter 3
Multirate signal processing Periodically Shift-Varying Systems A discrete-time system T is called periodically shift-varying of order (๐ฟ๐ฟ,๐๐) when,
for any integer ๐๐ and input ๐ฅ๐ฅ,
That is, if I shift the input by ๐ฟ๐ฟ, I shift the output by ๐๐
23ECE 6534, Chapter 3
๐ฆ๐ฆ = ๐๐ ๐ฅ๐ฅ โ ๐ฆ๐ฆโฒ = ๐๐ ๐ฅ๐ฅ๐ฅ
๐ฅ๐ฅ๐๐โฒ = ๐ฅ๐ฅ๐๐โ๐ฟ๐ฟ๐๐ ๐ฆ๐ฆ๐๐โฒ = ๐ฆ๐ฆ๐๐โ๐๐๐๐
Multirate signal processing Downsampling by 2 Periodically shift-varying of order (2,1)
[if I shift the input by 2, I shift the output by 1]
24ECE 6534, Chapter 3
โฎ๐ฆ๐ฆ โ1๐ฆ๐ฆ 0๐ฆ๐ฆ 1๐ฆ๐ฆ 2โฎ
=
โฑ โฑ โฑ โฑ โฑ โฑ โฑ โฑโฑ 1 0 0 0 0 0 โฑโฑ 0 0 1 0 0 0 โฑโฑ 0 0 0 0 1 0 โฑโฑ 0 0 0 0 0 0 โฑโฑ โฑ โฑ โฑ โฑ โฑ โฑ โฑ
โฎ๐ฅ๐ฅ โ2๐ฅ๐ฅ โ1๐ฅ๐ฅ 0๐ฅ๐ฅ 1๐ฅ๐ฅ 2๐ฅ๐ฅ[3]โฎ
๐ท๐ท2
Multirate signal processing Question: When I downsampleโฆ What occurs in time?
What occurs in frequency?
25ECE 6534, Chapter 3
Multirate signal processing Question: When I downsampleโฆ What occurs in time?
โ Answer: Condense in time (effectively)
What occurs in frequency? โ Answer: Expand in frequency (with possible aliasing)
26ECE 6534, Chapter 3
Multirate signal processing Downsampling by 2 Periodically shift-varying of order (2,1)
[if I shift the input by 2, I shift the output by 1]
27ECE 6534, Chapter 3
Image from Martin Vertelliโs notes
Downsample by 2
Multirate signal processing Downsampling by N Periodically shift-varying of order (N,1)
[if I shift the input by N, I shift the output by 1]
28ECE 6534, Chapter 3
๐ฆ๐ฆ๐๐ = ๐ฅ๐ฅ๐๐๐๐
๐๐ ๐ง๐ง =1๐๐๏ฟฝ๐๐=0
๐๐โ1
๐๐ ๐๐๐๐๐๐๐ง๐ง1/๐๐
๐ฆ๐ฆ = ๐ท๐ท๐๐๐ฅ๐ฅ
Multirate signal processing Upsampling by 2 Periodically shift-varying of order (1,2)
[if I shift the input by 1, I shift the output by 2]
29ECE 6534, Chapter 3
โฎ๐ฆ๐ฆ โ2๐ฆ๐ฆ โ1๐ฆ๐ฆ 0๐ฆ๐ฆ 1๐ฆ๐ฆ 2๐ฆ๐ฆ 3โฎ
=
โฑ โฑ โฑ โฑ โฑ โฑโฑ 1 0 0 0 โฑโฑ 0 0 0 0 โฑโฑ 0 1 0 0 โฑโฑ 0 0 0 0 โฑโฑ 0 0 1 0 โฑโฑ 0 0 0 0 โฑโฑ โฑ โฑ โฑ โฑ โฑ
โฎ๐ฅ๐ฅ โ2๐ฅ๐ฅ โ1๐ฅ๐ฅ 0๐ฅ๐ฅ 1๐ฅ๐ฅ 2๐ฅ๐ฅ[3]โฎ
๐๐2
Multirate signal processing Question: When I upsamplingโฆ What occurs in time?
What occurs in frequency?
30ECE 6534, Chapter 3
Multirate signal processing Question: When I upsamplingโฆ What occurs in time?
โ Answer: Expand in time (effectively)
What occurs in frequency? โ Answer: Condense in frequency
31ECE 6534, Chapter 3
Multirate signal processing Upsampling by 2 Periodically shift-varying of order (1,2)
[if I shift the input by 1, I shift the output by 2]
32ECE 6534, Chapter 3
Image from Martin Vertelliโs notes
Upsample by 2
Multirate signal processing Upsampling by N Periodically shift-varying of order (1,N)
[if I shift the input by 1, I shift the output by N]
33ECE 6534, Chapter 3
๐ฆ๐ฆ๐๐ = ๏ฟฝ๐ฅ๐ฅ๐๐/๐๐ , for๐๐๐๐โ โค
0 , otherwise
๐๐ ๐ง๐ง = ๐๐ ๐ง๐ง๐๐
๐ฆ๐ฆ = ๐๐๐๐๐ฅ๐ฅ
Multi-rate Signal ProcessingUpsampling and downsampling
Multirate signal processing Question: What is the adjoint of downsampling?
What is the adjoint of upsampling?
35ECE 6534, Chapter 3
Multirate signal processing Question: What is the adjoint of downsampling?
โ Answer: ๐ท๐ท๐๐โ = ๐๐๐๐
What is the adjoint of upsampling? โ Answer: ๐๐๐๐โ = ๐ท๐ท๐๐
36ECE 6534, Chapter 3
Multirate signal processing Question: What is the ๐ท๐ท๐๐๐ท๐ท๐๐โ ๐ฅ๐ฅ = ? (reminder: matrix operations are right to left)
What does the result mean?
37ECE 6534, Chapter 3
Multirate signal processing Question: What is the ๐ท๐ท๐๐๐ท๐ท๐๐โ ๐ฅ๐ฅ = ? (reminder: matrix operations are right to left)
โ Answer: ๐ท๐ท๐๐๐ท๐ท๐๐โ๐ฅ๐ฅ = ๐ท๐ท๐๐๐๐๐๐๐ฅ๐ฅ = ๐ฅ๐ฅ
What does the result mean?โ Answer:โ ๐๐๐๐ is the right inverse of ๐ท๐ท๐๐โ ๐ท๐ท๐๐โ is the right inverse of ๐ท๐ท๐๐โ ๐ท๐ท๐๐ is a 1-tight frame
38ECE 6534, Chapter 3
Multirate signal processing Properties of Downsampling and Upsampling Relationship between upsampling and downsampling
Upsampling followed by downsampling
39ECE 6534, Chapter 3
๐๐๐๐ = ๐ท๐ท๐๐โ
๐ท๐ท๐๐๐๐๐๐ = ๐ผ๐ผ
๐ฅ๐ฅ
๐๐2๐ฅ๐ฅ
๐ท๐ท2๐๐2๐ฅ๐ฅ
Multirate signal processing Properties of Downsampling and Upsampling Relationship between upsampling and downsampling
Downsampling followed by upsampling
40ECE 6534, Chapter 3
๐๐๐๐ = ๐ท๐ท๐๐โ
๐๐๐๐๐ท๐ท๐๐ = ๐๐ (projection operator)
๐ฅ๐ฅ
๐ท๐ท2๐ฅ๐ฅ
๐๐2๐ท๐ท2๐ฅ๐ฅ
Multirate signal processing Properties of Downsampling and Upsampling Relationship between upsampling and downsampling
Downsampling followed by upsampling
41ECE 6534, Chapter 3
๐๐๐๐ = ๐ท๐ท๐๐โ
๐๐๐๐๐ท๐ท๐๐ = ๐๐ (projection operator)
๐ฅ๐ฅ
๐ท๐ท2๐ฅ๐ฅ
๐๐2๐ท๐ท2๐ฅ๐ฅ
Multirate signal processing Properties of Downsampling and Upsampling Upsampling by N and downsampling by M commute when N and M have no
common factors (i.e., N = 3 and M = 2)
42ECE 6534, Chapter 3
๐ฅ๐ฅ
๐๐3๐ฅ๐ฅ
๐ท๐ท2๐๐3๐ฅ๐ฅ
๐ฅ๐ฅ
๐ท๐ท2๐ฅ๐ฅ
๐๐3๐ท๐ท2๐ฅ๐ฅ
Multi-rate Signal ProcessingFiltering with downsampling and upsampling
Multirate signal processing Question Why incorporate filtering?
44ECE 6534, Chapter 3
Multirate signal processing Example (from Martin Veterlliโs notes)
45ECE 6534, Chapter 3
Original signal Downsampled by 4 (aliasing)
Multirate signal processing Example (from Martin Veterlliโs notes)
46ECE 6534, Chapter 3
Downsampled THEN filtered (aliasing)
Filtered THEN downsampled
Multirate signal processing Properties of Downsampling and Upsampling Filtering followed by downsampling
47ECE 6534, Chapter 3
๐ฆ๐ฆ =
โฑ โฑ โฑ โฑ โฑ โฑ โฑ โฑโฑ 1 0 0 0 0 0 โฑโฑ 0 0 1 0 0 0 โฑโฑ 0 0 0 0 1 0 โฑโฑ 0 0 0 0 0 0 โฑโฑ โฑ โฑ โฑ โฑ โฑ โฑ โฑ
โฑ โฑ โฑ โฑ โฑ โฑ โฑ โฑโฑ ๐๐ 1 ๐๐ 0 0 0 0 0 โฑโฑ ๐๐ 2 ๐๐ 1 ๐๐ 0 0 0 0 โฑโฑ ๐๐ 3 ๐๐ 2 ๐๐ 1 ๐๐ 0 0 0 โฑโฑ 0 ๐๐ 3 ๐๐ 2 ๐๐ 1 ๐๐ 0 0 โฑโฑ โฑ โฑ โฑ โฑ โฑ โฑ โฑ
โฎ๐ฅ๐ฅ โ3๐ฅ๐ฅ โ2๐ฅ๐ฅ โ1๐ฅ๐ฅ 0๐ฅ๐ฅ 1๐ฅ๐ฅ 2โฎ
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง โ 2
Downsample across columns of G
Multirate signal processing Properties of Downsampling and Upsampling Filtering followed by downsampling
48ECE 6534, Chapter 3
๐ฆ๐ฆ =
โฑ โฑ โฑ โฑ โฑ โฑ โฑ โฑโฑ ๐๐ 1 ๐๐ 0 0 0 0 0 โฑโฑ ๐๐ 3 ๐๐ 2 ๐๐ 1 ๐๐ 0 0 0 โฑโฑ 0 0 ๐๐ 3 ๐๐ 2 ๐๐ 1 ๐๐ 0 โฑโฑ 0 0 0 0 ๐๐ 3 ๐๐ 2 โฑโฑ โฑ โฑ โฑ โฑ โฑ โฑ โฑ
โฎ๐ฅ๐ฅ โ3๐ฅ๐ฅ โ2๐ฅ๐ฅ โ1๐ฅ๐ฅ 0๐ฅ๐ฅ 1๐ฅ๐ฅ 2โฎ
= ๐ป๐ป๐ฅ๐ฅ = ๐๐ฮ๐๐โ1๐ฅ๐ฅ
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง โ 2
No longer a DFT matrix
Multirate signal processing Example (from Martin Veterlliโs notes)
49ECE 6534, Chapter 3
Original
Upsampledby 4
Upsampledby 4 THEN
filtered
Multirate signal processing Properties of Downsampling and Upsampling Upsampling followed by filtering
50ECE 6534, Chapter 3
๐ฆ๐ฆ =
โฑ โฑ โฑ โฑ โฑ โฑ โฑ โฑโฑ ๐๐ 1 ๐๐ 0 0 0 0 0 โฑโฑ ๐๐ 2 ๐๐ 1 ๐๐ 0 0 0 0 โฑโฑ ๐๐ 3 ๐๐ 2 ๐๐ 1 ๐๐ 0 0 0 โฑโฑ 0 ๐๐ 3 ๐๐ 2 ๐๐ 1 ๐๐ 0 0 โฑโฑ โฑ โฑ โฑ โฑ โฑ โฑ โฑ
โฑ โฑ โฑ โฑ โฑ โฑโฑ 1 0 0 0 โฑโฑ 0 0 0 0 โฑโฑ 0 1 0 0 โฑโฑ 0 0 0 0 โฑโฑ 0 0 1 0 โฑโฑ 0 0 0 0 โฑโฑ โฑ โฑ โฑ โฑ โฑ
โฎ๐ฅ๐ฅ โ3๐ฅ๐ฅ โ2๐ฅ๐ฅ โ1๐ฅ๐ฅ 0๐ฅ๐ฅ 1๐ฅ๐ฅ 2โฎ
Upsample across columns of x
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
Multirate signal processing Properties of Downsampling and Upsampling Upsampling followed by filtering
51ECE 6534, Chapter 3
๐ฆ๐ฆ =
โฑ โฑ โฑ โฑ โฑ โฑ โฑ โฑโฑ ๐๐ 1 ๐๐ 0 0 0 0 0 โฑโฑ ๐๐ 2 ๐๐ 1 ๐๐ 0 0 0 0 โฑโฑ ๐๐ 3 ๐๐ 2 ๐๐ 1 ๐๐ 0 0 0 โฑโฑ 0 ๐๐ 3 ๐๐ 2 ๐๐ 1 ๐๐ 0 0 โฑโฑ โฑ โฑ โฑ โฑ โฑ โฑ โฑ
โฎ๐ฅ๐ฅ โ2
0๐ฅ๐ฅ โ1
0๐ฅ๐ฅ 0
0๐ฅ๐ฅ 1โฎ
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
Multirate signal processing Properties of Downsampling and Upsampling Upsampling followed by filtering
52ECE 6534, Chapter 3
๐ฆ๐ฆ =
โฑ โฑ โฑ โฑ โฑ โฑ โฑ โฑโฑ ๐๐ 1 ๐๐ 0 0 0 0 0 โฑโฑ ๐๐ 2 ๐๐ 1 ๐๐ 0 0 0 0 โฑโฑ ๐๐ 3 ๐๐ 2 ๐๐ 1 ๐๐ 0 0 0 โฑโฑ 0 ๐๐ 3 ๐๐ 2 ๐๐ 1 ๐๐ 0 0 โฑโฑ โฑ โฑ โฑ โฑ โฑ โฑ โฑ
โฑ โฑ โฑ โฑ โฑ โฑโฑ 1 0 0 0 โฑโฑ 0 0 0 0 โฑโฑ 0 1 0 0 โฑโฑ 0 0 0 0 โฑโฑ 0 0 1 0 โฑโฑ 0 0 0 0 โฑโฑ โฑ โฑ โฑ โฑ โฑ
โฎ๐ฅ๐ฅ โ3๐ฅ๐ฅ โ2๐ฅ๐ฅ โ1๐ฅ๐ฅ 0๐ฅ๐ฅ 1๐ฅ๐ฅ 2โฎ
Downsample across rows of G
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
Multirate signal processing Properties of Downsampling and Upsampling Upsampling followed by filtering
53ECE 6534, Chapter 3
๐ฆ๐ฆ =
โฑ โฑ โฑ โฑ โฑ โฑโฑ ๐๐ 1 0 0 0 โฑโฑ ๐๐ 2 ๐๐ 0 0 0 โฑโฑ 0 ๐๐ 1 0 0 โฑโฑ 0 ๐๐ 2 ๐๐ 0 0 โฑโฑ 0 0 ๐๐ 1 0 โฑโฑ 0 0 ๐๐ 2 ๐๐ 0 โฑโฑ โฑ โฑ โฑ โฑ โฑ
โฎ๐ฅ๐ฅ โ2๐ฅ๐ฅ โ1๐ฅ๐ฅ 0๐ฅ๐ฅ 1๐ฅ๐ฅ 2โฎ
= ๐ป๐ป๐ฅ๐ฅ = ๐๐ฮ๐๐โ1๐ฅ๐ฅ
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
Multirate signal processing Properties of Downsampling and Upsampling Upsampling and downsampling with filters
How is this used?
54ECE 6534, Chapter 3
๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2๐ฅ๐ฅ ๐ป๐ป ๐ง๐ง โ 2
๐ป๐ป ๐ง๐ง
Multi-rate Signal ProcessingRe-ordering downsampling and upsampling
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
56ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง2 โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
Upsample the filter
๐ฆ๐ฆ = ๐ป๐ป๐ท๐ท2๐ฅ๐ฅ
My notation
๐ฆ๐ฆ = ๐ท๐ท2๐ป๐ปโ2๐ฅ๐ฅ
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
How is this work linear algebraically? ๐ฆ๐ฆ = ๐ป๐ป๐ท๐ท2๐ฅ๐ฅ = ๐ป๐ป๐๐2โ๐ฅ๐ฅ = ๐ท๐ท2๐๐2๐ป๐ป๐๐2โ๐ฅ๐ฅ
57ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง2 โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
Upsample the filter
Downsample x Upsampleacross rows of G
Identity
๐ฆ๐ฆ = ๐ป๐ป๐ท๐ท2๐ฅ๐ฅ ๐ฆ๐ฆ = ๐ท๐ท2๐ป๐ปโ2๐ฅ๐ฅ = ๐ท๐ท2 ๐๐2๐ป๐ป๐๐2โ ๐ฅ๐ฅ
My notation
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
How is this work linear algebraically? ๐ฆ๐ฆ = ๐ป๐ป๐ท๐ท2๐ฅ๐ฅ = ๐ป๐ป๐๐2โ๐ฅ๐ฅ = ๐ท๐ท2๐๐2๐ป๐ป๐๐2โ๐ฅ๐ฅ
58ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง2 โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
Upsample the filter
๐ฆ๐ฆ = ๐ป๐ป๐ท๐ท2๐ฅ๐ฅ
My notation
Downsample x Upsampleacross rows of G
๐ฆ๐ฆ = ๐ท๐ท2๐ป๐ปโ2๐ฅ๐ฅ = ๐ท๐ท2 ๐๐2๐ป๐ป๐๐2โ ๐ฅ๐ฅ
Upsample across columns of G
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
How is this work linear algebraically?
๐ฆ๐ฆ = 1 22 1
1 0 0 00 0 1 0
1234
= 1 22 1
13 = 1 0 2 0
2 0 1 0
1234
59ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง2 โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
Upsample the filter
๐ฆ๐ฆ = ๐ป๐ป๐ท๐ท2๐ฅ๐ฅ
My notation
๐ฆ๐ฆ = ๐ท๐ท2๐ป๐ปโ2๐ฅ๐ฅ = ๐ท๐ท2 ๐๐2๐ป๐ป๐๐2โ ๐ฅ๐ฅ
Downsample x Upsampled across rows of G
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
How is this work linear algebraically?
๐ฆ๐ฆ = 1 0 2 02 0 1 0
1234
60ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง2 โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
Upsample the filter
๐ฆ๐ฆ = ๐ป๐ป๐ท๐ท2๐ฅ๐ฅ
My notation
๐ฆ๐ฆ = ๐ท๐ท2๐ป๐ปโ2๐ฅ๐ฅ = ๐ท๐ท2 ๐๐2๐ป๐ป๐๐2โ ๐ฅ๐ฅ
Upsampled across rows of G
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
How is this work linear algebraically?
๐ฆ๐ฆ = 1 0 0 00 0 1 0
1 00 00 10 0
1 0 2 02 0 1 0
1234
61ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง2 โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
Upsample the filter
๐ฆ๐ฆ = ๐ป๐ป๐ท๐ท2๐ฅ๐ฅ
My notation
๐ฆ๐ฆ = ๐ท๐ท2๐ป๐ปโ2๐ฅ๐ฅ = ๐ท๐ท2 ๐๐2๐ป๐ป๐๐2โ ๐ฅ๐ฅ
Upsampled across rows of GIdentity
Upsample across columns of ๐ป๐ป๐ท๐ท2
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
How is this work linear algebraically?
๐ฆ๐ฆ = 1 0 0 00 0 1 0
1 0 2 00 0 0 02 0 1 00 0 0 0
1234
62ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง2 โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
Upsample the filter
๐ฆ๐ฆ = ๐ป๐ป๐ท๐ท2๐ฅ๐ฅ
My notation
๐ฆ๐ฆ = ๐ท๐ท2๐ป๐ปโ2๐ฅ๐ฅ = ๐ท๐ท2 ๐๐2๐ป๐ป๐๐2โ ๐ฅ๐ฅ
Upsampled across rows and columns of G
Downsample by 2
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
63ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ ๐๐=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ ๐๐
Upsample the filter
๐ฆ๐ฆ = ๐ป๐ป๐ท๐ท๐๐๐ฅ๐ฅ ๐ฆ๐ฆ = ๐ท๐ท๐๐๐ป๐ปโ๐๐๐ฅ๐ฅ = ๐ท๐ท๐๐ ๐๐๐๐๐ป๐ป๐๐๐๐โ ๐ฅ๐ฅ
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
64ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง โ ๐๐ = ๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐โ ๐๐
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ ๐๐=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ ๐๐
Upsample the filter
๐ฆ๐ฆ = ๐ป๐ป๐ท๐ท๐๐๐ฅ๐ฅ = ๐ท๐ท๐๐ ๐๐๐๐๐ป๐ป๐๐๐๐โ ๐ฅ๐ฅ = ๐ท๐ท๐๐๐ป๐ปโ๐๐๐ฅ๐ฅ
๐ฆ๐ฆ = ๐๐๐๐๐ป๐ป๐ฅ๐ฅ = ๐๐๐๐๐ป๐ป๐๐๐๐โ ๐๐๐๐๐ฅ๐ฅ = ๐ป๐ปโ๐๐๐๐๐๐๐ฅ๐ฅ
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
Why is this useful? What does it do?
65ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง โ ๐๐ = ๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐โ ๐๐
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ ๐๐=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ ๐๐
Upsample the filter
๐ฆ๐ฆ = ๐ป๐ป๐ท๐ท๐๐๐ฅ๐ฅ = ๐ท๐ท๐๐ ๐๐๐๐๐ป๐ป๐๐๐๐โ ๐ฅ๐ฅ = ๐ท๐ท๐๐๐ป๐ปโ๐๐๐ฅ๐ฅ
๐ฆ๐ฆ = ๐๐๐๐๐ป๐ป๐ฅ๐ฅ = ๐๐๐๐๐ป๐ป๐๐๐๐โ ๐๐๐๐๐ฅ๐ฅ = ๐ป๐ปโ๐๐๐๐๐๐๐ฅ๐ฅ
Multirate signal processing Properties of Downsampling and Upsampling Computationally inefficient
Computationally efficient
This concept is also used in the design of polyphase filters
66ECE 6534, Chapter 3
๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2๐ฅ๐ฅ ๐ป๐ป ๐ง๐ง โ 2
๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ 2๐ฅ๐ฅ ๐ป๐ป ๐ง๐ง๐๐โ 2
Example 1
67
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
68ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
๐๐2
๐๐ 3๐๐2
2๐๐๐๐2
๐๐3๐๐2
2๐๐
๐๐ ๐๐
1
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
69ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
๐๐2
๐๐ 3๐๐2
2๐๐๐๐2
๐๐3๐๐2
2๐๐
๐๐๐๐ ๐๐๐ฅ๐ฅ๐๐
0.5
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
70ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
๐๐2
๐๐ 3๐๐2
2๐๐๐๐2
๐๐3๐๐2
2๐๐
๐๐๐๐ ๐๐๐ฅ๐ฅ๐๐
Filter (gain: 1)0.5
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
71ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
๐๐2
๐๐ 3๐๐2
2๐๐๐๐2
๐๐3๐๐2
2๐๐
๐๐ ๐๐๐ฅ๐ฅ๐๐
0.5
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
72ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
๐๐2
๐๐ 3๐๐2
2๐๐๐๐2
๐๐3๐๐2
2๐๐
๐๐ ๐๐
1
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
73ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
๐๐2
๐๐ 3๐๐2
2๐๐๐๐2
๐๐3๐๐2
2๐๐
๐๐ ๐๐
1 Filter (gain: 1)
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
74ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
๐๐2
๐๐ 3๐๐2
2๐๐๐๐2
๐๐3๐๐2
2๐๐
๐๐๐๐ ๐๐๐ฅ๐ฅ๐๐
1
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
75ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
๐๐2
๐๐ 3๐๐2
2๐๐๐๐2
๐๐3๐๐2
2๐๐
๐๐ ๐๐๐ฅ๐ฅ๐๐
0.5
Example 2
76
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
77ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
๐๐2
๐๐ 3๐๐2
2๐๐๐๐2
๐๐3๐๐2
2๐๐
๐๐ ๐๐
1
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
78ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
๐๐2
๐๐ 3๐๐2
2๐๐๐๐2
๐๐3๐๐2
2๐๐
๐๐๐๐ ๐๐๐ฅ๐ฅ๐๐
0.5
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
79ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
๐๐2
๐๐ 3๐๐2
2๐๐๐๐2
๐๐3๐๐2
2๐๐
๐๐๐๐ ๐๐๐ฅ๐ฅ๐๐
0.5
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
80ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
๐๐2
๐๐ 3๐๐2
2๐๐๐๐2
๐๐3๐๐2
2๐๐
๐๐๐๐ ๐๐๐ฅ๐ฅ๐๐
Filter (gain: 1)0.5
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
81ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
๐๐2
๐๐ 3๐๐2
2๐๐๐๐2
๐๐3๐๐2
2๐๐
๐๐ ๐๐๐ฅ๐ฅ๐๐
0.5
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
82ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
๐๐2
๐๐ 3๐๐2
2๐๐๐๐2
๐๐3๐๐2
2๐๐
๐๐ ๐๐
1
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
83ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
๐๐2
๐๐ 3๐๐2
2๐๐๐๐2
๐๐3๐๐2
2๐๐
๐๐ ๐๐
1 Filter (gain: 1)
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
84ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
๐๐2
๐๐ 3๐๐2
2๐๐๐๐2
๐๐3๐๐2
2๐๐
๐๐๐๐ ๐๐
1
๐ฅ๐ฅ๐๐
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
85ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
๐๐2
๐๐ 3๐๐2
2๐๐๐๐2
๐๐3๐๐2
2๐๐
๐๐ ๐๐๐ฅ๐ฅ๐๐
0.5
Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
86ECE 6534, Chapter 3
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