multi-rate signal processing - university of utah
TRANSCRIPT
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Lecture #10Multi-rate Signal Processing
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The DTFT
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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
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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
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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
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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
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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
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Exercise Exercise: An allpass filter satisfies
๐ป๐ป ๐๐๐๐๐๐ = 1
What property must by matrix satisfy to be an allpass filter?
9ECE 6534, Chapter 3
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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
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The DFT
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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
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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
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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
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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)
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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
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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
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The Graph Fourier Transform
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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
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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
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Multi-rate Signal ProcessingDownsampling and Upsampling
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Multirate signal processing Question: What is multirate signal processing?
22ECE 6534, Chapter 3
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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
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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
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Multirate signal processing Question: When I downsampleโฆ What occurs in time?
What occurs in frequency?
25ECE 6534, Chapter 3
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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
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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
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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
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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
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Multirate signal processing Question: When I upsamplingโฆ What occurs in time?
What occurs in frequency?
30ECE 6534, Chapter 3
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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
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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
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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
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Multi-rate Signal ProcessingUpsampling and downsampling
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Multirate signal processing Question: What is the adjoint of downsampling?
What is the adjoint of upsampling?
35ECE 6534, Chapter 3
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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
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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
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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
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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
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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
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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
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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
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Multi-rate Signal ProcessingFiltering with downsampling and upsampling
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Multirate signal processing Question Why incorporate filtering?
44ECE 6534, Chapter 3
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Multirate signal processing Example (from Martin Veterlliโs notes)
45ECE 6534, Chapter 3
Original signal Downsampled by 4 (aliasing)
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Multirate signal processing Example (from Martin Veterlliโs notes)
46ECE 6534, Chapter 3
Downsampled THEN filtered (aliasing)
Filtered THEN downsampled
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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
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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
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Multirate signal processing Example (from Martin Veterlliโs notes)
49ECE 6534, Chapter 3
Original
Upsampledby 4
Upsampledby 4 THEN
filtered
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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
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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
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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
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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
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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
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Multi-rate Signal ProcessingRe-ordering downsampling and upsampling
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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Example 1
67
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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
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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
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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
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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
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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
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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)
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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
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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
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Example 2
76
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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
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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
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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
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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
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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
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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
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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)
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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
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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
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Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling
86ECE 6534, Chapter 3
๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐ง๐๐ โ 2=๐ฅ๐ฅ ๐ฆ๐ฆ๐ป๐ป ๐ง๐งโ 2
๐๐2
๐๐ 3๐๐2
2๐๐๐๐2
๐๐3๐๐2
2๐๐
๐๐ ๐๐๐ฅ๐ฅ๐๐
0.5