Properties of Downsampling and Upsampling

Lecture #10 Multi-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

0 1 2 3 1 ⋱ ⋱1 ⋱0 ⋱1 ⋱2 ⋱ ⋮0 = = = ⋱ ℎ 2 ℎ −1 ℎ −0 ℎ −1 ⋱ 𝑥𝑥 −1 ⋱ ℎ ℎ ℎ − ℎ − ⋱ 𝑥𝑥 −1 𝑦𝑦 3 2 1 0 2 𝐻𝐻𝐻𝐻 𝑈𝑈Λ𝑈𝑈 𝑥𝑥 ⋱ ℎ ℎ ℎ ℎ − ⋱ 𝑥𝑥 ⋱ ℎ ℎ ℎ ℎ ⋱ 𝑥𝑥 ECE 6534, Chapter 3 ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋮ 4 The DTFT

 Eigenvalue decomposition . The eigenvalue decomposition of a Toeplitz matrix is

0 1 2 3 1 ⋱ ⋱1 ⋱0 ⋱1 ⋱2 ⋱ ⋮0 = = = ⋱ ℎ 2 ℎ −1 ℎ −0 ℎ −1 ⋱ 𝑥𝑥 −1 ⋱ ℎ ℎ ℎ − ℎ − ⋱ 𝑥𝑥 −1 𝑦𝑦 3 2 1 0 2 𝐻𝐻𝐻𝐻 𝑈𝑈Λ𝑈𝑈 𝑥𝑥 ⋱ ℎ ℎ ℎ ℎ − ⋱ 𝑥𝑥 ⋱ ℎ ℎ ℎ ℎ ⋱ 𝑥𝑥 ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋮

= The DTFT Operator

𝑈𝑈 = −1 ∗ 𝑈𝑈 𝑈𝑈

ECE 6534, Chapter 3 5 The DTFT

 Eigenvalue decomposition . The eigenvalue decomposition of a Toeplitz matrix is

0 1 2 3 1 ⋱ ⋱1 ⋱0 ⋱1 ⋱2 ⋱ ⋮0 = = = ⋱ ℎ 2 ℎ −1 ℎ −0 ℎ −1 ⋱ 𝑥𝑥 −1 ⋱ ℎ ℎ ℎ − ℎ − ⋱ 𝑥𝑥 −1 𝑦𝑦 3 2 1 0 2 𝐻𝐻𝐻𝐻 𝑈𝑈Λ𝑈𝑈 𝑥𝑥 ⋱ ℎ ℎ ℎ ℎ − ⋱ 𝑥𝑥 ⋱ ℎ ℎ ℎ ℎ ⋱ 𝑥𝑥 . = ⋱ =⋱ ⋱ ⋱ ⋱ ⋱= ⋮ . −1 −1 −1 −1 𝑧𝑧So what𝐻𝐻𝐻𝐻𝐻𝐻 is ?𝑈𝑈 Λ𝐻𝐻𝑈𝑈 𝑈𝑈ΛG𝑈𝑈 𝑈𝑈ΛV𝑈𝑈 𝑈𝑈ΛHΛ𝐺𝐺Λ𝑉𝑉𝑈𝑈 Λ

ECE 6534, Chapter 3 6 The DTFT

 Eigenvalue decomposition . The eigenvalue decomposition of a Toeplitz matrix is

0 1 2 3 1 ⋱ ⋱1 ⋱0 ⋱1 ⋱2 ⋱ ⋮0 = = = ⋱ ℎ 2 ℎ −1 ℎ −0 ℎ −1 ⋱ 𝑥𝑥 −1 ⋱ ℎ ℎ ℎ − ℎ − ⋱ 𝑥𝑥 −1 𝑦𝑦 3 2 1 0 2 𝐻𝐻𝐻𝐻 𝑈𝑈Λ𝑈𝑈 𝑥𝑥 ⋱ ℎ ℎ ℎ ℎ − ⋱ 𝑥𝑥 ⋱ ℎ ℎ ℎ ℎ ⋱ 𝑥𝑥 . When⋱ does⋱ the inverse⋱ of ⋱filter exist?⋱ ⋱ ⋮ . How do you compute the pseudo𝑯𝑯-inverse of ? . How do you compute the Weiner deconvolution𝑯𝑯 of ? 𝑯𝑯

ECE 6534, Chapter 3 7 The DTFT

 Eigenvalue decomposition . The eigenvalue decomposition of a Toeplitz matrix is

0 1 2 3 1 ⋱ ⋱1 ⋱0 ⋱1 ⋱2 ⋱ ⋮0 = = = ⋱ ℎ 2 ℎ −1 ℎ −0 ℎ −1 ⋱ 𝑥𝑥 −1 ⋱ ℎ ℎ ℎ − ℎ − ⋱ 𝑥𝑥 −1 𝑦𝑦 3 2 1 0 2 𝐻𝐻𝐻𝐻 𝑈𝑈Λ𝑈𝑈 𝑥𝑥 ⋱ ℎ ℎ ℎ ℎ − ⋱ 𝑥𝑥 ⋱ ℎ ℎ ℎ ℎ ⋱ 𝑥𝑥 . When⋱ does⋱ the inverse⋱ of ⋱filter exist?⋱ ⋱ ⋮ — When the frequency domain is all non-zero values 𝑯𝑯 . How do you compute the pseudo-inverse of ? — Only invert non-zero values in the frequency domain 𝑯𝑯 . How do you compute the Weiner deconvolution of ? — Perform a Tikhonov regularized inverse in frequency domain 𝑯𝑯 ECE 6534, Chapter 3 8 Exercise

 Exercise: An allpass filter satisfies

= 1 𝑗𝑗𝑗𝑗 What property must by matrix satisfy𝐻𝐻 to𝑒𝑒 be an allpass filter?

ECE 6534, Chapter 3 9 Exercise

 Exercise: An allpass filter satisfies

= 1 𝑗𝑗𝑗𝑗 What property must by matrix satisfy𝐻𝐻 to𝑒𝑒 be an allpass filter?

Answer: The magnitudes of the eigenvalues must be equal to 1.

ECE 6534, Chapter 3 10 The DFT The DFT

 Diagonalization of a shift

… 2 1 0 1 2 3 …

𝑥𝑥 − 𝑥𝑥 − 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 Toeplitz Matrix DTFT Operator 0 0 0 0 1 ⋱ ⋱1 ⋱0 ⋱0 ⋱0 ⋱ ⋮0 = = = ⋱ 0 1 0 0 ⋱ 𝑥𝑥 −1 ⋱ ⋱ 𝑥𝑥 −1 𝑦𝑦 0 0 1 0 2 𝐻𝐻𝐻𝐻 𝑈𝑈Λ𝑈𝑈 𝑥𝑥 ⋱ ⋱ 𝑥𝑥 ⋱ ⋱ 𝑥𝑥 ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋮ ECE 6534, Chapter 3 12 The DFT

 Diagonalization of a circular shift

0 1 2 3 4 5

𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 𝑥𝑥 Circulant Matrix DFT Matrix 0 0 0 0 0 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 𝑥𝑥 2 = = = 0 0 1 0 0 0 𝑥𝑥 3 𝑥𝑥 −1 𝑦𝑦 0 0 0 1 0 0 4 𝐻𝐻𝐻𝐻 𝑈𝑈Λ𝑈𝑈 𝑥𝑥 0 0 0 0 1 0 𝑥𝑥 5 𝑥𝑥 𝑥𝑥 ECE 6534, Chapter 3 13 The DFT

 Circular convolution

= ,

𝑥𝑥 ∗ ℎ 𝑛𝑛 � 𝑥𝑥𝑘𝑘ℎ𝑚𝑚𝑚𝑚𝑚𝑚 𝑛𝑛−𝑘𝑘 𝑁𝑁 𝑘𝑘∈ℤ

DFT Matrix [0] [5] [4] [3] [2] [1] 0 [1] [0] [5] [4] [3] [2] 1 ℎ[2] ℎ[1] ℎ[0] ℎ[5] ℎ[4] ℎ[3] 𝑥𝑥 2 = = = ℎ[3] ℎ[2] ℎ[1] ℎ[0] ℎ[5] ℎ[4] 𝑥𝑥 3 ℎ ℎ ℎ ℎ ℎ ℎ 𝑥𝑥 −1 𝑦𝑦 [4] [3] [2] [1] [0] [5] 4 𝐻𝐻𝐻𝐻 𝑈𝑈Λ𝑈𝑈 𝑥𝑥 ℎ[5] ℎ[4] ℎ[3] ℎ[2] ℎ[1] ℎ[0] 𝑥𝑥 5 ℎ ℎ ℎ ℎ ℎ ℎ 𝑥𝑥 ℎ ℎ ℎ ℎ ℎ ℎ 𝑥𝑥

ECE 6534, Chapter 3 14 The DFT

 The DFT Matrix

1 1 1 1 1 1 1 1 2 3 ⋯ 𝑁𝑁−1 = 1 𝑊𝑊2 𝑊𝑊4 𝑊𝑊6 ⋯ 𝑊𝑊2 𝑁𝑁−1 𝑊𝑊 𝑊𝑊 𝑊𝑊 ⋯ 𝑊𝑊 𝐹𝐹 3 6 9 3 𝑁𝑁−1 𝑁𝑁 1 𝑊𝑊 𝑊𝑊 𝑊𝑊 ⋯ 𝑊𝑊( ) ⋮ 𝑁𝑁⋮ −1 2 ⋮𝑁𝑁−1 3 ⋮𝑁𝑁−1 ⋱ 𝑁𝑁−1⋮ 𝑁𝑁−1 𝑊𝑊 𝑊𝑊 𝑊𝑊 ⋯ 𝑊𝑊 Makes matrix unitary ( = ) ∗ −1 = 𝑈𝑈 𝑈𝑈 2𝜋𝜋 −𝑗𝑗 𝑁𝑁 𝑊𝑊 𝑒𝑒 ECE 6534, Chapter 3 15 Exercise

 Question: What property must matrices (filters) satisfy to have a zero group delay (i.e., zero phase)? Show this with matrices.

ECE 6534, Chapter 3 16 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 — = ∗ 𝐻𝐻 𝑈𝑈Λ𝑈𝑈

Real if symmetric

ECE 6534, Chapter 3 17 The Graph Fourier Transform Graph Spectrum

 For a given graph, there exists a shift matrix

𝑥𝑥1 𝑥𝑥4

𝑥𝑥5 Graph Fourier Transform 𝑥𝑥2 𝑥𝑥3 0 0 1 0 0 0 1 1 0 0 0 0 1 2 = 0 1 0 0 0 0 𝑥𝑥 3 = 0 1 0 0 0 0 𝑥𝑥 4 −1 𝑦𝑦 0 0 0 0 1 0 𝑥𝑥 5 𝑈𝑈Λ𝑈𝑈 𝑥𝑥 𝑥𝑥

ECE 6534, Chapter 3 𝑥𝑥 19 Graph Spectrum

 Question: What are graph frequency components?

𝑥𝑥1 𝑥𝑥4

ECE 6534, Chapter 3 𝑥𝑥 20 Multi-rate Signal Processing Downsampling and Upsampling Multirate signal processing

 Question: What is multirate signal processing?

ECE 6534, Chapter 3 22 Multirate signal processing

 Periodically Shift-Varying Systems . A discrete-time system T is called periodically shift-varying of order ( , ) when, for any integer and input , 𝐿𝐿 𝑀𝑀 𝑘𝑘 𝑥𝑥 = = ′ 𝑦𝑦 =𝑇𝑇 𝑥𝑥 ⇒ 𝑦𝑦 = 𝑌𝑌 𝑥𝑥푥 ′ ′ 𝑥𝑥𝑛𝑛 𝑥𝑥𝑛𝑛−𝐿𝐿𝐿𝐿 𝑦𝑦𝑛𝑛 𝑦𝑦𝑛𝑛−𝑀𝑀𝑀𝑀 . That is, if I shift the input by , I shift the output by

𝐿𝐿 𝑀𝑀

ECE 6534, Chapter 3 23 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]

2 𝐷𝐷2 1 1 0 0 0 0 0 ⋮ 1 ⋮0 ⋱ ⋱0 ⋱0 ⋱1 ⋱0 ⋱0 ⋱0 ⋱ 𝑥𝑥 −0 = 𝑦𝑦 −1 ⋱ 0 0 0 0 1 0 ⋱ 𝑥𝑥 −1 𝑦𝑦 2 ⋱ 0 0 0 0 0 0 ⋱ 𝑥𝑥 2 𝑦𝑦 ⋱ ⋱ 𝑥𝑥[3] 𝑦𝑦 ⋱ ⋱ 𝑥𝑥 ⋮ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ 𝑥𝑥 ⋮ ECE 6534, Chapter 3 24 Multirate signal processing

 Question: When I downsample… . What occurs in time? . What occurs in frequency?

ECE 6534, Chapter 3 25 Multirate signal processing

 Question: When I downsample… . What occurs in time? — Answer: Condense in time (effectively) . What occurs in frequency? — Answer: Expand in frequency (with possible aliasing)

ECE 6534, Chapter 3 26 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]

Downsample by 2

Image from Martin Vertelli’s notes

ECE 6534, Chapter 3 27 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]

𝑦𝑦 = 𝐷𝐷𝑁𝑁𝑥𝑥 𝑛𝑛 𝑁𝑁𝑁𝑁 𝑦𝑦 𝑥𝑥1 / = 𝑁𝑁−1 𝑘𝑘 1 𝑁𝑁 𝑌𝑌 𝑧𝑧 � 𝑋𝑋 𝑊𝑊𝑁𝑁 𝑧𝑧 𝑁𝑁 𝑛𝑛=0

ECE 6534, Chapter 3 28 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]

2 2 1 0𝑈𝑈 0 0 2 ⋮ 1 ⋱ ⋱0 ⋱0 ⋱0 ⋱0 ⋱ ⋮ 1 𝑦𝑦 −0 ⋱ 0 1 0 0 ⋱ 𝑥𝑥 −0 = 𝑦𝑦 −1 ⋱ 0 0 0 0 ⋱ 𝑥𝑥 −1 𝑦𝑦 2 ⋱ 0 0 1 0 ⋱ 𝑥𝑥 2 𝑦𝑦 3 ⋱ 0 0 0 0 ⋱ 𝑥𝑥[3] 𝑦𝑦 ⋱ ⋱ 𝑥𝑥 𝑦𝑦 ⋱ ⋱ 𝑥𝑥 ⋮ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋮ ECE 6534, Chapter 3 29 Multirate signal processing

 Question: When I upsampling… . What occurs in time? . What occurs in frequency?

ECE 6534, Chapter 3 30 Multirate signal processing

 Question: When I upsampling… . What occurs in time? — Answer: Expand in time (effectively) . What occurs in frequency? — Answer: Condense in frequency

ECE 6534, Chapter 3 31 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]

Upsample by 2

Image from Martin Vertelli’s notes

ECE 6534, Chapter 3 32 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]

𝑁𝑁 𝑦𝑦 𝑈𝑈 𝑥𝑥 , for = / 0 , otherwise𝑛𝑛 𝑥𝑥𝑛𝑛 𝑁𝑁 ∈ ℤ 𝑦𝑦𝑛𝑛 � 𝑁𝑁 = 𝑁𝑁 𝑌𝑌 𝑧𝑧 𝑋𝑋 𝑧𝑧

ECE 6534, Chapter 3 33 Multi-rate Signal Processing Upsampling and downsampling Multirate signal processing

 Question: . What is the adjoint of downsampling?

. What is the adjoint of upsampling?

ECE 6534, Chapter 3 35 Multirate signal processing

 Question: . What is the adjoint of downsampling? — Answer: = ∗ 𝐷𝐷𝑁𝑁 𝑈𝑈𝑁𝑁

. What is the adjoint of upsampling? — Answer: = ∗ 𝑈𝑈𝑁𝑁 𝐷𝐷𝑁𝑁

ECE 6534, Chapter 3 36 Multirate signal processing

 Question: . What is the = ? (reminder: matrix operations are right to left) ∗ 𝐷𝐷𝑁𝑁𝐷𝐷𝑁𝑁𝑥𝑥

. What does the result mean?

ECE 6534, Chapter 3 37 Multirate signal processing

 Question: . What is the = ? (reminder: matrix operations are right to left) — Answer: ∗ = = 𝐷𝐷𝑁𝑁𝐷𝐷𝑁𝑁𝑥𝑥 ∗ 𝐷𝐷𝑁𝑁𝐷𝐷𝑁𝑁𝑥𝑥 𝐷𝐷𝑁𝑁𝑈𝑈𝑁𝑁𝑥𝑥 𝑥𝑥

. What does the result mean? — Answer: — is the right inverse of — is the right inverse of 𝑈𝑈𝑁𝑁 𝐷𝐷𝑁𝑁 — ∗ is a 1-tight frame 𝐷𝐷𝑁𝑁 𝐷𝐷𝑁𝑁 𝐷𝐷𝑁𝑁

ECE 6534, Chapter 3 38 Multirate signal processing

 Properties of Downsampling and Upsampling . Relationship between upsampling and downsampling

= ∗ . Upsampling followed by downsampling𝑈𝑈𝑁𝑁 𝐷𝐷𝑁𝑁

𝐷𝐷𝑁𝑁𝑈𝑈𝑁𝑁 𝐼𝐼

𝑥𝑥

𝑈𝑈2𝑥𝑥

𝐷𝐷2𝑈𝑈2𝑥𝑥 ECE 6534, Chapter 3 39 Multirate signal processing

 Properties of Downsampling and Upsampling . Relationship between upsampling and downsampling

= ∗ . Downsampling followed by upsampling𝑈𝑈𝑁𝑁 𝐷𝐷𝑁𝑁

= (projection operator)

𝑈𝑈𝑁𝑁𝐷𝐷𝑁𝑁 𝑃𝑃

𝑥𝑥

𝐷𝐷2𝑥𝑥

𝑈𝑈2𝐷𝐷2𝑥𝑥 ECE 6534, Chapter 3 40 Multirate signal processing

 Properties of Downsampling and Upsampling . Relationship between upsampling and downsampling

= ∗ . Downsampling followed by upsampling𝑈𝑈𝑁𝑁 𝐷𝐷𝑁𝑁

= (projection operator)

𝑈𝑈𝑁𝑁𝐷𝐷𝑁𝑁 𝑃𝑃

𝑥𝑥

𝐷𝐷2𝑥𝑥

𝑈𝑈2𝐷𝐷2𝑥𝑥 ECE 6534, Chapter 3 41 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)

𝑥𝑥

𝑈𝑈3𝑥𝑥

𝐷𝐷2𝑈𝑈3𝑥𝑥

𝑥𝑥

𝐷𝐷2𝑥𝑥

ECE 6534,3 Chapter2 3 𝑈𝑈 𝐷𝐷 𝑥𝑥 42 Multi-rate Signal Processing Filtering with downsampling and upsampling Multirate signal processing

 Question . Why incorporate filtering?

ECE 6534, Chapter 3 44 Multirate signal processing

 Example (from Martin Veterlli’s notes)

Original signal Downsampled by 4 (aliasing)

ECE 6534, Chapter 3 45 Multirate signal processing

 Example (from Martin Veterlli’s notes)

Downsampled THEN Filtered THEN filtered (aliasing) downsampled

ECE 6534, Chapter 3 46 Multirate signal processing

 Properties of Downsampling and Upsampling . Filtering followed by downsampling

3 1 0 0 0 0 0 1 0 0 0 0 0 ⋮ 2 ⋱ ⋱0 ⋱0 ⋱1 ⋱0 ⋱0 ⋱0 ⋱ ⋱ ⋱2 ⋱1 ⋱0 ⋱0 ⋱0 ⋱0 ⋱ 𝑥𝑥 −1 = ⋱ 0 0 0 0 1 0 ⋱ ⋱ 𝑔𝑔 3 𝑔𝑔 2 1 0 0 0 ⋱ 𝑥𝑥 −0 ⋱ ⋱ ⋱ 𝑔𝑔 𝑔𝑔 𝑔𝑔 ⋱ 𝑥𝑥 − 𝑦𝑦 0 0 0 0 0 0 0 3 2 1 0 0 1 ⋱ ⋱ ⋱ 𝑔𝑔 𝑔𝑔 𝑔𝑔 𝑔𝑔 ⋱ 𝑥𝑥 2 ⋱ ⋱ ⋱ 𝑔𝑔 𝑔𝑔 𝑔𝑔 𝑔𝑔 ⋱ 𝑥𝑥 ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ 𝑥𝑥 Downsample across columns of G ⋮

𝑥𝑥 𝐺𝐺 𝑧𝑧 ↓ 𝑦𝑦 ECE 6534, Chapter 3 47 Multirate signal processing

 Properties of Downsampling and Upsampling . Filtering followed by downsampling

3 1 0 0 0 0 0 ⋮ 2 ⋱ ⋱3 ⋱2 ⋱1 ⋱0 ⋱0 ⋱0 ⋱ 𝑥𝑥 −1 = = = ⋱ 𝑔𝑔0 𝑔𝑔0 3 2 1 0 ⋱ 𝑥𝑥 −0 ⋱ 𝑔𝑔 𝑔𝑔 𝑔𝑔 𝑔𝑔 ⋱ 𝑥𝑥 − −1 𝑦𝑦 0 0 0 0 3 2 1 𝐻𝐻𝐻𝐻 𝑈𝑈Λ𝑈𝑈 𝑥𝑥 ⋱ 𝑔𝑔 𝑔𝑔 𝑔𝑔 𝑔𝑔 ⋱ 𝑥𝑥 2 No longer ⋱ 𝑔𝑔 𝑔𝑔 ⋱ 𝑥𝑥 a DFT matrix ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ 𝑥𝑥 ⋮

𝑥𝑥 𝐺𝐺 𝑧𝑧 ↓ 𝑦𝑦 ECE 6534, Chapter 3 48 Multirate signal processing

 Example (from Martin Veterlli’s notes)

Upsampled by 4 Upsampled by 4 THEN Original filtered

ECE 6534, Chapter 3 49 Multirate signal processing

 Properties of Downsampling and Upsampling . Upsampling followed by filtering

1 0 0 0 3 1 0 0 0 0 0 ⋱ ⋱0 ⋱0 ⋱0 ⋱0 ⋱ ⋮ 2 ⋱ ⋱2 ⋱1 ⋱0 ⋱0 ⋱0 ⋱0 ⋱ ⋱ 0 1 0 0 ⋱ 𝑥𝑥 −1 = ⋱ 𝑔𝑔 3 𝑔𝑔 2 1 0 0 0 ⋱ ⋱ 0 0 0 0 ⋱ 𝑥𝑥 −0 ⋱ 𝑔𝑔 𝑔𝑔 𝑔𝑔 ⋱ ⋱ ⋱ 𝑥𝑥 − 𝑦𝑦 0 3 2 1 0 0 0 0 1 0 1 ⋱ 𝑔𝑔 𝑔𝑔 𝑔𝑔 𝑔𝑔 ⋱ ⋱ 0 0 0 0 ⋱ 𝑥𝑥 2 ⋱ 𝑔𝑔 𝑔𝑔 𝑔𝑔 𝑔𝑔 ⋱ ⋱ ⋱ 𝑥𝑥 ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ 𝑥𝑥 ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋮ Upsample across columns of x

𝑥𝑥 ↑ 𝐺𝐺 𝑧𝑧 𝑦𝑦 ECE 6534, Chapter 3 50 Multirate signal processing

 Properties of Downsampling and Upsampling . Upsampling followed by filtering

2 1 0 0 0 0 0 0⋮ 𝑥𝑥 −1 ⋱ ⋱2 ⋱1 ⋱0 ⋱0 ⋱0 ⋱0 ⋱ = ⋱ 𝑔𝑔 𝑔𝑔 ⋱ 0 3 2 1 0 0 0 𝑥𝑥 − ⋱ 𝑔𝑔 𝑔𝑔 𝑔𝑔 ⋱ 0 𝑦𝑦 0 3 2 1 0 0 0 ⋱ 𝑔𝑔 𝑔𝑔 𝑔𝑔 𝑔𝑔 ⋱ 𝑥𝑥 1 ⋱ 𝑔𝑔 𝑔𝑔 𝑔𝑔 𝑔𝑔 ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ 𝑥𝑥 ⋮

𝑥𝑥 ↑ 𝐺𝐺 𝑧𝑧 𝑦𝑦 ECE 6534, Chapter 3 51 Multirate signal processing

 Properties of Downsampling and Upsampling . Upsampling followed by filtering

1 0 0 0 3 1 0 0 0 0 0 ⋱ ⋱0 ⋱0 ⋱0 ⋱0 ⋱ ⋮ 2 ⋱ ⋱2 ⋱1 ⋱0 ⋱0 ⋱0 ⋱0 ⋱ ⋱ 0 1 0 0 ⋱ 𝑥𝑥 −1 = ⋱ 𝑔𝑔 3 𝑔𝑔 2 1 0 0 0 ⋱ ⋱ 0 0 0 0 ⋱ 𝑥𝑥 −0 ⋱ 𝑔𝑔 𝑔𝑔 𝑔𝑔 ⋱ ⋱ ⋱ 𝑥𝑥 − 𝑦𝑦 0 3 2 1 0 0 0 0 1 0 1 ⋱ 𝑔𝑔 𝑔𝑔 𝑔𝑔 𝑔𝑔 ⋱ ⋱ 0 0 0 0 ⋱ 𝑥𝑥 2 ⋱ 𝑔𝑔 𝑔𝑔 𝑔𝑔 𝑔𝑔 ⋱ ⋱ ⋱ 𝑥𝑥 ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ 𝑥𝑥 ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋮ Downsample across rows of G

𝑥𝑥 ↑ 𝐺𝐺 𝑧𝑧 𝑦𝑦 ECE 6534, Chapter 3 52 Multirate signal processing

 Properties of Downsampling and Upsampling . Upsampling followed by filtering

1 0 0 0 2 ⋱ ⋱2 ⋱0 ⋱0 ⋱0 ⋱ ⋱ 𝑔𝑔 ⋱ ⋮ 1 0 1 0 0 𝑥𝑥 − = ⋱ 𝑔𝑔 𝑔𝑔 ⋱ 0 = = 0 2 0 0 𝑥𝑥 − ⋱ 𝑔𝑔 ⋱ 1 −1 𝑦𝑦 0 0 1 0 𝑥𝑥 𝐻𝐻𝐻𝐻 𝑈𝑈Λ𝑈𝑈 𝑥𝑥 ⋱ 𝑔𝑔 𝑔𝑔 ⋱ 2 0 0 2 0 𝑥𝑥 ⋱ 𝑔𝑔 ⋱ 𝑥𝑥 ⋱ 𝑔𝑔 𝑔𝑔 ⋱ ⋮ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱

𝑥𝑥 ↑ 𝐺𝐺 𝑧𝑧 𝑦𝑦 ECE 6534, Chapter 3 53 Multirate signal processing

 Properties of Downsampling and Upsampling . Upsampling and downsampling with filters

2 2

𝑥𝑥 ↑ 𝐺𝐺 𝑧𝑧 𝐺𝐺 𝑧𝑧 ↓ 𝑦𝑦

𝐻𝐻 𝑧𝑧  How is this used?

ECE 6534, Chapter 3 54 Multi-rate Signal Processing Re-ordering downsampling and upsampling Multirate signal processing

 Properties of Downsampling and Upsampling . Flipping filtering with downsampling and upsampling Upsample the filter

2 = 2 2 𝑥𝑥 ↓ 𝐺𝐺 𝑧𝑧 𝑦𝑦 𝑥𝑥 𝐺𝐺 𝑧𝑧 ↓ 𝑦𝑦 = =

2 2 ↑2 𝑦𝑦 𝐺𝐺𝐷𝐷 𝑥𝑥 𝑦𝑦 My 𝐷𝐷notation𝐺𝐺 𝑥𝑥

ECE 6534, Chapter 3 56 Multirate signal processing

 Properties of Downsampling and Upsampling . Flipping filtering with downsampling and upsampling Upsample the filter

2 = 2 2 𝑥𝑥 ↓ 𝐺𝐺 𝑧𝑧 𝑦𝑦 𝑥𝑥 𝐺𝐺 𝑧𝑧 ↓ 𝑦𝑦 = = = ∗ 2 2 ↑2 2 2 2 𝑦𝑦 𝐺𝐺𝐷𝐷 𝑥𝑥 𝑦𝑦 My 𝐷𝐷notation𝐺𝐺 𝑥𝑥 𝐷𝐷 𝑈𝑈 𝐺𝐺𝑈𝑈 𝑥𝑥  How is this work linear algebraically? . = = = ∗ ∗ 𝑦𝑦 𝐺𝐺𝐷𝐷2𝑥𝑥 𝐺𝐺𝑈𝑈2𝑥𝑥 𝐷𝐷2𝑈𝑈2𝐺𝐺𝑈𝑈2𝑥𝑥 Identity Downsample x Upsample across rows of G

ECE 6534, Chapter 3 57 Multirate signal processing

 Properties of Downsampling and Upsampling . Flipping filtering with downsampling and upsampling Upsample the filter

2 = 2 2 𝑥𝑥 ↓ 𝐺𝐺 𝑧𝑧 𝑦𝑦 𝑥𝑥 𝐺𝐺 𝑧𝑧 ↓ 𝑦𝑦 = = = ∗ 2 2 ↑2 2 2 2 𝑦𝑦 𝐺𝐺𝐷𝐷 𝑥𝑥 𝑦𝑦 My 𝐷𝐷notation𝐺𝐺 𝑥𝑥 𝐷𝐷 𝑈𝑈 𝐺𝐺𝑈𝑈 𝑥𝑥  How is this work linear algebraically? . = = = ∗ ∗ 𝑦𝑦 𝐺𝐺𝐷𝐷2𝑥𝑥 𝐺𝐺𝑈𝑈2𝑥𝑥 𝐷𝐷2𝑈𝑈2𝐺𝐺𝑈𝑈2𝑥𝑥 Upsample across Downsample x Upsample columns of G across rows of G

ECE 6534, Chapter 3 58 Multirate signal processing

 Properties of Downsampling and Upsampling . Flipping filtering with downsampling and upsampling Upsample the filter

2 = 2 2 𝑥𝑥 ↓ 𝐺𝐺 𝑧𝑧 𝑦𝑦 𝑥𝑥 𝐺𝐺 𝑧𝑧 ↓ 𝑦𝑦 = = = ∗ 2 2 ↑2 2 2 2 𝑦𝑦 𝐺𝐺𝐷𝐷 𝑥𝑥 𝑦𝑦 My 𝐷𝐷notation𝐺𝐺 𝑥𝑥 𝐷𝐷 𝑈𝑈 𝐺𝐺𝑈𝑈 𝑥𝑥  How is this work linear algebraically? 1 1 1 2 1 0 0 0 2 1 2 1 1 0 2 0 2 . = = = 2 1 0 0 1 0 3 2 1 3 2 0 1 0 3 𝑦𝑦 4 4

Downsample x Upsampled across rows of G ECE 6534, Chapter 3 59 Multirate signal processing

 Properties of Downsampling and Upsampling . Flipping filtering with downsampling and upsampling Upsample the filter

2 = 2 2 𝑥𝑥 ↓ 𝐺𝐺 𝑧𝑧 𝑦𝑦 𝑥𝑥 𝐺𝐺 𝑧𝑧 ↓ 𝑦𝑦 = = = ∗ 2 2 ↑2 2 2 2 𝑦𝑦 𝐺𝐺𝐷𝐷 𝑥𝑥 𝑦𝑦 My 𝐷𝐷notation𝐺𝐺 𝑥𝑥 𝐷𝐷 𝑈𝑈 𝐺𝐺𝑈𝑈 𝑥𝑥  How is this work linear algebraically? 1 1 0 2 0 2 . = 2 0 1 0 3 𝑦𝑦 4

Upsampled across rows of G ECE 6534, Chapter 3 60 Multirate signal processing

 Properties of Downsampling and Upsampling . Flipping filtering with downsampling and upsampling Upsample the filter

2 = 2 2 𝑥𝑥 ↓ 𝐺𝐺 𝑧𝑧 𝑦𝑦 𝑥𝑥 𝐺𝐺 𝑧𝑧 ↓ 𝑦𝑦 = = = ∗ 2 2 ↑2 2 2 2 𝑦𝑦 𝐺𝐺𝐷𝐷 𝑥𝑥 𝑦𝑦 My 𝐷𝐷notation𝐺𝐺 𝑥𝑥 𝐷𝐷 𝑈𝑈 𝐺𝐺𝑈𝑈 𝑥𝑥  How is this work linear algebraically? 1 0 1 1 0 0 0 0 0 1 0 2 0 2 . = 0 0 1 0 0 1 2 0 1 0 3 𝑦𝑦 0 0 4 Upsampled across rows of G Identity Upsample across columns of ECE 6534, Chapter 3 𝐺𝐺𝐷𝐷2 61 Multirate signal processing

 Properties of Downsampling and Upsampling . Flipping filtering with downsampling and upsampling Upsample the filter

2 = 2 2 𝑥𝑥 ↓ 𝐺𝐺 𝑧𝑧 𝑦𝑦 𝑥𝑥 𝐺𝐺 𝑧𝑧 ↓ 𝑦𝑦 = = = ∗ 2 2 ↑2 2 2 2 𝑦𝑦 𝐺𝐺𝐷𝐷 𝑥𝑥 𝑦𝑦 My 𝐷𝐷notation𝐺𝐺 𝑥𝑥 𝐷𝐷 𝑈𝑈 𝐺𝐺𝑈𝑈 𝑥𝑥  How is this work linear algebraically? 1 0 2 0 1 1 0 0 0 0 0 0 0 2 . = 0 0 1 0 2 0 1 0 3 𝑦𝑦 0 0 0 0 4

Downsample by 2 Upsampled across rows ECE 6534, Chapter 3 and columns of G 62 Multirate signal processing

 Properties of Downsampling and Upsampling . Flipping filtering with downsampling and upsampling Upsample the filter = 𝑁𝑁 𝑥𝑥 ↓ 𝑁𝑁 𝐺𝐺 𝑧𝑧 𝑦𝑦 𝑥𝑥 𝐺𝐺 𝑧𝑧 ↓ 𝑁𝑁 𝑦𝑦 = = = ∗ 𝑦𝑦 𝐺𝐺𝐷𝐷𝑁𝑁𝑥𝑥 𝑦𝑦 𝐷𝐷𝑁𝑁𝐺𝐺↑𝑁𝑁𝑥𝑥 𝐷𝐷𝑁𝑁 𝑈𝑈𝑁𝑁𝐺𝐺𝑈𝑈𝑁𝑁 𝑥𝑥

ECE 6534, Chapter 3 63 Multirate signal processing

 Properties of Downsampling and Upsampling . Flipping filtering with downsampling and upsampling Upsample the filter = 𝑁𝑁 𝑥𝑥 ↓ 𝑁𝑁 𝐺𝐺 𝑧𝑧 𝑦𝑦 𝑥𝑥 ↓ 𝑁𝑁 𝑦𝑦 = = = 𝐺𝐺 𝑧𝑧 ∗ 𝑦𝑦 𝐺𝐺𝐷𝐷𝑁𝑁𝑥𝑥 𝐷𝐷𝑁𝑁 𝑈𝑈𝑁𝑁𝐺𝐺𝑈𝑈𝑁𝑁 𝑥𝑥 𝐷𝐷𝑁𝑁𝐺𝐺↑𝑁𝑁𝑥𝑥 = 𝑁𝑁 𝑥𝑥 𝐺𝐺 𝑧𝑧 ↑ 𝑁𝑁 𝑦𝑦 𝑥𝑥 ↑ 𝑁𝑁 𝐺𝐺 𝑧𝑧 𝑦𝑦 = = = ∗ 𝑦𝑦 𝑈𝑈𝑁𝑁𝐺𝐺𝐺𝐺 𝑈𝑈𝑁𝑁𝐺𝐺𝑈𝑈𝑁𝑁 𝑈𝑈𝑁𝑁𝑥𝑥 𝐺𝐺↑𝑁𝑁𝑈𝑈𝑁𝑁𝑥𝑥

ECE 6534, Chapter 3 64 Multirate signal processing

 Properties of Downsampling and Upsampling . Computationally inefficient

2 2

. 𝑥𝑥Computationally↑ efficient𝐺𝐺 𝑧𝑧 𝐺𝐺 𝑧𝑧 ↓ 𝑦𝑦

2 2 𝑁𝑁 𝑁𝑁 𝑥𝑥 𝐺𝐺 𝑧𝑧 ↑ ↓ 𝐺𝐺 𝑧𝑧 𝑦𝑦 . This concept is also used in the design of polyphase filters

ECE 6534, Chapter 3 66 Example 1

67 Multirate signal processing