Multi-rate Signal Processing - University of Utah

Lecture #10Multi-rate Signal Processing

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

𝑦𝑦 =

⋱ ⋱ ⋱ ⋱ ⋱ ⋱⋱ ℎ 0 ℎ −1 ℎ −2 ℎ −3 ⋱⋱ ℎ 1 ℎ 0 ℎ −1 ℎ −2 ⋱⋱ ℎ 2 ℎ 1 ℎ 0 ℎ −1 ⋱⋱ ℎ 3 ℎ 2 ℎ 1 ℎ 0 ⋱⋱ ⋱ ⋱ ⋱ ⋱ ⋱

⋮𝑥𝑥 −1𝑥𝑥 0𝑥𝑥 1𝑥𝑥 2⋮

= 𝐻𝐻𝑥𝑥 = 𝑈𝑈Λ𝑈𝑈−1𝑥𝑥

The DTFT Eigenvalue decomposition The eigenvalue decomposition of a Toeplitz matrix is


𝑦𝑦 =




𝑈𝑈 = The DTFT Operator

𝑈𝑈−1 = 𝑈𝑈∗


𝑧𝑧 = 𝐻𝐻𝐻𝐻𝐻𝐻 = 𝑈𝑈Λ𝐻𝐻𝑈𝑈−1𝑈𝑈ΛG𝑈𝑈−1𝑈𝑈ΛV𝑈𝑈−1 = 𝑈𝑈ΛHΛ𝐺𝐺Λ𝑉𝑉𝑈𝑈−1

So what is Λ?


𝑦𝑦 =





When does the inverse of filter 𝑯𝑯 exist?

How do you compute the pseudo-inverse of 𝑯𝑯?

How do you compute the Weiner deconvolution of 𝑯𝑯?


𝑦𝑦 =





When does the inverse of filter 𝑯𝑯 exist?— When the frequency domain is all non-zero values

How do you compute the pseudo-inverse of 𝑯𝑯?— Only invert non-zero values in the frequency domain

How do you compute the Weiner deconvolution of 𝑯𝑯?— Perform a Tikhonov regularized inverse in frequency domain


𝑦𝑦 =




Exercise Exercise: An allpass filter satisfies

𝐻𝐻 𝑒𝑒𝑗𝑗𝑗𝑗 = 1

What property must by matrix satisfy to be an allpass filter?


Exercise Exercise: An allpass filter satisfies

𝐻𝐻 𝑒𝑒𝑗𝑗𝑗𝑗 = 1

What property must by matrix satisfy to be an allpass filter?

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


The DFT

The DFT Diagonalization of a shift


……𝑥𝑥 0𝑥𝑥 −1𝑥𝑥 −2 𝑥𝑥 1 𝑥𝑥 2 𝑥𝑥 3

𝑦𝑦 =

⋱ ⋱ ⋱ ⋱ ⋱ ⋱⋱ 0 0 0 0 ⋱⋱ 1 0 0 0 ⋱⋱ 0 1 0 0 ⋱⋱ 0 0 1 0 ⋱⋱ ⋱ ⋱ ⋱ ⋱ ⋱



DTFT OperatorToeplitz Matrix

The DFT Diagonalization of a circular shift


𝑥𝑥 0 𝑥𝑥 1 𝑥𝑥 2 𝑥𝑥 3 𝑥𝑥 4 𝑥𝑥 5

𝑦𝑦 =

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

𝑥𝑥 0𝑥𝑥 1𝑥𝑥 2𝑥𝑥 3𝑥𝑥 4𝑥𝑥 5


Circulant Matrix DFT Matrix

The DFT Circular convolution

𝑥𝑥 ∗ ℎ 𝑛𝑛 = �𝑘𝑘∈ℤ

𝑥𝑥𝑘𝑘ℎ𝑚𝑚𝑚𝑚𝑚𝑚 𝑛𝑛−𝑘𝑘,𝑁𝑁


𝑦𝑦 =

ℎ[0] ℎ[5] ℎ[4] ℎ[3] ℎ[2] ℎ[1]ℎ[1] ℎ[0] ℎ[5] ℎ[4] ℎ[3] ℎ[2]ℎ[2] ℎ[1] ℎ[0] ℎ[5] ℎ[4] ℎ[3]ℎ[3] ℎ[2] ℎ[1] ℎ[0] ℎ[5] ℎ[4]ℎ[4] ℎ[3] ℎ[2] ℎ[1] ℎ[0] ℎ[5]ℎ[5] ℎ[4] ℎ[3] ℎ[2] ℎ[1] ℎ[0]

𝑥𝑥 0𝑥𝑥 1𝑥𝑥 2𝑥𝑥 3𝑥𝑥 4𝑥𝑥 5


DFT Matrix

The DFT The DFT Matrix


𝐹𝐹 =1𝑁𝑁

1 1 1 1 ⋯ 11 𝑊𝑊 𝑊𝑊2 𝑊𝑊3 ⋯ 𝑊𝑊𝑁𝑁−1

1 𝑊𝑊2 𝑊𝑊4 𝑊𝑊6 ⋯ 𝑊𝑊2 𝑁𝑁−1

1 𝑊𝑊3 𝑊𝑊6 𝑊𝑊9 ⋯ 𝑊𝑊3 𝑁𝑁−1

⋮ ⋮ ⋮ ⋮ ⋱ ⋮1 𝑊𝑊𝑁𝑁−1 𝑊𝑊2 𝑁𝑁−1 𝑊𝑊3 𝑁𝑁−1 ⋯ 𝑊𝑊(𝑁𝑁−1) 𝑁𝑁−1

𝑊𝑊 = 𝑒𝑒−𝑗𝑗2𝜋𝜋𝑁𝑁

Makes matrix unitary (𝑈𝑈∗ = 𝑈𝑈−1)

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


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

The Graph Fourier Transform

Graph Spectrum For a given graph, there exists a shift matrix


𝑥𝑥1

𝑥𝑥2𝑥𝑥3

𝑥𝑥4

𝑥𝑥5

𝑦𝑦 =

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

𝑥𝑥 1𝑥𝑥 2𝑥𝑥 3𝑥𝑥 4𝑥𝑥 5

= 𝑈𝑈Λ𝑈𝑈−1𝑥𝑥

Graph Fourier Transform

Graph Spectrum Question: What are graph frequency components?


𝑥𝑥1

𝑥𝑥2𝑥𝑥3

𝑥𝑥4

𝑥𝑥5

𝑦𝑦 =

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

𝑥𝑥 1𝑥𝑥 2𝑥𝑥 3𝑥𝑥 4𝑥𝑥 5

= 𝑈𝑈Λ𝑈𝑈−1𝑥𝑥

Graph Fourier Transform

Multi-rate Signal ProcessingDownsampling and Upsampling

Multirate signal processing Question: What is multirate signal processing?


Multirate signal processing Periodically Shift-Varying Systems A discrete-time system T is called periodically shift-varying of order (𝐿𝐿,𝑀𝑀) when,

for any integer 𝑘𝑘 and input 𝑥𝑥,

That is, if I shift the input by 𝐿𝐿, I shift the output by 𝑀𝑀


𝑦𝑦 = 𝑇𝑇 𝑥𝑥 ⇒ 𝑦𝑦′ = 𝑌𝑌 𝑥𝑥𝑥

𝑥𝑥𝑛𝑛′ = 𝑥𝑥𝑛𝑛−𝐿𝐿𝑘𝑘 𝑦𝑦𝑛𝑛′ = 𝑦𝑦𝑛𝑛−𝑀𝑀𝑘𝑘

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]


⋮𝑦𝑦 −1𝑦𝑦 0𝑦𝑦 1𝑦𝑦 2⋮

=

⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱⋱ 1 0 0 0 0 0 ⋱⋱ 0 0 1 0 0 0 ⋱⋱ 0 0 0 0 1 0 ⋱⋱ 0 0 0 0 0 0 ⋱⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱

⋮𝑥𝑥 −2𝑥𝑥 −1𝑥𝑥 0𝑥𝑥 1𝑥𝑥 2𝑥𝑥[3]⋮

𝐷𝐷2

Multirate signal processing Question: When I downsample… What occurs in time?

What occurs in frequency?


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)


Multirate signal processing Downsampling by 2 Periodically shift-varying of order (2,1)



Image from Martin Vertelli’s 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]


𝑦𝑦𝑛𝑛 = 𝑥𝑥𝑁𝑁𝑛𝑛

𝑌𝑌 𝑧𝑧 =1𝑁𝑁�𝑛𝑛=0

𝑁𝑁−1

𝑋𝑋 𝑊𝑊𝑁𝑁𝑘𝑘𝑧𝑧1/𝑁𝑁

𝑦𝑦 = 𝐷𝐷𝑁𝑁𝑥𝑥

Multirate signal processing Upsampling by 2 Periodically shift-varying of order (1,2)



⋮𝑦𝑦 −2𝑦𝑦 −1𝑦𝑦 0𝑦𝑦 1𝑦𝑦 2𝑦𝑦 3⋮

=

⋱ ⋱ ⋱ ⋱ ⋱ ⋱⋱ 1 0 0 0 ⋱⋱ 0 0 0 0 ⋱⋱ 0 1 0 0 ⋱⋱ 0 0 0 0 ⋱⋱ 0 0 1 0 ⋱⋱ 0 0 0 0 ⋱⋱ ⋱ ⋱ ⋱ ⋱ ⋱

⋮𝑥𝑥 −2𝑥𝑥 −1𝑥𝑥 0𝑥𝑥 1𝑥𝑥 2𝑥𝑥[3]⋮

𝑈𝑈2

Multirate signal processing Question: When I upsampling… What occurs in time?

What occurs in frequency?


Multirate signal processing Question: When I upsampling… What occurs in time?

— Answer: Expand in time (effectively)

What occurs in frequency? — Answer: Condense in frequency


Multirate signal processing Upsampling by 2 Periodically shift-varying of order (1,2)



Image from Martin Vertelli’s 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]


𝑦𝑦𝑛𝑛 = �𝑥𝑥𝑛𝑛/𝑁𝑁 , for𝑛𝑛𝑁𝑁∈ ℤ

0 , otherwise

𝑌𝑌 𝑧𝑧 = 𝑋𝑋 𝑧𝑧𝑁𝑁

𝑦𝑦 = 𝑈𝑈𝑁𝑁𝑥𝑥

Multi-rate Signal ProcessingUpsampling and downsampling

Multirate signal processing Question: What is the adjoint of downsampling?

What is the adjoint of upsampling?


Multirate signal processing Question: What is the adjoint of downsampling?

— Answer: 𝐷𝐷𝑁𝑁∗ = 𝑈𝑈𝑁𝑁

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


Multirate signal processing Question: What is the 𝐷𝐷𝑁𝑁𝐷𝐷𝑁𝑁∗ 𝑥𝑥 = ? (reminder: matrix operations are right to left)

What does the result mean?


Multirate signal processing Question: What is the 𝐷𝐷𝑁𝑁𝐷𝐷𝑁𝑁∗ 𝑥𝑥 = ? (reminder: matrix operations are right to left)

— Answer: 𝐷𝐷𝑁𝑁𝐷𝐷𝑁𝑁∗𝑥𝑥 = 𝐷𝐷𝑁𝑁𝑈𝑈𝑁𝑁𝑥𝑥 = 𝑥𝑥

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


Multirate signal processing Properties of Downsampling and Upsampling Relationship between upsampling and downsampling

Upsampling followed by downsampling


𝑈𝑈𝑁𝑁 = 𝐷𝐷𝑁𝑁∗

𝐷𝐷𝑁𝑁𝑈𝑈𝑁𝑁 = 𝐼𝐼

𝑥𝑥

𝑈𝑈2𝑥𝑥

𝐷𝐷2𝑈𝑈2𝑥𝑥


Downsampling followed by upsampling



𝑈𝑈𝑁𝑁𝐷𝐷𝑁𝑁 = 𝑃𝑃 (projection operator)

𝑥𝑥

𝐷𝐷2𝑥𝑥

𝑈𝑈2𝐷𝐷2𝑥𝑥


Downsampling followed by upsampling



𝑈𝑈𝑁𝑁𝐷𝐷𝑁𝑁 = 𝑃𝑃 (projection operator)

𝑥𝑥

𝐷𝐷2𝑥𝑥


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


Multi-rate Signal ProcessingFiltering with downsampling and upsampling

Multirate signal processing Question Why incorporate filtering?


Multirate signal processing Example (from Martin Veterlli’s notes)


Original signal Downsampled by 4 (aliasing)



Downsampled THEN filtered (aliasing)

Filtered THEN downsampled

Multirate signal processing Properties of Downsampling and Upsampling Filtering followed by downsampling


𝑦𝑦 =

⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱⋱ 1 0 0 0 0 0 ⋱⋱ 0 0 1 0 0 0 ⋱⋱ 0 0 0 0 1 0 ⋱⋱ 0 0 0 0 0 0 ⋱⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱

⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱⋱ 𝑔𝑔 1 𝑔𝑔 0 0 0 0 0 ⋱⋱ 𝑔𝑔 2 𝑔𝑔 1 𝑔𝑔 0 0 0 0 ⋱⋱ 𝑔𝑔 3 𝑔𝑔 2 𝑔𝑔 1 𝑔𝑔 0 0 0 ⋱⋱ 0 𝑔𝑔 3 𝑔𝑔 2 𝑔𝑔 1 𝑔𝑔 0 0 ⋱⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱

⋮𝑥𝑥 −3𝑥𝑥 −2𝑥𝑥 −1𝑥𝑥 0𝑥𝑥 1𝑥𝑥 2⋮

𝑥𝑥 𝑦𝑦𝐻𝐻 𝑧𝑧 ↓ 2

Downsample across columns of G

Multirate signal processing Properties of Downsampling and Upsampling Filtering followed by downsampling


𝑦𝑦 =

⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱⋱ 𝑔𝑔 1 𝑔𝑔 0 0 0 0 0 ⋱⋱ 𝑔𝑔 3 𝑔𝑔 2 𝑔𝑔 1 𝑔𝑔 0 0 0 ⋱⋱ 0 0 𝑔𝑔 3 𝑔𝑔 2 𝑔𝑔 1 𝑔𝑔 0 ⋱⋱ 0 0 0 0 𝑔𝑔 3 𝑔𝑔 2 ⋱⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱ ⋱



𝑥𝑥 𝑦𝑦𝐻𝐻 𝑧𝑧 ↓ 2

No longer a DFT matrix



Original

Upsampledby 4

Upsampledby 4 THEN

filtered

Multirate signal processing Properties of Downsampling and Upsampling Upsampling followed by filtering


𝑦𝑦 =


⋱ ⋱ ⋱ ⋱ ⋱ ⋱⋱ 1 0 0 0 ⋱⋱ 0 0 0 0 ⋱⋱ 0 1 0 0 ⋱⋱ 0 0 0 0 ⋱⋱ 0 0 1 0 ⋱⋱ 0 0 0 0 ⋱⋱ ⋱ ⋱ ⋱ ⋱ ⋱


Upsample across columns of x

𝑥𝑥 𝑦𝑦𝐻𝐻 𝑧𝑧↑ 2



𝑦𝑦 =


⋮𝑥𝑥 −2

0𝑥𝑥 −1

0𝑥𝑥 0

0𝑥𝑥 1⋮




𝑦𝑦 =


⋱ ⋱ ⋱ ⋱ ⋱ ⋱⋱ 1 0 0 0 ⋱⋱ 0 0 0 0 ⋱⋱ 0 1 0 0 ⋱⋱ 0 0 0 0 ⋱⋱ 0 0 1 0 ⋱⋱ 0 0 0 0 ⋱⋱ ⋱ ⋱ ⋱ ⋱ ⋱


Downsample across rows of G




𝑦𝑦 =

⋱ ⋱ ⋱ ⋱ ⋱ ⋱⋱ 𝑔𝑔 1 0 0 0 ⋱⋱ 𝑔𝑔 2 𝑔𝑔 0 0 0 ⋱⋱ 0 𝑔𝑔 1 0 0 ⋱⋱ 0 𝑔𝑔 2 𝑔𝑔 0 0 ⋱⋱ 0 0 𝑔𝑔 1 0 ⋱⋱ 0 0 𝑔𝑔 2 𝑔𝑔 0 ⋱⋱ ⋱ ⋱ ⋱ ⋱ ⋱

⋮𝑥𝑥 −2𝑥𝑥 −1𝑥𝑥 0𝑥𝑥 1𝑥𝑥 2⋮



Multirate signal processing Properties of Downsampling and Upsampling Upsampling and downsampling with filters

How is this used?


𝑦𝑦𝐻𝐻 𝑧𝑧↑ 2𝑥𝑥 𝐻𝐻 𝑧𝑧 ↓ 2

𝐻𝐻 𝑧𝑧

Multi-rate Signal ProcessingRe-ordering downsampling and upsampling

Multirate signal processing Properties of Downsampling and Upsampling Flipping filtering with downsampling and upsampling


𝑥𝑥 𝑦𝑦𝐻𝐻 𝑧𝑧2 ↓ 2=𝑥𝑥 𝑦𝑦𝐻𝐻 𝑧𝑧↓ 2

Upsample the filter

𝑦𝑦 = 𝐻𝐻𝐷𝐷2𝑥𝑥

My notation

𝑦𝑦 = 𝐷𝐷2𝐻𝐻↑2𝑥𝑥


How is this work linear algebraically? 𝑦𝑦 = 𝐻𝐻𝐷𝐷2𝑥𝑥 = 𝐻𝐻𝑈𝑈2∗𝑥𝑥 = 𝐷𝐷2𝑈𝑈2𝐻𝐻𝑈𝑈2∗𝑥𝑥



Upsample the filter

Downsample x Upsampleacross rows of G

Identity

𝑦𝑦 = 𝐻𝐻𝐷𝐷2𝑥𝑥 𝑦𝑦 = 𝐷𝐷2𝐻𝐻↑2𝑥𝑥 = 𝐷𝐷2 𝑈𝑈2𝐻𝐻𝑈𝑈2∗ 𝑥𝑥

My notation


How is this work linear algebraically? 𝑦𝑦 = 𝐻𝐻𝐷𝐷2𝑥𝑥 = 𝐻𝐻𝑈𝑈2∗𝑥𝑥 = 𝐷𝐷2𝑈𝑈2𝐻𝐻𝑈𝑈2∗𝑥𝑥



Upsample the filter


My notation

Downsample x Upsampleacross rows of G

𝑦𝑦 = 𝐷𝐷2𝐻𝐻↑2𝑥𝑥 = 𝐷𝐷2 𝑈𝑈2𝐻𝐻𝑈𝑈2∗ 𝑥𝑥

Upsample across columns of G


How is this work linear algebraically?

𝑦𝑦 = 1 22 1

1 0 0 00 0 1 0

1234

= 1 22 1

13 = 1 0 2 0

2 0 1 0

1234



Upsample the filter


My notation


Downsample x Upsampled across rows of G



𝑦𝑦 = 1 0 2 02 0 1 0

1234



Upsample the filter


My notation


Upsampled across rows of G



𝑦𝑦 = 1 0 0 00 0 1 0

1 00 00 10 0

1 0 2 02 0 1 0

1234



Upsample the filter


My notation


Upsampled across rows of GIdentity

Upsample across columns of 𝐻𝐻𝐷𝐷2



𝑦𝑦 = 1 0 0 00 0 1 0

1 0 2 00 0 0 02 0 1 00 0 0 0

1234



Upsample the filter


My notation


Upsampled across rows and columns of G

Downsample by 2



𝑥𝑥 𝑦𝑦𝐻𝐻 𝑧𝑧𝑁𝑁 ↓ 𝑁𝑁=𝑥𝑥 𝑦𝑦𝐻𝐻 𝑧𝑧↓ 𝑁𝑁

Upsample the filter

𝑦𝑦 = 𝐻𝐻𝐷𝐷𝑁𝑁𝑥𝑥 𝑦𝑦 = 𝐷𝐷𝑁𝑁𝐻𝐻↑𝑁𝑁𝑥𝑥 = 𝐷𝐷𝑁𝑁 𝑈𝑈𝑁𝑁𝐻𝐻𝑈𝑈𝑁𝑁∗ 𝑥𝑥



𝑥𝑥 𝑦𝑦𝐻𝐻 𝑧𝑧 ↑ 𝑁𝑁 = 𝑥𝑥 𝑦𝑦𝐻𝐻 𝑧𝑧𝑁𝑁↑ 𝑁𝑁


Upsample the filter

𝑦𝑦 = 𝐻𝐻𝐷𝐷𝑁𝑁𝑥𝑥 = 𝐷𝐷𝑁𝑁 𝑈𝑈𝑁𝑁𝐻𝐻𝑈𝑈𝑁𝑁∗ 𝑥𝑥 = 𝐷𝐷𝑁𝑁𝐻𝐻↑𝑁𝑁𝑥𝑥

𝑦𝑦 = 𝑈𝑈𝑁𝑁𝐻𝐻𝑥𝑥 = 𝑈𝑈𝑁𝑁𝐻𝐻𝑈𝑈𝑁𝑁∗ 𝑈𝑈𝑁𝑁𝑥𝑥 = 𝐻𝐻↑𝑁𝑁𝑈𝑈𝑁𝑁𝑥𝑥


Why is this useful? What does it do?


𝑥𝑥 𝑦𝑦𝐻𝐻 𝑧𝑧 ↑ 𝑁𝑁 = 𝑥𝑥 𝑦𝑦𝐻𝐻 𝑧𝑧𝑁𝑁↑ 𝑁𝑁


Upsample the filter

𝑦𝑦 = 𝐻𝐻𝐷𝐷𝑁𝑁𝑥𝑥 = 𝐷𝐷𝑁𝑁 𝑈𝑈𝑁𝑁𝐻𝐻𝑈𝑈𝑁𝑁∗ 𝑥𝑥 = 𝐷𝐷𝑁𝑁𝐻𝐻↑𝑁𝑁𝑥𝑥

𝑦𝑦 = 𝑈𝑈𝑁𝑁𝐻𝐻𝑥𝑥 = 𝑈𝑈𝑁𝑁𝐻𝐻𝑈𝑈𝑁𝑁∗ 𝑈𝑈𝑁𝑁𝑥𝑥 = 𝐻𝐻↑𝑁𝑁𝑈𝑈𝑁𝑁𝑥𝑥

Multirate signal processing Properties of Downsampling and Upsampling Computationally inefficient

Computationally efficient

This concept is also used in the design of polyphase filters


𝑦𝑦𝐻𝐻 𝑧𝑧↑ 2𝑥𝑥 𝐻𝐻 𝑧𝑧 ↓ 2

𝑦𝑦𝐻𝐻 𝑧𝑧𝑁𝑁 ↑ 2𝑥𝑥 𝐻𝐻 𝑧𝑧𝑁𝑁↓ 2

Example 1

67



𝑥𝑥 𝑦𝑦𝐻𝐻 𝑧𝑧𝑁𝑁 ↓ 2=𝑥𝑥 𝑦𝑦𝐻𝐻 𝑧𝑧↓ 2

𝜋𝜋2

𝜋𝜋 3𝜋𝜋2

2𝜋𝜋𝜋𝜋2

𝜋𝜋3𝜋𝜋2

2𝜋𝜋

𝑋𝑋 𝜔𝜔

1




𝜋𝜋2

𝜋𝜋 3𝜋𝜋2

2𝜋𝜋𝜋𝜋2

𝜋𝜋3𝜋𝜋2

2𝜋𝜋

𝑋𝑋𝑚𝑚 𝜔𝜔𝑥𝑥𝑚𝑚

0.5




𝜋𝜋2

𝜋𝜋 3𝜋𝜋2

2𝜋𝜋𝜋𝜋2

𝜋𝜋3𝜋𝜋2

2𝜋𝜋


Filter (gain: 1)0.5




𝜋𝜋2

𝜋𝜋 3𝜋𝜋2

2𝜋𝜋𝜋𝜋2

𝜋𝜋3𝜋𝜋2

2𝜋𝜋

𝑌𝑌 𝜔𝜔𝑥𝑥𝑚𝑚

0.5




𝜋𝜋2

𝜋𝜋 3𝜋𝜋2

2𝜋𝜋𝜋𝜋2

𝜋𝜋3𝜋𝜋2

2𝜋𝜋

𝑋𝑋 𝜔𝜔

1




𝜋𝜋2

𝜋𝜋 3𝜋𝜋2

2𝜋𝜋𝜋𝜋2

𝜋𝜋3𝜋𝜋2

2𝜋𝜋

𝑋𝑋 𝜔𝜔

1 Filter (gain: 1)




𝜋𝜋2

𝜋𝜋 3𝜋𝜋2

2𝜋𝜋𝜋𝜋2

𝜋𝜋3𝜋𝜋2

2𝜋𝜋

𝑋𝑋𝑓𝑓 𝜔𝜔𝑥𝑥𝑓𝑓

1




𝜋𝜋2

𝜋𝜋 3𝜋𝜋2

2𝜋𝜋𝜋𝜋2

𝜋𝜋3𝜋𝜋2

2𝜋𝜋

𝑌𝑌 𝜔𝜔𝑥𝑥𝑓𝑓

0.5

Example 2

76




𝜋𝜋2

𝜋𝜋 3𝜋𝜋2

2𝜋𝜋𝜋𝜋2

𝜋𝜋3𝜋𝜋2

2𝜋𝜋

𝑋𝑋 𝜔𝜔

1




𝜋𝜋2

𝜋𝜋 3𝜋𝜋2

2𝜋𝜋𝜋𝜋2

𝜋𝜋3𝜋𝜋2

2𝜋𝜋


0.5




𝜋𝜋2

𝜋𝜋 3𝜋𝜋2

2𝜋𝜋𝜋𝜋2

𝜋𝜋3𝜋𝜋2

2𝜋𝜋


Filter (gain: 1)0.5




𝜋𝜋2

𝜋𝜋 3𝜋𝜋2

2𝜋𝜋𝜋𝜋2

𝜋𝜋3𝜋𝜋2

2𝜋𝜋

𝑌𝑌 𝜔𝜔𝑥𝑥𝑚𝑚

0.5




𝜋𝜋2

𝜋𝜋 3𝜋𝜋2

2𝜋𝜋𝜋𝜋2

𝜋𝜋3𝜋𝜋2

2𝜋𝜋

𝑋𝑋 𝜔𝜔

1




𝜋𝜋2

𝜋𝜋 3𝜋𝜋2

2𝜋𝜋𝜋𝜋2

𝜋𝜋3𝜋𝜋2

2𝜋𝜋

𝑋𝑋 𝜔𝜔

1 Filter (gain: 1)




𝜋𝜋2

𝜋𝜋 3𝜋𝜋2

2𝜋𝜋𝜋𝜋2

𝜋𝜋3𝜋𝜋2

2𝜋𝜋

𝑋𝑋𝑓𝑓 𝜔𝜔

1

𝑥𝑥𝑓𝑓




𝜋𝜋2

𝜋𝜋 3𝜋𝜋2

2𝜋𝜋𝜋𝜋2

𝜋𝜋3𝜋𝜋2

2𝜋𝜋


0.5

Multi-rate Signal Processing - University of Utah

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