![Page 1: ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS … · ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS VIA ALGEBRAIC THEORETICAL SIGNAL MODELS Doru Balcan and Satashu Goel Carnegie](https://reader030.vdocuments.us/reader030/viewer/2022040510/5e59d02dade4cc12c1613804/html5/thumbnails/1.jpg)
ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS
VIA ALGEBRAIC THEORETICAL SIGNAL MODELS
Doru Balcan and Satashu GoelCarnegie Mellon University
![Page 2: ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS … · ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS VIA ALGEBRAIC THEORETICAL SIGNAL MODELS Doru Balcan and Satashu Goel Carnegie](https://reader030.vdocuments.us/reader030/viewer/2022040510/5e59d02dade4cc12c1613804/html5/thumbnails/2.jpg)
Outline
Algebraic Signal Processing Theory
From finite to infinite models: issue of convergence
Termwise to normwise convergence
Convergence using a direct approach
Conclusion
![Page 3: ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS … · ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS VIA ALGEBRAIC THEORETICAL SIGNAL MODELS Doru Balcan and Satashu Goel Carnegie](https://reader030.vdocuments.us/reader030/viewer/2022040510/5e59d02dade4cc12c1613804/html5/thumbnails/3.jpg)
Algebraic signal models
Signal model:
Space models: visualization
As n →∞, does the finite model converge to infinite model? What does convergence mean?
Algebra of filters Module of signals ‘z-transform’
… …
Finite: left and right bc Infinite: only left bc
![Page 4: ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS … · ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS VIA ALGEBRAIC THEORETICAL SIGNAL MODELS Doru Balcan and Satashu Goel Carnegie](https://reader030.vdocuments.us/reader030/viewer/2022040510/5e59d02dade4cc12c1613804/html5/thumbnails/4.jpg)
Space marks, shift
Shift matrix: captures operation of shift on basis
e.g., for DTTs,
. . .
Definition of a signal modelshift
Space marks
![Page 5: ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS … · ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS VIA ALGEBRAIC THEORETICAL SIGNAL MODELS Doru Balcan and Satashu Goel Carnegie](https://reader030.vdocuments.us/reader030/viewer/2022040510/5e59d02dade4cc12c1613804/html5/thumbnails/5.jpg)
Filter matrices
If ,
Key property: Fourier transform diagonalizes filter matrices
Finite transform applied to infinite filter matrixKey question:
for large n?
![Page 6: ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS … · ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS VIA ALGEBRAIC THEORETICAL SIGNAL MODELS Doru Balcan and Satashu Goel Carnegie](https://reader030.vdocuments.us/reader030/viewer/2022040510/5e59d02dade4cc12c1613804/html5/thumbnails/6.jpg)
Finite sections of an infinite matrix
Ideally, if transforms are asymptotically equivalent
Infinite matrix!But finite Finite section (nxn)
![Page 7: ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS … · ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS VIA ALGEBRAIC THEORETICAL SIGNAL MODELS Doru Balcan and Satashu Goel Carnegie](https://reader030.vdocuments.us/reader030/viewer/2022040510/5e59d02dade4cc12c1613804/html5/thumbnails/7.jpg)
Types of convergence
Is termwise convergence enough?Performance degradation is function of weak normFor matrix , weak norm is defined as
Example:
Termwise:
Normwise:
![Page 8: ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS … · ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS VIA ALGEBRAIC THEORETICAL SIGNAL MODELS Doru Balcan and Satashu Goel Carnegie](https://reader030.vdocuments.us/reader030/viewer/2022040510/5e59d02dade4cc12c1613804/html5/thumbnails/8.jpg)
Normwise Convergence
Modify criterion: use weak norm
Fourier Transform Difference in filter matrices
![Page 9: ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS … · ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS VIA ALGEBRAIC THEORETICAL SIGNAL MODELS Doru Balcan and Satashu Goel Carnegie](https://reader030.vdocuments.us/reader030/viewer/2022040510/5e59d02dade4cc12c1613804/html5/thumbnails/9.jpg)
Outline
Algebraic Signal Processing Theory
From finite to infinite models: issue of convergence
Termwise to normwise convergence
Convergence using a direct approach
Conclusion
![Page 10: ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS … · ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS VIA ALGEBRAIC THEORETICAL SIGNAL MODELS Doru Balcan and Satashu Goel Carnegie](https://reader030.vdocuments.us/reader030/viewer/2022040510/5e59d02dade4cc12c1613804/html5/thumbnails/10.jpg)
Termwise to normwise convergence
Given: Termwise convergence
Decay of “Fourier coefficients”
To show:
Equivalently:
![Page 11: ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS … · ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS VIA ALGEBRAIC THEORETICAL SIGNAL MODELS Doru Balcan and Satashu Goel Carnegie](https://reader030.vdocuments.us/reader030/viewer/2022040510/5e59d02dade4cc12c1613804/html5/thumbnails/11.jpg)
Key intuition for proof
and can be bounded separately, in different regions
Infinite matrixwith entries
Choose s.t.
Choose N s.t.
![Page 12: ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS … · ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS VIA ALGEBRAIC THEORETICAL SIGNAL MODELS Doru Balcan and Satashu Goel Carnegie](https://reader030.vdocuments.us/reader030/viewer/2022040510/5e59d02dade4cc12c1613804/html5/thumbnails/12.jpg)
Outline
Algebraic Signal Processing Theory
From finite to infinite models: issue of convergence
Termwise to normwise convergence
Convergence using a direct approach
Conclusion
![Page 13: ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS … · ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS VIA ALGEBRAIC THEORETICAL SIGNAL MODELS Doru Balcan and Satashu Goel Carnegie](https://reader030.vdocuments.us/reader030/viewer/2022040510/5e59d02dade4cc12c1613804/html5/thumbnails/13.jpg)
New Question
Q: Can we approximate infinite signal modelsby related, finite signal models?
A: In what sense?
![Page 14: ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS … · ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS VIA ALGEBRAIC THEORETICAL SIGNAL MODELS Doru Balcan and Satashu Goel Carnegie](https://reader030.vdocuments.us/reader030/viewer/2022040510/5e59d02dade4cc12c1613804/html5/thumbnails/14.jpg)
Normwise Convergence
Modify criterion: use weak norm
Fourier Transform Difference in filter matrices
![Page 15: ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS … · ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS VIA ALGEBRAIC THEORETICAL SIGNAL MODELS Doru Balcan and Satashu Goel Carnegie](https://reader030.vdocuments.us/reader030/viewer/2022040510/5e59d02dade4cc12c1613804/html5/thumbnails/15.jpg)
Theorem Let be the space signal model corresponding to the infinite C-transform, and . For , let be the signal model corresponding to related finite C-transform, and be the “n-prefix” of . Then,
•“smoothness”:
•stronger than L2
•neither stronger, nor weaker than L1
![Page 16: ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS … · ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS VIA ALGEBRAIC THEORETICAL SIGNAL MODELS Doru Balcan and Satashu Goel Carnegie](https://reader030.vdocuments.us/reader030/viewer/2022040510/5e59d02dade4cc12c1613804/html5/thumbnails/16.jpg)
Filter matrix (infinite)example:T-transform
![Page 17: ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS … · ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS VIA ALGEBRAIC THEORETICAL SIGNAL MODELS Doru Balcan and Satashu Goel Carnegie](https://reader030.vdocuments.us/reader030/viewer/2022040510/5e59d02dade4cc12c1613804/html5/thumbnails/17.jpg)
Filter matrix (finite)example: DCT-3
![Page 18: ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS … · ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS VIA ALGEBRAIC THEORETICAL SIGNAL MODELS Doru Balcan and Satashu Goel Carnegie](https://reader030.vdocuments.us/reader030/viewer/2022040510/5e59d02dade4cc12c1613804/html5/thumbnails/18.jpg)
Filter matrix (skew)example: DCT-4(r)
![Page 19: ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS … · ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS VIA ALGEBRAIC THEORETICAL SIGNAL MODELS Doru Balcan and Satashu Goel Carnegie](https://reader030.vdocuments.us/reader030/viewer/2022040510/5e59d02dade4cc12c1613804/html5/thumbnails/19.jpg)
Outline
Algebraic Signal Processing Theory
From finite to infinite models: issue of convergence
Termwise to normwise convergence
Convergence using a direct approach
Conclusion
![Page 20: ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS … · ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS VIA ALGEBRAIC THEORETICAL SIGNAL MODELS Doru Balcan and Satashu Goel Carnegie](https://reader030.vdocuments.us/reader030/viewer/2022040510/5e59d02dade4cc12c1613804/html5/thumbnails/20.jpg)
Conclusions
We investigated asymptotic decorrelation properties of DTTs / skew DTTsfixed bug in old proof
Adopted the Algebraic Signal Processing Theoretical point of viewidentify signal model for transform:
filter algebra, signal module, “z-transform”, shift, b.c., etc.
New perspective: (“nice” finite models)(n) infinite models
How about models for other types of signal extension?k-monomial, sparse polynomial
n →∞
![Page 21: ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS … · ASYMPTOTIC PROPERTIES OF TRIGONOMETRIC TRANSFORMS VIA ALGEBRAIC THEORETICAL SIGNAL MODELS Doru Balcan and Satashu Goel Carnegie](https://reader030.vdocuments.us/reader030/viewer/2022040510/5e59d02dade4cc12c1613804/html5/thumbnails/21.jpg)