Fractional Delay Filters Based on <br />Generalized Cardinal Splines <br />
Good Morning!<br />Lunch is Far Away!!!<br />
Philosophy of this talk—Should or Shouldn’t be Eclectic? <br /> First person eats the pie!<br />⌘ You have to depend on l...
Outline<br />Is Fractional Delay Filtering important?<br />Have you heard about Generalized Interpolation?<br />B(eautiful...
Fractional <br />Delay Filters<br />
Given: Samples/Digital Sequence!<br /><ul><li>Producing delays that are multiples of sampling time—No Big Deal!
What if you felt like delaying your samples by non-integer time delays? </li></li></ul><li>	Synchronization in Modems<br /...
Problem Statement:<br />What if you felt like delaying your samples by half-sample time? <br />—Keep sampling rate unchang...
FDF—Bandlimited Interpolation<br />Quick Remedy—Bandlimited interpolation.<br />Fractional Delay<br />Given Sequence<br />
The (In)Famous Sinus Cardinalis<br />Slow decay—Infinite support—Slow convergence of its shifted sums<br />
Other Remedies<br />
Remedies Should Meet Practical Requirements<br />
Splines<br />are the way!<br />
CITATIONS<br />For your eyes only!<br />Carl de Boor<br />Grace Wahba<br />Fen<br />Schumaker<br />Barlets<br />
Hmm!!!<br />Studying Splines Must Be Very Interesting!!!<br />CITATIONS<br />Ground breaking literature and Splines<br />1...
Generalized Interpolation<br />Shift-invariant Space<br />
Generalized Interpolation<br />Generic Interpolants! <br />Spline Coefficients<br />
Exponentially decaying function (compared to sinc). The advantages will be visible soon!<br />Generalized Interpolation—Fi...
Generalized Interpolation—Example<br />
Generalized Interpolation—Example<br />n<br />n<br />n<br />V<br />(t) : de la Vallée-Poussin Kernel<br />(t): Windowed de...
 Filter Coefficients and Linear Combination of Basis Functions<br />1.5<br />1<br />0.5<br />Basis for n=2<br />0.8<br />0...
So How To Get Suitable Basis Functions?<br />— Riesz Basis Conditions <br />— Partition of Unity <br />— Approximation Ord...
Polynomial B(eautiful)-Splines<br />B-spline of degree n:<br />—Piecewise Polynomials<br />—Positive<br />—Symmetric<br />...
Generalized Cardinal Exponential Splines<br />Tools of trade! <br />Continuous domain Derivative<br />Discrete Derivative<...
The L-spline<br />Some pictures first! <br />
The L-spline<br />A function is called a cardinal L-spline if and only if:<br /><ul><li>Cardinality
Generalization</li></ul>Discrete Derivative<br />Continuous  Derivative<br />
Splines and Green Functions<br />GreenFunctions:<br />Cardinal (uniform knot)L-Splines:<br />
L is a Linear/Translation Invariant Operator<br />Shift-Invariance<br />Linearity<br />
Green’s Functions and E-Splines<br />
Green’s Functions and E-Splines<br />Possible??<br />
Green’s Functions and E-Splines<br />Null space of <br />Belongs to Null-space! This leads to a non-unique solution of thi...
GenCESP<br />
GenCESP—Frequency Domain<br />
FDF—Generalized Interpolation<br />
FDF—Generalized Interpolation<br />Great! We know that this function is separable for Splines and furthermore, it is alway...
FDF—GenCESP<br />
B-splines: m-Scale Relationship<br />Dilation properties are important!<br />Dirichlet Series….<br />
GenCESP—Q-Scale Relationship<br />
FDF—GenCESP<br />
FDF—GenCESP: Design Example<br />
FDF—GenCESP: Design Example<br />Cascade of First Order Causal and Anti-Causal Filters, time-symmetric, exponentially deca...
Magnitude Response and Phase Delay Characteristics<br />
Phase Delay Characteristics: GenCESP—Lagrange—Sinc<br />
Magnitude Response: GenCESP—Lagrange—Sinc<br />
(Absolute) Interpolation Error — Lower the Better!<br />
What/Why/How are we selling…<br />
Wrapping Up the Talk…<br />
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  • SIS : [Vleft( varphi
    ight) = left{ {sleft( t
    ight) = sumlimits_{k = - infty }^{ + infty } {cleft[ k
    ight]varphi left( {t - k}
    ight):c in ell _2 } }
    ight}]
  • [sleft( t
    ight) = sumlimits_{k = - infty }^{ + infty } {cleft[ k
    ight]varphi left( {t - k}
    ight)} xrightarrow{{{ ext{sampling}}}}sleft( n
    ight) = cleft[ n
    ight] * varphi left( n
    ight)][ Rightarrow c = frac{1}{{varphi ^n }} * s][sleft( t
    ight) = sumlimits_{k = - infty }^{ + infty } {left( {sleft[ k
    ight] * frac{1}{{varphi left( k
    ight)}}}
    ight)varphi left( {t - k}
    ight)} overset {} longleftrightarrow sumlimits_{k = - infty }^{ + infty } {sleft[ k
    ight]underbrace {left( {frac{1}{{varphi left( k
    ight)}} * varphi left( {t - k}
    ight)}
    ight)}_{{ ext{new - interpolant}}}} ]
  • Eq1 eta _ + ^n left( t
    ight) = underbrace {eta _ + ^n * eta _ + ^n * cdots * eta _ + ^n }_{left( {n + 1}
    ight){ ext{ times}}}left( t
    ight)Eq2[ Rightarrow eta _ + ^n left( t
    ight) = left( {eta _ + ^0 * eta _ + ^{n - 1} }
    ight)left( t
    ight)]
  • D[D = frac{partial }{{partial t}}overset {FT} longleftrightarrow jomega ]d[Delta overset {FT} longleftrightarrow 1 - e^{ - jomega } ][Delta _lambda ^n (x) = left| lambda
    ight|sumlimits_{k = 0}^n {leftlangle {n} mathrel{left | {vphantom {n k}}
    ight. kern-
    ulldelimiterspace} {k}
    ight
    angle left( { - 1}
    ight)^k delta left( {x - lambda k}
    ight)} ]
  • [sleft( t
    ight) = sumlimits_{n in mathbb{Z}} {sleft[ n
    ight]eta ^0 left( {t - n}
    ight)} ][Dleft{ {sleft( t
    ight)}
    ight} = sumlimits_{n in mathbb{Z}} {Delta left{ {sleft[ n
    ight]}
    ight}delta left( {t - n}
    ight)} ]
  • [Lleft{ {sleft( t
    ight)}
    ight} = sumlimits_{n in mathbb{Z}} {cleft[ n
    ight]delta left( {t - n}
    ight)} ][eta ^0 left( t
    ight)]
  • [delta left( t
    ight) Rightarrow]
  • [delta left( {t - t_d }
    ight)][sum {cleft[ n
    ight]
    ho left( {t - n}
    ight)} ]
  • [delta left( {t - t_d }
    ight)][sum {cleft[ n
    ight]
    ho left( {t - n}
    ight)} ]
  • [delta left( {t - t_d }
    ight)][sum {cleft[ n
    ight]
    ho left( {t - n}
    ight)} ]
  • [delta left( {t - t_d }
    ight)][sum {cleft[ n
    ight]
    ho left( {t - n}
    ight)} ]
  • [delta left( {t - t_d }
    ight)][sum {cleft[ n
    ight]
    ho left( {t - n}
    ight)} ]
  • [delta left( {t - t_d }
    ight)][sum {cleft[ n
    ight]
    ho left( {t - n}
    ight)} ]
  • [delta left( {t - t_d }
    ight)][sum {cleft[ n
    ight]
    ho left( {t - n}
    ight)} ]
  • [delta left( {t - t_d }
    ight)][sum {cleft[ n
    ight]
    ho left( {t - n}
    ight)} ]
  • as GenCESP is a library of many other well known classes of splines.precise fractional delays due to linear phase characteristics. Furthermore, we do not rely on windowing or Taylor series based methods. While the former suffers with phase-shift problem, the later leads to inaccurate designs as one has to truncate the series to $n$-terms.Causal anti—causal!Depends on parameters!
  • as GenCESP is a library of many other well known classes of splines.precise fractional delays due to linear phase characteristics. Furthermore, we do not rely on windowing or Taylor series based methods. While the former suffers with phase-shift problem, the later leads to inaccurate designs as one has to truncate the series to $n$-terms.Causal anti—causal!Depends on parameters!
  • as GenCESP is a library of many other well known classes of splines.precise fractional delays due to linear phase characteristics. Furthermore, we do not rely on windowing or Taylor series based methods. While the former suffers with phase-shift problem, the later leads to inaccurate designs as one has to truncate the series to $n$-terms.Causal anti—causal!Depends on parameters!
  • Talk Gencesp

    1. 1. Fractional Delay Filters Based on <br />Generalized Cardinal Splines <br />
    2. 2. Good Morning!<br />Lunch is Far Away!!!<br />
    3. 3. Philosophy of this talk—Should or Shouldn’t be Eclectic? <br /> First person eats the pie!<br />⌘ You have to depend on luck!<br />⏏ Only marginal improvements on performance! <br />Application <br />e.g. filtering<br />Space of (Useful) <br />Functions<br /> If you nail one problem you make a considerable difference!<br />Finding the corresponding application that matches your tools may not be a straight forward task!<br />
    4. 4. Outline<br />Is Fractional Delay Filtering important?<br />Have you heard about Generalized Interpolation?<br />B(eautiful)-spline Signal Processing.<br />Designing Generalized Cardinal Spline based Filters.<br />Performance Analysis and Conclusions—It works! <br />
    5. 5. Fractional <br />Delay Filters<br />
    6. 6. Given: Samples/Digital Sequence!<br /><ul><li>Producing delays that are multiples of sampling time—No Big Deal!
    7. 7. What if you felt like delaying your samples by non-integer time delays? </li></li></ul><li> Synchronization in Modems<br />Sampling Rate Conversion<br /> Speech Coding<br />
    8. 8. Problem Statement:<br />What if you felt like delaying your samples by half-sample time? <br />—Keep sampling rate unchanged! <br />—Approximate the delay as closely as possible.<br />—The problem is related to interpolation in multirate signal processing and filter design techniques.<br />
    9. 9. FDF—Bandlimited Interpolation<br />Quick Remedy—Bandlimited interpolation.<br />Fractional Delay<br />Given Sequence<br />
    10. 10. The (In)Famous Sinus Cardinalis<br />Slow decay—Infinite support—Slow convergence of its shifted sums<br />
    11. 11. Other Remedies<br />
    12. 12. Remedies Should Meet Practical Requirements<br />
    13. 13. Splines<br />are the way!<br />
    14. 14. CITATIONS<br />For your eyes only!<br />Carl de Boor<br />Grace Wahba<br />Fen<br />Schumaker<br />Barlets<br />
    15. 15. Hmm!!!<br />Studying Splines Must Be Very Interesting!!!<br />CITATIONS<br />Ground breaking literature and Splines<br />1970s - 2006<br />
    16. 16. Generalized Interpolation<br />Shift-invariant Space<br />
    17. 17. Generalized Interpolation<br />Generic Interpolants! <br />Spline Coefficients<br />
    18. 18. Exponentially decaying function (compared to sinc). The advantages will be visible soon!<br />Generalized Interpolation—Finally…<br />
    19. 19. Generalized Interpolation—Example<br />
    20. 20. Generalized Interpolation—Example<br />n<br />n<br />n<br />V<br />(t) : de la Vallée-Poussin Kernel<br />(t): Windowed de la Vallée-Poussin<br />1<br />1<br />0.8<br />0.8<br />0.6<br />0.6<br />0.4<br />0.4<br />0.2<br />0.2<br />0<br />0<br />-5<br />-4<br />-3<br />-2<br />-1<br />0<br />1<br />2<br />3<br />4<br />5<br />-5<br />-4<br />-3<br />-2<br />-1<br />0<br />1<br />2<br />3<br />4<br />5<br />time<br />time<br />
    21. 21. Filter Coefficients and Linear Combination of Basis Functions<br />1.5<br />1<br />0.5<br />Basis for n=2<br />0.8<br />0<br />0.6<br />-0.5<br />0.4<br />-6<br />-4<br />-2<br />0<br />2<br />4<br />time<br />0.2<br />InterpolatingValleePoussin Filter and its Sinc counterpart<br />0<br />1<br />-0.2<br />0.8<br />-6<br />-4<br />-2<br />0<br />2<br />4<br />6<br />time<br />0.6<br />0.4<br />0.2<br />0<br />-0.2<br />-5<br />0<br />5<br />time<br />
    22. 22. So How To Get Suitable Basis Functions?<br />— Riesz Basis Conditions <br />— Partition of Unity <br />— Approximation Order aka Strang Fix Stuff<br />— Smoothness/Holder Continuity <br />— Ease of Implementation…etc.<br />How about the Gaussian Function?<br />
    23. 23. Polynomial B(eautiful)-Splines<br />B-spline of degree n:<br />—Piecewise Polynomials<br />—Positive<br />—Symmetric<br />—Recursive Implementation <br />—Holder Continuous of order n<br />Spline!<br />
    24. 24. Generalized Cardinal Exponential Splines<br />Tools of trade! <br />Continuous domain Derivative<br />Discrete Derivative<br />
    25. 25. The L-spline<br />Some pictures first! <br />
    26. 26. The L-spline<br />A function is called a cardinal L-spline if and only if:<br /><ul><li>Cardinality
    27. 27. Generalization</li></ul>Discrete Derivative<br />Continuous Derivative<br />
    28. 28. Splines and Green Functions<br />GreenFunctions:<br />Cardinal (uniform knot)L-Splines:<br />
    29. 29. L is a Linear/Translation Invariant Operator<br />Shift-Invariance<br />Linearity<br />
    30. 30. Green’s Functions and E-Splines<br />
    31. 31. Green’s Functions and E-Splines<br />Possible??<br />
    32. 32. Green’s Functions and E-Splines<br />Null space of <br />Belongs to Null-space! This leads to a non-unique solution of this equation. <br />This can be avoided by imposing boundary conditions.<br />
    33. 33. GenCESP<br />
    34. 34. GenCESP—Frequency Domain<br />
    35. 35. FDF—Generalized Interpolation<br />
    36. 36. FDF—Generalized Interpolation<br />Great! We know that this function is separable for Splines and furthermore, it is always an FIR filter!!!<br />
    37. 37. FDF—GenCESP<br />
    38. 38. B-splines: m-Scale Relationship<br />Dilation properties are important!<br />Dirichlet Series….<br />
    39. 39. GenCESP—Q-Scale Relationship<br />
    40. 40. FDF—GenCESP<br />
    41. 41. FDF—GenCESP: Design Example<br />
    42. 42. FDF—GenCESP: Design Example<br />Cascade of First Order Causal and Anti-Causal Filters, time-symmetric, exponentially decaying filters.<br />Causal<br />Anti—Causal<br />
    43. 43. Magnitude Response and Phase Delay Characteristics<br />
    44. 44. Phase Delay Characteristics: GenCESP—Lagrange—Sinc<br />
    45. 45. Magnitude Response: GenCESP—Lagrange—Sinc<br />
    46. 46. (Absolute) Interpolation Error — Lower the Better!<br />
    47. 47. What/Why/How are we selling…<br />
    48. 48. Wrapping Up the Talk…<br />
    49. 49. Thanks!<br />Questions | Remarks | Comments <br />

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