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Talk Gencesp
- 1. Fractional Delay Filters Based on Generalized Cardinal Splines
- 2. Good Morning! Lunch is Far Away!!!
- 3. Philosophy of this talk—Should or Shouldn’t be Eclectic? First person eats the pie! ⌘ You have to depend on luck! ⏏ Only marginal improvements on performance! Application e.g. filtering Space of (Useful) Functions If you nail one problem you make a considerable difference! Finding the corresponding application that matches your tools may not be a straight forward task!
- 4. Outline Is Fractional Delay Filtering important? Have you heard about Generalized Interpolation? B(eautiful)-spline Signal Processing. Designing Generalized Cardinal Spline based Filters. Performance Analysis and Conclusions—It works!
- 5. Fractional Delay Filters
- 8. Problem Statement: What if you felt like delaying your samples by half-sample time? —Keep sampling rate unchanged! —Approximate the delay as closely as possible. —The problem is related to interpolation in multirate signal processing and filter design techniques.
- 9. FDF—Bandlimited Interpolation Quick Remedy—Bandlimited interpolation. Fractional Delay Given Sequence
- 10. The (In)Famous Sinus Cardinalis Slow decay—Infinite support—Slow convergence of its shifted sums
- 11. Other Remedies
- 12. Remedies Should Meet Practical Requirements
- 13. Splines are the way!
- 14. CITATIONS For your eyes only! Carl de Boor Grace Wahba Fen Schumaker Barlets
- 15. Hmm!!! Studying Splines Must Be Very Interesting!!! CITATIONS Ground breaking literature and Splines 1970s - 2006
- 17. Generalized Interpolation Generic Interpolants! Spline Coefficients
- 18. Exponentially decaying function (compared to sinc). The advantages will be visible soon! Generalized Interpolation—Finally…
- 20. Generalized Interpolation—Example n n n V (t) : de la Vallée-Poussin Kernel (t): Windowed de la Vallée-Poussin 1 1 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 -5 -4 -3 -2 -1 0 1 2 3 4 5 -5 -4 -3 -2 -1 0 1 2 3 4 5 time time
- 21. Filter Coefficients and Linear Combination of Basis Functions 1.5 1 0.5 Basis for n=2 0.8 0 0.6 -0.5 0.4 -6 -4 -2 0 2 4 time 0.2 InterpolatingValleePoussin Filter and its Sinc counterpart 0 1 -0.2 0.8 -6 -4 -2 0 2 4 6 time 0.6 0.4 0.2 0 -0.2 -5 0 5 time
- 22. So How To Get Suitable Basis Functions? — Riesz Basis Conditions — Partition of Unity — Approximation Order aka Strang Fix Stuff — Smoothness/Holder Continuity — Ease of Implementation…etc. How about the Gaussian Function?
- 23. Polynomial B(eautiful)-Splines B-spline of degree n: —Piecewise Polynomials —Positive —Symmetric —Recursive Implementation —Holder Continuous of order n Spline!
- 24. Generalized Cardinal Exponential Splines Tools of trade! Continuous domain Derivative Discrete Derivative
- 25. The L-spline Some pictures first!
- 27. GeneralizationDiscrete Derivative Continuous Derivative
- 28. Splines and Green Functions GreenFunctions: Cardinal (uniform knot)L-Splines:
- 29. L is a Linear/Translation Invariant Operator Shift-Invariance Linearity
- 30. Green’s Functions and E-Splines
- 31. Green’s Functions and E-Splines Possible??
- 32. Green’s Functions and E-Splines Null space of Belongs to Null-space! This leads to a non-unique solution of this equation. This can be avoided by imposing boundary conditions.
- 33. GenCESP
- 36. FDF—Generalized Interpolation Great! We know that this function is separable for Splines and furthermore, it is always an FIR filter!!!
- 37. FDF—GenCESP
- 38. B-splines: m-Scale Relationship Dilation properties are important! Dirichlet Series….
- 40. FDF—GenCESP
- 42. FDF—GenCESP: Design Example Cascade of First Order Causal and Anti-Causal Filters, time-symmetric, exponentially decaying filters. Causal Anti—Causal
- 43. Magnitude Response and Phase Delay Characteristics
- 44. Phase Delay Characteristics: GenCESP—Lagrange—Sinc
- 46. (Absolute) Interpolation Error — Lower the Better!
- 47. What/Why/How are we selling…
- 48. Wrapping Up the Talk…
- 49. Thanks! Questions | Remarks | Comments
Editor's Notes
- 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!