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!
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.
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!
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.
42. FDF—GenCESP: Design Example Cascade of First Order Causal and Anti-Causal Filters, time-symmetric, exponentially decaying filters. Causal Anti—Causal
[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!