IEEE ICASSP’00, Istanbul, Turkey, June 2000 HELSINKI UNIVERSITY OF TECHNOLOGY Principles of Fractional Delay Filters Vesa Välimäki 1 and Timo I. Laakso 2 Helsinki University of Technology 1 Laboratory of Acoustics and Audio Signal Processing 2 Signal Processing Laboratory (Espoo, Finland) Välimäki and Laakso 2000 1 HELSINKI UNIVERSITY OF TECHNOLOGY Principles of Fractional Delay Filters 1. Motivation 2. Ideal FD Filter and Its Approximations 3. FD Filters for Very Small Delay 4. Time-Varying FD Filters 5. Resampling of Nonuniformly Sampled Signals 6. Conclusions Välimäki and Laakso 2000 2
HELSINKI UNIVERSITY OF TECHNOLOGY 1. Motivation: The Importance of Sampling at the Right Time a) Uniform sampling problems • Fine-tune sampling rate and/or instant 1) Constant delay : accurate time delays 2) Time-varying delay : resampling on a nonuniform grid b) Nonuniform sampling problems • Sampling instants determined, e.g., by physical constraints • Resample on a uniform grid Välimäki and Laakso 2000 3 HELSINKI UNIVERSITY OF TECHNOLOGY 1. Motivation: Many Applications (2) • Sampling rate conversion – Especially conversion between incommensurate rates, e.g., between standard audio sample rates 48 and 44.1 kHz • Music synthesis using digital waveguides – Comb filters using fractional-length delay lines • Doppler effect in virtual reality • Synchronization of digital modems • Speech coding and synthesis • Beamforming • etc. Välimäki and Laakso 2000 4
HELSINKI UNIVERSITY OF TECHNOLOGY 2. Ideal FD Filter and Approximations • FD filter = digital version of a continuous time delay • An ideal lowpass filter with a time shift: Impulse response is a sampled and shifted sinc function : sinc( n – D ) = sin[p( n – D )]/p( n – D ) where n is the time index; D is delay in samples Välimäki and Laakso 2000 5 HELSINKI UNIVERSITY OF TECHNOLOGY 2. Ideal FD Filter and Approximations (2) Sampled Sinc Function ( D = 0) When D integer: 1 Sampled at zero- 0.5 crossings 0 (no fractional delay) -0.5 -5 0 5 Sampled & Shifted Sinc ( D = 0.3) When D non-integer: 1 Sampled between 0.5 zero-crossings 0 ⇒ Infinite-length -0.5 -5 0 5 impulse response Time in Samples Välimäki and Laakso 2000 6
HELSINKI UNIVERSITY OF TECHNOLOGY 2. FIR FD Approximations • FD must be approximated using FIR or IIR filters (see, e.g., Laakso et al., IEEE SP Magazine , 1996) • FIR FD filters have asymmetric impulse response but they aim at having linear phase • Approximation of complex-valued frequency response (magnitude and phase) ⇒ traditional linear-phase methods not applicable • Most popular technique: Lagrange interpolation Välimäki and Laakso 2000 7 HELSINKI UNIVERSITY OF TECHNOLOGY 2. Lagrange Interpolation • Polynomial curve fitting = max. flat approximation • Closed-form formula for coefficients: N − ∏ D k = h n ( ) for n = 0, 1, 2,..., N − n k = k 0 ≠ k n where D is delay and N is the filter order • Linear interpolation is obtained with N = 1: h (0) = 1 – D , h (1) = D • Good approximation at low frequencies only Välimäki and Laakso 2000 8
HELSINKI UNIVERSITY OF TECHNOLOGY 2. IIR FD approximations • Allpass filters are well suited to FD approximations, since their magnitude response is exactly flat • The easiest choice is the Thiran a N x ( n ) y ( n ) allpass filter (Fettweis, 1972): − − N 1 1 − + z z N ∏ D N n − a a = − k − N 1 1 a ( 1 ) k − + + k D N k n = n 0 Μ Μ for n = 0, 1, 2, ..., N − − 1 1 z z − a a − 1 N 1 • Close relative to Lagrange: Max. flat approximation at 0 Hz − − 1 1 z z − a N Välimäki and Laakso 2000 9 HELSINKI UNIVERSITY OF TECHNOLOGY 3. FD Filters for Very Small Delays • Very small delays required, e.g., in feedback loops and control applications – We consider the case of D < 1 • There is always inherent delay in good-quality FD filters – Total delay about N /2 for FIR and about N for allpass filters – Allpass FD filters are stable only for D > N – 1 • Thiran all-pole filter (Thiran, 1971) provides small delay – Lowpass-type magnitude response cannot be controlled • FIR filters can approximate small delays but the quality gets low Välimäki and Laakso 2000 10
HELSINKI UNIVERSITY OF TECHNOLOGY 3. FD Filters for Very Small Delays (2) • Comparison of various FD filters for a delay D = 0.5 20 Frequency response error (dB) 0 Lagrange ( N = 1) Lagrange ( N = 9) -20 Thiran allpass ( N = 1) Thiran all-pole ( N = 1) Thiran all-pole ( N = 10) -40 (Fig. 4 of the paper) -60 0 0.1 0.2 0.3 0.4 0.5 Normalized frequency Välimäki and Laakso 2000 11 HELSINKI UNIVERSITY OF TECHNOLOGY 4. Time-Varying FD Filters • Many applications need tunable FD filters • Three principles to change the coefficients: 1) Recomputing of coefficients 2) Table lookup 3) Polynomial approximation of coefficients – Farrow structure (Farrow, 1988) • FIR filters better suited to TV filtering than IIR filters – Time-varying recursive filters suffer from transients (we proposed a solution at ICASSP’98 ; see also IEEE Trans. SP , Dec. 1998) Välimäki and Laakso 2000 12
HELSINKI UNIVERSITY OF TECHNOLOGY 4. Time-Varying FD Filters (2) • Farrow (1988) structure for FIR FD filters – Direct control of filter properties by delay parameter D x ( n ) C N ( z ) C N − 1 z ( ) C 2 z ( ) C 1 z ( ) C 0 z ( ) . . . D D D D y ( n ) – Polynomial interpolation filters can be directly implemented – Vesma and Saramäki (1996) have proposed a modified Farrow structure and general methods to design the filters C k ( z ) Välimäki and Laakso 2000 13 HELSINKI UNIVERSITY OF TECHNOLOGY 5. Polynomial Resampling of Nonuniformly Sampled Signals • When sampling is nonuniform and sampling instants are known accurately, uniform resampling is possible – Problem: traditional sinc series LS fitting computationally intensive and numerically problematic • Alternative: polynomial signal model for smooth (low- frequency) signals – Extension of nonuniform Lagrange interpolation – Suppress noise also instead of exact reconstruction – See: Laakso et al. , Signal Processing , vol. 80, no. 4, 2000 Välimäki and Laakso 2000 14
HELSINKI UNIVERSITY OF TECHNOLOGY 5. Examples of Nonuniform Reconstruction • 2 sinusoids plus noise (SNR 3 dB); pol. order 5; filter order 6 • Noise reduction: 3.70 dB (LS reconstruction), 3.43 dB (0th-order appr.) and 3.82 dB (2nd-order appr.) JITTERED RANDOM SAMPLING JITTERED RANDOM SAMPLING 2 2 1 1 AMPLITUDE AMPLITUDE 0 0 -1 -1 -2 -2 10 20 30 40 50 10 20 30 40 50 TIME TIME Välimäki and Laakso 2000 15 HELSINKI UNIVERSITY OF TECHNOLOGY 6. Conclusions • Fractional delay filters provide a link between uniform and nonuniform sampling • Useful in numerous signal processing tasks – Sampling rate conversion, synchronization of digital modems, time delay estimation, music synthesis, ... • Resampling of nonuniformly sampled signals on a uniform grid • MATLAB tools for FD filter design available at: http://www.acoustics.hut.fi/software Välimäki and Laakso 2000 16
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