OLA and FBS Duality Review

Transcription

1 OLA and FBS Duality Review MUS421/EE367B Lecture 10A Review of OverLap-Add (OLA) and Filter-Bank Summation (FBS) Interpretations of Short-Time Fourier Analysis, Modification, and Resynthesis Julius O. Smith III Center for Computer Research in Music and Acoustics (CCRMA) Department of Music, Stanford University Stanford, California March 24, 2014 We now review Filter Bank Summation (FBS) and OverLap Add (OLA), stressing their Fourier duality. For simplicity, R will be taken as 1. Remember that in OLA, R is the hop size, or the distance (in time samples) between successive overlap-add frames. In FBS, R is interpreted as a downsampling factor. The common starting point for both OLA and FBS is the : m (ω k ) = w(n m)e jω kn The point of departure is the interpretation of the above summation. 1 2

2 Analysis For each frame: OLA m (ω k ) = Sample 2π/N (DTFT(x m )) = [w(n m)]e jω kn = = F k {x m (n)} x m (n)e jω kn where x m (n) is the m th windowed frame of x. Hence we have the following interpretation: Window the signal Take the Fourier Transform w 0 w 1 w M 1 DFT n (ω 0 ) n (ω 1 ) n (ω N 1 ) Windowing 3 4

3 FBS FBS Analysis Filter Bank m (ω k ) = [e jωkn ]w(n m) = = (x k w)(m) x k (n) w(m n) e jω 0n n (ω 0 ) n (ω 1 ) e jω 1n where x k is x modulated by e jω kn, and w =. Hence we have the following interpretation: For each analysis frequency: n (ω N 1 ) Heterodyne signal by e jω kn Filter with w e jω N 1n 5 6

4 OLA To resynthesize the signal: Re-Synthesis OLA Analysis and Synthesis w 0 w 1 w M 1 Windowing Inverse Transform Overlap and add shifted frames DFT n (ω 0 ) n (ω 1 ) n (ω N 1 ) x = m= COLA constraint: Shift mr (DFT 1 (Sample2π N,ω k (DTFT( m H)))) m= w(n mr) = c, where c is any constant W(ω l ) = 0, l = 1,2,...,R 1, ω l = 2πl/R, i.e., Window transform is 0 at all harmonics of the frame rate IDFT Overlap Add Buffer 7 8

5 Filter-Bank Summation (FBS) n (ω 0 ) To resynthesize the signal: Heterodyne each subband (shift in frequency) Sum e jω 0n e jω 1n n (ω 1 ) e jω 0n + e jω 1n Window constraint: x(m) = N 1 k=0 m (ω k )e jω km w(rn) = 0, r = 1,2,... This is equivalent to a constant overlap constraint in frequency (dual of Poisson summation formula) The filterbank summation performs an overlap add in frequency. e jω N 1n n (ω N 1 ) e jω N 1n Unmodified FBS is COLA in the frequency domain, Nyquist(N) in the time domain Unmodified OLA is COLA in the time domain, Nyquist(2πL/M) in the frequency domain 9 10

6 OLA Fixed Spectral Modification Perform multiplicative spectral modification before resynthesis Must zero pad in time, and window must meet COLA constraint Results in the convolution of the signal and the impulse response of the filter FBS Perform multiplicative spectral modification before resynthesis Results in convolution of the signal and the windowed impulse response of the filter e jω0n e jω1n n (ω 0 ) n (ω 1 ) H 0 H 1 e jω0n + e jω1n w 0 w 1 w M 1 DFT n (ω 0 ) n (ω 1 ) n (ω N 1 ) H 0 H 1 H M 1 Y n (ω 0 ) Y n (ω 1 ) Y n (ω N 1 ) IDFT Windowing Spectral Modification Modified Spectrum e jωn 1n n (ω N 1 ) H(N-1) Modification e jωn 1n Overlap Add Buffer 11 12

7 Time Varying Spectral Modification OLA Perform time varying multiplicative spectral modification before resynthesis Must zero pad, and window must meet COLA constraints Results in the convolution of the signal and a window-filtered version of the time varying impulse response (no length limit) w 0 w 1 w M 1 n (ω 0 ) n (ω 1 ) H n (0) H n (1) H n (N 1) Y n (ω 0 ) Y n (ω 1 ) DFT IDFT mn- YmN- Windowing Time Varying Modification Modified Spectrum Overlap Add Buffer FBS Perform multiplicative time varying spectral modification before resynthesis Results in convolution of the signal and the windowed time varying impulse response of the filter Spectral changes occur immediately Discontinuities likely in time domain when filter changes 13 14

8 OLA more practical for time-varying filter implementation n (ω 0 ) e jω0n H n (0) e jω0n n (ω 1 ) + e jω1n H n (1) e jω1n n (ω N 1 ) e jωn 1n H n (N 1) e jωn 1n Time Varying Modification 15