Digital sound processing FFT Fast Convolution

1. Digital sound processingFFTFast Convolution

2. FFT IFFT Frequency spectrum (32 bands + DC) The inverse transform is also possible (from frequency to time) The FFT Algorithm The number of points in the time block must be a power of 2 – for example: 4096, 8192, 16384, etc. Time signal (64 points)

3. Elaborazione numerica del suono Complexspectrum, autospectrum • FFT yields a complexspectrum, ateveryfrequencyweget a value made of a real and an imaginaryparts(Pr, Pi), or, equivalently, by modulus and phase • In manycases the phaseisconsideredmeaningless,and only the magnitude of the spectrum is plotted in dB: • The second version of the formula contains the definition of the Autospectrum, that is the product, at every frequency, of the spectral complex number P(f) with its complex conjugate P’(f)

4. Complexspectrum, autospectrum In othercases, also the phase information isrelevant, and ischartedseparately (mainlywhen the FFT isapplied to an impulseresponse).

5. Leakage Theoretical spectrum “leakage” and “windows” • One of the assumptions of Fourier analysis is that the time-segment analysed represents a complete period of a periodic waveform • This is generally UNTRUE: the imperfect connection of the end of a segment with the beginning of the next one (identical, as the signal is assumed to be periodic), causes a “click”, which produces a wide-band “white noise”, contaminating the whole spectrum (“leakage”):

6. “leakage” and “windows” • If we want to analyze a generic, aperiodic signal, we need to “window” the signal inside the block being analyzed, bringing it to zero at both ends • To this purpose, many differnet types of “windows” are used, named “Hanning”, “Hamming”, “Blackmann”, “Kaizer”, “Bartlett”, “Parzen”, etc.

7. Windowoverlapping • The problemisthateventsoccurringnear the ends of twoadjacentblocks are substantiallynotanalyzed • To avoidthisloss of information, instead of shifting the analysiswindow by onewholeblock, weneed to analyzepartially-overlappedblocks, with atleast50% overlapping, usuallyoveralppedat75% or even more Block 1 Window FFT Block 2 Window FFT Block 3 Window FFT

8. Averaging, waterfall, spectrogram • Once a sequence of FFT spectra is obtained, we can average them either exponentially (Fast, Slow) o linearly (Leq), emulating a SLM • Alternatively, we can visualize how the spectrum changes over time, by two graphical representations called “waterfall” and “spectrogram” (or “sonogram”)

9. FFT X(k) x(n) x(n)  h(n) X(k)  H(k) IFFT Y(k) y(n) y(n) • The wholelenght of signal must be recordedbeforebeingprocessed Problems • ifNis large, a lot of memoryisrequired. Solution • “Overlap & Save”Algorithm Fast FIR filtering with FFT Convolution is signifcantly faster if performed in frequency domain:

10. xm(n) FFT N-point Select last N – Q + 1 samples IFFT Xm(k)H(k) x h(n) FFT N-point • Excessive processing latencybetween input and output Problems • IfNis large, a lot of memoryisstillrequired Append to y(n) Solution • “uniformly-partitioned Overlap & Save” Overlap & Save Convoluzione veloce FFT con Overlap & Save (Oppenheim & Shafer, 1975):

11. Filter’s impulse response h(n) is also partitioned in a number of blocks Now latency and memory occupation reduce to the length of one single block 1st block 3rd block 4th block 2nd block UniformlyPartitionedOverlap & Save