2. | Page 1 09.11.2012 Angelo Farina UNIPR | All Rights Reserved | Confidential Digital sound processing Convolution Digital Filters FFT Fast Convolution

3. | Page 2 09.11.2012 Angelo Farina UNIPR | All Rights Reserved | Confidential Sampling  Sampling an electrical signal means capturing its value repeatedly at a constant, very fast rate.  Sampling frequency (f s ) is defined as the number of samples captured per second  The sampled value is known with finite precision, given by the “number of bits” of the analog-to-digital converter, which is limited (typically ranging between 16 and 24 bits) Of consequence, in a time-amplitude chart, the analog waveform is approximated by a sequence of points, which lye in the knots of a lattice, as both time and amplitude are integer multiplies of small “sampling units” of time and amplitude Of consequence, in a time-amplitude chart, the analog waveform is approximated by a sequence of points, which lye in the knots of a lattice, as both time and amplitude are integer multiplies of small “sampling units” of time and amplitude

4. | Page 3 09.11.2012 Angelo Farina UNIPR | All Rights Reserved | Confidential Time/frequency discretization  VVVV Analog signal (true) Digital signal (sampled)

5. | Page 4 09.11.2012 Angelo Farina UNIPR | All Rights Reserved | Confidential Fidelity of sampled signals Can a sampled digital signal represent faithfully the original analog one? YES, but only if the following “Shannon theorem” is true: “Sampling frequency must be at least twice of the largest frequency in the signal being sampled” A frequency equal to half the sampling frequency is named the “Nyquist frequency”– for avoiding the presence of signals at frequencies higher than the Nyquist’s one, an analog low-pass filter is inserted before the sampler. It is called an “anti Aliasing” filter.

6. | Page 5 09.11.2012 Angelo Farina UNIPR | All Rights Reserved | Confidential Common cases  CD audio – f s = 44.1 kHz – discretization = 16 bit Nyquist frequency is 22.05 kHz, the anti-aliasing starts at 20 kHz, so that at 22.05 kHz the signal is already attenuated by at least 80 dB. Hence the filter is very steep, causing a lot of artifacts in time domain (ringing, etc.)  DAT recorder – fs = 48 kHz – discretization = 16 bit Nyquist frequency is 24 kHz, the anti-aliasing starts at 20 kHz, so that at 24 kHz the signal is already attenuated by at least 80 dB. Now the filter is less steep, and the time-domain artifacts are almost gone.  DVD Audio – f s = 96 kHz – discretization = 24 bit Nyquist frequency is 48 kHz, but the anti-aliasing starts around 24 kHz, with a very gentle slope, so that at 48 kHz the signal is attenuated by more than 120 dB.  Such a gentle filter is very “short” in time domain, hence there are virtually no time-domain artifacts.

7. | Page 6 09.11.2012 Angelo Farina UNIPR | All Rights Reserved | Confidential Impulse Response System under test Unit pulse  System’s impulse response Time of flight Direct sound Early reflections Reverberant tail

8. | Page 7 09.11.2012 Angelo Farina UNIPR | All Rights Reserved | Confidential A simple linear system CD player AmplifierLoudspeakerMicrophone Real-world system (one input, one output) Block diagram x(  ) h(  ) y(  ) Input signal System’s Impulse Response (Transfer function) Output signal “SYSTEM” Analyzer

9. | Page 8 09.11.2012 Angelo Farina UNIPR | All Rights Reserved | Confidential FIR Filtering (Finite Impulse Response) The effect of the linear system h on the signal x passing through it is described by the mathematical operation called “convolution”, defined by: This “sum of products” is also called FIR filtering, and models accurately any kind of linear systems. This is usually written, in compact notation, as: “convolution” operator

10. | Page 9 09.11.2012 Angelo Farina UNIPR | All Rights Reserved | Confidential IIR Filtering (Infinite Impulse Response) Alternatively, the filtering caused by a linear system can also be described by the following recursive formula: In practice, the filter is computed not only from the input samples x, but also as a function of the output samples y, obtained at the previous time steps.In practice, the filter is computed not only from the input samples x, but also as a function of the output samples y, obtained at the previous time steps. In many cases, this method allows for representinging faithfully the behaviour of the system with a much smaller number of coefficients than when employing FIR filtering.In many cases, this method allows for representinging faithfully the behaviour of the system with a much smaller number of coefficients than when employing FIR filtering. However, modern algorithms on fast computers make FIR filtering preferable and even fasterHowever, modern algorithms on fast computers make FIR filtering preferable and even faster

11. | Page 10 09.11.2012 Angelo Farina UNIPR | All Rights Reserved | Confidential The FFT Algorithm  The Fast Fourier Transform (FFT) is often employed in Acoustics, with two goals: ► Performing spectral analysis with constant bandwidth ► Fast FIR filtering  FFT transforms a segment of time-domain data in the corresponding spectrum, with constant frequency resolution, starting at 0 Hz (DC) up to Nyquist frequency (which is half of the sampling frequency)  The longer the time segment, the narrower will be the frequency resolution: [N sampled points in time] = > [N/2+1 frequency bands] (the +1 represents the band at frequency 0 Hz, that is the DC component – but in acoustics, this is always with zero energy…)

12. | Page 11 09.11.2012 Angelo Farina UNIPR | All Rights Reserved | Confidential 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) FFT Frequency spectrum (32 bands + DC) IFFT The inverse transform is also possible (from frequency to time)

13. | Page 12 09.11.2012 Angelo Farina UNIPR | All Rights Reserved | Confidential Elaborazione numerica del suono Complex spectrum, autospectrum  FFT yields a complex spectrum, at every frequency we get a value made of a real and an imaginary parts (Pr, Pi), or, equivalently, by modulus and phase  In many cases the phase is considered meaningless,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)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)

14. | Page 13 09.11.2012 Angelo Farina UNIPR | All Rights Reserved | Confidential Complex spectrum, autospectrum In other cases, also the phase information is relevant, and is charted separately (mainly when the FFT is applied to an impulse response).

15. | Page 14 09.11.2012 Angelo Farina UNIPR | All Rights Reserved | Confidential “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”): Theoretical spectrum Leakage

16. | Page 15 09.11.2012 Angelo Farina UNIPR | All Rights Reserved | Confidential “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.

17. | Page 16 09.11.2012 Angelo Farina UNIPR | All Rights Reserved | Confidential Window overlapping  The problem is that events occurring near the ends of two adjacent blocks are substantially not analyzed  To avoid this loss of information, instead of shifting the analysis window by one whole block, we need to analyze partially-overlapped blocks, with at least 50% overlapping, usually overalpped at 75% or even more Block 1 WindowFFT Block 2 WindowFFT Block 3 WindowFFT

18. | Page 17 09.11.2012 Angelo Farina UNIPR | All Rights Reserved | Confidential 17 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”)

19. | Page 18 09.11.2012 Angelo Farina UNIPR | All Rights Reserved | Confidential Convolution is signifcantly faster if performed in frequency domain: Problems The whole lenght of signal must be recorded before being processed The whole lenght of signal must be recorded before being processed if N is large, a lot of memory is required. if N is large, a lot of memory is required. x(n)X(k)FFT X(k)  H(k) Y(k)y(n) IFFT x(n)  h(n) y(n) Solution “Overlap & Save”Algorithm “Overlap & Save”Algorithm Fast FIR filtering with FFT

20. | Page 19 09.11.2012 Angelo Farina UNIPR | All Rights Reserved | Confidential FFT N-point FFT x IFFTXm(k)H(k) Select last N – Q + 1 samples Append to y(n) x m (n) h(n) Convoluzione veloce FFT con Overlap & Save (Oppenheim & Shafer, 1975): Problems Excessive processing latency between input and output Excessive processing latency between input and output If N is large, a lot of memory is still required If N is large, a lot of memory is still required Solution “uniformly-partitioned Overlap & Save” “uniformly-partitioned Overlap & Save” Overlap & Save

21. | Page 20 09.11.2012 Angelo Farina UNIPR | All Rights Reserved | Confidential 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 1 st block 2 nd block 3 rd block 4 th block Uniformly Partitioned Overlap & Save