Digital sound processing Convolution Digital Filters FFT Fast Convolution

Sampling Sampling an electrical signal means capturing its value repeatedly at a constant, very fast rate. Sampling frequency (fs) 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 ADSP-2116x Workshop

Time/frequency discretization DV Digital signal (sampled) Analog signal (true) Dt

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. ADSP-2116x Workshop

Common cases CD audio – fs = 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 – fs = 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. ADSP-2116x Workshop

System’s impulse response Time of flight Direct sound Early reflections Reverberant tail System under test Unit pulse d System’s impulse response

A simple linear system Real-world system (one input, one output) CD player Amplifier Loudspeaker Microphone Analyzer “SYSTEM” Block diagram x(t) h(t) y(t) Input signal System’s Impulse Response (Transfer function) Output signal

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

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 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 faster

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…) ADSP-2116x Workshop

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) IFFT The inverse transform is also possible (from frequency to time) FFT Frequency spectrum (32 bands + DC) ADSP-2116x Workshop

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) Elaborazione numerica del suono ADSP-2116x Workshop

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). ADSP-2116x Workshop

“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 ADSP-2116x Workshop

“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. ADSP-2116x Workshop

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 Window FFT Block 2 Window FFT Block 3 Window FFT ADSP-2116x Workshop

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”) ADSP-2116x Workshop

Fast FIR filtering with FFT Convolution is signifcantly faster if performed in frequency domain: x(n) X(k) FFT X(k)  H(k) Y(k) y(n) IFFT x(n)  h(n) Problems The whole lenght of signal must be recorded before being processed if N is large, a lot of memory is required. Solution “Overlap & Save”Algorithm ADSP-2116x Workshop

Convoluzione veloce FFT con Overlap & Save (Oppenheim & Shafer, 1975): N-point x IFFT Xm(k)H(k) Select last N – Q + 1 samples Append to y(n) xm(n) h(n) Problems Excessive processing latency between input and output If N is large, a lot of memory is still required Solution “uniformly-partitioned Overlap & Save” ADSP-2116x Workshop

Uniformly Partitioned Overlap & Save 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 2nd block 3rd block 4th block ADSP-2116x Workshop