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

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 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 Sampling frequency (f s ) is defined as the number of samples captured per second The sampled value is known with finite precision, given by thenumber of bits of the analog-to-digital converter, which is limited (typically ranging between 16 and 24 bits) The sampled value is known with finite precision, given by thenumber 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 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 Nyquists 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.) 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. 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. 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. 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 Unit pulse Systems 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 Systems Impulse Response (Transfer function) Output signal SYSTEMSYSTEM 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 operatorconvolution 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: 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) 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…) 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…)