1. Decibel values: sum and difference

2. Sound level summation in dB (1): Incoherent (energetic) sum of two different sounds: Lp 1 = 10 log (p 1 /p rif ) 2 (p 1 /p rif ) 2 = 10 Lp1/10 Lp 2 = 10 log (p 2 /p rif ) 2 (p 2 /p rif ) 2 = 10 Lp2/10 (p T /p rif ) 2 = (p 1 /p rif ) 2 + (p 2 /p rif ) 2 = 10 Lp1/10 + 10 Lp2/10 Lp T = Lp 1 + Lp 2 = 10 log (p T /p rif ) 2 = 10 log (10 Lp1/10 + 10 Lp2/10 )

3. Sound level summation in dB (2): incoherent sum of two levels Example 1: L 1 = 80 dB L 2 = 85 dB L T = ? L T = 10 log (10 80/10 + 10 85/10 ) = 86.2 dB. Example 2: L 1 = 80 dB L 2 = 80 dB L T = 10 log (10 80/10 + 10 80/10 ) = L T = 80 + 10 log 2 = 83 dB.

4. Sound level subtraction in dB (3): Level difference Example 3: L 1 = 80 dB L T = 85 dB L 2 = ? L 2 = 10 log (10 85/10 - 10 80/10 ) = 83.35 dB

5. Sound level summation in dB (4): Coherent sum of two (identical) sounds: Lp 1 = 20 log (p 1 /p rif )(p 1 /p rif ) = 10 Lp1/20 Lp 2 = 20 log (p 2 /p rif ) (p 2 /p rif ) = 10 Lp2/20 (p T /p rif ) = (p 1 /p rif )+ (p 2 /p rif ) = 10 Lp1/20 + 10 Lp2/20 Lp T = Lp 1 + Lp 2 = 10 log (p T /p rif ) 2 = 20 log (10 Lp1/20 + 10 Lp2/20 )

6. Level summation in dB (5): coherent sum of levels Example 4: L 1 = 80 dB L 2 = 85 dB L T = ? L T = 20 log (10 80/20 + 10 85/20 ) = 88.9 dB. Example 5: L 1 = 80 dB L 2 = 80 dB L T = 20 log (10 80/20 + 10 80/20 ) = L T = 80 + 20 log 2 = 86 dB.

7. Frequency analysis

8. Sound spectrum The sound spectrum is a chart of SPL vs frequency. Simple tones have spectra composed by just a small number of spectral lines, whilst complex sounds usually have a continuous spectrum. a)Pure tone b)Musical sound c)Wide-band noise d)White noise

9. Time-domain waveform and spectrum: a)Sinusoidal waveform b)Periodic waveform c)Random waveform

10. Analisi in bande di frequenza: A practical way of measuring a sound spectrum consist in employing a filter bank, which decomposes the original signal in a number of frequency bands. Each band is defined by two corner frequencies, named higher frequency f hi and lower frequency f lo. Their difference is called the bandwidth f. Two types of filterbanks are commonly employed for frequency analysis: constant bandwidth (FFT); constant percentage bandwidth (1/1 or 1/3 of octave).

11. Constant bandwidth analysis: narrow band, constant bandwidth filterbank: f = f hi – f lo = constant, for example 1 Hz, 10 Hz, etc. Provides a very sharp frequency resolution (thousands of bands), which makes it possible to detect very narrow pure tones and get their exact frequency. It is performed efficiently on a digital computer by means of a well known algorithm, called FFT (Fast Fourier Transform)

12. Constant percentage bandwidth analysis: Also called octave band analysis The bandwidth f is a constant ratio of the center frequency of each band, which is defined as: f hi = 2 f lo 1/1 octave f hi = 2 1/3 f lo 1/3 octave Widely employed for noise measurments. Typical filterbanks comprise 10 filters (octaves) or 30 filters (third-octaves), implemented with analog circuits or, nowadays, with IIR filters

13. Nominal frequencies for octave and 1/3 octave bands: 1/1 octave bands 1/3 octave bands

14. Octave and 1/3 octave spectra: 1/3 octave bands 1/1 octave bands

15. Narrowband spectra: Linear frequency axis Logaritmic frequency axis

16. White noise and pink noise White Noise: Flat in a narrowband analysis Pink Noise: flat in octave or 1/3 octave analysis

17. Critical Bands (BARK): The Bark scale is a psychoacoustical scale proposed by Eberhard Zwicker in 1961. It is named after Heinrich Barkhausen who proposed the first subjective measurements of loudness Third octave bands

18. Critical Bands (BARK): Comparing the bandwidth of Barks and 1/3 octave bands Barks 1/3 octave bands