Sound waves Sound is created by elastic vibrations or oscillations of particles in a particular medium. The vibrations are transmitted from particles to (neighbouring) particles: sound wave. Sound waves propagate in the propagation medium (air), not in the empty space. septembre 2014 Audio signals and systems 1
Frequency Periodic vibrations are characterized by their frequency = number of oscillation periods per second (Hertz or Hz). Humans can hear frequencies between 20 Hz and 20 khz INFRASOUNDS SPEECH BASS TONES TREBLE ULTRASOUNDS 20 Hz 20 khz septembre 2014 Audio signals and systems 2
Pure tones (harmonic vibrations) 880 Hz 1320 Hz 440 Hz 1760 Hz 2200 Hz septembre 2014 Audio signals and systems 3
Transverse propagation The vibrations of the particles are perpendicular to the direction of propagation (x). Displacement of the particle A as a function of time. If the vibration is harmonic (pure tone), the wavelength l expresses its periodicity in the space domain. septembre 2014 Audio signals and systems 4
Wavelength: definition The wavelength is the distance travelled by the sound wave during one period of oscillations (pure tone). at 100 Hz : 3,4 m at 1000 Hz : 0,34 m (34 cm) at 10 000 Hz : 0,034 m (3,4 cm) l=c/f Frequency (Hz) Sound speed (celerity): 340 m/s septembre 2014 Audio signals and systems 5
Acoustic pressure Air pressure Dp = 2.10-5 to 20 Pa Atmospheric pressure (without sound) 10 5 Pa time Time evolution of the air pressure at a given location in space. Acoustic vibrations create perturbations in the propagation medium (air). In particular, the atmospheric pressure is slightly modified by a sound wave. Regions of air compression and rarefaction are created in the medium. septembre 2014 Audio signals and systems 6
Acoustic pressure Air pressure Dp = 2.10-5 to 20 Pa Atmospheric pressure (without sound) 10 5 Pa time Time evolution of the air pressure at a given location in space. The acoustic pressure is defined as the difference between the instantaneous air pressure and the atmospheric pressure existing at the same location without sound. septembre 2014 Audio signals and systems 7
Acoustic pressure (Pa) is a function of time and space The variations of air pressure can be recorded by a microphone, giving an electric signal (the audio signal). septembre 2014 Audio signals and systems 8
Acoustic pressure (Pa) is a function of time and space p(x,t) = A(x) cos(wt-kx) Amplitude A(x) related to loudness. Frequency w=2pf related to the pitch (musical note). Wavenumber k = 2p/l : expresses the periodicity in space (along the direction of propagation). septembre 2014 Audio signals and systems 9
The acoustic pressure in time and frequency For a periodic signal, the Fourier series theorem gives the amplitude of the sound contributions at each discrete frequency. jw p( x, t) m( x) e m wm 2pmf m t 0 septembre 2014 Audio signals and systems 10
Time and frequency representations of a (non-sinusoidal) periodic signal septembre 2014 Audio signals and systems 11
The acoustic pressure in time and frequency For a non-periodic signal, the Fourier transform gives the amplitude of the sound contributions at each frequency (continuous spectrum). p( x, t) 2 1 p P ( x, w ) e j w t d w septembre 2014 Audio signals and systems 12
Time and frequency representations of a non-periodic signal septembre 2014 Audio signals and systems 13
The octave band system In acoustics, the frequency analysis of sounds if often performed by grouping frequencis in octave bands. Octave bands are frequency intervals whose width is proportional to their central frequency. Octave number Central frequency (Hz) 2 3 4 5 6 7 8 9 62,5 125 250 500 1000 2000 4000 8000 Width (Hz) 44 89 177 353 707 1414 2829 5657 septembre 2014 Audio signals and systems 14
The octave band system Examples of the white noise and the pink noise 80 70 60 50 40 30 white noise pink noise 20 10 0 31,25 62,5 125 250 500 1K 2K 4K 8K 16K septembre 2014 Audio signals and systems 15
Acoustic power and intensity Acoustic intensity I (W/m 2 ) = Flow of sound energy through unit cross-sectional area, per second. I is often proportional to (p rms )² Acoustic power (watts) = Energy that is measured or observed in one second. Example : the acoustic power of a sound source (a loudspeaker). septembre 2014 Audio signals and systems 16
Sound pressure level in decibels (db) RMS value of the acoustic pressure Hearing threshold at 1 khz : 20 µpa (rms) Threshold of pain at 1 khz : 20 Pa (rms) p rms 1 T T 0 p 2 ( t) dt Definition of Lp in db Hearing threshold at 1 khz : 0 db Threshold of pain at 1 khz : 120 db L p 20 log10 p ( Pa) rms 0.00002 septembre 2014 Audio signals and systems 17
Addition of acoustic pressures If the pressure signals are uncorrelated (independent): p 2 p 2 p 2 rms, tot rms,1 rms,2 In decibels: 80 db + 80 db = 83 db! If intensity (power) is doubled, then +3dB! If acoustic pressure is doubled, then +6dB! Lp, tot L p 10 log 10 10 p ( Pa) rms 20 log10 0.00002 L p /10 L 1 p 10 2 /10 septembre 2014 Audio signals and systems 18
Addition of acoustic pressures Addition of «N» uncorrelated pressure levels with the same amplitude: L p, tot L p /10 L p 10 log 10 10 1 10 + 10 log(n) 2 /10... N 10 log(n) 2 +3 db 4 +6 db 8 +9 db 10 +10 db 100 +20 db septembre 2014 Audio signals and systems 19
Relation amplitude/loudness -6dB +6dB Moins fort (-6 db) Plus fort + 6 db septembre 2014 Audio signals and systems 20
Time and frequency representations of some typical audio signals. septembre 2014 Audio signals and systems 21
Pure tones (440 Hz) septembre 2014 Audio signals and systems 22
Non-sinusoidal periodic signal (f 0 =440 Hz) septembre 2014 Audio signals and systems 23
Beat : 440 Hz and 445 Hz Pure tone 440 Hz Time (sec) Beat Frequency (Hz) septembre 2014 Audio signals and systems 24
Beat : 440 Hz and 540 Hz - pure tone 440 Hz - beat septembre 2014 Audio signals and systems 25
Spectral lines (addition of pure tones) - harmonics (periodic signal) - non-periodic signal septembre 2014 Audio signals and systems 26
Periodic signal Harmonics Time (sec) Frequency (Hz) septembre 2014 Audio signals and systems 27
Boat foghorn Time (sec) Non-periodic signal Frequency (Hz) septembre 2014 Audio signals and systems 28
Musical instruments - pure tone 440 Hz - viola (A440) A ADSR contour (envelope) D S R - violin (A440) t - cello (A220) Attack-Decay- Sustain-Release septembre 2014 Audio signals and systems 29
Viola A 440 Violin A 440 Time (sec) Time (sec) Frequency (Hz) A D S R Frequency (Hz) t Attack-Decay- Sustain-Release septembre 2014 Audio signals and systems 30
Cello A 220 A D S R t Attack-Decay- Sustain-Release Time (sec) Frequency (Hz) septembre 2014 Audio signals and systems 31
Noise signals - white noise 80 70 60 - pink noise 50 40 30 white noise pink noise 20 10 0 31,25 62,5 125 250 500 1K 2K 4K 8K 16K septembre 2014 Audio signals and systems 32
Combination of noise and Factory noise 1 harmonics Factory noise 2 Time (sec) Time (sec) Frequency (Hz) Frequency (Hz) septembre 2014 Audio signals and systems 33
Properties of audio systems Frequency response and bandwith: bandwith at +/- 1dB Noise measurement: Zin (=0) Audio system Vout SNR = 20 log(v rms,signal /V rms, noise ) septembre 2014 Audio signals and systems 34
Signal-to-noise ratio (SNR) - original -SNR 50 db - SNR 90 db - SNR 20 db SNR = 20 log(v rms,signal /V rms, noise ) septembre 2014 Audio signals and systems 35
Typical transfer function of an audio system Clipping Noise septembre 2014 Audio signals and systems 36
Harmonic distorsion 3kHz sinusoid 3kHz clipped sinusoid septembre 2014 Audio signals and systems 37
Harmonic distorsion Clipping (saturation) creates harmonic distorsion. THD = total harmonic distorsion THD is the ratio of the total power contained in the harmonics, at the system s output, to the power contained in the fundamental. V 0,rms septembre 2014 Audio signals and systems 38
Harmonic distorsion Clipping (saturation) creates harmonic distorsion. THD+N = total harmonic distorsion + noise THD+N is the ratio of the output voltage without the fundamental component to the output voltage of the fundamental. V HDN,rms = output voltage without fundamental (notch filter) septembre 2014 Audio signals and systems 39
Typical variation of the THD+N vs the amplitude of the input signal septembre 2014 Audio signals and systems 40
Intermodulation distorsion 250 Hz + 8 khz septembre 2014 Audio signals and systems 41
Intermodulation distorsion Clipped :250 Hz + 8 khz septembre 2014 Audio signals and systems 42