SLIDE 1
CTP431- Music and Audio Computing Digital Audio
Graduate School of Culture Technology KAIST Juhan Nam
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Digital Representations
Sound Image Text
… 0 1 1 0 1 1 0 … … 1 0 0 1 1 0 1 … … 0 0 1 1 0 1 1 …
CTP431- Music and Audio Computing Digital Audio Graduate School of - - PowerPoint PPT Presentation
CTP431- Music and Audio Computing Digital Audio Graduate School of - - PowerPoint PPT Presentation
CTP431- Music and Audio Computing Digital Audio Graduate School of Culture Technology KAIST Juhan Nam 1 Digital Representations 0 1 1 0 1 1 0 Sound 1 0 0 1 1 0 1 Image 0 0 1 1 0 1 1 Text Digital
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SLIDE 3
Digital Representations
§ Sampling and Quantization
– Sound (samples) – Image (pixels)
§ Trade-off
– Resolution (quality) and data size
SLIDE 4
Digital Audio Chain
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…0 0 1 0 1 0 …
SLIDE 5
Sampling
- Convert continuous-time signal to discrete-time signal by periodically
picking up the instantaneous values
– Represented as a sequence of numbers; pulse code modulation (PCM) – Sampling period (Ts): the amount of time between samples – Sampling rate ( fs =1/Ts )
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Ts
x(t) → x(nTs)
Signal notation
SLIDE 6
Sampling Theorem
§ What is an appropriate sampling rate?
– Too high: increase data rate – Too low: become hard to reconstruct the original signal
§ Sampling Theorem
– In order for a band-limited signal to be reconstructed fully, the sampling rate must be greater than twice the maximum frequency in the signal – Half the sampling rate is called Nyquist frequency ( )
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fs > 2⋅ fm
fs 2
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Sampling in Frequency Domain
§ Sampling in time creates imaginary content of the original at every fs frequency § Why?
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fm
- fm
fm
- fm
fs+fm fs-fm fs
- fs
𝑦 𝑢 = sin 2𝜌𝑔
*𝑢
𝑦 𝑜 = sin 2𝜌𝑔
*𝑜𝑈
- = sin 2𝜌𝑔
*𝑜/𝑔
- = sin 2𝜌𝑔
*𝑜/𝑔
- ± 2𝜌𝑙𝑜 = sin 2𝜌𝑜(𝑔
* ± 𝑙𝑔
- )/𝑔
- Nyquist Frequency
- fs +fm
Audible range Audible range
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Aliasing
§ If the sampling rate is less than twice the maximum frequency, the high- frequency content is folded over to lower frequency range
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0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 x 10
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−0.8 −0.6 −0.4 −0.2 0.2 0.4 0.6 0.8 1
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Aliasing in Frequency Domain
§ Sampling in time creates imaginary content of the original at every fs frequency § The frequency that we hear is 𝑔
- − 𝑔
*
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fm
- fm
fm
- fm
fs+fm fs-fm fs
- fs
- fs +fm
Audible range Audible range
fm < fs − fm
In order to avoid aliasing
SLIDE 10
Aliasing in Frequency Domain
§ For general signals, high-frequency content is folded over to lower frequency range
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fm
- fm
fs+fm fs-fm fs
- fs
Audible range
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Avoid Aliasing
§ Increase sampling rate § Use lowpass filters before sampling
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fs/2 fs
- fs
- fs/2
fs > 2⋅ fm
fm
- fm
fs+fm fs-fm fs
- fs
Lowpass Filter
SLIDE 12
Example of Aliasing
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Frequency sweep of the trivial sawtooth wave
Time (s) Frequency (Hz) 1 1.5 2 2.5 3 3.5 4 4.5 0.5 1 1.5 2 x 10
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Bandlimited sawtooth wave spectrum
5 10 15 20 −60 −40 −20 Frequency (kHz) Magnitude (dB)
5 10 15 20 −60 −40 −20 Frequency (kHz) Magnitude (dB)
Trivial sawtooth wave spectrum
SLIDE 13
Example of Aliasing
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https://www.youtube.com/watch?v=jHS9JGkEOmA Aliasing in Video
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Sampling Rates
§ Determined by the bandwidth of signals or hearing limits
– Consumer audio product: 44.1 kHz (CD) – Professional audio gears: 48/96/192 kHz – Speech communication: 8/16 kHz
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Quantization
§ Discretizing the amplitude of real-valued signals
– Round the amplitude to the nearest discrete steps – The discrete steps are determined by the number of bit bits
- Audio CD: 16 bits (-215 ~ 215-1) ß B bits (-2B-1 ~ 2B-1-1)
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Quantization Error
§ Quantization causes noise
– Average power of quantization noise: obtained from the probability density function (PDF) of the error
§ Signal to Noise Ratio (SNR)
– Based on average power – Based on the max levels
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1/2
- 1/2
P(e)
1
x2p(e)dx
−1/2 1/2
∫
= 112
Root mean square (RMS) of noise 20log10 Srms Nrms = 20log10 2B−1 / 2 112 = 6.02B+1.76 dB
(With 16bits, SNR = 98.08dB)
20log10 Smax Nmax = 20log10 2B−1 12 = 6.02B dB (With 16bits, SNR = 96.32 dB) RMS of full-scale sine wave
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Dynamic Range
§ Dynamic range
– The ratio between the loudest and softest levels
§ Human ear’s dynamic range
– Depending on frequency band
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20log10 Srms,max Srms,min = 20log10 2B−1 / 2 1/ 2 = 6.02B − 6 (With 16bits, DR = 90.31 dB) Equal Loudness Curve
Again, RMS of full-scale sine wave for both loudest and softest
SLIDE 18
Clipping and Headroom
§ Clipping
– Non-linear distortion that occurs when a signal is above the max level
§ Headroom
– Margin between the peak level and the max level
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0 dB
- 98.08 dB
Noise floor (By quantization) Max level Head room Clipping Min level
- 90.31 dB
B = 16 bits In digital audio, 0dB is regarded as the maximum level