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• (Pohlmann pg. 27)
Proof in Couch pg. 90-91
• (Pohlmann pg. 27)
Proof in Couch pg. 90-91
• (Pohlmann pg. 27)
Proof in Couch pg. 90-91
• High fs: Large guard band, Allows varispeed
Low fs: Reduces transmission and storage BW
Critical Sampling: When a signal is sampled at exactly twice its highest frequency. Never done in audio
• High fs: Large guard band, Allows varispeed
Low fs: Reduces transmission and storage BW
Critical Sampling: When a signal is sampled at exactly twice its highest frequency. Never done in audio
• High fs: Large guard band, Allows varispeed
Low fs: Reduces transmission and storage BW
Critical Sampling: When a signal is sampled at exactly twice its highest frequency. Never done in audio

• A band limited waveform amplitude modulates an impulse train. The spectrum of an impulse train is sinewaves @ multiples of Fs. Modulated spectrum is waveform spectrum (bandlimited) repeated around multiples of Fs (with upper and lower sidebands). If impulses have some width, then the total spectrum is superimposed with the |Sin (x)/x| curve.

• giving a &amp;#x2018;quantity&amp;#x2019;
• giving a &amp;#x2018;quantity&amp;#x2019;
• giving a &amp;#x2018;quantity&amp;#x2019;

• 6dB of dynamic range per bit
• 6dB of dynamic range per bit
• 6dB of dynamic range per bit
• 6dB of dynamic range per bit
• 6dB of dynamic range per bit

• Error is +/- 1/2 Q with a rectangular PDF (equal chance)
• Error is +/- 1/2 Q with a rectangular PDF (equal chance)
• Error is +/- 1/2 Q with a rectangular PDF (equal chance)
• Error is +/- 1/2 Q with a rectangular PDF (equal chance)
• Distortion produces harmonics which can alias
Multiple input freq. can cause intermodulation distortion
Quantization error can create Aliasing (frequencies not present in source) even though it occurs after the sample process
• Distortion produces harmonics which can alias
Multiple input freq. can cause intermodulation distortion
Quantization error can create Aliasing (frequencies not present in source) even though it occurs after the sample process

• encodes low-level signals via PWM
ear averages PWM signal to resolve signal
With dither, resolution is below least significant bit!
• encodes low-level signals via PWM
ear averages PWM signal to resolve signal
With dither, resolution is below least significant bit!

Transcript

• 1. MUSC 365 Basics of Digital Audio Module (revised)
• 2. Analog = continuous
• 3. Analog = continuous • sound or audio waveform
• 4. Analog = continuous • sound or audio waveform • continuous time and amplitude
• 5. Analog = continuous • sound or audio waveform • continuous time and amplitude • inﬁnite possibilities within a given range
• 6. Digital = discrete
• 7. Digital = discrete • time and amplitude are discrete
• 8. Digital = discrete • time and amplitude are discrete • only certain values are allowed
• 9. Continuous vs. Discrete Continuous Discrete All numbers (including fractions) Only integers distance down my street number of houses on my street time it takes to cook an egg number of eggs a chicken lays volume of applesauce number of apples in a basket
• 10. Sampling
• 11. Sampling • the process of making discrete time
• 12. Sampling • the process of making discrete time • Amplitude of a waveform is captured (sampled) at regularly spaced intervals
• 13. Sampling • the process of making discrete time • Amplitude of a waveform is captured (sampled) at regularly spaced intervals • the rate of repeat of this regularly spaced interval is called the sample rate
• 14. Sampling Rate (ƒs)
• 15. Sampling Rate (ƒs) • The sample rate determines the bandwidth of the system
• 16. Sampling Rate (ƒs) • The sample rate determines the bandwidth of the system • A signal of bandwidth BW may be LOSSLESSLY sampled if the sampling rate ƒs ≥ 2 • BW
• 17. Sampling Rate (ƒs) • The sample rate determines the bandwidth of the system • A signal of bandwidth BW may be LOSSLESSLY sampled if the sampling rate ƒs ≥ 2 • BW • Input must be bandlimited to half the sampling rate
• 18. Sampling Rate (ƒs)
• 19. Sampling Rate (ƒs) • Common sample rates:
• 20. Sampling Rate (ƒs) • Common sample rates: • 44.1kHz for audio only (CD, MP3, etc)
• 21. Sampling Rate (ƒs) • Common sample rates: • 44.1kHz for audio only (CD, MP3, etc) • 48kHz for video/ﬁlm (DVD, etc)
• 22. Sampling Rate (ƒs) • Common sample rates: • 44.1kHz for audio only (CD, MP3, etc) • 48kHz for video/ﬁlm (DVD, etc) • double (2x) and quadruple (4x) those rates
• 23. Nyquist Frequency
• 24. Nyquist Frequency • half the sampling frequency (ƒs / 2)
• 25. Nyquist Frequency • half the sampling frequency (ƒs / 2) • Highest frequency possible in a digital system
• 26. Nyquist Frequency • half the sampling frequency (ƒs / 2) • Highest frequency possible in a digital system • 22.05kHz for ƒs = 44.1kHz
• 27. Sampling Process - Input
• 28. Sampling Process - Input • Initial audio input
• 29. Sampling Process - Input • Initial audio input • frequencies above ƒs / 2 have been removed
• 30. Sampling Process
• 31. Sampling Process • waveform is periodically sampled
• 32. Sampling Process
• 33. Sampling Process • the sampled signal
• 34. Sampling Process - Reconstruction
• 35. Sampling Process - Reconstruction • the sampled signal is reconstructed (made continuous)
• 36. Sampling Process - Reconstruction • the sampled signal is reconstructed (made continuous) • “connecting-the-dots”
• 37. Sampling Process
• 38. Sampling Process • There is only waveform that satisﬁes 2 conditions:
• 39. Sampling Process • There is only waveform that satisﬁes 2 conditions: • it passes thru all the sample points, and
• 40. Sampling Process • There is only waveform that satisﬁes 2 conditions: • it passes thru all the sample points, and • it does not have frequencies above ƒs / 2
• 41. Aliasing
• 42. Aliasing • Input signal must be bandlimited (frequencies above ƒs / 2 removed)
• 43. Aliasing • Input signal must be bandlimited (frequencies above ƒs / 2 removed) • If it is not, frequencies above ƒs / 2 are folded back into audio band
• 44. Aliasing • Input signal must be bandlimited (frequencies above ƒs / 2 removed) • If it is not, frequencies above ƒs / 2 are folded back into audio band • This artifact is called aliasing
• 45. Aliasing
• 46. Aliasing • a high frequency waveform is put into the sampler
• 47. Aliasing
• 48. Aliasing • the waveform is periodically sampled
• 49. Aliasing
• 50. Aliasing • the sampled signal
• 51. Aliasing
• 52. Aliasing • after the signal is reconstructed to a continuous waveform,
• 53. Aliasing • after the signal is reconstructed to a continuous waveform, • but it is not what was input!
• 54. Aliasing
• 55. Aliasing • Aliasing also happens in visual media...
• 56. Aliasing • Aliasing also happens in visual media... • If you watch a ﬁlm of a spinning wheel, it can seem to stop or go backwards
• 57. Quantization
• 58. Quantization • the process of making discrete Amplitude
• 59. Quantization • the process of making discrete Amplitude • the amplitude range is broken up into a ﬁxed number of level, also called quantization intervals
• 60. Quantization • the process of making discrete Amplitude • the amplitude range is broken up into a ﬁxed number of level, also called quantization intervals • amplitude is measured and assigned to the closest interval
• 61. Quantization Process
• 62. Quantization Process • sampled waveform
• 63. Quantization Process • sampled waveform • any amplitude is possible
• 64. Quantization Process
• 65. Quantization Process • amplitude is rounded to the closest quantization interval
• 66. Quantization Process
• 67. Quantization Process • this close-up shows that there is some error in the quantization process
• 68. Quantization Process
• 69. Quantization Process • the smaller the intervals, the smaller the error will be
• 70. Quantization Process • the smaller the intervals, the smaller the error will be • since the range (maximum to minimum) is ﬁxed,
• 71. Quantization Process • the smaller the intervals, the smaller the error will be • since the range (maximum to minimum) is ﬁxed, • more levels will mean smaller levels
• 72. Quantization
• 73. Quantization • number of levels based on word length (number of bits per sample)
• 74. Quantization • number of levels based on word length (number of bits per sample) • # of levels = 2 # of bits
• 75. Quantization • number of levels based on word length (number of bits per sample) • # of levels = 2 # of bits • Adding bit doubles the number of levels,
• 76. Quantization • number of levels based on word length (number of bits per sample) • # of levels = 2 # of bits • Adding bit doubles the number of levels, • which cuts the error in half
• 77. Quantization • number of levels based on word length (number of bits per sample) • # of levels = 2 # of bits • Adding bit doubles the number of levels, • which cuts the error in half • reduces error by 6dB
• 78. Dynamic Range
• 79. Dynamic Range • Loudest to quietest
• 80. Dynamic Range • Loudest to quietest • 6dB of dynamic range per bit
• 81. Dynamic Range
• 82. Dynamic Range • 8 bits = 28 = 256 = 48dB
• 83. Dynamic Range • 8 bits = 28 = 256 = 48dB • 12 bits = 212 = 4,096 = 72dB
• 84. Dynamic Range • 8 bits = 28 = 256 = 48dB • 12 bits = 212 = 4,096 = 72dB • 16 bits = 216 = 65,536 = 96dB
• 85. Dynamic Range • 8 bits = 28 = 256 = 48dB • 12 bits = 212 = 4,096 = 72dB • 16 bits = 216 = 65,536 = 96dB • 20 bits = 220 = 1,048,576 = 120dB
• 86. Dynamic Range • 8 bits = 28 = 256 = 48dB • 12 bits = 212 = 4,096 = 72dB • 16 bits = 216 = 65,536 = 96dB • 20 bits = 220 = 1,048,576 = 120dB • 24 bits = 224 = 16,777,216 = 144dB
• 87. Incredible accuracy
• 88. Incredible accuracy • Imagine a stack of paper 22 feet high.
• 89. Incredible accuracy • Imagine a stack of paper 22 feet high. • The thickness of a sheet of paper is the accuracy of a 16-bit quantization interval!
• 90. Incredible accuracy • Imagine a stack of paper 22 feet high. • The thickness of a sheet of paper is the accuracy of a 16-bit quantization interval! • Now imagine a stack of paper a mile high. The thickness of a sheet of paper is the accuracy of a 24-bit quantization interval!
• 91. Quantization Error
• 92. Quantization Error • Distortion power relative to number of intervals, independent of amplitude of signal
• 93. Quantization Error • Distortion power relative to number of intervals, independent of amplitude of signal • Error changes perceptively with input level
• 94. Quantization Error • Distortion power relative to number of intervals, independent of amplitude of signal • Error changes perceptively with input level • High level signal has un-correlated error (random noise)
• 95. Quantization Error • Distortion power relative to number of intervals, independent of amplitude of signal • Error changes perceptively with input level • High level signal has un-correlated error (random noise) • Low level signal has correlated error – distortion, not noise-like
• 96. Quantization Error
• 97. Quantization Error • Quantization noise is not random, but
• 98. Quantization Error • Quantization noise is not random, but • based on signal
• 99. Dither
• 100. Dither • Noise added to the signal to de-correlate the signal from the quantizer
• 101. Pros
• 102. Pros • Randomizes granulation distortion,
• 103. Pros • Randomizes granulation distortion, • changing it to random noise
• 104. Con
• 105. Con • Raises noise ﬂoor slightly
• 106. Analog vs. Digital Deterioration
• 107. Analog vs. Digital Deterioration • In Analog, noise steadily deteriorates the signal-to-noise ratio
• 108. Analog vs. Digital Deterioration • In Analog, noise steadily deteriorates the signal-to-noise ratio • In Digital, audio quality is independent of transmission/storage quality
• 109. Analog vs. Digital Deterioration • In Analog, noise steadily deteriorates the signal-to-noise ratio • In Digital, audio quality is independent of transmission/storage quality • Until we reach a point of catastrophic failure, when the data can no longer be received correctly
• 110. Analog vs. Digital Deterioration