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by: Destiney Wintheiser


Destiney Wintheiser
GPA 3.83


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Class Notes
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This 2 page Class Notes was uploaded by Destiney Wintheiser on Monday September 28, 2015. The Class Notes belongs to ESE 250 at University of Pennsylvania taught by Staff in Fall. Since its upload, it has received 19 views. For similar materials see /class/215410/ese-250-university-of-pennsylvania in Electrical Engineering at University of Pennsylvania.

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Date Created: 09/28/15
ESE250 Spring 2010 January 21 2010 Big Idea Week 2 Signals Symbols and In nities Sampling Quantization and Noise A sound is a eld of acoustic local atmospheric pressure waveforms structured variations in space and time A computer is a symbol processor with a nite memory and limited computational power respecting both its model of arithmetic and the speed with which it performs that arithmetic Digital audio engineering starts with the problem of acquiring and storing in the computer s memory a symbolic representation of those continuous waveforms amenable to processing minimally internal copying and reconstruction via external speakers using its model of arithmetic The nearly universal solution to this rst problem found in contemporaiy digital audio technology is recourse to pulse code modulation PCM the process of converting acoustic sound pressure waves into analog voltage signals using analog electro mechanical components and then sampling those voltage signals in time and value using analog to digital electronics The question now arises as to what level of delity to the original sound can be expected from this electromechanically transformed and double digitally both in time and value sampled signal That is how much noise does a particular sampling rate or quantization interval introduces compared to the original signal More formally we introduce the notion of a quantization function QuantizeLz RoundL zL7 l where Round is the function that rounds real numbers to the nearest integer Using this formalism we can de ne the PCM function as a composition of quantization steps as follows A sampler quantizes time to convert a continuous signal 5t to one varying discretely between constant intervals of duration T 5Tt de ned as 5Tt SampleTst 5Quantize1Tt 5T RoundtT 2 The PCM function follows the time sampled signal with a quantization in value to some level L yielding 5LTt de ned as 5LTt PCMLTst QuantizeL SampleTst QuantizeL lt5Quantize1Ttgt 3 The noise introduced in sampling and quantization can then be formally de ned as nt 5t 7 POMWT 4 Since there are an uncountably in nite number of 5t s but only a countable number of sdt 3 this double digital sampling process always discards information introducing non zero noise However if we have a model for the structure that exists within 5t or at least the structure that we care about in 5t and this structure is appropriately restricted we can represent 5t perfectlyior at least so that the noise University of Pennsylvania ESE250 Spring 2010 January 21 2010 is irrelevant to our application For sound we will be exploiting a model of the human auditory system that allows us to differentiate the noise characteristics that humans can and cannot hear An empirical answer to this question can be found in the last decade s digital audio technology eg compact discs and digital audio tapes Only very very rare human beings can discern any difference perceive any noise between the sound recorded and played back on the best analog audio tape player and the same sound 116 the transformed voltage signal sampled in time every 144 10 3 227 10 5 seconds and sampled in value with 216 65536 different voltage levels at each time step University of Pennsylvania


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