Block, SkipCaptcha, apieditor, Automoderated users, Bots, edit, ratelimit-exempt
58,758
edits
Changes
Jump to navigation
Jump to search
Line 4:
Line 4: − + − The beginning of this series is:+ − + − + +
→The Solution: Spelling/Grammar Check, typos fixed: convertors → converters
Sound waves are continuous and can have any [[frequency]], while digital sampling only records the value at discrete intervals. For example, a sampling frequency of two kilo[[hertz]] records the value of the sound wave 2,000 times per [[second]]. If the original sound wave has a frequency of 500 hertz, that means that its value will be recorded four times per [[period]], or cycle.
Sound waves are continuous and can have any [[frequency]], while digital sampling only records the value at discrete intervals. For example, a sampling frequency of two kilo[[hertz]] records the value of the sound wave 2,000 times per [[second]]. If the original sound wave has a frequency of 500 hertz, that means that its value will be recorded four times per [[period]], or cycle.
Unfortunately, multiple [[sine]] waves can fit these points. For example, <tt>sin(2πt)</tt> and <tt>sin(10πt)</tt> cross at five points in each cycle. When digitalized with a frequency of 5t, they will create identical digital records. A CD player playing the digital record will not be able to tell which sound was recorded. These two waves are therefore called ''aliases'' of each other.
Unfortunately, multiple [[sine]] waves can fit these points. For example, <tt>sin(2πt)</tt> and <tt>sin(10πt)</tt> cross at six points in each cycle. When digitalized with a frequency of 6t, they will create identical digital records. A CD player playing the digital record will not be able to tell which sound was recorded. These two waves are therefore called ''aliases'' of each other.
===Analyzing Aliases===
===Analyzing Aliases===
In general, the sequence of alias frequencies is <tt>n*f<sub>s</sub>±g</tt>, where <tt>g</tt> is the lowest alias frequency, <tt>f<sub>s</sub></tt> is the frequency of sampling, and <tt>n</tt> is any [[natural number]].
In general, the sequence of alias frequencies is <tt>n*f<sub>s</sub>±g</tt>, where <tt>g</tt> is the lowest alias frequency, <tt>f<sub>s</sub></tt> is the frequency of sampling, and <tt>n</tt> is any [[natural number]].
This series begins:
<tt>g</tt>, <tt>f<sub>s</sub>-g</tt>, <tt>f<sub>s</sub>+g</tt>, <tt>2f<sub>s</sub>-g</tt>, <tt>2f<sub>s</sub>+g</tt>, and so on to infinity.
<tt>g</tt>, <tt>f<sub>s</sub>-g</tt>, <tt>f<sub>s</sub>+g</tt>, <tt>2f<sub>s</sub>-g</tt>, <tt>2f<sub>s</sub>+g</tt>. It continues to infinity.
==The Solution==
==The Solution==
The '''Nyquist-Shannon Sampling Criterion''', formulated by [[Harry Nyquist]] in 1928 and proven by [[Claude Shannon]] in 1949, states that each digital sample can be converted to a unique analog signal if the sampling rate is more than twice as large as the highest frequency in the signal. While Shannon's proof requires [[Fourier transform]]s, it can be deduced from the sequence of alias frequencies above. If <tt>g < 1/2 f<sub>s</sub></tt>, then f<sub>s</sub> is, by definition, greater than two times the sample's frequency. If <tt>g > 1/2 f<sub>s</sub></tt>, then <tt>f<sub>s</sub>-g</tt>, which is another alias, is less than one-half the frequency of sampling. Therefore, there is always one unique alias frequency less than 1/2 f<sub>s</sub>.
The '''Nyquist-Shannon Sampling Criterion''', formulated by [[Harry Nyquist]] in 1928 and proven by [[Claude Shannon]] in 1949, states that each digital sample can be converted to a unique analog signal if the sampling rate is more than twice as large as the highest frequency in the signal. While Shannon's proof requires [[Fourier transform]]s, it can be deduced from the sequence of alias frequencies above. If <tt>g < 1/2 f<sub>s</sub></tt>, then f<sub>s</sub> is, by definition, greater than two times the sample's frequency. If <tt>g > 1/2 f<sub>s</sub></tt>, then <tt>f<sub>s</sub>-g</tt>, which is another alias, is less than one-half the frequency of sampling. Therefore, there is always one unique alias frequency less than 1/2 f<sub>s</sub>.
Of course, every frequency always has aliases. However, digital-to-analog convertors assume that the sound has been recorded according to the Nyquist Criterion; therefore, they always play back the lowest alias.
Of course, every frequency always has aliases. However, digital-to-analog converters assume that the sound has been recorded according to the Nyquist Criterion; therefore, they always play back the lowest alias.
[[Category:Physics]]