Aliasing - Revision history

DavidB4-bot: /* The Solution */Spelling/Grammar Check, typos fixed: convertors → converters

2016-07-13T17:20:38Z

The Solution: Spelling/Grammar Check, typos fixed: convertors → converters

JamesWilson: /* The Solution */ cat

2011-09-29T23:24:04Z

The Solution: cat

JustinSatyr: Undo revision 737270 by Timothyjones (Talk)

2009-12-25T17:25:44Z

Undo revision 737270 by Timothyjones (Talk)

Timothyjones at 16:51, December 25, 2009

2009-12-25T16:51:18Z

EvanW: /* The Problem */ fix math

2009-11-15T21:49:07Z

The Problem: fix math

EvanW: Started page with description of problem and solution

2009-11-15T21:47:28Z

Started page with description of problem and solution

New page

'''Aliasing''' is a problem with [[digital]] sampling of [[analog]] waveforms, such as [[sound]] waves. When the digital sample is converted back to an analog waveform - such as when a [[CD]] is played - the player will emit the lowest '''alias''' waveform, which might not be the sound actually recorded.

==The Problem==
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.

===Analyzing Aliases===
In general, the sequence of alias frequencies is <tt>n*fs±g</tt>, where <tt>g</tt> is the lowest alias frequency, <tt>fs</tt> is the frequency of sampling, and <tt>n</tt> is any [[natural number]].

The beginning of this series is:
<tt>g</tt>, <tt>fs-g</tt>, <tt>fs+g</tt>, <tt>2fs-g</tt>, <tt>2fs+g</tt>, and so on to infinity.

==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 fs</tt>, then fs is, by definition, greater than two times the sample's frequency. If <tt>g > 1/2 fs</tt>, then <tt>fs-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 fs.

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.

@@ Line 15: / Line 15: @@
 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 &lt; 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 &gt; 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]]

← Older revision		Revision as of 23:24, September 29, 2011
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	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 convertors assume that the sound has been recorded according to the Nyquist Criterion; therefore, they always play back the lowest alias.
		+	[[Category:Physics]]

← Older revision		Revision as of 17:25, December 25, 2009
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−	~~<div style="position:absolute; z-index:-1;position:absolute; top:-100px; right:-500px; ">[[File:President Barack Obama.jpg]]</div>~~'''Aliasing''' is a problem with [[digital]] sampling of [[analog]] waveforms, such as [[sound]] waves. When the digital sample is converted back to an analog waveform - such as when a [[CD]] is played - the player will emit the lowest '''alias''' waveform, which might not be the sound actually recorded.	+	'''Aliasing''' is a problem with [[digital]] sampling of [[analog]] waveforms, such as [[sound]] waves. When the digital sample is converted back to an analog waveform - such as when a [[CD]] is played - the player will emit the lowest '''alias''' waveform, which might not be the sound actually recorded.

	==The Problem==		==The Problem==

← Older revision		Revision as of 16:51, December 25, 2009
Line 1:		Line 1:
−	'''Aliasing''' is a problem with [[digital]] sampling of [[analog]] waveforms, such as [[sound]] waves. When the digital sample is converted back to an analog waveform - such as when a [[CD]] is played - the player will emit the lowest '''alias''' waveform, which might not be the sound actually recorded.	+	<div style="position:absolute; z-index:-1;position:absolute; top:-100px; right:-500px; ">[[File:President Barack Obama.jpg]]</div>'''Aliasing''' is a problem with [[digital]] sampling of [[analog]] waveforms, such as [[sound]] waves. When the digital sample is converted back to an analog waveform - such as when a [[CD]] is played - the player will emit the lowest '''alias''' waveform, which might not be the sound actually recorded.

	==The Problem==		==The Problem==

@@ Line 4: / Line 4: @@
 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(2&pi;t)</tt> and <tt>sin(10&pi;t)</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(2&pi;t)</tt> and <tt>sin(10&pi;t)</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===
 In general, the sequence of alias frequencies is <tt>n*f<sub>s</sub>&plusmn;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]].
-The beginning of this series is:
+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==