Changes

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(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===

===Analyzing Aliases===

In general, the sequence of alias frequencies is <tt>n*f_s&plusmn;g</tt>, where <tt>g</tt> is the lowest alias frequency, <tt>f_s</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_s&plusmn;g</tt>, where <tt>g</tt> is the lowest alias frequency, <tt>f_s</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_s-g</tt>, <tt>f_s+g</tt>, <tt>2f_s-g</tt>, <tt>2f_s+g</tt>, and so on to infinity.

<tt>g</tt>, <tt>f_s-g</tt>, <tt>f_s+g</tt>, <tt>2f_s-g</tt>, <tt>2f_s+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 &lt; 1/2 f_s</tt>, then  f_s is, by definition, greater than two times the sample's frequency.  If <tt>g &gt; 1/2 f_s</tt>, then <tt>f_s-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_s.

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_s</tt>, then  f_s is, by definition, greater than two times the sample's frequency.  If <tt>g &gt; 1/2 f_s</tt>, then <tt>f_s-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_s.

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.