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→The Problem: fix math
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==