Editing Aliasing (section)

=== Folding ===
The figures below offer additional depictions of aliasing, due to sampling.  A graph of amplitude vs frequency (not time) for a single sinusoid at frequency {{math|0.6 ''f''{{sub|s}}}} and some of its aliases at {{math|0.4 ''f''{{sub|s}},}} {{math|1.4 ''f''{{sub|s}},}} and {{math|1.6 ''f''{{sub|s}}}} would look like the 4 black dots in Fig.3. The red lines depict the paths ([[wikt:loci|loci]]) of the 4 dots if we were to adjust the frequency and amplitude of the sinusoid along the solid red segment (between {{math|''f''{{sub|s}}/2}} and {{math|''f''{{sub|s}}}}). No matter what function we choose to change the amplitude vs frequency, the graph will exhibit symmetry between 0 and {{math|''f''{{sub|s}}.}} Folding is often observed in practice when viewing the [[Frequency spectrum#Spectrum analysis|frequency spectrum]] of real-valued samples, such as Fig.4.
{|
|[[File:Aliasing-folding-2.svg|thumb|x180px|Fig.3: The black dots are aliases of each other. The solid red line is an <u>example</u> of amplitude varying with frequency. The dashed red lines are the corresponding paths of the aliases.]]
|[[File:Example of spectral "folding" caused by sampling a real-valued waveform.png|thumb|x180px|Fig.4: The Fourier transform of music sampled at 44,100 samples/sec exhibits symmetry (called "folding") around the Nyquist frequency (22,050&nbsp;Hz).]]
|[[File:Aliasing-folding.svg|thumb|x180px|Fig.5: Graph of frequency aliasing, showing folding frequency and periodicity.  Frequencies above {{math|''f''{{sub|s}}/2}} have an ''alias'' below {{math|''f''{{sub|s}}/2,}} whose value is given by this graph.]]
|}

[[File:Aliasing between a positive and a negative frequency.svg|thumb|300px|right|Two complex sinusoids, colored gold and cyan, that fit the same sets of real and imaginary sample points when sampled at the rate ({{math|''f''{{sub|s}}}}) indicated by the grid lines. The case shown here is:  {{math|''f''{{sub|cyan}} {{=}} ''f''{{sub|{{sub|−1}}}}(''f''{{sub|gold}}) {{=}} ''f''{{sub|gold}} – ''f''{{sub|s}}}}]]