Update

aliasing.md

@@ -1,6 +1,6 @@
 # Aliasing
 
-Aliasing is a certain typically undesirable phenomenon that distorts signals (such as sounds or images) when they are [sampled](sampling.md) [discretely](discrete.md) (captured at single points, usually at periodic intervals) -- this can happen e.g. when capturing sound with digital recorders or when [rendering](rendering.md) computer graphics. There exist [antialiasing](antialiasing.md) methods for suppressing or even eliminating aliasing. Aliasing can be often seen on small checkerboard patterns as a moiré pattern (spatial aliasing), or maybe more famously on rotating wheels or helicopter rotor blades that in a video look like standing still or rotating the other way (temporal aliasing, caused by capturing images at intervals given by the camera's [FPS](fps.md)).
+Aliasing is a certain typically undesirable phenomenon that distorts [signals](signal.md) (such as sounds or images) when they are [sampled](sampling.md) [discretely](discrete.md) (captured at single points, usually at periodic intervals) -- this can happen e.g. when capturing sound with digital recorders or when [rendering](rendering.md) computer graphics. There exist [antialiasing](antialiasing.md) methods for suppressing or even eliminating aliasing. Aliasing can be often seen on small checkerboard patterns as a moiré pattern (spatial aliasing), or maybe more famously on rotating wheels or helicopter rotor blades that in a video look like standing still or rotating the other way (temporal aliasing, caused by capturing images at intervals given by the camera's [FPS](fps.md)).
 
 A simple example showing how sampling at discrete points can quite dramatically alter the recorded result:
 
@@ -48,7 +48,7 @@ The explanation above shows why a helicopter's rotating blades look to stand sti
 
 Of course this doesn't only happen with perfect sine waves. [Fourier transform](fourier_transform.md) shows that any signal can be represented as a sum of different sine waves, so aliasing can appear anywhere.
 
-**Nyquist–Shannon sampling theorem** says that aliasing can NOT appear if we sample with at least twice as high frequency as that of the highest frequency in the sampled signal. This means that we can eliminate aliasing by using a [low pass filter](low_pass.md) before sampling which will eliminate any frequencies higher than the half of our sampling frequency. This is why audio is normally sampled with the rate of 44100 Hz -- from such samples it is possible to correctly reconstruct frequencies up to about 22000 Hz which is about the upper limit of human hearing.
+**Nyquist-Shannon sampling theorem** says that aliasing can NOT appear if we sample with at least twice as high frequency as that of the highest frequency in the sampled signal. This means that we can eliminate aliasing by using a [low pass filter](low_pass.md) before sampling which will eliminate any frequencies higher than the half of our sampling frequency. This is why audio is normally sampled with the rate of 44100 Hz -- from such samples it is possible to correctly reconstruct frequencies up to about 22000 Hz which is about the upper limit of human hearing.
 
 Aliasing is also a common problem in [computer graphics](computer_graphics.md). For example when rendering textured 3D models, aliasing can appear in the texture if that texture is rendered at a smaller size than its resolution (when the texture is enlarged by rendering, aliasing can't appear because enlargement decreases the frequency of the sampled signal and the sampling theorem won't allow it to happen). (Actually if we don't address aliasing somehow, having lower resolution textures can unironically have beneficial effects on the quality of graphics.) This happens because texture samples are normally taken at single points that are computed by the texturing algorithm. Imagine that the texture consists of high-frequency details such as small checkerboard patterns of black and white pixels; it may happen that when the texture is rendered at lower resolution, the texturing algorithm chooses to render only the black pixels. Then when the model moves a little bit it may happen the algorithm will only choose the white pixels to render. This will result in the model blinking and alternating between being completely black and completely white (while it should rather be rendered as gray).