Engineers, mathematicians, and physicists all love the Fourier transform. When we hear about the Transformers movie, we keep thinking about Fourier Transformers, with Oscilloscope Prime leading the noble Autocorrelationbots against the nefarious Decepticonvolutions. Look heroic (and nerdy) with our Oscilloscope Prime math shirt!
The Fourier transform converts functions of time into their frequency components. In other words, it expresses the function as a superposition of a continuum of complex waves of pure frequencies.
If you clap your hands near a piano, how will this excite the strings? You can imagine sustaining a single note on a flute near a piano, and this might resonate with just the string at that frequency while leaving most of the other strings of the piano largely unaffected. The Fourier transform says that the clap can be viewed as a superposition of many flutes playing different pitches at different amplitudes. If we understand how the piano reacts to each flute, then we can understand the reaction to the clap.
Convolution is a natural but complicated operation. The convolution of two probability distributions is the distribution of the sum of independent random variables with those distributions. When you multiply two numbers, you convolve the digits, and then carry. If the digits are small enough, then there are no carries and you only have to convolve the digit string. When you multiply 1011 by 203, you get the convolution 205233. The Fourier transform of the convolution of f and g is the pointwise product of the convolutions of f and g. While convolution is a complicated operation on functions, it is simple on their Fourier transforms.
Noise such as static in an audio recording or scratches on a photograph may affect primarily the higher frequencies. To improve the signal-to-noise ratio, one approach is to dampen the higher frequencies relative to the lower ones. To do this, we can convolve with a function g whose Fourier transform is 1 on low frequencies, and 0 on high frequencies. Since the Fourier transform applied twice returns the original function multiplied by -1, we can find this useful g by taking the Fourier transform of a function which is 1 when -1< t < 1, and 0 outside this interval. The Fourier transform of this function is the function sinc 2x, so convolving with a sinc function is one way to reduce noise.
Our picture of Oscilloscope Prime shows a rectangular wave displayed on his oscilloscope form, and its Fourier transform, a sinc function displayed on his chest. This math shirt can be purchased via our nerdy shirt site.