TITLE: Fourier - Are sinusoidals strictly required?
QUESTION [33 upvotes]: We can define all signals as a sum of sinusoidals by taking fourier transform of the signal. Thats OK. My question is, why sinusoidals.? Can there be an another transform like Fourier somewhere in the universe that can explain all signals as sum of rectangulars or triangulars(or any periodic shape)?

REPLY [10 votes]: Can there be an another transform like Fourier somewhere in the
  universe that can explain all signals as sum of rectangulars or
  triangulars(or any periodic shape)?

Yes, there are several popular transforms similar to Fourier transforms, except using something other than both sines and cosines.
Every day people watch digitized video or listen to digitized audio
that uses several such transforms, usually without even realizing it.
For example,

The Fourier transform decomposes a signal into a sum of both infinitely-long sine waves and infinitely-long cosine waves.
The discrete cosine transform (DCT) is used in many image compression and video compression systems, including JPEG, MPEG, and Theora. The DCT decomposes a signal into a sum of cosine waves.
The modified discrete cosine transform (MDCT) is used in a few more recent image and video compression systems and many audio compression systems, such as MP3, AAC, and Vorbis, since it reduces block-boundary artifacts typical of systems based on DCT. The MDCT also decomposes a signal into a sum of cosine waves.
The discrete sine transform decomposes a signal into a sum of sine waves. I am unaware of any practical use.
The Hartley transform and discrete Hartley transform (DHT) decomposes a signal into a sum of cas waves, where cas(t) = cos(t) + sin(t). It seems to have several advantages over the Fourier transform, but I am unaware of any practical use.
The Walsh transform -- aka Hadamard transform -- decomposes a signal into a sum of infinitely-long even square waves and odd square waves. It is used in JPEG XR and H.264 and also seems to be useful for quantum computing.

Time-frequency transforms are useful for making waterfall plots and spectrograms which are used in a variety of fields -- speech training; studies of animal sounds; radar; seismology; ham radio operators use them to discover what frequencies and protocols people are using and to decode very slow Morse code "by eye"; etc.

The short-time Fourier transform is a time-frequency transformation that decomposes a signal into a sum of short pieces of sine and cosine waves, each piece localized in time.
Various wavelet transforms is an even more popular time-frequency transformation. The simplest is the Haar transform, which uses a mother wavelet composed of 2 square pulses.
chirplet transform is another time-frequency transform.
the fractional Fourier transform (FRFT), which I find quite unexpected and surprising that such a thing is even possible, but I am unaware of any practical use.