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Where is the Fourier sine/cosine series ever used in the real world? - Quora

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Clipped on: 2016-08-08

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11 Answers
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Paul King, Computational Neuroscientist, Software Entrepreneur
The Fourier transform, which converts a signal into a weighted sum of sine waves and then back again, is the secret ingredient that makes all lossy media compression possible!

MP3 is a lossy audio compression algorithm, as is JPEG for still images and MPEG for movies. These centerpiece of these algorithms is the conversion of the audio or image signal into 1D or 2D sine waves, then deleting the parts that "won't be noticed", and then later converting the data back into the original signal with the deleted frequencies missing. (The loss is deliberate and strategically applied to reduce data without reducing perceived quality.)

The Fourier transform is also the key enabling technique for all spread-spectrum wireless data technology, including WiFi, digital cell phone broadcasts, 4G mobile data broadcast, and dynamic bandwidth regulating DSL.

A key difference between the ideal Fourier transform and the way it is applied is that the ideal Fourier transform turns infinitely-long signals into infinitely long sine waves, which on a practical scale is impossible and not useful. The applied Fourier transform uses "wavelets" which are sine waves that quickly taper to zero, representing, for example, a few milliseconds of sound (in MP3) or 8x8 pixel image blocks (in JPEG). An example wavelet:
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Bill McDonald, Computer arithmetic makes my head hurt.
On your stereo if it has a graphic equalizer. This is a physical hardware implementation of the Fourier Transform.

The idea that an arbitrary waveform can be decomposed into sine waves, or that sine waves can be combined to make an arbitrary waveform, is so prevalent in electrical engineering that it is hard to notice. (sort of like the faint lines on graph paper)  It's just the way things are done, and thought about.  To use a software analogy, it's part of the <>  (standard input & output) module for electronics.

It is used in almost every modem in the DSP, and in DSL systems as discrete multi-tone modulation.

Fourier's idea shows up once again when we look at a stereo speaker which has a woofer (bass), mid-range, and tweeter (high notes) - the combination of the sounds in the air reproduces the complex music waveform.
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Rupert Baines, CEO at a startup.Semiconductors, wireless, comms, start-ups.
Your lecturer is oddly ignorant. This is hugely important and is used just about everywhere in signal processing, advanced telecoms.

The idea of the Fourier Transform is at the heart of WiFi (specifally of OFDM used in 802.11 g & n), ADSL & VDSL, of WiMAX and LTE.

[EDIT  After Yunquing's comment I have updated my answer to be more complete, rather than comment to a comment. So his comment may look odd, this is why]

Fourier Transorm is the mechanism for turning one signal (time domain) into its frequency domain equivalent - which is Fourier Series.

The FFT works on a set of samples which is finite time (say 250us) usually called 'a frame'. It creates the periodic equivalent to that set (only that set) as if it were periodic by adding a cyclic prefix that helps the FFT into thinking that sample repeats ie a periodic waveform with period 250us (4KHz -- I use that example because that is the frame duration/symbol period of ADSL).

You do the FFT of that 'periodic' waveform to create sin/cos frequency domain (Fourier series) as your OFDM symbol set - each sin (nW) corresponds to a different frequency, which is a different bin or subtone, which you modulate and transmit. So you have taken a time domain (250us) and tranformed it to frerquency domain which is your transmitted symbol.

The next sample set (next 250us) will have a different pattern (as you say, it appears random) so the sin/cos output series will be different, but uses same process. And so on.

eg for ADSL

The downstream and upstream data channels are synchronized to the 4 kHz ADSL DMT (Discrete Multi Tone) symbol rate, and multiplexed into two separate data buffers (fast and interleaved).
ADSL uses the superframe structure shown in figure . Each superframe is composed of 68 ADSL data frames, which are encoded and modulated into DMT symbols. From the bit-level and user data perspective, the DMT symbol rate is 4000 baud (period = 250
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s). Because of the sync symbol inserted to the end of each superframe, the transmitted DMT symbol rate is 69/68 * 4000 baud.
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Rob Weir, Former Member, American Daffodil Society
2.7k Views · Rob has 60+ answers in Mathematics
The Fourier series essentially tells how any periodic signal can be decomposed into a weighted sum of simple sine waves.  So you can have two equivalent representations of the same function:  one in the "time domain" (amplitude as a function of time) and one in the "frequency domain" (power as a function of frequency).  You can transform back and forth between these representation using the Fourier Transform.  The power comes from the fact that there are important operations that can be done very easily in the frequency domain.  For example, to filter out certain frequencies in a sound clip is almost trivial to do in the frequency domain.  So you have applications where a signal is transformed from time domain to frequency domain, then modified in the frequency domain, and then transformed back to time domain.  Filtering, noise elimination, etc.

Similarly, this generalizes to 2D functions as well.  So you can have 2D Fourier transforms of images.   This can be used for powerful techniques, such as deconvolution, where an image that is blurred can be restored.

In a world of digital data, in sound, image and video, the Fourier series is almost everywhere, most commonly via an algorithm called the Fast Fourier Transform or FFT.
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Aviv Keshet, I pretend to do Physics which pretends to do Math.
Your lecturer was nuts if he said that.

You cannot use or understand Fourier Transforms of unbounded-time signals without first understanding Fourier Series decomposition of bounded-time signals. In fact, the Fourier Transform is really just a generalization of Fourier Series applied to time-unbounded signals.

Fourier Transforms are so utterly fundamental and ubiquitous in every aspect of modern communications technology that it's too ridiculous to point out their every individual use.
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Christopher Reiss, BS in math, Putnam Rank 254 in 1990
1.2k Views · Christopher has 30+ answers in Mathematics
To take a special case of Rob Weir's answer, one of the simplest and most impressive applications of the Fourier series is in musical synthesizers.   (See Moog Synthesizer and Fourier Synthesizer

What's really cool is that it doesn't take too many superimposed sine waves to get close to a real world sound, like a violin string.

There's even an "app for that" :

This can be an engaging way to introduce students to the topic.

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