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fourier_transform.md

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 # Fourier Transform
 
-Fourier Transform (FT) is one of the most important transformations/[algorithms](algorithm.md) in [signal](signal.md) processing (and really in [computer science](compsci.md) and [mathematics](math.md) in general), which enables us to express and manipulate a signal (such as a sound or picture) in terms of [frequencies](frequency.md) it is composed of (rather than in terms of individual [samples](samples.md)). It is so important because frequencies (basically [sine](sine.md) waves) are actually THE important thing in signals, they allow us to detect things (voices, visual objects, chemical elements, ...), [compress](compression.md) signals, modify them in useful ways (e.g. filter out [noise](noise.md) in specific frequency band, enhance specific frequency bands, change audio speed without changing its pitch, pitch-shift a song without changing its speed and so on). There also exists a related algorithm called **[Fast Fourier Transform](fft.md)** (FFT) which is able to compute one specific version of FT very quickly and so is often used in practice.
+Fourier Transform (FT) is one of the most prominent transformations/[algorithms](algorithm.md) in [signal](signal.md) processing (and really in [computer science](compsci.md) and [mathematics](math.md) in general), which enables us to express and manipulate a [signal](signal.md) (such as a sound or picture) in terms of [frequencies](frequency.md) it is composed of (rather than in terms of individual [samples](samples.md)). It is so important because frequencies (basically [sine](sine.md) waves) are actually THE important essence in signals, they allow us to detect things (voices, visual objects, chemical elements, ...), [compress](compression.md) signals, modify them in useful ways (e.g. filter out [noise](noise.md) in specific frequency band, enhance specific frequency bands, change audio speed without changing its pitch, pitch-shift a song without changing its speed and so on). There also exists a related algorithm called **[Fast Fourier Transform](fft.md)** (FFT) which is able to compute one specific version of FT very quickly and so is often used in practice.
 
-For newcomers FT is typically not easy to understand, it takes time to wrap one's head around it. There is also a very confusing terminology; there exist slightly different kinds of the Fourier Transform that are called by similar names, or sometimes all simply just "Fourier Transform" -- what programmers usually mean by FT is DFT or FFT. There also exist Fourier Transforms in higher dimensions (2D, 3D, ...) -- the base case is called one dimensional (because our input signal has one coordinate). All this will be explained below.
+FT is typically not easy to understand right away, it takes time to wrap one's head around it. Confusing terminology doesn't help in this regard either; there exist slightly different kinds of Fourier Transform bearing similar names and sometimes they're all just called "Fourier Transform" -- what programmers usually mean by FT is DFT or FFT. There also exist Fourier Transforms in higher dimensions (2D, 3D, ...) -- the base case is called one dimensional (because our input signal has one coordinate). All this mess will be explained below.
 
-**What FT does in essence**: it transforms an input signal (which can also be seen as a [function](function.md)) from **time (also space) domain**, i.e. the usual representation that for each *x* says the sample value *f(x)*, to **frequency domain**, another function that for each frequency *f* says "how much of the frequency is present" (amplitude and phase). For example an FT of a simple sine wave will be a function with a single spike at the frequency of the sine wave. There is also an **inverse Fourier Transform** that does the opposite (transforms the signal from frequencies back to time samples). The time and frequency representations are EQUIVALENT in that either one can be used to represent the signal -- it turns out that even "weird" looking functions can be decomposed into just sums of many differently shifted and scaled sine waves. In the frequency domain we can usually do two important things we cannot do in time domain: firstly analyze what frequencies are present (which can help e.g. in [voice recognition](voice_recognition.md), [spectral analysis](spectral_analysis.md), earthquake detection, [music](music.md) etc.) and also MODIFY them (typicaly example is e.g. music [equalizer](equalizer.md) or [compression](compression.md) that removes or quantizes some frequencies); if we modify the frequencies, we may use the inverse FT to get back the "normal" (time representation) signal back. Some things are also easier to do in the frequency domain, for example [convolution](convolution.md) becomes mere multiplication.
+**What FT does in essence**: it transforms an input signal (which can also be though of as a [function](function.md)) from **time (also space) domain**, i.e. the usual representation that for each *x* says the sample value *f(x)*, to **frequency domain**, another function that for each frequency *f* says "how much of the frequency is present" (amplitude and phase). For example an FT of a simple sine wave will be a function with a single spike at the frequency of the sine wave. There is also an **inverse Fourier Transform** that does the opposite (transforms the signal from frequencies back to time samples). The time and frequency representations are EQUIVALENT in that either one can be used to represent the signal -- it turns out that even "weird" looking functions can be decomposed into just sums of many differently shifted and scaled sine waves. In the frequency domain we can usually do two important things we cannot do in time domain: firstly analyze what frequencies are present (which can help e.g. in [voice recognition](voice_recognition.md), [spectral analysis](spectral_analysis.md), earthquake detection, [music](music.md) etc.) and also MODIFY them (typicaly example is e.g. music [equalizer](equalizer.md) or [compression](compression.md) that removes or quantizes some frequencies); if we modify the frequencies, we may use the inverse FT to get back the "normal" (time representation) signal back. Some things are also easier to do in the frequency domain, for example [convolution](convolution.md) becomes mere multiplication.
 
 FT is actually just one of many so called **[integral transforms](integral_transform.md)** that are all quite similar -- they always transform the signal to some other domain and back, they use similar equation but usually use a different kind of [function](function.md). Other integral transforms are for example **[discrete cosine transformation](dct.md)** (DCT) or **[wavelet transform](wavelet_transform.md)**. DCT is actually a bit simpler than FT, so if you are having hard time with FT, go check out DCT.
 
-**If you know [linear algebra](linear_algebra.md), this may help you understand what (D)FT really does:** Imagine the signal we work with is a POINT (we can also say a [vector](vector.md)) in many [dimensional](dimension.md) space; if for example we have a recorded sound that has 1000 samples, it is really a 1000 dimensional vector, a point in 1000 dimensional space, expressed as an "array" of 1000 numbers (vector components). A short note: since we consider a finite number of discrete samples here, we are actually dealing with what's called DISCRETE FT here, not the "vanilla" FT, but for now let's not digress. (D)FT does nothing more than transforming from one vector [basis](basis.md) ("system of coordinates", "frame of reference") to another basis; i.e. by default the signal is expressed in time domain (our usual vector basis), the numbers in the sound "array" are such because we are viewing them from the time "frame of reference" -- (D)FT will NOT do anything with to the signal itself (it is a vector/point in space, which will stay where it is, the recorded sound itself will not change), it will merely express this same point/vector from a different "point of view"/"frame of reference" (set of basis vectors) -- that of frequencies. That's basically how all the integral transforms work, they just have to ensure the basis they are transforming to is orthogonal (i.e. kind of legit, "usable") of course. In addition the FT equation is nothing complex, it literally just uses a **[dot product](dot_product.md)** of the whole input signal with each checked frequency wave to find out how similar the signal is to that particular frequency, as dot product simply says "how similar two vectors are" -- really, think about the equation and you will see it's really doing just that.
+**If you know [linear algebra](linear_algebra.md), this may help you understand what (D)FT really does:** Picture the signal we work with as a POINT (we can also say a [vector](vector.md)) in many [dimensional](dimension.md) space; if for instance we have a recorded sound of 1000 samples, it is really a 1000 dimensional vector, a point in 1000 dimensional space, expressed as an "array" of 1000 [numbers](number.md) (vector components). A short note: since we assume a finite number of discrete samples here, we are actually dealing with what's called DISCRETE FT here, not the "vanilla" FT, but for now let's not digress. (D)FT does nothing more than transforming from one vector [basis](basis.md) ("system of coordinates", "frame of reference") to another basis; i.e. by default the signal is expressed in time domain (our usual vector basis), the numbers in the sound "array" are such because we are viewing them from the time "frame of reference" -- (D)FT will NOT do anything with to the signal itself (it is a vector/point in space, which will stay where it is, the recorded sound itself will not change), it will merely express this same point/vector from a different "point of view"/"frame of reference" (set of basis vectors) -- that of frequencies. That's basically how all the integral transforms work, they just have to ensure the basis they are transforming to is orthogonal (i.e. kind of legit, "usable") of course. In addition the FT equation is nothing complex, it literally just uses a **[dot product](dot_product.md)** of the whole input signal with each checked frequency wave to find out how similar the signal is to that particular frequency, as dot product simply says "how similar two vectors are" -- really, think about the equation and you will see it's really doing just that.
 
 TODO: alternatives (like performing FIR filtering without actually doing FT etc.)
 
 ## Details
 
-First let's make clearer the whole terminology around FT:
+First and foremost let's clarify the whole terminology around FT a bit:
 
 - **Fourier Series (FS)**: Transforms a PERIODIC (repeating) signal into a DISCRETE (non-continuous) spectrum. We can see this spectrum also as an infinite SERIES of coefficients *c0*, *c1*, *c2*, etc. The input signal can generally be [complex](complex_number.md), in which case the output spectrum also has negative part (*c-1*, *c-2*, *c-3* etc.) and shows us COMPLEX EXPONENTIALS (i.e. not mere sine waves) of the input signal; however if the input signal is [real](real_number.md) (probably most signals we practically deal with), the spectrum's negative part is symmetric to the positive part and the corresponding positive and negative complex exponentials always together give a sine wave, so for "normal" signals we can see the spectrum only being in the non-negative part and showing us the sine waves of the signal.
 - **Fourier Transform (FT)**: Generalization of FS to work on any signal, not just periodic ones, i.e. FT takes a NON-PERIODIC signal and transforms it into a CONTINUOUS spectrum. This is achieved simply by considering the period of the signal to be infinite -- the spectrum now becomes continuous exactly because the input is non-periodic (this is a relationship that generally holds); since the output is continuous, we now rather see it as a function rather than a series. Same as with FS the input can be [complex](complex_number.md) (in which case the same implications apply), but we usually work with [real](real.md) signals.
@@ -24,7 +24,7 @@ First let's make clearer the whole terminology around FT:
 - **Discrete Fourier Transform (DFT)** (not to be confused with DTFT!): Uses DFS to transform a FINITE DISCRETE signal to a FINITE DISCRETE spectrum (with the same period as the input) by simply "pretending" the finite input signal is actually repeating over and over and then, after the transform, only leaving in the first period of the result (since the rest is just repeating). **This is actually what programmers usually mean by Fourier Transform** because in computers we practically always only deal with finite discrete signals (i.e. [arrays](array.md) of data).
 - **Fast Fourier Transform (FFT)**: Computes DFT (NOT FT!) that's faster than the [naive](naive.md) implementation, i.e. computing the equation that defines DFT as it's written has time complexity O(n^2) while FFT improves this to O(n * log(n)).
 
-From now on we will implicitly be talking about DFT of a real function (we'll ignore the possibility of complex input), the most notable transform here.
+From now on we will implicitly be talking about DFT of a real function (we'll ignore the possibility of complex input), the most noteworthy transform here.
 
 The input to DFT is a real function, i.e. the time domain representation of the signal. The output is a complex valued function of frequency, i.e. the spectrum -- for each frequency it says a complex number whose magnitude and phase say the magnitude and phase of that frequency (a sine wave) in the signal (many programs will visualize just the magnitude part as that's usually the important thing, however keep in mind there is always also the phase part as well).