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fourier_transform.md
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@ -8,7 +8,7 @@ For newcomers FT is typically not easy to understand, it takes time to wrap one'
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 diverge. (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:** 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.
TODO: alternatives (like performing FIR filtering without actually doing FT etc.)