# Imaginary Numbers and DFT

I'm not a math guy per se, but i am trying to understand the DFT. I get to the point where imaginary numbers are used with Euler's formula. What I don't understand is why we need an imaginary plane or "i" to begin with since we can plot any point on a real plane using sin and cos ? I am using this site and I get to the Euler's formula page before I crash. I am a Musician/Programmer trying to broaden my knowledge.

https://jackschaedler.github.io/circles-sines-signals/

I think that in the sense you are asking, you don't in fact "need" the complex plane. You can just use the real plane with pairs of coordinates. That's choosing to write $$(\cos \theta , \sin \theta)$$ instead of $$e^{i \theta} .$$ But if you do eschew the complex number formalism you miss out on some compact notation and some algebra that helps you understand how the DFT works. In particular, the identity $$e^{i (\sigma + \theta)} = e^{i \sigma} e^{i \theta}$$ is much more suggestive than the addition formulas for $$\sin$$ and $$\cos$$.

It's true that in the end you can use the real Fourier series or the complex one and they are equal. I can think of one argument (which I think is very important) why the complex Fourier series is important:

Consider some function $$f$$ for which you want to find the real Fourier series. To do that you have to compute all the coefficients $$a_i, b_i$$ in the series $$f(x) = a_0 + \Sigma_{i = 1}^{\infty}a_icos(ix) + b_isin(ix)$$: The coefficients can be calculated as integrals, for the $$a_k$$s: $$a_k = 2/T\int_{-T/2}^{T/2}f(t)cos(\frac{kt2\pi}{T})dt$$ where $$T$$ denotes the fundamental period of the function $$f$$. Sometimes these integrals are hard to calculate but it is easier to calculate the integrals which are needed for the complex Fourier series coefficients (using the residue theorem of complex analysis for example).

Of course you can compute the real coefficients once you found the complex coefficients. But the thing is that for the sake of computation the complex coefficients are often very useful.

Since you are a programmer: In order to understand libraries which implement the DFT you will certainly have to learn about the complex Fourier series.

Since you are a musician: The real Fourier series is more interesting here because it's coefficients are easily interpreted. The sound of a particular instrument is determined by its waveform. Its waveform is a periodic function for which you can find the real Fourier series. In this representation the coefficient tell you which overtones are the most relevant for this instrument. For example: You are playing an A($$440$$Hz) on your instrument, hence your fundamental period is $$1/440s$$. Now assume that all the $$a_k$$'s are zero and that $$b_2$$ is significantly larger than other coefficients in the real Fourier series representation for your waveform. This means that the period $$1/880s$$ is important meaning that the overtone at $$880$$Hz is strongly represented in your instrument.

You don't need to use the complex Fourier series for many applications. The Wikipedia article Discrete cosine transorm describes some of them as follows:

DCTs are important to numerous applications in science and engineering, from lossy compression of audio (e.g. MP3) and images (e.g. JPEG) (where small high-frequency components can be discarded), to spectral methods for the numerical solution of partial differential equations

The article goes on to state that among the variants of the transform are

the discrete sine transform (DST), which is equivalent to a DFT of real and odd functions

Later on in more detail it states

The obvious distinction between a DCT and a DFT is that the former uses only cosine functions, while the latter uses both cosines and sines (in the form of complex exponentials).

You can read the article for more technical details and pointers to references for more information. The summary is that the complex Fourier series is elegant, but for many real world applications, there are clear advantages to using DCT. This does not negate the great importance of the DFT for theoretical purposes which is based on the use of Euler's formula to model simple harmonic motion. In other words, simple harmonic motion is closely related to circular motion using the important identity $$\, \sin(t)^2 + \cos(t)^2 = 1\,$$ and that $$\, \frac{d}{dt}\sin(t) = \cos(t), \: \frac{d}{dt} \cos(t) = -\sin(t).\,$$ The fact that these are naturally modeled by complex numbers is one of the most striking results in all of mathematics.