Complex Oscillation

Euler¶

Euler's formula is crucial when applying complex numbers in digital signal processing, like in the Discrete Fourier Transform. With Euler's Number $e \approx 2.71828$ - the base of the natural logarithm and exponential function - and the imaginary unit $j$, this complex exponential function is defined as:

$$ e^{j\varphi} = \cos(\varphi) + j \sin(\varphi)$$

The above equation states that we can use a combination of sine and cosine to express a complex number. This is not only fundamental to DSP applications, but is also considered one of the most fundamental equations in mathematics...

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Side Note on the Exponential Function

$$ y(x) = e^x $$

The exponential function with $e$ is the counterpart to the natural logarithm:

$$ x = ln(exp(x))$$

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Visualization¶

We can visualize Euler's equation in the complex domain with the exponential representation of a complex number:

$$ z = r e^{j\varphi}$$

Real and imaginary part are the projections on the x-axes and the y-axes, respectively:

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Harmonic Functions with Complex Numbers¶

With the help of Euler, we can also express basic harmonic functions like sine and cosine through complex numbers:

$$ \cos{\varphi} = \frac{1}{2} (e^{j \varphi} + e^{-j \varphi})$$

$$ \sin{\varphi} = \frac{1}{2j} (e^{j \varphi} - e^{-j \varphi})$$

This relation is important when calculating the Fourier transform of specific signals.