Euler's Formula
\(\(\LARGE e^{ix}=cos(x)+isin(x)\)\)
- Euler's formula basically states that a exponential raised to a imaginary value has both a real and a imaginary component
- If we map the axis such that vertical is Imag. and horizontal is real, we find that our exponential raised to a imaginary value will appear to rotate CCW with increasing constant values, hence the sine and cosine
- The magnitude of the complex value is constant. In other words, the only thing that is changing is the phase.
- Remember that complex magnitude discards j: \(\sqrt{cos^2(x)+sin^2(x)}=\sqrt{1}=1\), such that the magnitude of the the exponential will be 1 if there is no coefficient.
- The exponential rotates with a periodicity of \(x=2\pi\)
Complex Sinusoid
Here is our DSP Complex Sinusoid:
\(\(\LARGE x[n]=Xe^{j2\pi fn}\)\)
- X is a complex constant
- \(f\) is Normalized Frequency
- Note that the magnitude of the exponential term is always 1, so the magnitude at any n will be equal to X. In other words, X is the complex magnitude of the sinusoid
- Since X does not vary with n, the magnitude of this complex sinusoid will be constant!
- Notice that \(\large e^{-j \frac{\pi}{2}}=cos(\frac{\pi}{2})+jsin(\frac{\pi}{2})=0+j\)
A goal without a plan is just a wish.
— Larry Elder