I'm stumped on an equation from a coursera course (Intro to DSP) that has to do with complex exponential multiplication:
In line 2, by simple rules of complex multiplication, it should be (h+k). I understand it being (h-k) has to do with complex exponential conjugation but not really sure why/how.
Any help understanding would be appreciated.
The complex conjugate of a complex number written as $re^{i\theta}$ where $r,\theta \in \mathbb{R}$ is given by $(re^{i\theta})^* = re^{-i\theta}$. Since we're taking the complex conjugate before multiplying, we have $$(e^{j\frac{2\pi}{N}nk})^*e^{j\frac{2\pi}{N}nh} = e^{-j\frac{2\pi}{N}nk}e^{j\frac{2\pi}{N}nh} = e^{j\frac{2\pi}{N}(h-k)n}. $$