Frequency Response of the First-Difference Filter

Figure 1: Magnitude and phase of the frequency response of the first-difference filter. Below is a derivation of the analytical expression for the frequency response:

$$ H(e^{j\hat\omega})=\sum\limits_{k=0}^{M} b_k e^{-j\hat\omega} = 1-e^{-j\hat\omega} $$