# How to show that this filter is a bandpass filter [duplicate]

i got a system with transfer function given by:

$$H(\omega)=1-e^{-j\omega}$$ I already plot it, and that seems to be a periodic function with $$H(0)=0$$ $$H(\pi)=2$$ $$H(\infty)=1$$ is that enough to show that this is a FIR bandpass filter. thank you very much

You can rewrite the transfer function as $$H(\omega) = 2je^{-j\omega/2}\sin(\omega/2)$$ which gives $$|H(\omega)| = 2|\sin(\omega/2)|$$ I assume we're talking about a discrete-time system, so there is no frequency $\omega=\infty$. We're just considering the range $\omega\in [0,\pi]$. This means that the system is a highpass filter. It's magnitude response increases from $$|H(0)|=0$$ to $$|H(\pi)|=2.$$ Also note that $H(0)$ and $H(\pi)$ can be easily computed directly from the filter coefficients: $$H(0) = \sum_{k=0}^{n-1}h_k = 1 - 1 = 0$$ $$H(\pi) = \sum_{k=0}^{n-1}(-1)^kh_k = 1 - (-1) = 2$$