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I'm wondering that does only filter coefficients can represent it's own filter characteristic?

I found filter design tool as FILTERDESIGNER of matlab. that can make filter coefficients of low pass filter or band pass filter or high pass filter etc as the below. enter image description here

But I want to know what if there is only information of coefficients without information which is Response Type such as low or high and band filter.

Can we know that coefficients was made originally for low pass filter with h[k]*x[n-k] equation?

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You can compute the filter's frequency response to check the behavior of the filter:

$$H(\omega)=\sum_{n=0}^{N-1}h[n]e^{-jn\omega}\tag{1}$$

where $h[n]$ are the filter coefficients, and $N$ is the filter length (i.e., the number of coefficients). The frequency response $H(\omega)$ is a complex function. You can compute its magnitude and phase to evaluate the filter's magnitude and phase characteristics.

You can compute (and plot) the frequency response using Matlab's freqz command.

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