CT Frequency Response

First the system:

$ \sum_{k=0}^{N}a_k\frac{d^ky(t)}{dk^t}=\sum_{k=0}^{M}b_k\frac{d^kx(t)}{dt^k} $

Then the Fourier Transform:

$ \sum_{k=0}^Na_k(j\omega)^kY(\omega)=\sum_{k=0}^Mb_k(j\omega)^kX(\omega) $

Then the frequency response H(jw):

$ Y(\omega)=\frac{\sum_{k=0}^Mb_k(j\omega)^k}{\sum_{k=0}^Na_k(j\omega)^k}X(\omega) $

So:

$ H(j\omega)=\frac{\sum_{k=0}^Mb_k(j\omega)^k}{\sum_{k=0}^Na_k(j\omega)^k} $


DT Frequency Response

First the system:

$ \sum_{k=0}^Na_ky[n-k]=\sum_{k=0}^Mb_kx[n-k] $

Then the Fourier Transform:

$ \sum_{k=0}^Na_ke^{-jk\omega}Y(\omega)=\sum_{k=0}^Mb_ke^{-jk\omega}X(\omega) $

So the frequency response is:

$ H(e^{j\omega})=\frac{\sum_{k=0}^Mb_ke^{-jk\omega}}{\sum_{k=0}^Na_ke^{-jk\omega}} $

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Followed her dream after having raised her family.

Ruth Enoch, PhD Mathematics