Probably a stupid question, but I'm slowly making my way through Julius Smiths' filter book. The maths is slightly defeating me at points.
This function $H(e^{j\omega t})$, why is it a function of a formula?
All my functions at school a decade ago were of a single variable, and you took the thing in the brackets and substituted it in.. it was nice and easy. Here I'm confused.
I'm guessing here it's talking about complex frequency. But why is it written like this?
Answer
That is just a standard convention in the signal processing literature.
It is written that way because a DTFT is evaluating the z-transform on the unit circle in the complex z-plane: $$H(z)|_{z=e^{i\omega n}} = H(e^{i\omega n})$$
The same convention is followed for CTFT's because a CTFT is evaluating the Laplace transform on the imaginary axis in the complex s-plane: $$H(s)|_{s=i\Omega t} = H(i\Omega t)$$
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