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So lets say I have a transfer function for an FIR filter $H(z) = (z-a)...(z-n)$ where $a,...,n$ are points along the unit circle. But what about other points along the unit circle that correspond to frequencies we want to leave alone? Won't we end up multiplying these frequencies by random complex numbers? when we would really like them all to just be multiplied by one?

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    $\begingroup$ >Won't we end up multiplying these frequencies by random complex numbers? when we would really like them all to just be multiplied by one? Well, not random complex numbers, but certainly not the desired unity. Welcome to the real world where it is impossible to design ideal notch filters that zero out a selected set of frequencies while passing all others unchanged. $\endgroup$ – Dilip Sarwate May 15 '12 at 1:13
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    $\begingroup$ I resisted downvoting that the OP named an easily computable continuous and differentiable function as "random". Quite the opposite. $\endgroup$ – hotpaw2 May 15 '12 at 2:29
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The response of the filter at any frequency $\omega \in [-\pi, \pi)$ can be found by substituting $z = e^{-j\omega}$ (i.e. a position on the unit circle at an angle of $\omega$) into the transfer function. The resultant $H(e^{j\omega})$ is often referred to as a discrete-time system's frequency response. Most of the time, you're primarily concerned about the magnitude response of the system, so you will often see $|H(e^{j\omega)}|$ specifically considered in a system design.

When designing a filter, there will often be a region of frequencies that you would like "left alone;" this is called the filter passband. I wouldn't say that the presence of zeros elsewhere in the $z$-plane results in multiplying these frequencies by random complex numbers; indeed, the response at any frequency is well-defined. Design methods for digital filters that have flat passbands will yield a transfer function whose zeros are carefully placed such that the filter response is approximately unity in those regions (assuming no filter gain). The amount of allowed deviation from this ideal response is a design parameter that you often have at your disposal to control the required complexity of the filter.

So, to more pointedly answer your question, if you know all of the zeros of an FIR filter, then you know everything about its transfer function, at every point in the $z$-plane (and therefore everywhere on the unit circle).

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It's an existence problem. There exists no non-trivial linear time-invariant filter or transfer function that will "leave alone" any non-zero-size band of frequencies.

So either one deals with that characteristic of a linear system, or jumps into trying some non-linear or time-variable system which may be far less tractable and/or produce some really surprising weirdness for some inputs.

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