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What are the problems with designing an FIR filter using FFT?

For method 1, by taking IFFT of desired $H[\omega]$, you will likely get a impulse response $h[n]$ in complex numbers, which means the output of the filter $y[n]$ will be complex numbers too.
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Multiplication in frequency domain for a signal containing a whole frequency spectrum

Multiplication in the frequency domain is equivalent to convolution in the time domain. When the signal occupies multiple frequencies, describe the signal in the frequency domain and then do the ...
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Constructing an input signal whose response is determined by the impulse response

A more detailed proof based on Matt's answer. Let $n_0$ be the argument to $\max\{|y[n]|\}$, i.e. $\max\{|y[n]|\} = |y[n_0]|$. We can always find $x[n]$ as a series of +1's and -1's, s.t. $$|y[n_0]| = ...
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Constructing an input signal whose response is determined by the impulse response

You're right that you generally can't find a sequence $x[n]$ such that $$y[n]=\sum_{k=-\infty}^{\infty}\big|h[k]\big|\tag{1}$$ is satisfied for all values of $n$, but that's also not necessary. It is ...
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Validity of taking an inverse $\mathcal{Z}-$ transform instead of taking an inverse DTFT

Let's first clear up a misunderstanding: the inverse DTFT and the inverse $\mathcal{Z}$-transform are equally simple or difficult to compute; the integrals are the same. If you use the contour $|z|=1$ ...
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Convolution: graphing output y(t) with given input x(t) and impulse response h(t)

@Gunners You can work your problem in two parts. The first part is: the output with just the single +2 impulse input that occurs at time = 0 seconds. The second part is: the output with just the ...
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Convolution of squares / boxcars

I'll give this a whirl. Let's define the causal rectangular function $R(t)$ to be $$ R(t) = \left\{\begin{matrix} 1 & 0 < t < 1 \\ 0 & else\\ \end{matrix}\right.$$ Let's find $Y(t) = R(...
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What methods allow learning "changes" in data, and then generating synthetic output?

I think what you are describing here is basic system theory. As long as "something happened to data" is LTI (linear & time invariant), than the system can be fully characterized by it's ...
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Convolution of squares / boxcars

Apply convolution theorem: $$ |A||B|\ \mathrm{sinc}(A\omega/2) \ \mathrm{sinc}(B\omega/2) $$ WA gives for $\mathcal{F}^{-1}$, disregarding constants, $$ \begin{align} && (A + B - 2t) \ \...
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