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Nice question! It uses one of my favorite trig identities (which can also be used to show that quadrature modulation is actually simultaneous amplitude and phase modulation). The impulse response of the system described above is given by: Block diagram:


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This is an approximation, but you can make it as good as you like.


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I waited a bit to see if someone else takes the challenge, but since there are no answers yet, I'm providing mine now.


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Not quite as elegant as Matt L.'s answer, but also seems to work.


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White noise: sounds exactly the same if you play it back at double the speed Stack up diminished fifth starting at a low enough fundamental (say 20 Hz). Play back at double speed: sounds exactly the same. Stack octaves with equal amplitude starting with a low enough fundamental. Play back at twice the speed: sounds exactly the same Detune the octaves ...


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After reading Matt's resolution, I understood the puzzle (that was rather tricky by the way). I'll write a more formal demonstration for those who are interested: Let $\Gamma_L$ be a counter-clockwise contour that embraces the whole LHP. Let $\Gamma_R$ be a clockwise contour that embraces the whole RHP. Finally, let $\Gamma_T$ be a clockwise contour that ...


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Note that this trick only works well for filters with a relatively narrow pass band. Equi-ripple filters with wider pass bands usually do not show that irregularity in their impulse response.


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MBaz's answer is correct. I would just like to add another way of thinking about it, of course leading to the same result: Note that this system can be approximated quite well in a practical (discrete-time) implementation. Just take a well designed FIR linear-phase Hilbert transformer of length $2N+1$, and add a delay of $N$ samples to the other signal path....


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OK, this is a slightly constructed situation, but as far as I can see, the following is the only thing that makes sense, kind of ...


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! The answer by @cbos is correct in spirit but wrong in its details. In an answer on crypto.SE, I wrote "The Berlekamp-Massey algorithm is an iterative algorithm that solves the following problem. Given a sequence $s_0, s_1, s_2, \ldots$ of elements of a field, find the shortest linear feedback shift register (LFSR) that generates ...


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That's so easy, it's not even complex:


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There are a few pieces of background information that help in answering this question: Geometric series and in particular geometric series as applying to music A geometric series, $x_n$, is one where each successive term is generated by multiplying the previous term by some factor. For example: $x_{n+1} = a x_n$ Geometric series are everywhere in music, ...


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Actually you have asked two questions : Why is it that $H(e^{jw})$ can be obtained using the numerator and denominator of $H(z)$? Why do you need to compute that $H(z)$ at $z = e^{j2\pi \frac{f_k}{f_s}}$ for $k\in \{0,1,2,...,\frac{N}{2} \}$ ? It is true that z-transforms provide us the facility to figure out frequency response of a filter from the ...


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so Matt, i dunno why you don't think it's problematic comparing power signals to energy signals, but suppose we modify the definition of $f[n]$ slightly: $$ f[n] \triangleq \begin{cases} \ \tfrac12 e^{-\alpha n} \qquad & n \ge 0 \\ -\tfrac12 e^{ \alpha n} \qquad & n < 0 \\ \end{cases} $$ for some $\alpha > 0$. now we have finite ...


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I don't actually have an answer, just ideas, but since this is a riddle, I might of contribute these as (hidden as spoiler text) hints for others, which might have ideas that I miss: So, MSK is often introduced as         Oooh and I'd have an approach for the Gaussian pulse shaper, but that simply applies Sadly, that doesn't ...


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