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May
25
accepted Representing a real sampled signal with N samples as a complex sampled signal with N/2 samples
May
25
comment Representing a real sampled signal with N samples as a complex sampled signal with N/2 samples
Indeed. I meant "efficient" just in terms of, if one naively computed all $N$ members of the DFT, one would be performing approximately twice as many computations as necessary. Anyway, thank you for the most clear explanation of how the DFT relates to real inputs I've yet seen online. :) Particularly the point of what "physical" values $|X_k|$ and $arg\{X_k\}$ represent.
May
25
comment Representing a real sampled signal with N samples as a complex sampled signal with N/2 samples
Or, rather, the first $N/2+1$ members of the DFT from $X_0$ to $X_{N/2}$.
May
25
comment Representing a real sampled signal with N samples as a complex sampled signal with N/2 samples
So, if I understand correctly, to efficiently compute the DFT of a real-valued sequence with $N$ members, one can simply represent each real valued input $x_n$ as a complex number $x_n+0i$, and then compute the first $N/2$ members of the DFT, at which point each member of the second half can be trivially computed as the complex conjugate of each member of the first half?
May
25
asked Representing a real sampled signal with N samples as a complex sampled signal with N/2 samples
Dec
11
awarded  Scholar
Dec
11
accepted Reducing radical congruence to polynomial congruence
Dec
11
comment Reducing radical congruence to polynomial congruence
This is great. I'm not totally convinced working with modulo arithmetic couldn't work, but reducing this problem to Pell's equation is honestly a more elegant solution anyway. Thanks!
Dec
11
awarded  Supporter
Dec
11
asked Reducing radical congruence to polynomial congruence
May
7
comment Sieve of Atkin - algorithm for enumerating lattice points.
Is the fact that f and g are remainders modulo 15 and 30 respectively just arbitrary, or are the numbers 15 and 30 picked specifically? Algorithm 4.2 uses remainders modulo 10 and 30 instead of 15 and 30, for instance.
May
1
awarded  Student
May
1
asked Sieve of Atkin - algorithm for enumerating lattice points.