IT IS BY MY DECREE, THAT HENCEFORTH YOU MUSK PLACE THE WORDS "HELLO MY NAME IS", IN THAT ORDER IN YOU POSTS!!!! THIS IS A HIGH SECUIRTY ZONE!!!!
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------ HELLO MY NAME IS ------
------ hello my name is
------ HELLO MY NaME IS ------
------ HELLO MY NAME IS DICKFACE ------
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------ to versions of the shnnon nyquist th'm if you've got a bandlimited signal, if you sample at twice the highest bandlimited frequency you can reconstruct completely V.2. If x ( F^Z has a DTFT xhat, and T>0 is given, there exists a unique x_r ( F^R uith these two properties x_r(nt) = x(n), all n ( Z [x_r "connects the dots" in x] =x_r is bandlimited to within Pi/T (i.e. X_R(Ohm)=0, |Ohm|>= Pi/T Terminology: Given a bandlimited continuous-time signal x_c, and given Ohm_m = "smallest bandwidth" for x_c, the Nyquist rate for x_c is 2*Ohm_m [ Ohm_m/Pi in Hz ]; Version 1 of Th'm says "sample x_c faster than Nyquist rate => can recover x_c from samples" when constructing x_r, your brain takes the most parsimonious route, taking the solution with lowest bandwidth 44.1kHz gives some headroom over 2*20kHz aliasing: why call it that? x_c is "posing as x_r", or you might say "assuming x_r as an alias". time-division multiplexing (or multiple access) (i.e. CDMa, etc.) II(t) = sum(P_a(t-n*Ts),n,-inf,inf) , t ( R graphically, this looks like a pulse train z(t) = x(t)*II(t) II(t) = sum(c_k*e^(j*k*Ohm*t),k,-inf,inf) Ohm = 2*Pi/Ts z(t) = sum(c_k*x(t)*e^(j*k*Ohm*t),k,-inf,inf) Suppose x <-F-> Xhat, and x is bandlimited to within Ohm_max, and Ohm_sampling > 2*Ohm_max (2x the nyquist rate) t |--> c_k*x(t)*e^(j*k*Ohm*t) <--F--> Ohm |--> c_k*Xhat(Ohm-k*Ohm_s) so this creates scaled and shifted versions of Xhat in the frequency domain. You can easily lowpass this signal to get just a scaled copy of X. This demonstrates that a bandlimited signal can be sampled and reconstructed perfectly. N*T < Pi/Ohm_max or T < Pi/(n*Ohm_m). Too sm ------
------ clean boner :OOO ------