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The inverse Fourier transform of $H(f) = \operatorname{rect}\left(\frac{f}{2B}\right)$, the transfer function of the ideal lowpass filter of bandwidth $B$ Hz, is, as the OP says, $h(t) = 2B\operatorname{sinc}(2Bt)$. Now, the power transfer function of this ideal lowpass filter (ILPF) of bandwidth $B$ Hz is $|H(f)|^2$ to which we apply low-level math such as ...
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% create complex white normal noise
noise = randn(size(Signal)) + 1i*randn(size(Signal));
% calculate the gain for the noise
noiseGain = rms(Signal)./rms(noise)*exp(-SNR(i)*log(10)/20);
% add it
SignalN = Signal + noiseGain*noise;
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Is it really possible that a little recorder like this could record heart sounds even across a room?
No. There is no physical mechanism that could transfer enough of your heart beat for the recorder to pick it up.
Or is this noise from some other source that just happens to sound like a heartbeat and just happens to match my pulse?
Could be all sort of ...
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The noise is colored. If we are to talk of spectral densities (color) then we would best refer to phase noise and not jitter (jitter is the result of integrating the phase noise, and typically after a high pass function given by the typical cycle-cycle jitter measurement).
The PLL's phase noise would be a low pass filter of the phase noise from the reference ...
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