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Approaching The Sampling Theorem as Inner Product Space Preface There are many ways to derive the Nyquist Shannon Sampling Theorem with the constraint on the sampling frequency being 2 times the Nyquist Frequency. The classic derivation uses the summation of sampled series with Poisson SummationFormula. Let's introduce different approach which is more ...


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Yes you can bandpass filter an adequate portion of a sampled (ideal impulse modulated) signal spectrum and still retain the same information of the lowpass filtered version. As you have stated, the sampled signal has a spectrum which includes shifted and weighted copies of the original (possibly baseband) signal. Assuming no aliasing occured during the ...


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Not sure what you mean by "wrong". The Nyquist criteria simply requires you to have "two samples per Hz of bandwidth". It doesn't have to be $[-f_{max},-f_{max}]$, it can be any frequency range that includes at least $2 \cdot f_{max}$ of bandwidth. However, for real signals you need to figure out what to do with the negative frequencies. For more info ...


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It really boils down to aliasing. In continuous-time, if you have any two signals $x_1(t) = \sin(2 \pi F_1 t)$ and $x_2(t) = \sin(2 \pi F_2 t)$, then as long as $F_1$ and $F_2$ are distinct, the signals are, too. But consider sampling at some time interval $T_s$, so that the sampled signals are $x_1(k) = \sin(2 \pi F_1 T_s k)$ and $x_2(k) = \sin(2 \pi F_2 ...


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Beating just below the Nyquist frequency occurs when an attempt is made to reconstruct the time continuous signal without the use of sinc interpolation. This sinc reconstruction method and, in fact, requirement for the Nyquist–Shannon sampling theorem to hold true, is the Whittaker–Shannon interpolation formula.


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I think you're confusing two different (but related) terms. Nyquist says that in a channel of bandwidth $B$ you can transmit up to $2B$ orthogonal pulses per second. So, $R_p \leq 2B$, where $R_p$ is the pulse rate. To achieve $R_p = 2B$, the pulses need to be sinc-shaped. Other, more practical pulses achieve slightly less than that. For example, raised ...


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I don't think I've seen capacity defined like that before. In the "go-to" information theory book by Thomas Cover, capacity is defined as $C=\frac{1}{2}log_2(1+SNR)$ bits per channel use or $C=Wlog_2(1+SNR)$ bits per second. The bandwidth is the symbol rate so you could have a symbol represent multiple bits which is what happens in all digital communication ...


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Is the rate of 2B exclusive? Yes. The sampling theorem states that the signal must be band limited to half the sample rate. That implies that the energy at the Nyquist frequency must be zero. In practice you need a healthy margin between the highest usable frequency and the Nyquist frequency. There is always some "transition band" that you need to get the ...


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