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I'm confused by the Nyquist channel capacity formula. How can channel maximum capacity approach double the bandwidth.

$C = 2\times BW \times log_{2}(L)$ bits/sec

The way it was explained to me way back when when I took a telecommunications course was as follows.

C = Maximum channel capacity in bits/second

BW = (highest frequency - lowest frequency) that the channel can accommodate.

L = signal levels So in the figure below, where BW = f; the frequency of the sine wave; and L = 2 for -1 and + 1; the channel capacity is C = 2B because we can consider the positive part of the sine wave as = to a 1 bit for half the period, and as a 0 bit for the negative half of the period.

Now i'm not sure that this is correct, as that doesn't seem like "information", being that it doesn't seem to decrease entropy, or uncertainty?

so if that's not correct, what is an example of a signal with capacity twice the bandwidth?
Or am I just not understanding the entire concept?

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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 cosine pulses with 50% excess bandwidth achieve $R_p = 1.5B$.

If you send pulses with $L=2^k$ amplitude levels, then the bit rate is $R_b = kR_p$.

So far for Nyquist, who was only interested in maintaining orthogonality. Shannon wanted to find a limit for the bit rate of reliable communication, with probability of error $P_b \rightarrow 0$. This limit is the capacity $C = B \log_2(1+\text{SNR})$, assuming an AWGN channel and orthogonal pulses. Note that $L$ is not mentioned explicitly.

what is an example of a signal with capacity twice the bandwidth?

The capacity is a property of the channel, not the signal. A signal has a bandwidth, and implicitly a pulse rate and a bit rate. Regardless of that, the channel imposes a limit on the bit rate (its capacity), if you want reliable communication.

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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 systems. For example, in QPSK modulation a symbol represents two bits so this is how you can have bit rate twice the bandwidth.

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  • $\begingroup$ I should have pointed out that Nyquist is the maximum bound for a noiseless channel, but my understanding is that they are distinct though related. I've read the Shannon-Hartley was partially derived from Nyquist. $\endgroup$ – Frank Sep 30 at 20:13
  • $\begingroup$ Here's a couple of links. witestlab.poly.edu/blog/… $\endgroup$ – Frank Sep 30 at 20:14
  • $\begingroup$ And here: tutorialspoint.com/the-maximum-data-rate-of-a-channel $\endgroup$ – Frank Sep 30 at 20:15
  • $\begingroup$ @Frank maximum bound of what ? Nyquist rate is only stated for orthogonal system. You may be interested in faster than Nyquist techniques. For example, dsp.stackexchange.com/questions/52730/… $\endgroup$ – AlexTP Oct 1 at 13:20

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