Problem with Minimum Nyquist bandwidth formula and m-ary encoding

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SUMMARY

The discussion centers on the confusion surrounding the minimum Nyquist bandwidth formula as presented in Wayne Tomasi's "Electronic Communication Systems." The formula states that the minimum Nyquist bandwidth is given by fb=2B, where fb is the bit rate and B is the ideal Nyquist bandwidth. However, the subsequent equation for multilevel signaling, fb=2Blog2M, raises questions about the presence of the number 2 when rearranging to B=fb/log2M. Participants conclude that the omission of the 2 in the latter equation is likely a textbook error, as it cannot simply disappear through algebraic manipulation.

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  • Understanding of Nyquist bandwidth and frequency concepts
  • Familiarity with multilevel signaling and M-ary modulation
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  • Proficiency in algebraic manipulation of equations
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iampaul
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I am using the book Electronic Communication Systems by Wayne Tomasi:
I had a problem with a certain part of the book:
The book gives the formula for the minimum nyquist bandwidth as:

" The minimum theoretical bandwidth necessary to propagate a signal is called the minimum Nyquist bandwidth or sometimes the minimum nyquist frequency. Thus, fb=2B, where fb is the bit rate in bps and B is the ideal nyquist bandwidth."

Then it says that:
"Using multilevel signaling, the nyquist formultion for channel capacity is fb=2Blog2M. This equation can be rearranged to solve for the minimum bandwidth necessary to pass M-ary digitally modulated carriers. B=fb/log2M" ?

What happened to the number 2 in 2Blog2M?? I thought it was just a typo but then it used this equation for the next discussions.

Please reply, thanks in advance!
 
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I looked around online documents on the topic and can find no support for dropping the 2. My money's on it being a textbook blunder. (But considering my meagre bank balance, that isn't saying much!)

Have you raised the matter with your lecturer?
 
It seems to me the errror is that the 2's shouldn't be there.

The Nyquist bandwidth is half the sampling frequency. A sampled signal of 1, 0, 1, 0, ... is at the Nyquist frequency and requires 1 bit per sample, so I think the first equation should be bf = B.

log2M is the number of bits required to give M distinct sample levels. A single-bit signal needs 2 levels, not 1, and log22 = 1, so the 2 didn't appear or disappear because of that.

But even if the above comments are wrong, you certainly can't "lose" the 2 just by rearranging the equation fb=2Blog2M to get B=fb/log2M.

I guess the mistake(s) came from the author "knowing the right formula" but forgetting whether he/she was talking about the sampling frequency or the Nyquist frequency.
 

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