# Problem with Minimum Nyquist bandwidth formula and m-ary encoding

1. Jun 15, 2014

### iampaul

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.

2. Jun 17, 2014

### Staff: Mentor

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?

3. Jun 17, 2014

### AlephZero

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.