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I am trying to solve the below problem and I was not able to find the relation between bandwidth, Nyquist criteria and quantization. Can you please provide some hint?

Imagine that the possible transmissible symbols are 64. Whenever the transmitter needs to transmit a value=n, it sends n-1 bits to the channel, each one T/(n-1) seconds apart starting from the beginning of the same slot. What is the maximum bandwidth of an analog signal that can be transmitted without losing any information through this system if 4 bits are used for its quantization?

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    \$\begingroup\$ Saying "Nyquist Criteria" is vague. What do you mean by Nyquist Criteria? I'll give you some examples: Nyquist Stability Criterion, Nyquist Sampling, Nyquist ISI Criterion, Nyquist Frequency... etc. \$\endgroup\$
    – user103380
    Mar 14, 2018 at 3:37
  • \$\begingroup\$ Thanks @KingDuken for the quick comment to clarify the question. I am referring to Nyquist Criteria that states that a channel with bandwidth B Hz can be used to carry atmost 2B signal changes (symbols) per second. \$\endgroup\$ Mar 14, 2018 at 3:38
  • \$\begingroup\$ Okay, that's Nyquist Frequency :) \$\endgroup\$
    – user103380
    Mar 14, 2018 at 3:40
  • \$\begingroup\$ Thanks @KingDuken for the clarification. can you please let me know if there is any formula that relates bandwidth and quantization? \$\endgroup\$ Mar 14, 2018 at 3:43
  • \$\begingroup\$ With 4 bits quantization, what is your value for the SNR? And is that SNR a proper/meaningful number to use in the Nyquist equation for transmittable information per second? \$\endgroup\$ Mar 14, 2018 at 3:50

3 Answers 3

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Shannon-Hartley Theorom proved the maximum Channel capacity,C and Channel Bandwidth, B.

C/B=Log_2(1+S/N)=C/B

for BER of 1e-6 results in Ref indicate ;

BPSK needs 11 dB SNR @ C/B=1
64-QAM needs 27 dB SNR @ C/B=6.33
1024-QAM needs 39 dB SNR @ C/B=8 excluding 20% overhead for FEC.

Ref

https://www.csie.ntu.edu.tw/~hsinmu/courses/_media/wn_11fall/rate_adaptation.pdf
http://www.scte.org/TechnicalColumns/12-10-01%20spectral%20effieciency.pdf
https://www.embedded.com/print/4017668
http://www2.ensc.sfu.ca/people/faculty/cavers/ENSC805/classnotes/c4p3.pdf

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The first thing you need to do is understand how the data in the problem is encoded.

Imagine that the possible transmissible symbols are 64. Whenever the transmitter needs to transmit a value=n, it sends n-1 bits to the channel, each one T/(n-1) seconds apart starting from the beginning of the same slot.

This looks like each symbol consists of a time slot of time T, and that n-1 pulses are sent within that timeslot to represent the value n.

enter image description here

The problem says that the maximum value of n is 64. Therefore you can at most represent one number in the range of 0 to 63 within each time T.

What is the maximum bandwidth of an analog signal that can be transmitted without losing any information through this system if 4 bits are used for its quantization?

To find your answer you need to figure out the following.

  1. How many 4-bit values can encode in a number ranging from 0 to 63. Lets call that X. Hint: log2(64) = 6.

  2. If sending X 4-bit values takes T seconds, then how many 4-bit values can you send per second. Lets call that Y.

  3. Given that you can send Y 4-bit values per second. What is the maximum bandwidth of the signal that you can sample while still sampling at the minimum required rate to not loose information. Hint: The Nyquist sampling theorem says that you must sample at twice the signgal bandwidth prevent information loss.

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Any time a signal is transmitted through a system, INFORMATION will be lost.

Because noise will be added, and the SNR will be degraded, and INFORMATION will be lost.

To lose no information in your 64-symbol system, there can NEVER be any bit errors. But random noise will at some time add up to cause bit errors.

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