In: Computer Science
Show the calculation steps for each of the problems below.
a. Using Nyquist’ theorem, calculate the channel capacity C of a signal that has 18 different levels and a frequency of 50,000 Hz.
b. Using Shannon’s theorem, calculate the data transfer rate given the following information, signal frequency= 25,000 Hz; signal power = 5000 watts; noise power = 320 watts.
c. Using Nyquist’s theorem and given a frequency of 8000 Hz and a data rate of 90,000 bps, how many signal levels (L) will be needed to convey this data?
d. The signal-to-noise ratio is often given in decibels. Assume that SNRdB = 28 and the channel bandwidth is 8 GHz, what is the theoretical channel capacity?
Solution ::
a) Since the nature of channel(whether noisy or noiseless ) is not explicitly stated, it is assumed to be noiseless.
Capacity of channel C = 2*B*log2L
= 2*50000*log218 = 100000* 4.169925 = 416992.5 ~ 416992bits/second
b) In this case, the channel is noisy.
Shannon's theorem:-
SNR(Signal to Noise Ratio) :- Signal power/ Noise power = 5000/320 = 15.625
capacity:- B log2 (1+SNR) = 25000 * log2 (1+15.625) = 25000 * log2 (16.625) = 25000 * 4.055
= 101382.061 ~ 101382 bps
c) C = 2*B*log2L
90000 = 2*8000*log2L
log2L = 90000/16000 = 5.625
L = 25.625 = 49.35 ~ 50 levels (Answer)
Note:- Generally levels are taken to be power of two. In that case, 64 levels are required(power of 2 just greater than 50)
d)
C = B log2 (1+SNR) ..(1)
SNRdB = 10log10SNR
28 = 10log10SNR
log10SNR = 2.8
SNR = 102.8 = 630.957
Put this in (1)
C = 8*109 * log2(1+SNR) [Since 1 GHz = 109Hz]
C = 8*109 * log2(631.957)
C = 7.443 * 1010 bps
C = 74.43 * 109 bps
C = 74.43 Gbps (Answer)
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