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I'm encountering some troubles while trying to solve this exercise.

Let us consider a radio link between a geostationary satellite and a ground station. We send a binary signal (represented by two voltage levels), which got filtered by a rised cosine filter, frequency modulated to $f_p = 4~\text{GHz}$ with modulation index $I_f = 9$. We have to guarantee a signal-to-noise ratio after demodulation (let's call it SNRAD) of 15 dB.

The exercise asks to determine the bandwidth $B$ of the modulating signal known that:

  • At the receiver input the noise power, in the same bandwidth of the received signal, is $W_N = -100~\text{dBm}$;
  • The system temperature is $T_s = 290~\text{K}$;
  • The text of the exercise recalls that the bandwidth of the whole frequency modulated signal is 4 times the Carson's Bandwidth: $B_{tot} = 4B_c = 4B\left(I_f + 1\right)$.

Do someone have any suggestions about how to solve this exercise?

EDIT. I've tried to use the SNRAD information to obtain the SNR of the received signal, but I don't know if it is a correct way to use this information. By the way, as you will notice, the number that I obtain in this way seems incorrect since it is too low.

$SNRAD=3 I_f ^2 SNR = 3 ~ \cdot ~ 81 ~ \cdot ~ SNR = 243 ~ \cdot ~ SNR$

which in dB becomes

$ SNRAD_{dB} = 15 \text{dB} = 23.86 \text{dB} + SNR_{dB}$

from which I obtain something strange:

$ SNR_{dB} = 15 \text{dB} - 23.86 \text{dB} = -8,86 \text{dB}.$

A second approach I thought to calculate the bandwidth is to use the information given about the noise power level in the same bandwidth of the received signal. Again I don't know formulas which link the two things (noise power level and bandwidth) and the SNR obtained before seems too low.

EDIT2. I forget to try the classical formula which link the noise power level with the bandwidth considered. In this case we have:

$W_N = F k_B T B$

where $F = \frac{T_s + T_0}{T_s}$ . Now we have another problem, we have two unknown variables: the received signal power level (S) and the bandwidth of that signal (B) which need to find to solve the exercise. In fact:

$SNR_{dB} = -8.86 \text{dB} = S_{dB} - W_{N,dB}$ .

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