Communication Systems Numericals

1. An AM wave has carrier amplitude 20V and modulating signal amplitude 8V. What is the modulation index?

μ = A_m/A_c = 8/20 = 0.4

2. Calculate total power in AM transmission when carrier power is 400W and modulation index is 0.8.

P_t = P_c(1 + μ²/2) = 400(1 + 0.64/2) = 400 × 1.32 = 528W

3. Find bandwidth required for AM broadcast with maximum audio frequency 8kHz.

BW = 2f_m = 2 × 8kHz = 16kHz

4. A channel has bandwidth 10kHz and SNR=31. Calculate channel capacity.

C = B log₂(1 + SNR) = 10000 × log₂(32) = 10000 × 5 = 50kbps

5. An AM wave shows E_max = 30V and E_min = 10V. Calculate modulation index.

μ = (E_max - E_min)/(E_max + E_min) = (30-10)/(30+10) = 20/40 = 0.5

6. Carrier wave (1.2MHz) modulated by 6kHz signal. Find sideband frequencies.

USB = f_c + f_m = 1200 + 6 = 1206kHz

LSB = f_c - f_m = 1200 - 6 = 1194kHz

7. For 80% modulation, what should be ratio of modulating to carrier amplitude?

μ = A_m/A_c = 0.8 ⇒ A_m/A_c = 0.8

8. Total AM power is 1500W at μ=1. Find carrier power.

1500 = P_c(1 + 1²/2) ⇒ 1500 = 1.5P_c ⇒ P_c = 1000W

9. Signal power=5mW, noise power=20μW. Calculate SNR.

SNR = S/N = 0.005 / 0.00002 = 250

10. AM transmitter radiates 9kW with μ=0.5. Calculate carrier power.

9000 = P_c(1 + 0.25/2) ⇒ 9000 = 1.125P_c ⇒ P_c = 8000W

11. For μ=0.75 and P_c=1.2kW, find power in each sideband.

Total sideband power = P_cμ²/2 = 1200×(0.5625)/2 = 337.5W

Power per sideband = 337.5/2 = 168.75W

12. Channel with 3kHz bandwidth has SNR=7. Find maximum data rate.

C = B log₂(1 + SNR) = 3000 × log₂(8) = 3000 × 3 = 9kbps

13. AM wave has E_max=25V and μ=0.6. Find E_min.

μ = (E_max - E_min)/(E_max + E_min)

0.6 = (25 - E_min)/(25 + E_min) ⇒ E_min = 10V

14. Carrier frequency 96MHz modulated by 15kHz. Calculate bandwidth.

BW = 2f_m = 2 × 15 = 30kHz

15. In AM transmission with μ=1, what percentage of power is in carrier?

P_t = P_c(1 + 0.5) = 1.5P_c

% carrier power = (P_c/1.5P_c) × 100 = 66.67%

16. SNR=15dB. Convert to linear ratio.

SNR_linear = 10^(SNR_dB/10) = 10^1.5 ≈ 31.62

17. For μ=1, if carrier amplitude is 100V, find modulating amplitude.

μ = A_m/A_c = 1 ⇒ A_m = A_c = 100V

18. Channel capacity 24kbps with B=3kHz. Find minimum required SNR.

24 = 3 × log₂(1+SNR) ⇒ 8 = log₂(1+SNR) ⇒ 1+SNR=256 ⇒ SNR=255

19. AM wave with f_c=800kHz and f_m=4kHz. Find frequencies present.

Carrier: 800kHz

USB: 804kHz, LSB: 796kHz

20. Total AM power 1170W with μ=0.6. Find power saved in SSB-SC.

1170 = P_c(1 + 0.36/2) ⇒ P_c = 1000W

SSB-SC power = Sideband power = P_cμ²/4 = 1000×0.36/4 = 90W

Power saved = 1170 - 90 = 1080W

Formula Reference Table

Formula	Equation	Application
Modulation Index	μ = Am/Ac	AM depth calculation
Total AM Power	Pt = Pc(1 + μ²/2)	Transmitter power calculation
Bandwidth	BW = 2fm	Spectrum allocation
Shannon Capacity	C = B log2(1 + SNR)	Max data rate calculation
Sideband Frequencies	fUSB = fc + fm fLSB = fc - fm	Frequency spectrum analysis