עקרונות בתקשורת תקבילית וספרתית — אפנון AM ו-FM, דגימה וכימוי

λ = cf | l_ant = λ4 | SNR_dB = 10·log₁₀P_sP_n

1.00 MHz

1.00 kHz

×200

20 dB

40 m

—

λ = 3·10⁸ / 3·10³ = 100000 m = 100 km

l = 100 / 4 = 25 km

λ = 3·10⁸ / 10⁸ = 3 m ⟹ l = 0.75 m

25000 / 0.75 = 33333

2 mW / 8 µW = 2000 µW / 8 µW = 250 ⟹ 10·log₁₀(250) = 23.98 dB

s(t) = A_c[1 + m·cos ω_mt]·cos ω_ct | m = A_max − A_minA_max + A_min | P_t = P_c(1 + m²2)

0.60

5.0 V

4.0 kHz

10 kW

—

m = (8 − 2) / (8 + 2) = 6 / 10 = 0.6

A_c = (8 + 2)/2 = 5 V | A_m = (8 − 2)/2 = 3 V | 0.6 · 5 = 3 ✓

BW = 2 · 4 = 8 kHz

P_t = 10 · (1 + 0.6²/2) = 10 · 1.18 = 11.8 kW

P_sb = 11.8 − 10 = 1.8 kW ⟹ 1.8 / 11.8 = 0.1525 = 15.25 %

f_LO = f_s + f_IF | f_im = f_s + 2·f_IF | BW = f₀Q | α = √1 + Q²(ff₀ − f₀f)²

990 kHz

60

455 kHz

20 kW

55 %

10 ppm

—

f_LO = 990 + 455 = 1445 kHz

f_im = 990 + 2·455 = 1900 kHz | 1900 − 1445 = 455 ✓

1900/990 − 990/1900 = 1.9192 − 0.5211 = 1.3981 α = √(1 + 60²·1.3981²) = 83.89 ⟹ 20·log₁₀(83.89) = 38.5 dB

f_im = 990 + 200 = 1190 kHz ⟹ 1190/990 − 990/1190 = 0.3701 α = √(1 + 60²·0.3701²) = 22.23 ⟹ 26.9 dB

38.5 − 26.9 = 11.6 dB

s(t) = A·cos(ω_ct + β·sin ω_mt) | β = Δff_m | BW ≈ 2(Δf + f_m)

75 kHz

15 kHz

30 %

—

β = 75 / 15 = 5

BW = 2·(75 + 15) = 180 kHz

β = 75 / 5 = 15 | BW = 2·(75 + 5) = 160 kHz

15 > 5 | 160 < 180

BW_AM = 2·5 = 10 kHz ⟹ 160 / 10 = 16

f_s ≥ 2·f_max | Δ = V_pp2ⁿ | SNR_q ≈ 6.02n + 1.76 dB | R_b = f_s·n

3.4 kHz

8.0 kHz

8

—

2 · 3400 = 6800 Hz < 8000 Hz ✓

2⁸ = 256 | Δ = 2 V / 256 = 7.8125 mV

R_b = 8000 · 8 = 64000 bit/s = 64 kbit/s

SNR_q = 6.02 · 8 + 1.76 = 49.92 dB

R_b = 44100 · 16 = 705600 bit/s = 705.6 kbit/s SNR_q = 6.02 · 16 + 1.76 = 98.08 dB

BW_ASK = BW_PSK = 2·R_b | BW_FSK ≈ 2(Δf + R_b) | R_s = R_blog₂M

10 kbit/s

15 kHz

1

8

8 kHz

—

8 · 8 + 1 = 65 bit

R = 8000 · 65 = 520000 bit/s = 520 kbit/s

T_f = 1/8000 = 125 µs | T_slot = 125/8 = 15.625 µs | T_b = 125/65 = 1.923 µs

R_s = 520 / 2 = 260 kBaud

BW = 2 · 260 = 520 kHz ( במקום 1040 kHz )