a) Definition

Linear path loss refers to the reduction in power of a wireless signal as it propagates through space from a transmitter to a receiver. It is usually expressed as a ratio of transmitted power (Pt) to received power (Pr).

b) Why use dB?

Engineers often prefer decibels (dB) because:

• It converts multiplicative losses into additive values, making calculations simpler.
• It handles very large or small numbers conveniently.
• It is commonly used in specifications for transmitters, antennas, and receivers.

c) Calculations

Given:

• Transmitted power: Pt = 25 W
• Received power: Pr = 5 × 10⁻⁷ W

i) Linear Path Loss:

Substitute values:

L = 25 / (5 × 10⁻⁷) = 5 × 10⁷

ii) Path Loss in dB:

Substitute values:

L(dB) = 10 × log10(5 × 10⁷) ≈ 10 × 7.699 = 76.99 dB ≈ 77 dB

d) Interpretation

A path loss of 77 dB indicates significant attenuation. This is likely in a rural or urban environment at medium distances. Assuming free-space path loss, the distance can be estimated using the Friis formula. Such high attenuation implies the need for repeaters or higher-gain antennas for reliable communication.

a) Friis Transmission Equation

Where:

• Pr = received power (W)
• Pt = transmitted power (W)
• Gt = transmitter antenna gain (linear)
• Gr = receiver antenna gain (linear)
• λ = wavelength of the signal (m)
• d = distance between antennas (m)

b) Practical Situations

Free-space propagation is reasonable when there is a clear line-of-sight (LOS), e.g., satellite communication. It is not appropriate in urban environments with buildings and obstacles where multipath and shadowing occur.

c) Calculations

Given:

• Frequency: 2.4 GHz → f = 2.4 × 10⁹ Hz
• Transmitted power: Pt = 15 W
• Antenna gain: Gt = Gr = 6 dB → linear gain = 10^(6/10) ≈ 3.98
• Distance: d = 3 km = 3000 m

i) Signal wavelength:

Where c = 3 × 10⁸ m/s

λ = 3 × 10⁸ / 2.4 × 10⁹ ≈ 0.125 m

ii) Received Power:

Pr = 15 × 3.98 × 3.98 × (0.125 / (4 × π × 3000))²

Pr ≈ 15 × 15.84 × (0.125 / 37699)² ≈ 237.6 × (3.32 × 10⁻⁶)² ≈ 2.6 × 10⁻¹¹ W

iii) In dBm:

Pr(dBm) = 10 × log10(2.6 × 10⁻⁸ mW) ≈ -75.8 dBm

d) Discussion

The received power of –76 dBm is low but can still be usable with sensitive receivers. For practical cellular systems, additional techniques like amplifiers, coding, or diversity may be needed for reliable communication.

a) Why High FSPL?

Satellite links have very long distances (tens of thousands of kilometers) leading to high free-space path loss compared to terrestrial links.

b) Design Methods to Compensate

• Use of high-gain directional antennas.
• Increase transmitter power or use signal amplification/repeaters.

c) Calculations

Given:

• Distance: d = 36,000 km = 3.6 × 10⁷ m
• Frequency: f = 4 GHz → λ = c / f = 0.075 m
• Transmitter power: Pt = 250 W
• Antenna gains: Gt = 40 dB → linear = 10⁴, Gr = 45 dB → linear = 3.16 × 10⁴

i) Free space path loss:

FSPL = (4 × π × 3.6 × 10⁷ / 0.075)² ≈ (6.03 × 10⁹)² ≈ 3.64 × 10¹⁹

ii) Net path loss including antennas:

Net path loss = FSPL / (Gt × Gr) = 3.64 × 10¹⁹ / (10⁴ × 3.16 × 10⁴) ≈ 1.15 × 10¹¹

iii) Received Power:

Pr = Pt / Net path loss = 250 / 1.15 × 10¹¹ ≈ 2.17 × 10⁻⁹ W ≈ 2.17 nW

d) Importance of High-Gain Antennas

High-gain antennas focus the transmitted energy toward the receiver, reducing losses and improving link reliability, especially over extremely long distances.

a) Reference Distance (d₀)

Reference distance d₀ is a fixed distance where received signal power is measured or calculated to serve as a baseline for path loss modeling. It helps to simplify predictions at other distances.

b) Why different values?

Indoor systems have small d₀ (1–10 m) because distances are short. Outdoor cellular systems use larger d₀ (100 m – 1 km) to account for longer propagation and environmental effects.

c) Calculations

Given:

• P(d₀ = 100 m) = –35 dBm
• Free-space path loss, n = 2

i) At 500 m:

P(500) = –35 – 10 × 2 × log10(500/100) = –35 – 20 × log10(5) ≈ –35 – 20 × 0.699 ≈ –35 – 13.98 ≈ –48.98 ≈ –49 dBm

ii) At 1000 m:

P(1000) = –35 – 20 × log10(1000/100) = –35 – 20 × 1 = –55 dBm

d) Implication

As distance increases, signal power drops. Cell planning must ensure base stations are spaced to maintain acceptable received power levels, especially in urban areas.

a) Path Loss Exponent (n)

The path loss exponent n describes how quickly the signal power decays with distance. It is used in models: PL(d) = PL(d₀) + 10 × n × log10(d/d₀).

b) Typical values

• Free space: n = 2
• Urban: n = 3–5
• Indoor: n = 1.6–3

c) Calculations (Urban Example)

Given:

• P(d₁=200 m) = –60 dBm
• n = 4

i) At 400 m:

P(400) = –60 – 10 × 4 × log10(400/200) = –60 – 40 × log10(2) ≈ –60 – 40 × 0.301 ≈ –60 – 12.04 ≈ –72 dBm

ii) At 800 m:

P(800) = –60 – 40 × log10(800/200) = –60 – 40 × log10(4) = –60 – 40 × 0.602 ≈ –60 – 24.08 ≈ –84 dBm

d) Implication

Signal decays rapidly in urban areas. More base stations or smaller cells are needed for reliable coverage.

a) Definitions

• Thermal noise power: Pn = k × T × B, where k = Boltzmann constant, T = temperature in Kelvin, B = bandwidth in Hz.
• Noise figure (NF): The degradation of SNR caused by receiver components, NF(dB) = 10 × log10(SNR_in/SNR_out).

b) Effect of Noise Figure

Higher NF reduces receiver sensitivity, making weak signals harder to detect.

c) Calculations

Given:

• B = 200 kHz = 2 × 10⁵ Hz
• NF = 6 dB → linear = 10^(6/10) ≈ 3.98
• T = 290 K
• k = 1.38 × 10⁻²³ J/K

i) Total noise power:

Pn = k × T × B × NF = 1.38 × 10⁻²³ × 290 × 2 × 10⁵ × 3.98 ≈ 3.18 × 10⁻¹⁵ W

ii) In dBm:

Pn(dBm) = 10 × log10(3.18 × 10⁻¹² / 0.001) ≈ 10 × log10(3.18 × 10⁻¹²) + 30 ≈ –114.97 dBm ≈ –115 dBm

d) Practical Impact

Increasing bandwidth increases noise power, potentially degrading system performance unless SNR is maintained via higher power or coding.

a) Definition

SNR is the ratio of signal power (Ps) to noise power (Pn) at the receiver:

b) Importance

SNR indicates the quality of the link: higher SNR means better data reliability and lower bit error rate (BER).

c) Calculations

Given:

• Received signal: Ps = –85 dBm
• Noise power: Pn = –105 dBm

i) SNR in dB:

SNR = –85 – (–105) = 20 dB

ii) Check minimum requirement (12 dB): 20 dB > 12 dB → acceptable.

d) Interpretation

SNR of 20 dB ensures good voice quality and low error rate, making communication reliable.

a) Definitions

• System gain: Difference between transmitted power and minimum required received power: Gsys = Pt – Pr(min)
• Fade margin: Extra power to tolerate fading and shadowing: Fade Margin = Pr(actual) – Pr(min)

b) Importance

Fade margin ensures link remains reliable even under varying channel conditions.

c) Calculations

Given: Pt = 33 dBm, Pr(min) = –97 dBm, measured Pr = –80 dBm

i) System gain:

Gsys = 33 – (–97) = 130 dB

ii) Fade margin:

Fade Margin = –80 – (–97) = 17 dB

d) Practical reasoning

A 17 dB fade margin is sufficient to tolerate multipath fading and signal fluctuations, improving reliability in cellular systems.

a) Definitions

• Frequency reuse: Reuse of the same frequency in different cells separated by distance.
• Cluster size (N): Number of cells in a reuse pattern.
• Co-channel reuse ratio (Q): Distance between co-channel cells relative to cell radius.

b) Calculations

Given: cell radius R = 1.5 km, i = 2, j = 1

Distance between co-channel cells: D = R × √(3N) = 1.5 × √(21) ≈ 6.87 km

Frequency reuse ratio: Q = D / R ≈ 4.58

c) Implication

Higher cluster size reduces interference but lowers capacity. Proper balance is needed for efficient cellular design.

a) Definition

SIR is the ratio of desired signal power to co-channel interference power, unlike SNR which is signal-to-noise only.

b) Increasing power alone

Increasing transmit power raises interference to nearby cells as well, so it does not solve co-channel interference.

c) Calculations

Given: cell radius R = 1 km, cluster size N = 7, interference from 6 neighbors

Assuming n = 4 (urban environment) and 6 interferers (i₀ = 6), D ≈ R × √(3N) = 1 × √21 ≈ 4.583 km

SIR = (4.583 / 1)^4 / 6 ≈ 441 / 6 ≈ 73.5

SIR(dB) = 10 × log10(73.5) ≈ 18.66 dB

d) Evaluation

Minimum requirement is 18 dB → system just meets requirement. To improve: use sectoring, smaller cluster size adjustment, or directional antennas.

a) Definition

Link budget calculates received power accounting for gains and losses in the system. It ensures whether a wireless link is feasible.

b) Calculations

Given:

• Pt = 23 dBm
• Transmitter cable loss = 2 dB
• Transmitter antenna gain = 12 dBi
• Receiver antenna gain = 15 dBi
• Receiver cable loss = 2 dB
• Distance = 4 km, frequency = 2.4 GHz

i) Free Space Path Loss:

FSPL = 20 × log10(4) + 20 × log10(2400) + 32.44 ≈ 12.04 + 67.6 + 32.44 ≈ 112.08 dB

ii) Received Signal Level:

RSL = 23 – 2 + 12 + 15 – 2 – 112.08 ≈ –66.08 dBm

c) Feasibility

Receiver sensitivity = –85 dBm → RSL = –66 dBm > –85 dBm → link feasible and robust.

a) Definitions

• Receiver sensitivity: minimum power for acceptable performance.
• Link margin: difference between received power and receiver sensitivity.

b) Calculations

Given: Pt = 20 dBm, Gt = 10 dBi, Gr = 14 dBi, Tx/Rx cable loss = 3 dB each, distance = 6 km, f = 5 GHz

i) FSPL = 20 log10(d) + 20 log10(f) + 32.44 = 20 log10(6000) + 20 log10(5000) + 32.44 ≈ 135.56 dB

ii) RSL = Pt + Gt + Gr – TxLoss – RxLoss – FSPL = 20 + 10 + 14 – 3 – 3 – 135.56 ≈ –97.56 dBm

iii) Link margin = RSL – Receiver Sensitivity = –97.56 – (–90) ≈ –7.56 dB → insufficient

c) Interpretation

Negative link margin indicates unreliable link. Designers may increase antenna gain, reduce distance, or increase transmit power.

a) Explanation

Uplink (mobile to base station) and downlink (base station to mobile) often differ due to different transmit powers, antenna gains, and environmental factors.

b) Calculations

Given:

• Distance = 5 km, f = 2.4 GHz
• Access point: Pt = 20 dBm, Gt = 10 dBi
• Client: Pt = 15 dBm, Gt = 14 dBi
• Cable loss = 2 dB each side
• FSPL = 114 dB

i) Downlink RSL (AP → client):

RSL = Pt + Gt(AP) + Gr(Client) – TxLoss – RxLoss – FSPL = 20 + 10 + 14 – 2 – 2 – 114 ≈ –74 dBm

ii) Uplink RSL (Client → AP):

RSL = 15 + 14 + 10 – 2 – 2 – 114 ≈ –77 dBm

c) Interpretation

Client sensitivity = –82 dBm, AP sensitivity = –89 dBm → both links feasible. Uplink is closer to sensitivity limit, so it's more critical. Designers may increase client power or antenna gain to ensure robust uplink.