10.1 Signal Fundamentals, Decibel Calculations & Noise Analysis

10.1 Signal Fundamentals, Decibel Calculations & Noise Analysis

Board Examination Focus: Signal-to-noise ratio (SNR), logarithmic decibel conversions (dBm, dBW, dBmV), thermal noise spectral density ($P_n = kTB$), and multi-stage Noise Figure ($NF$) calculations using Friis' formula are heavily emphasized core topics in the PRC Electronics Engineering Licensure Examination (EST domain).

Electronic communications systems transmit information by converting electrical, optical, or acoustic signals into electromagnetic waves. A comprehensive understanding of signal power, decibel scale conversions, internal and external noise mechanisms, and receiver sensitivity metrics is fundamental to designing and analyzing analog and digital communication links.

The Decibel System in Communications

The decibel (dB) is a logarithmic unit used to express the ratio between two power or voltage levels. Because electronic communication systems involve power levels ranging from megawatts ($10^6\text{ W}$) at broadcast transmitters down to picowatts ($10^{-12}\text{ W}$) at receiver antennas, the logarithmic decibel scale compresses huge numerical ranges into manageable numbers and converts complex multiplication of stage gains into simple addition.

Power and Voltage Decibel Ratios

For two power levels, $P_1$ (input) and $P_2$ (output), the power ratio in decibels is defined as:

$$\text{dB} = 10 \log_{10}\left(\frac{P_2}{P_1}\right)$$

When dealing with voltage or current ratios across identical impedances ($R_1 = R_2 = R$), power is proportional to the square of voltage ($P = V^2 / R$). Applying logarithmic rules yields the voltage decibel ratio:

$$\text{dB} = 10 \log_{10}\left(\frac{V_2^2 / R}{V_1^2 / R}\right) = 20 \log_{10}\left(\frac{V_2}{V_1}\right)$$

Absolute Decibel Units: dBm, dBW, and dBmV

Decibels can express absolute power levels by establishing a standardized reference level in the denominator:

• dBm: Absolute power referenced to $1\text{ milliwatt } (1\text{ mW} = 10^{-3}\text{ W})$: $$\text{dBm} = 10 \log_{10}\left(\frac{P}{1\text{ mW}}\right) \implies P(\text{mW}) = 10^{\left(\frac{\text{dBm}}{10}\right)}$$
• dBW: Absolute power referenced to $1\text{ watt } (1\text{ W})$: $$\text{dBW} = 10 \log_{10}\left(\frac{P}{1\text{ W}}\right) = \text{dBm} - 30$$
• dBmV: Absolute voltage referenced to $1\text{ millivolt } (1\text{ mV})$ across a standard $75\ \Omega$ coaxial impedance (common in CATV systems): $$\text{dBmV} = 20 \log_{10}\left(\frac{V}{1\text{ mV}}\right)$$

Worked Example: Decibel Power Conversions

Problem: A mobile radio transmitter outputs $25\text{ Watts}$. Convert this power level to (a) $\text{dBm}$ and (b) $\text{dBW}$. If an inline RF attenuator reduces the signal by $6\text{ dB}$, what is the final output power in $\text{mW}$?

Solution:

1. Convert $25\text{ W}$ to milliwatts: $P = 25,000\text{ mW}$.
2. Calculate $\text{dBm}$: $$\text{dBm} = 10 \log_{10}(25,000) = 10 \times 4.3979 = 43.98\text{ dBm}$$
3. Calculate $\text{dBW}$: $$\text{dBW} = 43.98 - 30 = 13.98\text{ dBW}$$
4. Subtract the $6\text{ dB}$ attenuation: $$P_{\text{final}, \text{dBm}} = 43.98\text{ dBm} - 6\text{ dB} = 37.98\text{ dBm}$$
5. Convert back to milliwatts: $$P_{\text{final}} = 10^{(37.98 / 10)} = 10^{3.798} \approx 6,280\text{ mW} = 6.28\text{ W}$$

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Communication Noise Mechanics

Noise is defined as any unwanted electrical energy that degrades or interferes with the desired signal. Noise sources are categorized into external noise (originating outside the communication circuit) and internal noise (generated within the components of the circuit itself).

Classification of Noise Sources

1. External Noise:

   • Atmospheric Noise (Static): Caused by naturally occurring electrical discharges (lightning) in the atmosphere. It dominates at frequencies below $30\text{ MHz}$.
   • Extraterrestrial (Solar and Cosmic) Noise: Originates from the Sun and interstellar space. Solar noise fluctuates with the 11-year sunspot cycle, while cosmic noise radiates uniformly from the Milky Way galaxy ($8\text{ MHz}$ to $1.5\text{ GHz}$).
   • Man-Made (Industrial) Noise: Produced by spark discharges in automotive ignition systems, fluorescent lighting, high-voltage transmission lines, and switching power supplies ($1\text{ MHz}$ to $500\text{ MHz}$).

2. Internal Noise:

   • Thermal Agitation (Johnson-Nyquist) Noise: Random motion of free electrons inside conducting media due to heat.
   • Shot Noise: Caused by discrete arrival fluctuations of charge carriers (electrons and holes) across PN junctions in semiconductor diodes and transistors ($I_n = \sqrt{2 q I_{\text{dc}} B}$).
   • Transit-Time Noise: Occurs at high RF frequencies when the time taken for charge carriers to travel across a transistor junction approaches the period of the signal.
   • Flicker (1/f) Noise: Low-frequency semiconductor noise inversely proportional to frequency, becoming negligible above $1\text{ kHz}$.

Thermal (Johnson-Nyquist) Noise Equations

In 1928, J.B. Johnson and H. Nyquist demonstrated that thermal noise power ($P_n$) generated by any resistive conductor is directly proportional to absolute temperature ($T$) and signal bandwidth ($B$):

$$P_n = k T B$$

Where:

• $k = 1.3806 \times 10^{-23}\text{ Joules/Kelvin (J/K)}$ (Boltzmann's constant)
• $T = \text{Absolute temperature in Kelvin } (K = {^\circ}\text{C} + 273.15)$
• $B = \text{Equivalent noise bandwidth in Hertz } (\text{Hz})$

At standard reference room temperature ($T_0 = 290\text{ K}$, or $16.85{^\circ}\text{C}$):

$$k T_0 = (1.3806 \times 10^{-23}\text{ J/K}) \times 290\text{ K} = 4.0037 \times 10^{-21}\text{ Watts/Hz}$$

Expressed in decibels relative to $1\text{ mW}$ per Hertz of bandwidth ($\text{dBm/Hz}$):

$$\text{Noise Density Floor } (N_0) = 10 \log_{10}(4.0037 \times 10^{-18}\text{ mW/Hz}) \approx -174\text{ dBm/Hz}$$

The RMS open-circuit thermal noise voltage ($V_n$) developed across a resistor $R$ is given by:

$$V_n = \sqrt{4 k T B R}$$

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Signal-to-Noise Ratio, Noise Factor & Noise Figure

To evaluate receiver performance, engineers measure how much a circuit degrades the incoming Signal-to-Noise Ratio ($\text{SNR}$).

Signal-to-Noise Ratio (SNR)

$$\text{SNR (linear ratio)} = \frac{P_{\text{signal}}}{P_{\text{noise}}}$$ $$\text{SNR (dB)} = 10 \log_{10}\left(\frac{P_{\text{signal}}}{P_{\text{noise}}}\right) = 20 \log_{10}\left(\frac{V_{\text{signal}}}{V_{\text{noise}}}\right)$$

Noise Factor ($F$) and Noise Figure ($NF$)

The Noise Factor ($F$) is the ratio of input $\text{SNR}$ to output $\text{SNR}$ of an amplifier or system:

$$F = \frac{\text{SNR}_{\text{in (linear)}}}{\text{SNR}_{\text{out (linear)}}} = \frac{(P_{si} / P_{ni})}{(P_{so} / P_{no})}$$

Because any real amplifier adds internal thermal noise, $\text{SNR}_{\text{out}}$ is always lower than $\text{SNR}_{\text{in}}$, making $F \ge 1.0$. The Noise Figure ($NF$) is the decibel expression of Noise Factor:

$$NF = 10 \log_{10}(F) = \text{SNR}_{\text{in (dB)}} - \text{SNR}_{\text{out (dB)}}$$

Equivalent Noise Temperature ($T_e$)

An alternative measure of amplifier noisiness is equivalent noise temperature ($T_e$), representing a hypothetical resistive source at temperature $T_e$ that produces the same noise power as the amplifier internal sources:

$$T_e = T_0 (F - 1) = 290 (F - 1)$$

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Multi-Stage Systems & Friis' Noise Formula

When multiple amplifier and filter stages are connected in cascade, the overall system noise factor ($F_{\text{total}}$) is calculated using Friis' Formula:

$$F_{\text{total}} = F_1 + \frac{F_2 - 1}{A_{p1}} + \frac{F_3 - 1}{A_{p1} A_{p2}} + \frac{F_4 - 1}{A_{p1} A_{p2} A_{p3}} + \dots$$

Where $F_n$ represents the linear noise factor of stage $n$, and $A_{pn}$ represents the linear power gain of stage $n$.

Crucial Concept: All terms in Friis' formula MUST be converted from decibels to linear ratios ($F = 10^{NF/10}$ and $A_p = 10^{G_{\text{dB}}/10}$) before plugging into the formula.

Step-by-Step Worked Example: Multi-Stage Receiver Analysis

Problem: A satellite receiver front-end consists of three stages:

1. Low-Noise Amplifier (LNA): Gain $G_1 = 20\text{ dB}$, Noise Figure $NF_1 = 1.5\text{ dB}$
2. Bandpass RF Filter: Insertion Loss $L_2 = 3\text{ dB}$ ($G_2 = -3\text{ dB}$), Noise Figure $NF_2 = 3.0\text{ dB}$
3. Mixer-IF Stage: Gain $G_3 = 30\text{ dB}$, Noise Figure $NF_3 = 10.0\text{ dB}$

Calculate: (a) Total linear noise factor ($F_{\text{total}}$), (b) Total system noise figure ($NF_{\text{total}}$), and (c) Equivalent system noise temperature ($T_e$).

Solution:

1. Convert stage parameters to linear values:

   • Stage 1 (LNA): $A_{p1} = 10^{(20/10)} = 100$, $F_1 = 10^{(1.5/10)} = 1.4125$
   • Stage 2 (Filter): $A_{p2} = 10^{(-3/10)} = 0.5012$, $F_2 = 10^{(3/10)} = 1.9953$
   • Stage 3 (Mixer): $A_{p3} = 10^{(30/10)} = 1000$, $F_3 = 10^{(10/10)} = 10.0$

2. Apply Friis' Formula:

   $$F_{\text{total}} = F_1 + \frac{F_2 - 1}{A_{p1}} + \frac{F_3 - 1}{A_{p1} A_{p2}}$$ $$F_{\text{total}} = 1.4125 + \frac{1.9953 - 1}{100} + \frac{10.0 - 1}{100 \times 0.5012}$$ $$F_{\text{total}} = 1.4125 + \frac{0.9953}{100} + \frac{9.0}{50.12}$$ $$F_{\text{total}} = 1.4125 + 0.00995 + 0.17957 = 1.6020$$

3. Convert total linear noise factor to total Noise Figure:

   $$NF_{\text{total}} = 10 \log_{10}(1.6020) = 2.047\text{ dB}$$

4. Calculate equivalent noise temperature ($T_e$):

   $$T_e = 290 \times (F_{\text{total}} - 1) = 290 \times (1.6020 - 1) = 290 \times 0.6020 = 174.58\text{ K}$$

Notice how the LNA's high gain ($100$) strongly suppresses the noise contribution of subsequent stages, confirming why placing a high-gain, low-noise preamplifier right at the antenna feedhorn is critical in communication systems engineering.