The document discusses the addition of noise from multiple sources and noise in reactive circuits. It provides the following key points: - Total noise from components in series is the sum of the individual noise voltages. For components in parallel, the equivalent resistance is found first before calculating total noise. - For cascaded amplifiers, the noise of later stages is transformed to an equivalent input noise resistance of earlier stages by dividing by the gain. The total equivalent input noise resistance is the sum of these transformed resistances. - A tuned circuit passes the noise generated by its internal resistance unchanged at resonance. The noise voltage across the capacitor is equal to the quality factor Q times the input noise voltage. - Noise figure is

1. Basic Communication
Engineering

2. NOISE CALCULATIONS

3. Addition of Noise Due to Several Sources
• Let there are two sources of thermal noise R1 and R2 in series. Noise
generated by them-
𝑉𝑛1 = 4𝑘𝑇𝛿𝑓𝑅1
𝑉𝑛2 = 4𝑘𝑇𝛿𝑓𝑅2
• Total equivalent resistance = Rtot = R1 + R2
• Total noise voltage-
𝑉𝑛,𝑡𝑜𝑡 = 4𝑘𝑇𝛿𝑓𝑅𝑡𝑜𝑡
= 4𝑘𝑇𝛿𝑓 𝑅1 + 𝑅2
= 4𝑘𝑇𝛿𝑓𝑅1 + 4𝑘𝑇𝛿𝑓𝑅2
= 𝑉𝑛1
2
+ 𝑉𝑛2
2

4. Addition of Noise Due to Several Sources
• For noise sources in parallel- find equivalent resistance and then total
noise voltage.
• What will happen to total noise voltage and power?

5. Addition of Noise Due to Several Sources

6. Addition of noise due to several amplifier in cascade
• rms noise voltage at the output due to R3 –
𝑉𝑛3 = 4𝑘𝑇𝛿𝑓𝑅3
• Transferring R3 to the input of stage 2-
𝑉𝑛3
′
=
𝑉𝑛3
𝐴2
=
4𝑘𝑇𝛿𝑓𝑅3
𝐴2
=
4𝑘𝑇𝛿𝑓𝑅3
𝐴2
2 = 4𝑘𝑇𝛿𝑓𝑅3
′

7. Addition of noise due to several amplifier in cascade
𝑅3
′
=
𝑅3
𝐴2
2
• R’3 is the resistance which if placed at the input of the second stage
would produce same noise voltage at the output as does R3.
• Equivalent noise resistance at the input of second stage-
𝑅𝑒𝑞
′
= 𝑅2 + 𝑅3
′
= 𝑅2 +
𝑅3
𝐴2
2
• A resistor R’2 may be placed at the input of the first stage to replace
R’eq.

8. Addition of noise due to several amplifier in cascade
𝑅2
′
=
𝑅𝑒𝑞
′
𝐴1
2 =
𝑅2 +
𝑅3
𝐴2
2
𝐴1
2 =
𝑅2
𝐴1
2 +
𝑅3
𝐴1
2
𝐴2
2
• Equivalent noise resistance of the whole cascaded amplifier-
𝑅𝑒𝑞 = 𝑅1 + 𝑅2
′
= 𝑅1 +
𝑅2
𝐴1
2 +
𝑅3
𝐴1
2
𝐴2
2
Noise resistance located at the input of the first stage is the greatest
contributor to the total noise

9. Addition of noise due to several amplifier in cascade

10. Noise in reactive Circuit
• Noise generated by resistance is not affected by a subsequent
noiseless tuned circuit at resonant frequency.
• Bandwidth of noise is limited by tuned circuit at either side of
resonant.

11. Noise in reactive Circuit
• Noise current-
𝑖𝑛 =
𝑣𝑛
𝑍
𝑤ℎ𝑒𝑟𝑒, 𝑧 = 𝑅𝑠 + 𝑗 𝑋𝐿 − 𝑋𝐶

12. Noise in reactive Circuit
• At resonance-
𝑖𝑛 =
𝑣𝑛
𝑅𝑠
• Voltage appearing across the capacitor-
𝑣 = 𝑖𝑛𝑋𝐶 =
𝑣𝑛𝑋𝐶
𝑅𝑠
=
𝑉
𝑛𝑄𝑅𝑠
𝑅𝑠
= 𝑄𝑣𝑛
Where, Q = Quality or magnification factor

13. Noise in reactive Circuit
• Now-
• 𝑣2 = 𝑄2𝑣𝑛
2 = 𝑄2 4𝑘𝑇𝛿𝑓𝑅𝑠
2
= 4𝑘𝑇𝛿𝑓(𝑄2𝑅𝑠) = 4𝑘𝑇𝛿𝑓𝑅𝑝
• 𝑣 = 4𝑘𝑇𝛿𝑓𝑅𝑝
• v = noise voltage across a tuned circuit due to its internal resistance
• Rp = equivalent impedance of the tuned circuit at resonance.

14. NOISE FIGURE

15. Signal-to-Noise Ratio
• Signal-to-Noise Ratio (SNR) is the ratio of signal power to noise power
at the same point.
𝑆𝑁𝑅 =
𝑃𝑠
𝑃𝑛
=
𝑉
𝑠
2
𝑅
𝑉
𝑛
2
𝑅
=
𝑉
𝑠
𝑉
𝑛
2

16. Noise Figure
• Noise figure or noise factor is defined as the ratio of the SNR supplied
to the input terminals of a receiver of amplifier to the SNR supplied to
the output or load resistor.
𝐹 =
𝑖𝑛𝑝𝑢𝑡 𝑆𝑁𝑅
𝑂𝑢𝑡𝑝𝑢𝑡 𝑆𝑁𝑅
• For ideal receiver F = 1
• Alternate definition of noise figure-
𝐹 =
𝑖𝑛𝑝𝑢𝑡 𝑆𝑁𝑅
𝑂𝑢𝑡𝑝𝑢𝑡 𝑆𝑁𝑅
=
𝑂𝑢𝑡𝑝𝑢𝑡 𝑆𝑁𝑅 𝑜𝑓 𝑖𝑑𝑒𝑎𝑙 𝑟𝑒𝑐𝑒𝑖𝑣𝑒𝑟
𝑂𝑢𝑡𝑝𝑢𝑡 𝑆𝑁𝑅 𝑜𝑓 𝑝𝑟𝑎𝑐𝑡𝑖𝑐𝑎𝑙 𝑟𝑒𝑐𝑒𝑖𝑣𝑒𝑟
• Both are linear, working at same temperature, same BW, fed from
same source.

17. Noise Figure
• Ni = noise power at he amplifier input
• Nai = amplifier noise referred to input
• G = gain
• Na = internal noise power of amplifier
• 𝑁𝑎𝑖 =
𝑁𝑎
𝐺
,
• 𝑁0 = 𝐺𝑁𝑖 + 𝑁𝑎 = 𝐺 𝑁𝑖 +
𝑁𝑎
𝐺
= 𝐺 𝑁𝑖 + 𝑁𝑎𝑖
• 𝐹 =
𝑆𝑁𝑅𝑖𝑛
𝑆𝑁𝑅𝑜𝑢𝑡
=
𝑆𝑖
𝑁𝑖
𝐺 𝑆𝑖
𝐺(𝑁𝑖+𝑁𝑎𝑖)
=
𝑁𝑖+𝑁𝑎𝑖
𝑁𝑖
= 1 +
𝑁𝑎𝑖
𝑁𝑖

18. Noise Temperature