A mixer stage has a noise figure of 20 dB and is preceded by an amplifier that has a noise figure of 9 dB and an available power gain of 15 dB. The overall noise figure is :
10.44 dB
Friis's formula must be used with ratios, never with decibels — that is the whole discipline of this calculation.
\(F_{total}=F_{1}+\dfrac{F_{2}-1}{G_{1}}\)
Step 1 — convert everything out of dB.
| Quantity | dB | Ratio |
|---|---|---|
| F1 (amplifier) | 9 | \(10^{0.9}=7.943\) |
| G1 (amplifier) | 15 | \(10^{1.5}=31.62\) |
| F2 (mixer) | 20 | \(10^{2}=100\) |
Step 2 — apply Friis.
\(F_{total}=7.943+\dfrac{100-1}{31.62}=7.943+3.131=11.074\)
Step 3 — convert back to decibels.
\(F_{dB}=10\log_{10}(11.074)=10\times1.0443=10.44\ \text{dB}\)
which is option 3.
Notice what the arithmetic reveals. The mixer is a dreadful stage — 20 dB noise figure, a hundredfold degradation on its own — yet the system ends at only 10.44 dB, barely 1.4 dB worse than the amplifier alone. The 31.6-fold gain ahead of it divides the mixer's contribution down to 3.13, a small addition to the amplifier's 7.94. This is precisely why every sensitive receiver puts a low-noise amplifier first and tolerates a noisy mixer afterwards.
Option 4, 11.07 dB, is the deliberate trap — it is the correct ratio 11.074 mislabelled as decibels. Option 2, 11.44 dB, is what a careless dB-domain addition would give. Both are reminders that the only legitimate arithmetic is in ratios.
The general cascaded form extends the same idea down the chain:
\(F_{total}=F_{1}+\dfrac{F_{2}-1}{G_{1}}+\dfrac{F_{3}-1}{G_{1}G_{2}}+\cdots\)
Each stage's contribution is divided by the product of all the gains preceding it, so the influence of later stages decays geometrically. The corollary is unforgiving: anything lossy placed first — a long cable, a filter, a switch — has gain less than one and therefore multiplies the following stages' contributions, which is why the LNA is mounted at the antenna feed rather than at the receiver end of the cable.
Hence, the overall noise figure is 10.44 dB.
Similar Questions
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A two stage receiving system operates at 30 MHz bandwidth. Calculate noise figure (F) of the system for following values of G1 (dB) and arrange them in ascending order. Assuming F1 (dB) = 2 dB, F2 (dB) = 9 dB
A. G1 = 10 dB
B. G1 = 20 dB
C. G1 = 50 dB
D. G1 = 30 dB
E. G1 = 40 dB
Choose the correct answer from the options given below :
Read the following statements.
(a) Industrial noise is usually of impulse type.
(b) HF mixers are generally noisier than HF amplifiers.
(c) Thermal noise is independent of frequency at which it is measured.
(d) Impulse noise voltage is independent of the bandwidth.
Which of the above statements are correct ?
Assertion (A) : Noise figure is defined as the ratio of the signal to noise power at the input terminal to the signal to noise power at the output terminal of the system.
Reason (R) : The noise figure will be less than 1 for an ideal receiver, which introduces no noise of its own.
Select your answer using the codes given below :
List – I List – II a. Shot Noise i. Resistance b. Johnson Noise ii. Diode c. Current Noise iii. P-N junction d. Partition Noise iv. Triode Codes :
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