Understanding Noise Figure and Gain in Cascaded Amplifiers

When multiple amplifier stages are connected in series, forming a cascade amplifier, the overall performance, especially regarding noise, needs to be carefully evaluated. The noise figure (NF) is a crucial parameter that indicates how much the signal-to-noise ratio (SNR) degrades as a signal passes through a component or system. A lower noise figure means better performance. Gain, on the other hand, is a measure of how much the amplifier increases the power or amplitude of a signal.

In this problem, we are given a two-stage cascade amplifier where each stage has identical characteristics: a gain of 10 dB and a noise figure of 10 dB. Our goal is to determine the overall noise figure of this cascaded system.

Converting Decibels to Linear Values

Before we can use Friis's formula for cascaded systems, we must convert the given decibel (dB) values for gain and noise figure into their linear equivalents. The conversion formulas are as follows:

• Gain in linear units (\(G\)): \[G = 10^{\frac{G_{\text{dB}}}{10}}\]
• Noise Figure in linear units (\(NF\)): \[NF = 10^{\frac{NF_{\text{dB}}}{10}}\]

Let's apply these conversions to the given values for each stage:

• Gain of each stage (\(G_1 = G_2\)):

  Given \(G_{\text{dB}} = 10 \text{ dB}\)

  \[G_{\text{linear}} = 10^{\frac{10}{10}} = 10^1 = 10\] So, \(G_1 = 10\) and \(G_2 = 10\).
• Noise Figure of each stage (\(NF_1 = NF_2\)):

  Given \(NF_{\text{dB}} = 10 \text{ dB}\)

  \[NF_{\text{linear}} = 10^{\frac{10}{10}} = 10^1 = 10\] So, \(NF_1 = 10\) and \(NF_2 = 10\).

Applying Friis's Formula for Overall Noise Figure

For a two-stage cascaded system, the overall noise figure (\(NF_{\text{overall}}\)) is calculated using Friis's formula:

Where:

• \(NF_1\) is the linear noise figure of the first stage.
• \(G_1\) is the linear gain of the first stage.
• \(NF_2\) is the linear noise figure of the second stage.

Now, let's substitute the calculated linear values into Friis's formula:

• \(NF_1 = 10\)
• \(G_1 = 10\)
• \(NF_2 = 10\)

So, the calculation proceeds as follows:

Conclusion on Overall Noise Figure

The calculated overall noise figure for the two-stage cascade amplifier is 10.9. This value is in linear units, as Friis's formula operates with linear noise figures and gains. It's important to note that the noise contribution of subsequent stages is reduced by the gain of the preceding stages. In this specific case, because the gain of the first stage is quite high (10, or 10 dB), the noise contribution from the second stage is significantly attenuated.

Therefore, the overall Noise figure of a two-stage cascade amplifier is 10.9.