How Friis Formula Explains First Stage Noise Figure Dominance

This article provides a comprehensive overview of how Friis's formula mathematically proves that the first amplification stage dictates the overall noise performance and sensitivity of cascaded receiver electronics. By examining the linear relationship between individual component noise factors and their respective power gains, we will explore why a Low-Noise Amplifier (LNA) at the front end of a receiver effectively masks the noise generated by all subsequent components in the signal chain.

Understanding Friis's Formula for Noise

To understand how noise behaves in a receiver, engineers look at the chain of components—such as filters, amplifiers, and mixers—as a cascaded system. Each component introduces its own internal noise while amplifying or attenuating the incoming signal. Harald T. Friis developed a formula to calculate the total noise factor of such a cascaded system.

The formula is expressed using linear values (Noise Factor, \(F\), and Linear Gain, \(G\)), rather than decibels (Noise Figure, \(NF\)). The equation is:

\(F_{total} = F_1 + \frac{F_2 - 1}{G_1} + \frac{F_3 - 1}{G_1 G_2} + \dots + \frac{F_n - 1}{G_1 G_2 \dots G_{n-1}}\)

Where:

  • \(F_{total}\) is the total noise factor of the entire receiver chain.
  • \(F_1, F_2, F_n\) are the noise factors of the first, second, and \(n\)-th stages.
  • \(G_1, G_2, G_{n-1}\) are the linear power gains of the first, second, and \((n-1)\)-th stages.

The Mathematical Proof of First-Stage Dominance

Friis's formula clearly separates the noise contribution of the first stage from the rest of the system. Notice that the first term in the equation, \(F_1\), stands completely alone. It is not divided by any gain value. This means that 100% of the noise generated by the first stage is added directly to the total noise of the system.

Now, look at the second term in the equation: \(\frac{F_2 - 1}{G_1}\).

The noise contribution of the second stage (\(F_2 - 1\)) is divided by the linear gain of the first stage (\(G_1\)). If the first stage is a high-gain amplifier, \(G_1\) is a large number. Dividing the second stage's noise by this large number makes the resulting fraction very small.

As you move further down the chain to the third stage, its noise (\(F_3 - 1\)) is divided by the product of the gains of the first two stages (\(G_1 G_2\)). This makes the third stage's noise contribution practically negligible. Therefore, as long as \(G_1\) is sufficiently large, the total noise factor of the receiver is approximately equal to the noise factor of the first stage (\(F_{total} \approx F_1\)).

A Practical Calculation Example

To see this dominance in action, consider a simple two-stage receiver front end.

  • Stage 1 (Low-Noise Amplifier): Has a low Noise Factor (\(F_1\)) of 2, and a high linear Gain (\(G_1\)) of 100.
  • Stage 2 (Mixer): Has a poor Noise Factor (\(F_2\)) of 11, and a Gain (\(G_2\)) of 1.

Using Friis's formula to find the total noise factor:

\(F_{total} = 2 + \frac{11 - 1}{100}\) \(F_{total} = 2 + \frac{10}{100}\) \(F_{total} = 2 + 0.1 = 2.1\)

Even though the second stage is very noisy (\(F_2 = 11\)), it only increases the total system noise factor from 2 to 2.1. The high gain of the first stage (\(G_1 = 100\)) effectively suppressed the noise impact of the second stage.

If you were to reverse these components, placing the noisy mixer first and the amplifier second, the math changes drastically:

\(F_{total} = 11 + \frac{2 - 1}{1}\) \(F_{total} = 11 + 1 = 12\)

By failing to put the high-gain, low-noise stage first, the total system noise factor jumps from 2.1 to 12, ruining the receiver's performance.

Impact on Receiver Sensitivity

Receiver sensitivity is the ability of a system to detect weak signals. Sensitivity is fundamentally limited by the Signal-to-Noise Ratio (SNR). A receiver can only process a signal if that signal is sufficiently stronger than the background noise floor.

When a weak signal enters the first stage of a receiver, it is very close to the noise floor.

  1. The LNA's Role: By making the first stage a Low-Noise Amplifier (LNA), you add the absolute minimum amount of noise (\(F_1\)) while simultaneously boosting the signal power significantly (\(G_1\)).
  2. Protecting the SNR: When this boosted signal reaches the noisy second stage, the signal is now so large that the innate noise of the second stage (\(F_2\)) is tiny by comparison. The SNR established by the first stage is preserved.

The Danger of Front-End Losses

Friis's formula also demonstrates why placing passive components (like long cables, switches, or filters) before the first amplifier degrades sensitivity. Passive components do not amplify; they attenuate.

For a passive component, the gain (\(G_1\)) is a fraction less than 1. When you divide the second stage's noise (\(F_2 - 1\)) by a number less than 1, the resulting number gets larger, not smaller. Therefore, any loss before the first amplifier not only weakens the incoming signal, it also magnifies the noise impact of every component that follows it. To maximize sensitivity, the first active stage must have both the lowest possible noise figure and high gain, placed as close to the antenna as possible.