The Friis formula, named after Harald T. Friis, refers to two distinct formulas used in telecommunications engineering for calculating signal-to-noise ratios, specifically in multistage amplifier systems. These formulas, while both related to signal-to-noise performance, address different aspects of system analysis.
Understanding the Friis Formulas
The two formulas, though both linked to Friis's work, serve different purposes:
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Friis Transmission Formula (Power Received): This formula calculates the power received by an antenna given the transmitted power, antenna gains, and the distance between them and the signal wavelength. This is not a signal-to-noise formula.
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Friis Noise Formula (Noise Factor and Noise Figure): This formula determines the total noise factor (or noise figure) of a series of cascaded (connected) components, like amplifiers. This is a formula for determining the signal-to-noise ratio of multistage amplifiers, which was in the reference.
Focus on the Signal-to-Noise Formula:
The specific Friis formula that relates to signal-to-noise ratio is the Friis Noise Formula. This formula helps determine the degradation of the signal-to-noise ratio as a signal passes through a series of amplifiers and other components. This formula is key in the design of telecommunication and radio systems, as they often consist of multiple stages of amplifiers.
The Friis Noise Formula Equation
The formula for the overall noise factor (F) of n cascaded stages is:
F = F1 + (F2 - 1)/G1 + (F3 - 1)/(G1 * G2) + ... + (Fn - 1)/(G1 * G2 *... * G(n-1))Where:
- F is the overall noise factor of the cascaded system.
- Fi is the noise factor of the i-th stage.
- Gi is the gain of the i-th stage.
The noise figure is the decibel (dB) form of the noise factor. It is derived by:
Noise Figure (dB) = 10 * log10(F)Using the Friis Noise Formula
This formula indicates:
- First Stage Dominance: The noise factor of the first stage amplifier has the greatest impact on the overall noise figure because the following gains reduce the effect of subsequent stage noise factors.
- Low Noise Amplifiers (LNAs): It is crucial to use low-noise amplifiers in the first stage of a receiver to improve signal-to-noise performance.
Practical Implications
Here's a breakdown of practical uses and insights:
- Receiver Design: In radio receivers, the first amplifier should have a very low noise figure to capture weak signals clearly.
- System Optimization: By knowing the noise characteristics of individual components, the system design can be optimized by adjusting gains and component placement to minimize overall noise.
- Communication System Analysis: Enables engineers to predict how noise will impact signal quality as it passes through a communication system.
Examples
For a two-stage amplifier:
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Example 1: First stage gain G1=10 (10 dB) and noise factor F1=1.1 (0.41dB), second stage gain G2 =10 (10dB) and noise factor F2=2 (3.01 dB).
- Total noise factor: F = 1.1 + (2-1)/10 = 1.1 + 0.1 = 1.2
- Total noise figure: 10*log(1.2) = 0.79 dB
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Example 2: If you reverse the order of the amplifier. G1=10 and noise factor F1=2. Second stage G2 =10 and noise factor F2=1.1.
- Total noise factor: F = 2 + (1.1 -1)/10 = 2 + 0.01 = 2.01
- Total noise figure: 10*log(2.01) = 3.03 dB
By changing the first stage you reduced the noise of the amplifier system, this is why low noise amplifiers are used in front ends.
Conclusion
The Friis Noise Formula is a critical tool in telecommunications engineering, enabling the calculation of noise performance in multistage amplifier systems, which directly impacts signal-to-noise ratio. It is crucial for designing high-quality, reliable communication systems.
