Is this post of the study club incomplete? The formula part?
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This post was last edited by btty038 on 2020-1-15 10:39
The RF receiving front end includes LNA, Filter, Mixer and other components. From the perspective of noise factor cascade, it is hoped that the first stage of the receiving chain is a high-gain, low-noise factor amplifier in order to obtain a lower system noise factor and improve receiving sensitivity. In addition to LNA, the receiving chain also has a key component - the image rejection filter, which is located before the Mixer and is used to filter out image noise and image signals to improve the SNR and anti-image interference capability of the entire receiving chain.
The following will describe in detail the impact of whether to introduce an image rejection filter on the system noise floor through formula derivation.
1. With image rejection filter
Figure 1 shows a simplified RF front-end receiving link, including LNA, Filter and Mixer. Assume that the image rejection filter is ideal, that is, the insertion loss is 0dB, the out-of-band rejection is infinite, and the image noise can be completely suppressed. Assume that the room temperature is T 0 =290K, the receiving link is terminated with a 50 Ohm load, GA and FA are the gain and noise factor of the LNA, and GM and FM are the gain and noise factor of the Mixer.
Figure 1. Simplified RF receiver front end with image rejection filter.
The input noise power of the receiving front end is N in =kBT 0 , and the total output noise power is
(Formula 1) ?
(Formula 2) ?
From the perspective of system cascading, the noise factor and total gain of the entire receiving chain are
(Formula 3)
Because the current room temperature is 290K, the total output noise power is
(Formula 4) ?
Equation 2 and Equation 4 are consistent. This is the calculation process of the noise floor of the entire receiving link when an image rejection filter is present.
2. No image rejection filter
Figure 2 shows a simplified RF receiving front end without an image rejection filter. Due to the working characteristics of the mixer, when the receiving chain is working, not only the noise in the desired working frequency band will be converted to the intermediate frequency, but also the noise in the image frequency band will be converted to the intermediate frequency, which will cause the noise power of the system output to double compared to when an image rejection filter is included.
Figure 2. Simplified RF receive front end: no image rejection filter
Assume that the conversion loss of the mixer in the image frequency band is, and the gain and noise factor of the LNA in the image frequency band are and respectively. Here we only consider the simplest case, and the subsequent formula derivation and analysis will be based on the following assumptions:
(Formula 5) ?
The total output noise power of the entire receiving chain is
(Formula 6) ?
Substituting equation 5 into the above equation, we get
(Formula 7) ?
Further simplifying the above formula, we can get
(Formula 8) ?
From the perspective of system cascading, the noise factor and total gain of the entire receiving chain are
(Formula 9)
It is worth mentioning that, because there is no image rejection filter, for the mixer, the noise power fed into the previous LNA is twice that when there is an image rejection filter, so the above formula needs to be adjusted. It can be equivalent to the following structural block diagram with an image rejection filter.
Figure 3. Equivalent block diagram of the structure with an image rejection filter
In the figure above, the noise factor of LNA is equivalent to 2FA, and the gain remains unchanged; the gain of LNA can also be equivalent to 2GA, and the noise factor remains unchanged. For simplicity, it is still assumed that the image rejection filter is ideal.
At this time, the total noise factor and gain of the entire receiving chain are
(Formula 10)
Then the output noise power of the entire receiving chain is
(Formula 11) ?
Equation 11 is consistent with Equation 8. If the gain of the LNA is equivalent to 2GA and the noise factor remains unchanged, the total noise factor and gain of the entire receiving chain are
(Formula 12)
Then the output noise power of the entire receiving chain is
(Formula 13) ?
Equation 13 is consistent with Equation 8. This is the calculation process of the noise floor of the entire receiving link when there is no image rejection filter.
Conclusion: Comparing the noise floor in the two cases with and without the image rejection filter, it can be seen from Equation 4 and Equation 8 that if the gain of the LNA is very high, the noise floor without the image rejection filter is close to twice the noise floor with the image rejection filter. Comparing Equation 3 and Equation 10, it can be seen that when the gain of the LNA is very high, the total noise factor without the image rejection filter is close to twice the total noise factor with the image rejection filter.
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