Question 8.3: The input signal of a low-noise amplifier is a narrowband mo......
The input signal of a low-noise amplifier is a narrowband modulated signal x of complex envelope X(t) centered around f_0. Suppose that at a frequency f_1, close enough to f_0, a strong interferer is received by the input antenna and appears at the LNA input as a large sinusoid x_1 of phasor X_1. Discuss the amplifier behavior and show that the interfering signal negatively affects the amplifier response to the wanted signal, leading to the so-called desensitization or blocking of the receiver. Show that this effect would not be present in an ideally linear amplifier.
In the presence of the wanted signal only the amplifier amplification is A=\lim _{|X(t)| \rightarrow 0} F_P(|X(t)|), the small-signal amplification. Since the interferer is close in frequency to f_0, the amplifier response to it can be described by the same descriptive function, with an amplification A_1=F_P\left(\left|X_1\right|\right) \ll A. Considering both x and x_1 we have that the resulting signal x_2=x+x_1 can be written as:
x_2(t)=\operatorname{Re}\left[\left(X(t)+X_1 e ^{ \text{j} \left(\omega_1-\omega_0\right) t}\right) e ^{ \text{j} \omega_0 t}\right]=\operatorname{Re}\left[X_2 e ^{ \text{j} \omega_0 t}\right],
where:
X_2(t)=X(t)+X_1 e ^{ \text{j} \left(\omega_1-\omega_0\right) t} .
However, since the interferer is much stronger than the wanted signal, we have \left|X_2(t)\right| \approx \left|X_1\right| and therefore the system output will be a narrowband signal with complex envelope Y_2 such as:
\begin{aligned} Y_2(t) & =F_P\left(\left|X_2(t)\right|\right) X_2(t) \approx F_P\left(\left|X_1\right|\right) X_2(t) \\ & =A_1 X_1 e ^{ \text{j} \left(\omega_1-\omega_0\right) t}+A_1 X(t) \approx A_1 X_1 e ^{ \text{j} \left(\omega_1-\omega_0\right) t}. \end{aligned}
In other words, the gain compression associated with the interferer also affects the wanted signal, and the amplifier output is dominated by the interferer. Notice that in an ideally linear amplifier A_1=A and the amplifier response to the interferer can be eliminated, at least in principle, by filtering, since the gain of the wanted signal is not affected.