Noise Fundamentals

Thermal Noise

Fidelity of electromagnetic circuits is most commonly compromised by the noise generated by the components constituting the circuit, among other impairments,. For example, additive thermal white noise (also called Johnson-Nyquist noise) is generated by any electronic component whose temperature is above absolute 0 K. An ideal resistance of R ohms at temperature T Kelvin generates a thermal-noise RMS voltage v_n in a bandwidth B Hz, given by the following expression (see ^[1] and ^[2]).

where k = 1.38 × 10^-23 m²kg s^-2K^-1 is the Boltzmann constant.

The -174 dBm/Hz Noise Floor

Consider the following circuit diagram.

The power delivered to the load resistance R_L is obtained as shown in the following equation.

Maximum power is transferred to the load when the load impedance matches the source impedance, that is, when R_L = R.

Using v_n from (1), we get the expression for maximum noise power, P_MAX, delivered from a noisy resistor to a matched noiseless load.

Power spectral density, kT, is obtained as the ratio between delivered power and bandwidth. At a reference temperature of 290 K, this power spectral density expressed in dBm/Hz is obtained as shown:

This is the often cited -174 dBm/Hz noise floor of RF and microwave devices.

General Two Port Networks Noise Model

Consider the following noise model:

The terms used in the model and other related terms are defined as shown in the following table:

It can be shown that the noise factor of the signal analyzer when presented with a source with reflection coefficient, Γ_S is (see [3], [4])

where,

Γ_OPT is the optimal source impedance, resulting in the minimum possible noise factor for the signal analyzer. This is easily verified by evaluating F(Γ_S)/ Γ_S by setting Γ_S to Γ_OPT. Complete description of the noise factor of the signal analyzer using this equation requires estimation of the unknown parameters F_MIN, r_A, and Γ_OPT.

Refer to the Derivation: Noise Factor of Two Port Networks section for detailed derivation of these parameters.

Noise figure of the signal analyzer characterizes how the signal-to-noise ratio (SNR) degrades as a function of source impedance. However, for noise compensation, the noise power delivered to the signal analyzer load, denoted by admittance, Y_L, is the quantity of interest and not the degradation in SNR.

The total noise power delivered to the signal analyzer load is a superposition of the noise power delivered by the source noise current source (I_S), analyzer noise voltage (V_A), analyzer uncorrelated noise current (I_U), and analyzer correlated noise current I_C = V_AY_C.

We can write the noise current (I_IN) as a superposition of noise current due to the source (I_IN;S) and the analyzer (I_IN;A).

where,

and

Noise power delivered to the signal analyzer load(Y_L) that can be attributed to the signal analyzer is proportional to E{|I_IN;A|²}, where E{·} denotes the statistical expectation operator, and is expressed by the following equation:

Using the expressions for mean squared thermal noise currents and voltages measured over bandwidth B Hz, we get the following expression:

This shows that the noise power attributed to the signal analyzer, delivered to the load is a function of the source impedance.

For example, an RF short source (infinite admittance, Y_S = ∞) results in mean squared noise current:

and an RF open source (zero admittance, Y_S = 0) results in a mean squared noise current:

Note   Impact of impedance mismatch between the DUT and the signal analyzer is ignored in noise compensation.

Derivation: Noise Factor of Two Port Networks

For the noise model depicted in the previous section, we can see that the total noise current delivered to the analyzer is a superposition of the intrinsic noise current in the source, the intrinsic noise voltage of the analyzer, the intrinsic uncorrelated noise current of the analyzer and the intrinsic correlated noise current of the analyzer.

By substituting the correlated intrinsic noise current of the analyzer I_C = Y_CV_A, where Y_C is a fictitious admittance representing the correlation between I_C and V_A, we get the expression for the total input current:

Using the statistical expectation operator, E{·}, the average noise power measured by the analyzer is as shown in the following expression:

Using the expressions for mean squared thermal noise currents and voltages measured over bandwidth B Hz, we get

Similarly, in absence of any noise in the analyzer, the noise power delivered to the load from the source noise is given by the following expression:

By the definition of noise factor,

Selecting appropriate source conductance (G_S) and source susceptance (B_S) minimizes the analyzer noise factor, F.

Clearly, optimal B_S would be equal to -B_C.

Thus, B_OPT ≔ -B_C

To optimize noise factor with respect to G_S, set the partial derivative to zero.

Thus, optimal source conductance, G_S, for minimizing noise factor is

and the optimal source admittance, Y_S, is

With optimal source admittance, the minimum noise factor is

Eliminating G_U from the minimum noise factor (F_MIN) expression by re-writing the following expression for G_U:

Thus,

Re-writing noise factor, F, in terms of F_MIN:

Again, eliminating G_U,

or,

Normalizing with respect to characteristic impedance, noting that any normalized admittance, y, is

Thus, we arrive at the desired expression for noise factor of the analyzer as a function of source reflection coefficient, Γ_S:

References

^[1] Nyquist, Harry. "Thermal agitation of electric charge in conductors." Physical review 32.1 (1928): 110.

^[2] Abbott, Derek, et al. "Simple derivation of the thermal noise formula using window-limited Fourier transforms and other conundrums." Education, IEEE Transactions on 39.1 (1996): 1-13.

^[3] Harter, Alphonse. "LNA matching techniques for optimizing noise figures." RF design 2.03 (2003).

^[4]S. Long.  "Design of Low Noise Amplifiers (https://www.ece.ucsb.edu/~long/ece145a/LNAdesign.pdf)", UC Berkeley(2007).