Noise Figure


Noise Figure

Noise figure was introduced by H. T. Friis in 1944 [1]. It is a measure of degradation of signal’s SNR due to noise added by a circuit when the signal is passing through it. Noise figure(NF) is noise factor(F) expressed in decibel.

    \[NF = 10 \log_{10}(F) ~dB\]

Figure 1. Noise Figure

Figure 1. Noise Figure

Noise factor(F) of a circuit is defined as the ratio of SNR at input(SNR_i) to SNR at output(SNR_o) of the circuit.

(1)   \begin{eqnarray*} F &=& { SNR_i \over SNR_o} = {{P_i}/{N_i} \over {P_o}/{N_o}}\\ &=&{{P_i}/{N_i} \over {GP_i}/{(GN_i+N_A})} = 1 + \frac{N_A}{GN_i}\end{eqnarray*}

where,
P_i \rightarrow Signal power at the input of the circuit
P_o \rightarrow Signal power at the output of the circuit
N_i \rightarrow Noise power at the input of the circuit
N_o \rightarrow Total noise power at the output of the circuit
N_A \rightarrow Noise power added by the circuit referred to the output
G \rightarrow Power gain of the circuit

Units
Noise Factor (F) 1 1.25 1.414 1.6 2 4
Noise Figure (N) 0 1 1.5 2 3 6 dB

NF of a Resistive load

Figure 2. Noise figure computation of resistive load >> Norton equivalent representation >> Equivalent circuit for noise figure calculation

(2)   \begin{eqnarray*}F &=& { i_{n,R_s}^2 + i_{n,R_L}^2 \over i_{n,R_s}^2} \\ &=& {{4kT \over R_s}+{4kT \over R_s} \over {4kT \over R_s}} = 1 + {R_s \over R_L}\end{eqnarray*}

Alternatively,

    \[G=({v_o \over i_s})^2 = (R_s || R_L)^2\]

(3)   \begin{eqnarray*} F &=& 1 + {N_A \over G N_i} = 1 + { 4kT (R_s || R_L) \over (R_s || R_L)^2 . {4kT\over R_s}} \\ &=& 1 + {R_s \over R_L} \end{eqnarray*}

If the load impedance(R_L) is power matched for source resistance(R_s), then NF= 20 \log(2) = 6dB. Its conversion gain is -6dB.
For example a 6dB RF attenuator or pad has noise figure of 6dB and conversion gain of -6dB. If a signal enters into a attenuator or pad, then the signal is attenuated by 6dB while the noise floor remains constant. Therefore the signal to noise ratio through the pad is degraded by 6dB.

The Noise figure of a passive device is same as the of the conversion gain(in dB sense).

Noise Temperature

Thermal noise power of a passive device at temperature (T) is given by

(4)   \begin{equation*} N_i = k_B T B \end{equation*}

Noise temperature is another way of representing available noise power of a component or amplifier. Noise temperature (T_e) is the equivalent temperature at which a resistor connected at the input of the component (noiseless) produce the same noise as the real component or amplifier. Noise temperature is equivalent temperature but not the real temperature of the amplifier

By definition, noise factor of the amplfier is given by

(5)   \begin{equation*} F = 1 + {N_A \over G N_i} = 1 + {G_A k_B T_e B \over G_A k_B T_o B} = 1 + {T_e \over T_o} \end{equation*}

Therefore noise temperature in terms of noise figure and absolute temperature is \boxed{T_e = (F-1)T_o}

Cascaded Noise Figure

For a n-stage cascaded system shown in Figure 5, the total noise factor (F_{sys}) is given by

(6)   \begin{equation*} F_{sys} = 1 + \sum_{i=1}^{i=n} \frac{F_i-1}{\prod\limits_{j=0}^{i-1} G_j} \end{equation*}

where,
F_i \rightarrow Noise factor of i^{th}-stage
G_i \rightarrow Gain of i^{th}-stage; G_o = 1

Figure 4. Cascaded noise figure of a system

Figure 5. Cascaded noise figure of a n-stage system

The cascaded noise figure (NF_{sys}) is given by

(7)   \begin{equation*} NF_{sys} = 10 \log(F_{sys}) \end{equation*}

References

[1] H. T. Friis, “Noise Figure of Radio Receivers,” Proceeding of the IRE., vol. 32, pp. 419-422, July 1944.

 

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