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3.5 Noise Factor

The noise factor F of an amplifier, or any other network, is defined as:

Noise factor F = = [ ][ ] = Noise Figure (NF) = 10 [i.e. Noise Figure in terms of Noise Factor]

3.6 Noise Figure

1. To perform noise analysis at the receiver of a communication system, we need a convenient measure of the noise performance of a linear two-port device. One such measure is furnished by the so-called noise figure.

2. Consider a linear two-port device connected to a signal source of internal impedance Z(f) =R(f) + jX(f) at the input, as in Figure 1.

Figure 1. Linear two-port device 3. The noise voltage v(t) represents the thermal noise associated with the internal resistance

R(f) of the source. The output noise of the device is made up of two contributions, one due to the source and the other due to the device itself.

4. The available output noise power in a band of width centered at frequency f as the maximum average noise power in this band, obtainable at the output of the device.

5. The maximum noise power that the two port device can deliver to an external load is obtained when the load impedance is the complex conjugate of the output impedance of the device, that is, when the resistance is matched and the reactance is tuned out.

6. The noise figure of the two-port device as the ratio of the total available output noise power (due to the device and the source) per unit bandwidth to the portion thereof due solely to the source.

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7. Let the spectral density of the total available noise power of the device output be , and the spectral density of the available noise power due to the source at the device input be .

8. G(f) denote the available power gain of the two-port device, defined as the ratio of the available signal power at the output of the device to the available signal power of the source when the signal is a sinusoidal wave of frequency f.

9. The noise figure F of the device as, (1)

10. If the device were noise free, , and the noise figure would then be unity.

11. In a physical device, however, is larger than , so that the noise figure is always larger than unity. The noise figure is commonly expressed in decibels, that is, as .

12. The noise figure may also be expressed in an alternative form. Let denote the available signal power from the source, which is the maximum average signal power that can be obtained.

13. For the case of a source providing a single-frequency signal component with open-circuit voltage the available signal power is obtained when the load connected to the source is, (2) Where the asterisk denotes complex conjugation. Under this condition, we find that, [ ] = (3)

14. The available signal power at the output of the device is therefore, (4)

15. Then, multiplying both the numerator and denominator of the right-hand side of Equation.1 by we obtain (5)

Where, ;

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16. We refer to as the available signal-to-noise ratio of the source and to as the available signal-to-noise ratio at the device output, both measured in a narrow band of width centered at f.

17. Since the noise figure is always greater than unity, it follows from Equation(5) that the signal-to-noise ratio always decreases with amplification, which is a significant result.

18. The noise figure F is a function of the operating frequency f; it is therefore referred to as the spot noise figure.

19. In contrast, we may define an average noise figure of a two port device as the ratio of the total noise power at the device output to the output noise power due solely to the source. That is,

(6) It is apparent that in the case of thermal noise in the input circuit with R(f) constant and

constant gain throughout a fixed band with zero gain at other frequencies, the spot noise figure F and the average noise figure are identical.

3.7 Noise Temperature (or) Equivalent Noise Temprature 1. A disadvantage of the noise figure F is that when it is used to compare low-noise devices,

the values obtained are all close to unity, which makes the comparison rather difficult. In such cases, it is preferable to use the equivalent noise temperature.

2. Consider a linear two-port device whose input resistance is matched to the internal resistance of the source as shown in Figure 2. In this diagram, we have also included the noise voltage generator associated with the internal resistance of the source.

3. The mean-square value of this noise voltage is , where k is Boltzmann's constant. Hence, the available noise power at the device input is, (1) Let denote the noise power contributed by the two-port device to the total available output noise power . We define as,

(2) Where G is the available power gain of the device and is its equivalent noise

temperature. Then it follows that the total output noise power is, (3)

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Figure 2. Linear two-port device matched to the internal resistance of a source connected to the input.

4. The noise figure of the device is therefore (see the output port of Figure 2), (4) Solving for the equivalent noise temperature: (5)

The noise figure F is measured under matched input conditions, and with the noise source at temperature T. By convention the temperature T is taken as "room temperature," namely 290K, where K stands for "degree Kelvin."

3.7.1 Cascade Connection of Two-Port Networks 1. It is often necessary to evaluate the noise figure of a cascade connection of two-port

networks whose individual noise figures are known. Consider Figure 3, consisting of a pair of two-port networks of noise figures and and power gains and , connected in cascade.

Figure 3. A cascade of two noisy two-port networks

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2. It is assumed that the devices are matched, and that the noise figure of the second network is defined assuming an input noise power .

3. At the input of the first network, we have a noise power contributed by the source, plus an equivalent noise power contributed by the network itself.

4. The output noise power from the first network is therefore . Added to this noise power at the input of the second network, we have the equivalent extra power contributed by the second network itself.

5. The output noise power from this second network is therefore equal to + . We may consider the noise figure F as the ratio of the actual output noise power to the output noise power assuming the networks to be noiseless.

6. We may therefore express the overall noise figure of the cascade connection of Figure 3 as,

(1) The result may be readily extended to the cascade connection of any number of two-port

networks, as shown by,

(2)

Where are the individual noise figures, and , , . . .are the available power gains, respectively. Equation (2) shows that if the first stage of the cascade connection in Figure 3 has a high gain, the overall noise figure F is dominated by the noise figure of the first stage.

7. Correspondingly, we may express the overall equivalent noise temperature of the cascade connection of any number of noisy two-port networks as follows:

(3)

Where , , . . . are the equivalent noise temperatures of the individual networks, and , . . . are the available power gains, respectively. Equation (3) is known as the Friis formula. Here again we note that if the gain of the first stage is high, the equivalent noise temperature is dominated by that of the first stage.

+

+

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PROBLEMS 1) A mixer circuit has noise figure of 12 dB. It is preceded by an amplifier that has an

equivalent noise temperature of 200K and power gain of 30 dB. Calculate the equivalent noise temperature of the combination referred to the input of the amplifier.

Solution:

Fig 1. Cascade connection Given data: Let us first convert the quantities to their linear values:

G= 30 dB = 10 F= 12 dB = 10

Let the room temperature be 17 . Hence T= 273+17=290 Equivalent noise temperature of the mixer will be, Equivalent noise temperature of the cascade connection is given by, Here ; ; G= 1000

2) A receiver has a noise figure of 12 dB and it is fed by a low noise amplifier that has a gain of 50 dB a noise temperature of 90 K. Calculate the noise temperature of the receiver and the overall noise temperature of the receiving system. Take room temperature as 290K.

Solution:

Fig 2. Cascade connection

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Given Data: T=290K. = 10 => And = 12 dB = 10 =>

Equivalent noise temperature of the receiver will be, = (15.85-1) 290 = 4306.5 K Overall noise temperature of the receiving system will be, =

3) Consider two amplifiers are connected in cascade. First stage amplifier has gain and noise figure as 10 dB and 2 dB. Second stage has noise figure of 3 dB. Calculate total noise figure.

Solution:

Fig 3. Cascade connection

Given Data: = 10 => = 2 dB = 10 => = 3 dB = 10 => The total noise figure is given as, Or F =10 = 2.266 dB