We Now Want to Turn Our Attention to Noise. We Will Start with the Basic Definition Of

NOISE

We now want to turn our attention to noise. We will start with the basic definition of noise as used in radar theory and then discuss noise figure. The type of noise of interest in radar theory is termed thermal noise and is generated by the random motion of charges in resistive types of devices. One of the early attempts to characterize thermal noise was performed by Nyquist, and one of his theorems concerning thermal noise is that the mean-square voltage appearing across the terminals of a resistor of ohms at a temperature degrees Kelvin, in a frequency band Hertz, is given by[1]

(100)

where is Boltzman’s constant. The noise power associated with the resistor is

. (101)

The Thevenin equivalent circuit of a noisy resistor is as shown in Figure 18. It consists of a noise source with a voltage defined by (100) and a noiseless resistor with a value of .

If we connect the noisy resistor, , to a noiseless resistor, , we can find the power delivered to by using the equivalent circuit of Figure 18, computing the voltage across and then using this voltage to find the power delivered to . The resulting circuit is shown in Figure 19 and the voltage across is given by

. (102)

The power delivered to is

. (103)

If the load is matched to the source resistance, that is if , we have

(104)

Which is the familiar form used in the radar range equation.

If we have a network that consists of multiple noisy resistors we find the Thevenin equivalent circuit by using a modified version of superposition. To see this we consider the example of Figure 20. In the figure, the left schematic shows two, parallel noisy resistors and the center schematic shows the equivalent circuit based on Figure 18. The right schematic is the overall Thevenin equivalent circuit for the pair of resistors. To find we first consider one voltage source at a time and short all other sources. Thus, with only source we would get

(105)

and with only source we would get

. (106)

To get the total Thevenin equivalent voltage we must consider that and are noises. As such, we must add their squares. Thus, with this we get

. (107)

If we use and we find that

. (108)

We find the Thevenin equivalent resistance by the standard means of shorting all voltage sources and finding the equivalent resistance. The result of this is

. (109)

This leads to the Thevenin equivalent circuit represented by the right schematic of Figure 20.

Although not stated above, one of the assumptions we place on the noisy resistor is that its noise power density is constant over the bandwidth of . That is,

. (110)

In fact, although not realistic, we assume that the noise power density is constant for all frequencies and that the resistor is an ideal band pass filter with a bandwidth of . In other words, we assume that the noise is white. This is a good assumption in practice because radars are generally designed so that the noise spectrum into a device is flat over the bandwidth of the device. This is specifically done to assure that the white noise assumption can be invoked.

For passive devices such as resistive attenuators we can find the noise power delivered to a load by an extension of the technique used in the above example. For active devices this is not possible. For these devices the only way to determine the noise power delivered to a load is through measurement. In general, the noise power delivered to the load will depend upon the input noise power to the device and the internally generated noise. The standard method of representing this is to write the noise power delivered to the load as

(111)

where is the gain of the device, is the input noise power (in a bandwidth of ), is the noise power generated by the device (in a bandwidth of ) and is the equivalent noise temperature of the device. For resistors, the equivalent noise temperature is an actual temperature. For active devices the equivalent noise temperature is not an actual temperature. It is the temperature that would be necessary for a resistor to produce the same noise power as the active device. Both and can be measured. In the above equation, and in all calculations of noise to follow, we never specifically state the value of the bandwidth. We simply carry as it along as a required parameter.

An alternate to using gain and effective noise temperature to characterize the noise properties of devices is to use noise figure. The noise figure, , of a device is defined as

. (112)

In this definition it is assumed that the noise power into the device is given by

(113)

where .

To compute the noise out of the ideal device we assume that the device does not generate its own noise. Thus

. (114)

From (111), the actual noise power out of the device, when the input noise power is , is

. (115)

With this we can relate to as

. (116)

Alternately, we can solve for in terms of as

. (117)

An important point from (116) is that the minimum noise figure of a device is . Another important point in the above is that noise figure is always based on an assumption that the noise power into the device derives from a resistive noise source at the standard temperature of 290 ºK.

In working radar problems some people prefer noise figure and others prefer effective noise temperature. Most of the noise specifications of devices and radars are provided in terms of noise figure. However, as we will see shortly, effective noise temperature, and total noise temperature, are often of use when characterizing the combined effects of external noise sources and receiver noise.

For most devices, noise figure is determined by measurement. The exception to this is attenuators. For attenuators, the noise figure is the attenuation. Thus, if an attenuator has an attenuation of (a number greater than one) the noise figure is

. (118)

The rationale behind this is that if the attenuator is matched to the source and the load impedance, which are assumed the same, the noise power out of the attenuator is equal to the noise power input to the attenuator. There is a further, unstated, assumption that the noise temperature of the resistive elements that make-up the attenuator are at the same temperature as the noise source driving the attenuator.

With the above we can derive the noise figure of an attenuator as follows. If the attenuator is considered ideal, i.e. the resistive elements that make-up the attenuator do not generate noise, the noise power out of the attenuator is

. (119)

However, for an actual attenuator we have

. (120)

By the definition of (112) the noise figure of the attenuator is

. (121)

Since a typical radar has several devices that contribute to the overall noise figure of the radar we need a method of computing the noise figure of a cascade of components. To this end, we consider the block diagram of Figure 21. In this figure, the circle to the left is a noise source, which in a radar would be the antenna are other radar components. For the purpose of computing noise figure it is assumed that the noise source has an effective noise temperature of . The blocks following the noise source represent various radar components such as amplifiers, mixers, attenuators, etc. These blocks are represented by their gain, , and noise figure, . For purposes of computing noise figure, all of the devices are assumed to have the same bandwidth of .

To derive the equation for the overall noise figure of the N devices we will consider first device 1, then devices 1 and 2, then devices 1, 2, and 3, and so forth. This will allow us to develop a pattern that we can extend to N devices.

Since we have the noise figure of each device we can compute the effective noise temperature of each device via (117). Thus, the effective noise temperature of device is

. (122)

For Device 1, the input noise power is

. (123)

The noise power out of an ideal Device 1 is

. (124)

The actual noise power out of Device 1 is

. (125)

From (112) the system noise figure from the source through Device 1 is

(126)

where the last equality was a result of (116).

For Device 2, the input noise power is

. (127)

The noise power out of an ideal cascade of Devices 1 and 2 is

. (128)

The actual noise power out of Device 2 is

. (129)

The system noise figure from the source through Device 2 is

. (130)

Or, using (116)

. (131)

It is interesting to note that the noise figure of the second device is reduced by the gain of the first device. We will examine this again in an example to be presented shortly. For now we proceed to determine the system noise figure from the source through the third device.

The noise power out of an ideal cascade of Devices 1, 2 and 3 is

. (132)

The actual noise power out of Device 3 is

(133)

or, substituting for from (125),

. (134)

The system noise figure from the source through Device 3 is

. (135)

Or, again using (116)

. (136)

Here we note that the noise figure of Device 3 is reduced by the product of the gains of the preceding two devices.

With some thought we can extend (133) to write the system noise figure from the source through Device N as

. (137)

It will be left as a exercise to show that the effective noise temperature of the N devices is

. (138)

In the above we found the system noise figure between the input to Device 1 through the output of Device N. If we wanted the noise figure between the input of any other device, say Device k, to the output of some other succeeding device, say Device m, we would assume that the source of Figure 21 was connected to the input of Device k and we would include terms like those of (137) that would carry to the output of Device m. Thus, for example, the noise figure from the input of Device 2 to the output of Device 4 would be

. (139)

We now want to consider an example that illustrates why radar designers normally like to include an RF amplifier as the first element in a receiver. In this example we consider the two options of Figure 22. In the first option we have and amplifier followed by an attenuator and in the second option we reverse the order of the two components. The gains and noise figures of the two devices are the same in both configurations. For Option 1, the noise figure from the input of the first device to the output of the second device is

. (140)

For the second option the noise figure from the input of the first device to the output of the second device is

(141)

This represents a dramatic difference in noise figure of the combined devices. This difference is due to the aforementioned property that the noise contributed to the system by an individual device is a function of the noise figure of that device and the gains of all devices that precede the device. In general, if the preceding devices have a net gain, the noise contributed by a device will be reduced relative to its individual noise figure. If the preceding devices have a net loss, the noise contributed by the device will be increased relative to its individual noise figure.

In Option 1 of the above example, the noise figure of the two devices was close to the noise figure of the simply the amplifier. However, for the second option the noise figure was the combined noise figures of the two devices. This is why radar designers like to include an amplifier early in the receiver chain: it essentially sets the noise figure of the receiver.

In the above, we considered a source temperature of . We now want to examine how to compute the noise power out of a device when the source temperature is something other than . From (111) we have

(142)

where and is the noise temperature of the source. If we were to rewrite (142) using noise figure we would have

. (143)

If we use a cascade of N devices, is the combined gain of the N devices, is the effective noise temperature of the N devices (see (138)) and is the noise figure of the N devices (see (137)).

[1] From “Principles of Communications” by Ziemer and Tranter, Fourth Edition.