## GBPPR Tech Bulletin #5 - Receiver Noise, Sensitivity and Dynamic Range Relationships

Excerpt from Modern Electronic Communication, 4th Edition

Receiver Noise, Sensitivity and Dynamic Range Relationships

As will be seen, there are various trade-offs and relationships between noise figure, sensitivity and dynamic range when dealing with high-quality receiver systems.

To fully understand these relationships for a receiver, it is first necessary to recognize the factors limiting sensitivity.  In one word, the factor most directly limiting sensitivity is "noise".  Without noise it would only be necessary to provide enough amplification to receive any signal, no matter how small.  Unfortunately, noise is always present and must be understood and controlled as much as possible.

There are many sources of noise.  The overwhelming effect in a receiver is thermal noise caused by electron activity in a resistance.  The noise power (Pn) is:

`Pn = k * T * BW`

k = Boltzman's constant, 1.38 * 10-23 Joules/K

T = Resistor temperature in kelvin (K)

BW = Frequency bandwidth of the system being considered

For a 1 Hz bandwidth and at 290 K:

```Pn = 1.38 * 10-23 * 290 * 1

Pn = 4 * 10-21 Watts

Pn = -174 dBm
```

For a 1 Hz bandwidth and at 1 K:

```Pn = 1.38 * 10-23 Watts

Pn = -198 dBm
```

The preceding shows the temperature variable is of interest since it is possible to lower circuit temperature and decrease noise without changing other system parameters.  At 0 K there is no noise generated.  Unfortunately, it is very expensive and difficult to operate systems even near 0 K.  Most receiving systems are operated at ambient temperature.  The other possible means to lower thermal noise is to lower the bandwidth.  However, the designer has limited capability in this regard.

Noise and Receiver Sensitivity

What is the sensitivity of a receiver?  This question cannot be answered directly without making certain assumptions or knowing certain facts which will have an effect on the result.  Examination of the following formula illustrates the dependent factors in determining sensitivity (S).

`S = -174 dBm + NF + 10log10BW + desired S/N`

where -174 dBm is the thermal noise power at room temperature (290 K) in a 1 Hz bandwidth (BW).  It is the performance obtainable at room temperature if no other degrading factors are involved.  The 10log10BW factor represents the change in noise power due to change above a 1 Hz bandwidth.  The wider the bandwidth, the greater the noise power and the higher the noise floor.  S/N is the desired signal-to-noise ratio in dB.  It can be determined for the signal level which is barely detectable, or it may be considered to be the level allowing an output at various ratings or fidelity.  Often, a 0 dB S/N is used, which means that the signal and noise power at the output are equal.  The signal can therefore also be said to be equal to the noise floor of the receiver.  The receiver noise floor and the receiver output noise are one and the same thing.

Consider a receiver that has a 1 MHz bandwidth and a 20 dB noise figure.  If a S/N of 10 dB is desired, the sensitivity (S) is:

```S = -174 + 20 + 10log101,000,000 + 10

S = -84 dBm
```

It can be seen from this that if a lower S/N is required, better receiver sensitivity is necessary.  If a 0 dB S/N is used, the sensitivity would become -94 dBm.  The -94 dBm figure is the level at which the signal power equals noise power in the receiver's bandwidth.  If the bandwidth were reduced to 100 kHz while maintaining the same input signal level, the output S/N would be increased to 10 dB due to noise power reduction.

Dynamic Range

The dynamic range of an amplifier or receiver is the input power range over which it produces a useful output.  It should be stressed that a receiver's dynamic range and AGC range are ususally two different quanities.  The low power limit is essentially the sensitivity specifications discussed in the preceeding paragraphs.  It is a function of the noise.  The upper limit has to do with the point at which the system no longer provides the same linear increase as related to the input increase.  It also has to do with certain distortion componets and their degree of effect.

When testing a receiver (or amplifier) for the upper dynamic range limit, it is common to apply a single test frequency and determine the 1 dB compression point (P1dB).  As shown in Figure 1, this is the point on the input/output relationship where the output has just reached a level where is is 1 dB down from the ideal linear response.  The input power at that point is then specified as the upper power limit determination of dynamic range. Figure 1

In practice, certain distortion characteristics that affect the normally encountered multifrequency signals are often a major factor.  When two frequencies (f1 and f2) are amplified, the second-order distortion products are generally out of the system passband and are therefore not a problem.  They occur at 2f1, 2f2, f1 + f2 and f1 - f2.  Unfortunately, the third-order products at 2f1 + f2, 2f1 - f2, 2f2 - f1 and 2f2 + f1 usually have components in the system bandwidth.  The distortion thereby introduced, is called intermodulation distortion.  Intermodulation effects have such a major influence on the upper dynamic range of a receiver (or amplifier) that they are often specified via the third-order intercept point.  This is shown in Figure 1.  It is the input power at the point where straight-line extensions of desired and third-order input/output relationships meet.  It is about 20 dBm in Figure 1.  It is used only as a figure of merit.  The better the system is with respect to intermodulation distortion, the higher will be its third-order intercept.

The dynamic range of a system is usually approximated as:

`dynamic range (dB) = 2/3 * (third-order intercept - noise floor)`

Poor dynamic range causes problems, such as undesired interference and distortion, when a strong signal is received.  The current state of the art is a dynamic range of about 100 dB.

The greatest sensitivity can be realized by using a preamplifier with the lowest noise figure and highest available gain in order to mask the higher noise figure of the receiver.  It must be remembered that as gain increases, so does the chance of spurious signals and intermodulation distortion components from operating up into the nonlinear region.  A preamplifier used prior to a receiver input has the effect of decreasing the third-order intercept propotionally to the gain of the amplifier.  Therefore, to maintain a high dynamic range it is best only to use the amplification needed to obtain the desired noise figure.  It is not helpful in an overall sense to use excessive gain.