The universal headache

Dec 1, 2008 12:00 PM, By Harold Kinley

More From Test & Measurement

The amount of noise generated within a receiver system is one of the factors that sets the limits on such systems. Of particular importance is the noise present at the receiver front end. The higher the input noise level at the receiver, the greater the amount of signal required to achieve a desirable signal-to-noise ratio. This in turn sets the sensitivity of the receiver. In this article we will examine how noise affects the receiver system and how to determine noise levels.

A simple resistor will produce a noise voltage across its terminals. The amount of noise power produced by a resistor is dependent on the bandwidth in which the noise is measured and the temperature, but it is independent of the resistance value. The noise power produced by a resistor in a given bandwidth at a given temperature can be determined by the formula shown in Equation 1. The noise voltage is dependent on the resistance value, along with bandwidth and temperature.

Figure 1 depicts a resistor in series with a noise generator. The resistor is assumed to be noiseless, and the noise power developed in the resistor is found by using Equation 1. The noise power developed in a bandwidth of 1 Hz at a temperature of 298° K is equal to 4.1124 × 10^-21 watts. Comparing this to 0 dBm (1 mW) yields a figure of -174 dBm. This is the maximum available power that is deliverable to a matched load. Because a matched load would divide the source noise voltage by one-half, the formula for finding the root-mean-square (RMS) noise voltage across the resistor must be multiplied by a factor of 2, as shown in Equation 2. Assuming a noiseless resistor, the voltage of the noise generator in Figure 1 is equal to 9.069 × 10^-10 volts.

In Figure 2, a signal generator represented by E_S is connected in series with a noise generator represented by E_N and a 50-ohm resistor, R_S. The bandwidth is 10 kHz. The voltage level of the signal generator is 0.5 µV, and the voltage level of the noise generator is 0.09 µV. At the output terminals the voltage signal-to-noise ratio is 5.56:1, or 14.9 dB. This is the open-circuit, or no-load, signal-to-noise ratio.

Figure 3 shows the same circuit with a load resistor, R_L (equal to RS), connected across the output terminals. The noise generated by R_L is represented by the noise generator, E_N2, at a level of 0.09 µV. The bandwidth is still 10 kHz.

In Figure 4, the circuit depicted in Figure 3 is redrawn. The two noise generators, E_N1 and E_N2, are replaced by a single noise generator, E_NT. Notice that the output voltage level of the noise generator, E_NT, is equal to 0.127 µV — not 0.18 µV, as would be obtained by simply adding the two voltages. Because the noise voltages are at random phase relationships (noncoherent), the resultant of the two 0.09 µV generators is found by the root-sum-square (RSS) method, as shown in Equation 3.

Referring again to Figure 4, the signal voltage across the load resistor, R_L, is one-half the signal generator voltage, or 0.25 µV. The noise voltage across the load resistor, R_L, is equal to one-half the voltage of the noise generator, or 0.0636 µV. Therefore, the voltage signal-to-noise ratio across the load resistor, R_L, is 0.25/0.0636 = 11.9 dB. Connecting the matching load resistor across the generator circuit has reduced the signal-to-noise ratio from the open-circuit value of 14.9 dB to 11.9 dB under matched load conditions. Thus it can be stated that the noise figure of the matched load is 3 dB. That is, connecting two equal impedances in parallel as shown in Figure 4 results in the reduction of noise by 3 dB compared to the open-circuit condition, and the reduction in the signal by 6 dB compared to the open-circuit condition. Thus the signal-to-noise ratio is degraded by 3 dB, compared to the open-circuit condition.

Figure 5 shows the same circuit as in Figure 3, except that the load resistor, R_L, is 500 ohms. The noise voltage will be higher with the higher impedance. Using Equations 1 and 2 for a 10 kHz bandwidth at 500 ohms, the noise generated in R_L is 0.286 µV. Figure 6 shows the two noise generators, E_N1 and E_N2, replaced by a single generator, E_NT, at a level equal to the RSS of the two noise generators. The noise voltage of E_NT is 0.3 µV.

The signal voltage appearing across R_L is: 0.5 x 500/(500+50) = 0.5 x 0.91 = 0.455 µV

The noise appearing across R_L is: 0.3 x 500/(500+50) = 0.3 x 0.91 = 0.273 µV

The signal-to-noise ratio is then equal to: 20 log 0.455/0.273 = 4.44 dB

However, because the load impedance, R_L, is not equal to the source impedance, R_S, a correction factor must be applied to this signal-to-noise ratio. The correction factor is:

10 log R_L/R_S = 10 log 500/50 = 10 log 10 = 10 dB

Adding this 10 dB to 4.44 dB yields a signal-to-noise ratio of 14.44 dB. Comparing this figure to the open-circuit signal-to-noise ratio of 14.9 dB shows that the noise figure of this circuit is 14.9 - 14.44 = 0.46 dB.