Noise figure (NF) is a parameter commonly used by RF system designers to characterize noise in RF amplifiers, mixers, and other devices, and is widely used as a tool in radio receiver design. Many excellent communication and receiver design textbooks provide detailed descriptions of noise figure, and this article focuses on the application of this parameter to data converters.
There are now many wideband operational amplifiers and ADCs used in RF applications, making the noise figure of these devices important. Reference 2 discusses the applicable method for determining the noise figure of op amps. Not only must we know the voltage and current noise of the op amp, but we should also know the exact circuit conditions: closed-loop gain, gain setting resistance value, source resistance, bandwidth, etc. Calculating the noise figure of an ADC is more challenging, as you quickly become aware.
When RF engineers first calculate the noise figure of even the best low-noise, high-speed ADC, the results can be relatively higher than the noise figure of a typical RF gain block, low-noise amplifier, etc. In order to interpret the results correctly, it is necessary to understand the position of the ADC in the signal chain. Therefore, it is important to exercise caution when dealing with the noise figure of an ADC.
ADC noise figure definition
Figure 1 shows the basic model used to define the ADC noise figure. The noise factor F is the ratio of the total effective input noise power of the ADC to the noise power caused by the source resistance alone. Due to impedance matching, the noise power can be replaced by the square of the voltage noise. The noise figure NF is the noise factor expressed in dB, NF = 10log10F.
The model assumes that the input to the ADC comes from a signal source with a resistor of R, the input bandwidth is limited to fs/2, and the input has a filter with a noise bandwidth of fs/2. It is also possible to further limit the bandwidth of the input signal, resulting in oversampling and processing gain, which will be discussed later.
The model also assumes that the input impedance of the ADC is equal to the source resistance. Many ADCs have a high input impedance, so this termination resistor may be located outside the ADC or used in parallel with an internal resistor to produce an equivalent termination resistor with a value of R.
ADC noise figure derivation process
Full-scale input power refers to the power of a sine wave with a peak-to-peak amplitude that exactly fills the ADC's input range. The full-scale input sine wave given below has a peak-to-peak amplitude of 2VO, which corresponds to the peak-to-peak input range of the ADC:
v(t) = Vosin2πft 等式1
The full-scale power of this sine wave is:
Equation 2
This power is typically expressed as dBm (based on 1 mW):
Equation 3
The noise bandwidth B of the filter needs to be further discussed. The noise bandwidth of a non-ideal brick wall filter refers to the bandwidth required for an ideal brick wall filter to pass through the same noise power. As a result, the noise bandwidth of a filter is always greater than its 3dB bandwidth, and the ratio of the two depends on the sharpness of the filter cutoff region. Figure 2 shows the noise bandwidth of a Butterworth filter with up to 5 poles versus the 3dB bandwidth. Note: For the 2 poles, the noise bandwidth differs from the 3dB bandwidth by 11%; Beyond 2 poles, the two are basically equal.
The first step in the NF calculation is to calculate the effective input noise of the ADC based on its SNR. The ADC data sheet gives the SNR at different input frequencies to ensure that the value corresponding to the input frequency of the target IF is used. It should also be ensured that the harmonics of the fundamental signal are not included in the SNR values, as some ADC data sheets may confuse SINAD with SNR. Once the SNR is known, the equivalent input rms voltage noise can be calculated starting from the following equation:
Equation 4
Solution:
Equation 5
This is the total effective input rms noise voltage measured over the entire Nyquist bandwidth (DC to fs/2), note that this noise is included
Noise of the source resistance.
The next step is to actually calculate the noise figure. In Figure 3, note that the amount of input voltage noise caused by the source resistor is equal to the source resistor
The voltage noise of sqrt (4kTBR) divided by 2 is sqrt (kTBR) because the ADC input termination resistor forms a 2:1 attenuator.
The expression for the noise factor F can be written as:
Equation 6
Convert F to dB and simplify to get the noise figure:
NF = 10log10F = PFS(dBm) + 174 dBm – SNR – 10log10B,等式7
其中，SNR的单位为dB，B的单位为Hz，T = 300 K，k = 1.38 × 10–23 J/K。
Figure 3: ADC noise figure based on SNR, sample rate, and input power
其中SNR的单位是dB,带宽B是Hz,T=300K,k=1.38 × 10–23 J/K
Oversampling and digital filtering produce processing gain, which reduces the noise figure, as described above. For oversampling, the signal bandwidth B is lower than fs/2. Figure 4 gives the correction factor, so the noise figure is calculated as:
NF = 10log10F = PFS(dBm) + 174 dBm – SNR – 10 log10[fs/2B] – 10 log10 B.等式8
Figure 4: Effect of oversampling and processing gain on ADC noise figure
Example calculation of the AD9446 16-bit, 80/100 MSPS ADC
Figure 5 shows an example of NF computation for the AD9446, a 16-bit, 80/105 MSPS ADC. A 52.3 ohm resistor is connected in parallel with the 1 k ohm input impedance of the AD9446, resulting in a net input impedance equal to 50 ohm. The ADC operates under Nyquist conditions, and the SNR of 82 dB is the basis for the calculation using Equation 8 above, resulting in a noise figure of 30.1 dB.
Figure 5: The AD9446, a 16-bit 80/100 MSPS ADC
Example of noise figure calculation working under Nyquist conditions
Cascading noise figure
In a typical receiver, the ADC is preceded by at least one low noise amplifier (LNA) and mixing stage, which provides a signal gain high enough to minimize the impact of the ADC on the overall noise figure of the system.
This can be illustrated by Figure 6, which shows how the Friis equation can be used to calculate the noise factor of a cascading gain stage. Note that the high gain of the first stage reduces the effect of the noise factor of the second stage, so that the noise factor of the first stage dominates the overall noise figure.
Figure 6: Calculating the cascaded noise figure using the Friis equation
Figure 7 shows the effect of a high-gain (25 dB) low-noise (NF = 4 dB) level placed before a relatively high NF level (30 dB), with the noise figure of the second stage typical for a high-performance ADC. The overall noise figure is 7.53 dB, which is only 3.53 dB higher than the first stage noise figure (4 dB).
Figure 7: Example of a two-cascade network
Conclusion
When applying the noise figure concept to characterize wideband ADCs, special care must be taken to prevent misleading results. Trying to reduce the noise figure simply by changing the values in the equation can be counterproductive, resulting in an increase in the total noise of the circuit.
For example, according to the above equation, NF decreases as the source resistance increases, but increasing the source resistance increases the circuit noise. Another example has to do with the input bandwidth B of an ADC. According to the equation, raising B will reduce NF, but this is obviously contradictory, since increasing the ADC input bandwidth actually increases the effective input noise. In the above two examples, the total circuit noise increases, but the NF decreases. The reason for the decrease in NF is that the source noise accounts for a larger portion of the total noise as the source resistance or bandwidth increases. However, the total noise remains relatively stable, as the noise caused by the ADC is much greater than the noise of the signal source.
Therefore, according to the equation, the NF decreases, but the actual circuit noise increases. In light of this, NFs must be handled with care when handling ADCs. Valid results can be obtained using the equations in this article, but they can be misleading without a full understanding of the noise principles involved. From an isolated perspective, even low-noise ADCs have a higher noise figure than other RF devices such as LNAs or mixers. However, in a real-world system application, at least one low-noise gain block is placed in front of the ADC, which reduces the total noise contribution of the ADC to a very low level according to the Friis equation (see Figure 7).