Analyzing ADC Noise Impacts on Wireless System Performance

Analyzing ADC Noise Impacts on Wireless System Performance
It's commonly thought in the RF community that noise generated by an analog-to-digital converter will have little impact on the overall performance of their base station design. Think again. Noise generated from an ADC can have a big impact on channel performance and, as the number of ADCs in a wireless system increase, designers must effectively know what impact this noise will have.

In this article, we'll help designers understand techniques for determining the impact of noise on a wireless base station design. We'll also show the impact these measurement techniques will have on the development of cdma2000 systems.

Common Grounds
There are several figures of merit that are often assigned to data converters. The two most popular are signal-to-noise ratio (SNR) and effective number of bits (ENOB). Of the two, the SNR is the most useful measurement. However, given one, the other can be easily determined. Although the correct method for determining ENOB is through the use of the sinewave curvefit, a quick approximation of ENOB can be determined with the following equations.¹

As stated above, SNR is the preferred figure of merit when evaluating an ADC for a 3G base station design. For an ADC, SNR is defined as the log ratio of signal energy to the noise energy. This measurement is usually made in the frequency domain utilizing a Fast Fourier transform (FFT).

From the spectral information, the fundamental is located by sorting through the FFT bins to determine which bin has the largest energy content. Due to spectral leakage associated with windowing and other numerical artifacts, the energy in several adjacent bins around the main signal bin are usually added together to determine the total signal energy. The remaining energy not counted as signal, is technically considered noise.

There are two exceptions. First the energy at DC is usually not counted as noise because most converters have a fairly high DC offset and in a general sense, no information is carried at DC (this is not always true, but for AC coupled systems, it is generally considered correct). Second, harmonic energy is often separated from the remainder of the noise.

For many applications, it may be acceptable to include harmonics with noise while in others they are omitted and considered separately. For this discussion, the harmonics are considered part of the noise. Therefore, total noise energy is the summation of all non-signal FFT bins, except for DC. From the FFT bin summation, the SNR may be calculated using (Figure 1):

Figure 1: FFT with signal and noise highlighted.

Most ADC manufacturers do an adequate job specifying the SNR performance of their products either directly though tables or through graphs. So it should be only necessary to study the product data sheet to determine the SNR performance.

If SNR performance info is missing or not for the correct test conditions, SNR can easily be measured for most data converters using low-cost data-capture boards. These simple boards easily connect to a wide variety of data converters and easily interface to a standard PC through the printer port. The evaluation boards typically include the software to capture and measure data converter performance. No matter how it is determined, once the SNR is known the various noise models are easy to determine.

Input Referred Noise Voltage
The simplest and most common method of including ADC performance in overall wireless system analysis is through the use of the ADC's input referred noise voltage. If this voltage can be determined, the referred noise can then be added to the noise cascaded from earlier stages of the circuit. Since it is assumed that no additional noise will be added by the DSP, all of the noise can be referenced to the input of the ADC where all calculations will be made.

As with all ADC noise examples, designers looking to determine the input referred noise voltage must know the ADC's SNR. The fullscale voltage of the data converter must also be known. As with SNR, this is usually taken directly from the data sheet, but can also be measured by the application of test voltages to the input of the data converter that causes a fullscale indication.

With knowledge of SNR and the fullscale of the converter, the input referred noise voltage can be determined. The equation shown above for SNR is determined by examining the power within the spectral bins. However, an equally valid equation exists based on voltages. This equation is:

This equation can easily be solved for the input noise as shown in the following equation:

If the input signal for the SNR measurement is fullscale, then the fullscale of the data converter can be substituted in this equation. Normally, noise is measured in root-mean-square (rms) volts. Fullscale voltages, however, are measured peak-to-peak. Therefore, the input range must be appropriately scaled to rms, to produce noise rms. Otherwise, the noise voltage will be peak-to-peak and have only limited usefulness. Thus, the following equation accounts for the fullscale range of the converter and ensures that noise is determined in rms instead of peak-to-peak:

As an example, given an SNR of 78 dB and a fullscale range of 2 V, the noise is 89.02 μV rms.

In Figure 2, the front-end noise up to the data converter has been determined and is as shown. Likewise the noise from the ADC is given as determined above.

Figure 2: Simplified signal chain with noised expressed in volts.

As shown in Figure 2, the noise from the front portion of the circuit is determined to be 120 μV rms and the back end is determined to be 89.02 μV rms. Since these voltages are not correlated, they can be added by the square root of the sum of the squares as shown in:.

Evaluation of this expression shows that the total noise voltage on the input to the ADC is 149.41 μV rms.

Input Referred Noise Figure
The use of ADCs is very common in today's wireless architectures, and will continue in emerging 3G wireless systems. RF designers like to work with noise figures (NF). While data converters are not power devices as the use of NF implies, an equivalent NF can be computed for a data converter. Once this number is established, it may be used in computing the cascaded performance of a receiver strip, just as amplifiers, mixers and filters may be.

NF is easily used with noise density per hertz and has an advantage over the noise voltage method above. The noise voltage method assumes total integrated noise. When dealing with bandwidths that are smaller than Nyquist, the noise voltage must be scaled appropriately. Conversely, the NF method can easily be used with any bandwidth required as shown below.

NF is a figure of merit used to describe how much noise is added to a signal in the signal chain of a radio (or other low noise signal chain). Usually it is specified in dB although in the computation of noise figure, the numerical ratio (non-log) is used. The non-log value is called noise factor and is denoted by `F' and is defined below as:

Since SNR is defined as the ratio of Signal/Noise and since an ADC does not provide any gain (just numerical quantization), the output signal is the same as the input signal plus the quantization noise. With this in mind, the equation can be rewritten as:

Converting this to log form and at the same time converting to a common reference such as dBm gives the equation:

Although NF is not normally associated with a data converter, it can be calculated for a single set of operating conditions. If these are changed, the number is invalidated and must be recalculated. Specifically, the following must be known: SNR, sample rate, input voltage range, and input termination impedance (including both internal and external loading). Given this information along with operational temperature, NF can be determined.

From the noise factor equation above, F, is based on the ratio of output noise to input noise, the input noise is simply 'kT' noise. Output noise for all realizable ADCs is limited by its SNR performance. The most appropriate method is therefore a comparison of the noise density of the selected data converter with that of thermal noise. Therefore, the noise spectral density of the data converter must be determined. This can be determined by the following equation:²

This equation provides the equivalent input referred ADC thermal noise in dBm/Hz. Since a data converter has no gain, only quantization as stated above, the noise on the output also consists of input noise. When using the SNR term to compute output noise, it also includes the effects of input noise. Thus, ADC SNR includes output noise and input noise.

To determine the input thermal noise input to the ADC, the following equation can be used:

where k is Boltzmann's constant, T is absolute temperature, and B is bandwidth (1 Hz in this application). Therefore, in this application, thermal noise in 1 Hertz is -174 dBm/Hz.

Given an ADC with a fullscale input power of +4 dBm, a sample rate of 80 MSamples/s, and an SNR of 78 dB, the noise spectral density of the ADC is then -150 dBm/Hz. Using the equation above, NF is then the output noise less the input noise or 24 dB in this example.

In the RF signal chain shown in Figure 3, the same block diagram as shown in Figure 2 is examined. However, in this example, the noise figure of the system will be analyzed. Given the same ADC operating characteristics as defined above, the noise figure is 24 dB.

Figure 3: Simplified signal chain with noise expressed in noise figure.

Working with cascaded noise figures requires a little more math, but the equations are fairly simple. The following equation can be expanded for any number of required stages:²

In the equation above, the F represents the noise factor (non-log) stage noise and the G represents the non-log gain of the numbered stages. In this example, there are only two stages, therefore, the equation simplifies to:

In the simple figure above, F₁ is 4.467, G₁ is 100, and F₂ is 251.2. Substituting these into the equation above gives:

Expressed in log format gives a total cascaded noise figure of 8.4 dB.

How Many Bits Does it Take?
One of the most common questions asked about data converters is "How many bits do I need?" Armed with the discussion above and knowledge about the signal being digitized, this can be determined.

The key to answering this question lies in the spectral density of the desired signal. A good working knowledge of the desired signal is important. Often times, this signal can be modeled as a simple sinusoid. Other times, however, it may be modeled as Gaussian noise with a specified bandwidth.

As an example, consider a cdma2000 signal with a minimum power level into the data converter of -87 dBm spread across 1.25 MHz. The spectral density of this signal is:

which evaluates to -148 dBm/Hz. Because of the strong abilities of CDMA to correlate a signal with a poor SNR, an SNR of about -20 dB is required to recover the signal. Thus for this example, the noise spectral density of the signal chain could be as high as --128 dBm.

Although data converters are generally assumed to have a "white" noise floor, this is not always the case. To prevent the ADC from adding unnecessary spurious content (especially at low signal levels), it is ideal if the ADC noise floor is about 10 dB below the overall noise level. While this may not always be possible, this should be a goal.

Often times a compromise will be made. In the cdma2000 example above, the ADC noise floor will be placed 5 dB below the overall thermal noise. This compromise will prevent the ADC from dominating the noise, while at the same time prevent over specifying of the ADC.

Since it is assumed that the noise floor of an ADC is uniform, the integration of this noise across the Nyquist band of the ADC will provide the total noise on the output of the ADC. If the sample rate for the ADC is 61.44 MHz, the noise floor is integrated across the Nyquist band of 30.72 MHz. This gives a noise spectral density of:

which equates to -58.1 dBm. If the fullscale of the ADC is +5 dBm, then the required single tone SNR is 63.1 dB. From the equation presented earlier, this is an ENOB of 10.2 bits.

Wrap Up
Both analysis methods presented above have their uses and limitations. The choice of strategies will depend on the end application. No matter the system designing for, one key facet is that the ADC noise not be allowed to dominate overall noise. The implication is that other active and passive devices limit noise. The common assumption about data converters is that spectral noise is "white", but in fact, the ADC noise spectrum is not white. This is especially true at low signal levels (when small signals are of interest). Fortunately, where this is the case, it is often advantageous to introduce dither into the system to whiten up the ADC spectrum.

References

1. Brannon, Brad, "Calculate an ADC's Effective Bits," Test & Measurement World, May, 1996, pg 17.
2. Brannon, Brad, "Digital-Radio-Receiver Design require re-evaluation of parameters," EDN, November 5, 1998, pgs 163-170.
3. Brannon, Brad, "DNL and Some of its Effects on Converter Performance," Wireless Design & Development, June 2001, pg 11.
4. Brannon, Brad, "Overcoming Converter Nonlinearities with Dither", Analog Devices Applications Note AN-410, December 1995.

About the Author
Brad Brannon is a systems engineer at Analog Devices where he has worked for 19 years. During that time, he has functioned in a number of roles including test engineer, designer, and applications engineer. Brannon can be reach at brad.brannon@analog.com.