Dithering increases dynamic range in digital-radio system

Applications such as test equipment and digital radio for electronic intelligence are pushing the limits for both dynamic range and data rates. This drive is increasing the need for high-rate, multistage-converter implementations, so these systems can reach their full design capacity. Designers use dithering to increase SFDR (spurious-free dynamic range) in digital-radio systems that employ multistage A/D converters (Reference 1). In these systems, quantization errors in the conversion process stack up from stage to stage, often resulting in sizable spurs. Because designers cannot eliminate these errors, the design goal must be to randomize them so that the noise floor effectively absorbs them (Figure 1).

To accomplish dithering, it is necessary to have sufficient thermal noise present at the input of the ADC. Although design alternatives exist for ensuring this goal, the challenge is how to determine whether dithering is even necessary and, if so, to examine the alternative means of accomplishing it.

It is important to look at the ADC, as well as the architecture of the system in general. The first question to ask whether the signal contains enough thermal noise. If so, quantization spurs are not issues; otherwise, they are. Without a previous analysis, it is difficult to anticipate the magnitude of these quantization spurs. However, once a designer powers up the system, they will then become glaringly obvious.

A quick analysis saves design effort and time. Consider a typical high-speed ADC that might require a thermal-noise power of –35 to –40 dBm to randomize the quantization spurs. The designer can potentially use an analog or a digital predictive method. The analog method requires predicting the total received noise power. A quick analysis involves adding the receiver's noise figure to the ambient thermal noise power and multiplying it by any gain.

For example, look at a 70-MHz signal that is being undersampled with a receiver bandwidth of 5 MHz. Compute the ambient thermal noise power using Boltzmann's equation: N₁=kTB, where k is Boltzmann's constant, 1.380650×10^–23J/K; T is the effective temperature in Kelvin, and B is the receiver bandwidth. Therefore, N₁=(1.380650×10^–23J/K)×(290K)×(5 MHz)=2×10^–14W=2×10^–11 mW=–117 dBm. If the receiver has a noise figure of 10 dB and a gain of 30 dB, the total noise level, N, is –107 dBm+10 dB+30 dB=–67 dB, which is insufficient for dithering.

The digital method predicts noise power by examining expected or measured BER (bit-error rate) and the lowest expected signal-power amplitude, S. The modulation scheme can equate the BER to E_B/N₀ (bit-energy-to-noise-density ratio) by examining the waterfall curve (Figure 2 and Reference 2). By knowing the data rate, use S to determine E_B using E_B=S–10×log(data rate) (Reference 3). First, calculate E_B based on the lowest expected signal power. Then, from a known BER, find the corresponding N₀ value, where N₀ is the noise power normalized to a 1-Hz bandwidth. To calculate N, you must multiply this value by the noise bandwidth. A quick spectrum analysis of the modulated signal can be helpful in determining bandwidth. Figure 3 shows a QPSK (quadrature-phase-shift-keying) signal with a 2-Mbps data rate.

The 3-dB bandwidth of the signal can give a rough approximation to the associated noise bandwidth; from the plot, this bandwidth is 2.6 MHz. Calculating N from N₀ in decibel form, use N=N₀+10×log(BW), where BW is the bandwidth. Note that, even if the noise bandwidth answer is off by a factor of two, this error has only a 3-dB effect on N; therefore, crude estimates are fine for determining whether dithering is necessary. As an example, use an S value of –60 dBm; a data rate of 2 Mbps; a QPSK, three-quarter-rate, soft-decision modulation; and a BER of 10^–6. You then get the following: E_B=260 dBm–10 log (2 MHz)=–123 dBm/Hz; E_B/N₀=6 dB (from the waterfall curve). Therefore, N₀=–123 dBm/Hz–6 dB=–129 dBm/Hz; N=–129 dBm/Hz+10log (2.6 MHz); and N=–65 dBm, which is also insufficient noise to require dithering.

If this quick analysis predicts that, under all expected operating conditions, enough noise power enters the ADC, then extra circuitry is unnecessary. If, however, it appears that insufficient noise power exists, the designer must ensure sufficient noise power by amplifying the signal and its accompanying noise or adding noise to the signal from a noise source. For the first method, if the signal has a fairly narrow expected dynamic range, then a static gain stage can probably be effective. From a design standpoint, this method is easy. The noise figure of the gain stage is the figure of merit in this case. If the noise figure is high, then it counteracts the benefits of the dithering.

If your design has a wide dynamic range, then a static gain stage is probably unfeasible because the gain, which needs to be high enough to produce enough noise power for dithering, might be acceptable for weak signals but could result in compression or distortion of large incoming signals. A power amp with a 1-dB compression point to handle all cases would probably also have a high noise figure, as well as being large, inefficient, and costly. A possible approach is to use an AGC (automatic-gain-control) circuit, which tailors the effective gain to the input level. This method can involve a considerable design effort to ensure that the control loop is fast enough and to keep the overall noise figure in check.

The second method—adding noise to the signal from a noise source—is well-known but has caveats. You can easily add noise that is cospectral with the signal, but this technique increases the noise floor, which, again, counteracts the benefits of dithering. Generally, designers lowpass-filter the noise to avoid this outcome. The sampling rate determines the sharpness of the filtering, and it is a common practice to undersample the signal. This approach results in aliasing, which causes the digitized signal spectrum to fold over into lower frequencies, further squeezing the allowable band of the added noise. The greater the bandlimiting, the less the bandwidth available for adding noise, resulting in a higher noise spectral density to produce sufficient noise power for effective dithering.

Designers typically specify noise sources in units of spectral density, N₀, but also express them in ENR (excess-noise ratio). You can calculate N₀ from ENR by N₀=ENR–174 dBm/Hz. A typical noise diode produces a relatively small ENR, so a design may need a large gain to use such a diode as a noise source. Amplifying noise is sometimes difficult, because it is a broadband signal and also because thermal noise has a high peak-to-rms ratio. Without design diligence, unwanted resonance can crop up. The initial step is to calculate the needed gain.

Using the example of a 70-MHz, undersampled signal, the resulting maximum noise bandwidth that would be available without encroaching on the signal might be about 2 MHz. Using a noise diode that produces –144 dBm/Hz of noise output (30-dB ENR) and a desired integrated noise power for dithering of –40 dBm, N=N₀ +10×log(2 MHz)+G; G=–40 dBm–(–144 dBm/Hz)–63 dB; and G=41 dB, where G is gain. The next problem involves combining the noise and the signal. To minimize signal-path loss, also a figure of merit, a designer could use a 20-dB coupler, but this approach reduces the noise power by 20 dB. Thus, the required gain is actually 61 dB. Note also that some extra lowpass filtering may be necessary, again increasing the required gain.

Both methods require a large upfront design effort and potentially add expensive circuitry, which leads to the economic question: Is it worth it? An alternative is to use a self-contained dithering module, which can tip the scales toward employing dithering using the added-noise method. This type of device has a signal-input port and an output of signal plus dithering noise. The noise is sufficiently strong enough and filtered to not encroach on the signal, even in the presence of aliasing. The signal-insertion loss is less than 1 dB, making it comparable to a good low-noise amplifier. Having no active components in the signal path means that compression or distortion is not a problem.

There are always design alternatives to any problem. Looking at the big picture, a quick economic analysis is useful. How much will the upfront design effort cost? How much does the dithering circuitry add to the unit cost in production? How risky is the design of the dithering circuitry? What are the economic benefits of the program as a result of the increase in dynamic range that the dithering circuit yields? Whether it's a radio network requiring fewer base stations as a result of increased range or a piece of test equipment gaining a competitive advantage, dithering will become increasingly commonplace as its implementation becomes simple.