# Creating the Digital World: Step One

Age exposure alert! I have a textbook — a very good one I might add — from my undergraduate studies of communication systems called: *Modern Digital and Analog Communication Systems*. From this, and as far back as the 1980's, I devised Rob's first law — not to be confused with any of the somewhat more significant findings of Nyquist, Shannon, Hamming, or Viterbi. My law says: never use the word "modern" in the title of a book in the technology field.

While the book's principles are priceless, things evolved so quickly once digital became mainstream, that many classic texts needed a rewriting and a re-focus. But, to be fair, while we are endlessly reminded of our all-digital world, and while we watch incredulously as our children adopt a fluency with the resulting fruits the way that toddlers learn language skills, the fact is that all of our interactive senses remain, and will remain for a very long time, analog. However, to share and pass the content effortlessly about on MySpace and Facebook and Limewire and YouTube (and I did not need my daughter to come up with those applications!), we need them digitized. And this is where the analog-to-digital converter comes in.

To truly understand analog-to-digital (A/D) converters is a sign of well-rounded engineer, because the devices require fluency in several areas " analog design, digital design, signal processing, and for a hardware designer, PCB design. The clock speeds and precision require competence in high-speed digital, linear, RF, and in some cases instrumentation techniques to get the promised performance. Indeed, it is not unusual to find a vendor's evaluation board for a converter that needs special care and feeding to obtain those fantastic waveforms and spectrum snapshots on the data sheets. This month, we will step concept-by-concept through some common A/D parameters. We will continue with some more sophisticated topics next month.

**Aliasing**

One A/D idea that almost everyone has some familiarity with is aliasing. In **Figures 1** and **2**, Nyquist sampling and the aliasing concept are graphically depicted. The analog input spectrum is shown, along with the digitized version, for a properly sampled signal in Figure 1. In Figure 2, an undersampled version is shown. Clearly, the spectrum in Figure 2 does not allow a clear replication of the analog spectrum. Figure 1, by contrast, reveals a spectrum that maintains an undistorted version of the original spectrum, one that with proper filtering could be recreated exactly. The well-referenced, simplified, Nyquist criteria requires sampling at two-times the maximum input frequency. In practice, filtering constraints cause this to be slightly higher, thus spreading the repeated versions of the spectrum are well away from one another. The Nyquist criteria can be generalized to signal bandwidth instead, allowing the possibility of IF sampling, but we will save this for next month's discussion.

Note that aliasing works "both ways". An out-of-band signal in the analog domain can interfere in the digital spectrum if it is above Nyquist range and becomes "aliased" in-band of the digital version. Thus, "anti-alias" filters are typically used before A/D's. In our simplified discussion, these would generally be lowpass filters.

**Quantization and Noise**

The A/D converter provides a digitized version of the signal present at its input. Unlike the analog continuum of possible values, the A/D describes the signal amplitude in terms of a digital number, which uses discrete step increments. The resolution, or the smallest voltage step, is related to the number of bits used to encode, and the analog input amplitude range as defined by the converter. A high performance A/D may have a +/-1-V input range. The number of A/D bits varies, with 8 to 16 bits common (and an even number). Technology is constantly improving, with the 16-bits and higher now available at once impossible sample rates.

The most significant bit (MSB) is half of the full-scale input range, the next bit, (MSB-1), is half of that, and so on down to the LSB. Thus, the LSB for an 8-bit A/D represents a voltage increment of 2^{-8} of the full-scale input range, or to signal level -48 dB below full-scale. The LSB value in voltage is the converter's absolute resolution. You won't easily see such LSB variations on the oscilloscope, although it is reasonable to do so on a good spectrum analyzer or FFT processor interfaced to the digital side of the system. Based on the above relationship, each bit in the converter is worth 6 dB, with the LSBs becoming considerably more expensive when you need 14 bits or more to get to them.

It is apparent from the digitization process that digital samples will be assigned amplitude values that do not exactly match the analog version. This error energy is called quantization noise. Note that the quantization noise will be smaller as more bits of resolution are added (assuming a fixed input range on the A/D), since the step size shrinks. The rms noise voltage is given by [q/sqrt(12)], where q is the voltage weight of the LSB.

Statistically, quantization noise can be approximated as a Uniform distribution. This means that any noise value over the range (-q/2) to (q/2) is equally likely. This is not actually the case for a sine wave, where noise processes also become periodic, but is a good approximation for typical signal types. Unfortunately, the sinusoid is the full-scale waveform used to define the A/D signal-to-quantization-noise (SQNR) ratio, much as it is used as a reference signal type for other common analysis. Thus, on one hand, we represent SQNR using a full scale sine wave, but, in fact, the quantization noise type used in the calculation does not precisely describe the noise process incurred by use of a sine wave. This is one of those seeming paradoxical definitions that you just live with, as the practical nature of a real signal does in fact make the uniform assumption typically a good one.

Another item of note about the noise is that the Uniform probability density function (PDF) is one that is rarely seen in practice. It is best known for its role in academia as a model distribution to help illustrate the concepts of cumulative distributions and PDFs. The value of the variance (noise power) is well-known for this PDF.

**Quantifying SQNR**

The basic, well documented expression for SQNR, assuming a full scale sinusoidal input is:

_{(sample)}/ 2 f

_{(bandwidth)}] + 1.76 dB (1)

Here, N = number of A/D bits, which is why you may often hear about "6 dB per bit." The 1.76 dB is 10 log (3/2), a factor that shows up when you do the variance calculation of the Uniform PDF. Finally, the middle term emphasizes the benefits of over-sampling.

It is important to note that (1) is based on the full scale input. This is important because input levels to a communications or radar receiver — key A/D applications — can vary widely without compensation, and there are penalties for being too below or above full-scale. Effects somewhat mirror those of analog amplifiers in performance issues " the proper level is a trade-off of noise and distortion performance. However, in an A/D (or D/A), there are other more obscure forms of distortion. The SQNR and these other distortions together determine just how much resolution is really available, defined in specifications as Effective Number of Bits (ENOB). A 12-bit A/D may provide 11 "effective" bits of performance due to internal noise and distortions for a particular application.

**Oversampling**

We've discussed how sampling at the Nyquist rate is a theoretical necessity, and how sampling at something higher allows practical filtering. An advantage of sampling significantly higher is improved SQNR. The concept is illustrated in **Figure 3**, and is simple to comprehend. The rms quantization noise level, as described, is fixed by input range and number of bits, and is independent of bandwidth. It is spread evenly over the Nyquist bandwidth. When the Nyquist bandwidth is made wider by the higher sample rate, the noise density (watts/Hz) is lowered to maintain the same total power. Thus, within the desired signal bandwidth, the SQNR is improved. The relationship is directly related to the fractional bandwidth above Nyquist used, where it can be filtered, with improvement as quantified in (1).

**Backoff**

Consider an 8-bit A/D used on the front end of a QAM receiver, and oversampled by 1.5 times. This gives an SQNR over 50 dB. However, the SQNR term is based on a full scale sine wave reference input. In a real application, uncertainties at a receiver's input could be tens of dBs. If the level to the A/D is allowed to vary this much, the impact is dB-for-dB on the defined SQNR because quantization noise power is fixed. A 50-dB SQNR conversion may quickly become, for example, a 30 dB SQNR. Then, rather than being able to ignore the A/D noise impact, it may significantly affect the link budget. The degradation may be tolerable, but this is why automatic gain control (AGC) is commonplace ahead of an A/D circuit.

The second part of this perusal of these unique devices will delve into a few more complex topics, such as IF sampling, numerical representations, and dithering. Some of these advanced techniques have helped A/Ds play an even bigger role within digital world. Still, for us baby-boomer generation technology types, it is still comforting to know that, at its best, these techniques amount to just closer and closer approximations to the real analog content and ways to better move it. Then, my "modern" educational experience doesn't seem so obsolete!

**About the Author**

**Rob Howald** (rhowald@xytrans.com) is the vice president of engineering at Xytrans Inc. He is also the former director of systems engineering in the transmission network systems group of Motorola's Broadband Communications Sector in Horsham, Pa. He has a BSEE and an MSEE from Villanova University, a PhD from Drexel University, and an MBA from DeSales University.

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