## 5. Sampling issues

### 5.2 Dynamic range

A basic signal chain from analogue line input to line output in a networked audio system consists of an A/D converter, a distribution network (LAN), a DSP unit and a D/A converter:

A/D converter circuits are used to sample a continuous analogue (electronic) audio signal at a constant timeinterval, in broadcast and live sound applications normally 20.8 microseconds, corresponding with a sample rate of 48,000 samples per second. Each time interval, the sampling process produces a number to represent the analogue signal using a certain amount of binary digits: the bit depth. For linear A/D converters with a bit depth of 24 bits, the sample can cover 16,777,216 different values. After transport through the audio network and DSP processing, a D/A converter converts the digital samples back to a continuous analogue signal, in most cases with the same sample rate and bit depth.

D/A converter circuits basically consist of one or more highly accurate switched current sources producing an analogue output current as a result of a digital input code. To illustrate the basic concept of a D/A converter, an example of a simple 4-bit binary weighed D/A converter is presented in figure 502. The circuit uses individual resistors for each bit, connected to a summing point. Commonly available D/A converters use more complex high speed techniques such as delta-sigma modulation(*5J).

A/D converters are slightly more complex, using an AD logic component driving a D/A converter. The output of the D/A converter is compared to an analogue input, with the result (higher or lower) driving the AD logic component.

**Quantization error**

In a digital system’s A/D converter, the sampled waveform can never be represented accurately by the digital samples because the value representation of the least significant bit (LSB) is always a little off by up to plus or minus 1/2 LSB. Similar with A/D conversion, DSP MAC operations (Multiply and Accumulate*5K) and transformations from a high bit depth to a lower bit depth have to be rounded to up to plus or minus 1/2 LSB after each processing step. These errors are called quantization errors.

The level of the quantization error in an A/D converter depends on the sample rate, the bit depth and the sampled waveform. The quantization error in a DSP process depends also on the number of steps. As D/A converters are assumed to reproduce the digital representation of the signal - and not the original signal, they do not generate quantization errors. However, a D/A converter can never reach a higher resolution than 1 LSB. Figure 504 illustrates an A/D converter’s quantization error.

In binary numbers, each bit represents a factor of 2, which corresponds to a ratio of 6.02 dB. Although in reality it is much more complex(*5L), as a rule of thumb for A/D converters and DSP operations, a worst case quantization error noise floor of -6.02 dB times the bit depth is often used.

**Linearity error**

Ideal A/D and D/A converters use 100% accurate components - but in real life, electronic components always possess a certain tolerance causing nonlinearity. It is assumed that professional A/D and D/A converters possess a linearity error of less than 1 LSB. Digital transport, networked distribution, storage and DSP processes are all digital and therefore don’t generate linearity errors (unless compression is used eg. MP3).

**Dynamic range**

The internal resolution of dedicated audio DSP chips - the bit depth at which MAC operations are performed in the DSP core - is usually 40 bits or higher, providing enough internal resolution to keep the output of high DSP power algorithms (with many calculations) to stay well above 32 bits. If the audio network protocol used in the system is also 32 bits - eg. Dante in 32-bit mode, the dynamic range of the digital part of a networked audio system can be estimated using the rule of thumb of 6 dB per bit times 32 bits = 192 dB.

As this range is far greater than the theoretical 144 dB dynamic range limit of a system’s A/D and D/A converters, it can be assumed that a system’s dynamic range is mainly limited by just these components. Still, systems available on the market will not reach this value. Electronic circuits such as the power supplies, buffer amps and head amps before the A/D converter, as well as buffer amps and balancing circuits after the D/A converter, add electronic noise to the signal chain. Also, in most designs noise is added intentionally in the digital domain to improve the Performance at very low signal levels (dither)(*5M). A typical dynamic range of a signal chain in a networked audio system - excluding the power amplifier - is 108dB(*5N).