Analog and Digital Conversion/Resolution and Bitrate

Resolution

The resolution of a sampler is the number of bits that are used to represent each signal. For instance, a 12-bit sampler will output 12 bits of data for every sample. This means that there are 2¹² possible digital values that each sample can be converted to. In general, the more bits of resolution, the better (more faithful) the digital signal will be to the original. The resolution, n, is related to the number of steps, m, by the following formula:

2^{n}=m

Unipolar and Bipolar

Samplers come in two basic varieties: unipolar and bipolar. Basically, unipolar samplers only take positive values, and only output unsigned digital values. Bipolar converters can take positive and negative values, and output signed digital values. It is important to note that bipolar converters are generally symmetrical, that is that they have the same number of bits for expressing negative and positive numbers.

Sample Range

The range of possible samples is dependent on a number of factors, including the signed/unsigned number scheme in use by the converter, the resolution, and the step size.

Step Size: The step size of a sampler is the range of analog values that can be input before a bit is changed in the sampler. The step size (Δ) is given by the range (R) divided by the number of steps (m):

\Delta ={\frac {R}{m}}

The range of the sampler, R, is given by the difference between the minimum value, and the maximum value:

R=V_{max}-V_{min}

Note however, that bipolar converters are generally symmetric. That is, they have the same amount of range below zero as they do above zero. If we want a converter that goes from -5V to +25V, we are going to need to get a converter that can handle from -25V to +25V, which means we are wasting at least 2/5ths of the possible range of the device.

Example

For instance, let's say that we have a converter that only takes positive values (nothing below 0 volts), and we have a resolution of 2 bits. Also, we want to be able to handle all input values up to and including 10 volts. This means that our range is from 0 volts to 10 volts:

R=10-0=10

With 2 bits of resolution, we have a total number of steps of:

m=2^{2}=4

And we have a step size of:

\Delta ={\frac {10}{4}}=2.5

This means that the following analog values produce the following digital output patterns:

0V <= V < 2.5V → 00
2.5V <= V < 5.0V → 01
5.0V <= V < 7.5V → 10
7.5V <= V <= 10.0V → 11

Bitrate

The number of bits created per sample, times the sampling frequency, gives us the rate at which we are producing data bits. This rate is called the bitrate, and is frequently denoted as r_b, or simply r.

If we have a sampling time of T seconds, then the bitrate and the resolution are related as such:

r_{b}={\frac {n}{T}}

Where r is measured in units of bits/second, T is measured in seconds, and n is measured in bits.

Bandwidth

Bandwidth, denoted with a W, is the frequency range needed to transmit an analog or digital signal. This isn't directly related to the topic at hand, but is included for completeness. For more information about bandwidth, see Signals and Systems or Communication Systems. This text may use bandwidth as a constraint in example problems, but will not discuss it in any depth beyond this simple section.

Bandwidth is related to the bitrate as follows:

W=2r_{b}

This is for a bare, unmodulated bit stream. This value can change depending on what modulation scheme is used, if any.

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