3.4 Quantisation of a signal

When a continuous-time signal is sampled, the amplitude of each sampled point undergoes quantisation which means that it is forced to have only certain discrete values. The amplitude of each sample is represented by a digital binary code, and the word length of the code will be a fixed number of digital bits. Representing the amplitude of the samples in this way means the value is represented by a binary number, so it is truncated or quantised to its closest binary equivalent.

A 1-bit binary number can represent two levels because it can only take a value of 0 or a value of 1. A 2-bit binary number represents four levels, where each level takes one of the values 00, 01, 10 or 11. Figure 17 shows a discrete-time signal whose values are limited to a 3-bit binary number, which represents eight possible combinations of 0s and 1s – each possible combination is shown on the -axis of the figure. Note that each sampled, digitised signal is a binary representation of the analogue value, so in conversion to these binary representations some rounding of the signal values has occurred: some sampled values are just above and some just below the continuous-time signal.

Figure 17 Sampled and quantised signal

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This figure is a graph of x[n] against n. The vertical axis shows eight evenly spaced levels consisting of three-bit binary numbers ranging from 000 to 111, and horizontal lines are drawn across the graph to indicate these levels. A graph line representing the signal is shown fluctuating up and down. A series of evenly spaced vertical lines, representing the samples, extend from the horizontal axis to the horizontal level line to which the signal is nearest at that time. Each vertical line has a small filled circle at the point where the vertical line meets the horizontal level line.

Figure 17 Sampled and quantised signal

Any binary word will always give an even number of quantisation levels. In general, a binary word with bits gives quantisation levels – hence a 3-bit word gives 8 levels, a 4-bit word gives 16 levels, a 5-bit word gives 32 levels, etc.

Figure 18 shows a sine wave along with a set of quantisation levels. Here the 0 or midpoint of the sine wave occurs at a midpoint between the 011 and 100 levels. The gaps between the quantisation levels are called the quantisation intervals. Sometimes a quantisation level is assigned to the 0 or midpoint, which means that on one side of the 0 there is one more level than on the other side.

Figure 18 A sine wave and a set of quantisation levels

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This figure is a graph of signal value against time. The horizontal axis extends from 0 to 1 units of time in increments of 0.1. The vertical axis is divided evenly into the same eight levels as in the previous figure, each represented by a three-bit binary number from 000 to 111. The distance between two levels is the quantisation interval.

One cycle of a sine wave is shown on the graph: it starts midway between the 011 and 100 levels at time 0, rises to reach the maximum 111 level at time 0.25, falls to cross the midway point again at time 0.5, falls further to reach the minimum 000 level at time 0.75, then rises back to the midway point at time 1. Overlaid on the sine wave is another line consisting of a series of steps, showing what the sine wave would look like if it were quantised to the eight levels. This stepped line remains at each level until the sine wave crosses the halfway point between that level and the one above or below it, at which point the stepped line rises or falls vertically to the new level.

Figure 18 A sine wave and a set of quantisation levels

The larger the quantisation intervals, the more error will be introduced into the sampled signal. The quantisation error for a sample is the difference between the value of the input signal and the resultant quantised signal, with the maximum quantisation error being half the quantisation interval. Quantisation errors can be reduced by increasing the number of levels; however, as the number of levels increases, so does the number of bits needed to represent each sample.

In the next section you will return to digital filtering and how they work in the time domain.