Digital Number Representation

In addition to an understanding of modulation basics, it is also helpful to have at least an elementary understanding of numeric representation as employed in the digital world. The basic building block of digital numerics is the bit, which can only take on two values — 0 or 1. Bits may be concatenated to form larger numbers just like decimal digits. For example, a single digit in the decimal system can take on values from 0 to 9, but the number 148 uses three digits to represent the sum: 8(100) + 4(101) + 1(102) = 8 + 40 + 100 = 148. Each digit carries an increasing power of 10 weighting starting with the rightmost digit (ones, tens, hundreds…). Similarly, binary numbers are made up of bits carrying a power of two weighting starting with the rightmost bit. For example, 10010100 = 0(20) + 0(21) + 1(22) + 0(23) + 1(24) + 0(25) + 0(26) + 1(27) = 0 + 0 + 4 + 0 + 16 + 0 + 0 + 128 = 148.

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The benefit of binary number representation is not readily apparent. For instance, binary numbers can become very cumbersome to work with. Consider the number 1 million. It can be expressed with 7 decimal digits, whereas 20 binary digits are required; not very efficient in terms of notation. However, the real beauty of binary numbers comes from the fact that each digit can take on only two values. This is readily modeled by the state of a switch (on or off), which can be electronically implemented with a single transistor. Transistors, in turn, can be physically realized by the millions on a single silicon chip. The ability to place millions of binary switches on a single chip is what gives digital its advantage.

Returning to the concept of modulation, it was pointed out that modulation is applied in the context of sinusoids. Since a sinusoid is represented by a trigonometric function, it can take on positive or negative values. So, digital modulation will require the ability to represent negative binary values. From a notation point of view this is utterly simple — place a negative sign to the left of the leftmost bit. However, from the point of view of a physical implementation a negative sign does not exist.

To tackle the negative number problem, the concept of twos-complement binary representation was developed. In this system, the leftmost bit carries the positive/negative information. The leftmost bit is often referred to as the most significant bit or MSB. The remaining bits carry the magnitude information. Two's-complement numbers, in which the MSB is 0, are positive numbers, while those for which the MSB is 1 are negative numbers. When the MSB is 0 (positive numbers) the non-MSB bits are taken as an ordinary binary number. For example, the two's-complement number 0101 is +5 in decimal notation. When the MSB is 1 (negative numbers) the non-MSB bits are first inverted (i.e., 0s become 1s and vice versa) and then 1 is added to the result. For example, the two's-complement number 1101 is -3 in decimal notation. The MSB indicates a negative value, while the remaining bits (101) are inverted to yield 010, or 2 in decimal. Then 1 is added to the result to make 3. So, the end result is -3 as indicated by the MSB. Although this may seem complicated it is readily implemented in hardware using fundamental digital building blocks.

It should be noted that the digital implementation of an analog function requires a certain amount of compromise. For instance, an analog signal is not a numeric quantity, but a physical quantity. A digital signal, on the other hand, is a numeric quantity and serves only to model an analog signal.

Digital systems rely on a compromise between absolute numeric accuracy and sufficient numeric accuracy. For example, the number of amplitude values that constitute an analog sinusoidal signal is infinite. That is, we can think of an analog sinusoid as being made up of infinitely small steps in amplitude. If we elect to make the steps larger, we sacrifice accuracy for the luxury of requiring fewer numbers with which to represent the purely analog waveform.

In effect, we can trade an absolutely pure analog sinusoid for one made up of small (but finite) steps plus some noise (the deviation between each step and the ideal analog equivalent). The step size is directly related to digital resolution. Resolution is the number of bits used to represent the full range of amplitude of an analog signal. For example, a 10-bit binary number can represent an analog signal to an accuracy of 1 part in 210, or 1/1024 (approximately 0.1 percent).