# Analog and Digital Conversion/Signed and Unsigned Quantities

## Binary Number Representation

Any number can be represented using only "bits" (the digits 0 and 1); for example, the binary number 1101 represents thirteen. These bits, ordered in sequence, represent ascending powers of two.

${\displaystyle 1\times 2^{3}+1\times 2^{2}+0\times 2^{1}+1\times 2^{0}=thirteen}$

The bit on the right is called the "Least Significant Bit" (LSB) because it represents 1, the lowest power of 2; and the bit on the left is called the "Most Significant Bit" (MSB) because it represents the highest power of two.

## Unsigned Numbers

Unsigned numbers are represented in a straightforward manner, with the LSB on the right, and the MSB on the left. Unsigned numbers are always positive (or zero), because they don't have a sign for denoting negative numbers.

## Signed Numbers

To represent negative numbers, we need to implement signed numbers. There are two general schemes for representing signed numbers: sign–magnitude, or two's compliment. There are other schemes as well, such as one's compliment, but we won't discuss them here. For more information on the subject, see the relevant sections of Digital Circuits.

### Sign–Magnitude

Sign–magnitude numbers are the same as unsigned numbers, except with the addition of a "sign bit". If the sign bit is 0, the number is positive; if the sign bit is 1, the number is negative.

### Two's Complement

The digital logic required to implement two's complement is significantly more simple to implement than a sign–magnitude representation. Therefore, most computers store values in two's complement format. Two's compliment numbers follow these rules:

1. Half the numbers are positive, from 0 to (N/2)−1, where N is the number of expressible values (including zero) that are possible with the number of bits. If we have n bits, this value is:
${\displaystyle N=2^{n}}$
2. Negative numbers are obtained by inverting the bits in the positive number, and then adding 1 to it.

For instance, if we have the number 5 (0101) in a four-bit two's compliment number, we can get the representation for −5 by inverting the number (1010), and adding 1 to it (1011).

To get the positive value of a negative number, we reverse the process.

For example, the 4 bit signed numbers (3 data bits + 1 sign bit) are:

 0111 = +7
0110 = +6
0101 = +5
0100 = +4
0011 = +3
0010 = +2
0001 = +1
0000 = +0
1111 = −1
1110 = −2
1101 = −3
1100 = −4
1011 = −5
1010 = −6
1001 = −7
1000 = −8


### Questions

Q) Didn't you just finish telling me a few pages ago that 1101 means thirteen?

Yes. It could also be a lot of other things too. The bits 1101 form a bit pattern that can be interpreted to mean anything. When we're talking about unsigned arithmetic, 1101 is "13" in decimal numbers. When we're talking about signed arithmetic, 1101 might mean −3 if we're doing 4-bit signed arithmetic.

• In 16-bit signed arithmetic, 1101 is 13
• In sign–magnitude representation, 1101 is −5.
• In one's compliment, 1101 is −4
It's important to note that a single bit pattern might have different meanings depending on how you interpret it. Keeping the interpretation straight is therefore very important.

## Floating-Point Numbers

An additional type of binary numbers are called "Floating Point" numbers. These numbers allow fractional quantities to be expressed using bits. Floating-point numbers are beyond the scope of this book, but keep in mind that some samplers output values in floating point format.