Fixed point arithmetic  (Page 5/4)

Fixed-point arithmetic

This handout explains how numbers are represented in the fixed point TI C6211 DSP processor. Because hardware can only storeand process bits , all the numbers must be represented as a collection of bits. Each bit representseither "0" or "1", hence the number system naturally used in microprocessors is the binary system. This handout explainshow numbers are represented and processed in DSP processors for implementing DSP algorithms.

How numbers are represented

A collection of $N$ binary digits (bits) has $2^N$ possible states. This can be seen from elementary counting theory, which tells us that there are twopossibilities for the first bit, two possibilities for the next bit, and so on until the last bit, resulting in $$2\times 2\times 2=2^N$$ possibilities or states. In the most general sense, we can allow these states to represent anything conceivable. Thepoint is that there is no meaning inherent in a binary word, although most people are tempted to think of them aspositive integers. However, the meaning of an $N$ -bit binary word depends entirely on its interpretation .

Unsigned integer representation

The natural binary representation interprets each binary word as a positive integer. For example, weinterpret an 8-bit binary word $$b_7{b}_6{b}_5{b}_4{b}_3{b}_2{b}_1{b}_0$$ as an integer $$x=b_{7}2^7+b_{6}2^6++b_1\times 2+b_0=\sum_{i=0}^7 2^i{b}_i$$ This way, an $N$ -bit binary word corresponds to an integer between $0$ and $2^N-1$ . Conversely, all the integers in this range can be represented by an $N$ -bit binary word. We call this interpretation of binary words unsigned integer representation, because each word corresponds to a positive (or unsigned) integer.

We can add and multiply two binary words in a straightforward fashion. Because all the numbers are positive, the results ofaddition or multiplication are also positive.

However, the result of adding two $N$ -bit words in general results in an $N+1$ bits. When the result cannot be represented as an $N$ -bit word, we say that an overflow has occurred. In general, the result of multiplying two $N$ -bit words is a $2N$ bit word. Note that as we multiply numbers together, the number of necessary bits increasesindefinitely. This is undesirable in DSP algorithms implemented on hardware. So, later we will introduce the fractional interpretation of binary words, to overcome this problem.

Another problem of the unsigned integer representation is that it can only represent positive integers. Torepresent negative values, naturally we need a different interpretation of binary words, and we introduce the two's complement representation and corresponding operations to implement arithmetic on thenumbers represented in the two's complement format.

Two's complement integer representation

Using the natural binary representation, an $N$ -bit word can represent integers from $0$ to $2^N-1$ . However, to represent negative numbers as wellas positive integers, we can use the two's complement representation. In 2's complement representation, an $N$ -bit word represents integers from $-2^{(N-1)}$ to $2^{(N-1)}-1$ .