# Circuits and Arithmetic

Sunday, December 9, 2012

This is a continuation of Computing with Transistors, a series of blog posts describing how computers work from the ground up. The first post and namesake of the series is available as Computing with Transistors. In this post, I use Verilog to build arithmetic circuits, a language introduced in Digital Design Tools: Verilog and HDLs.

At the most fundamental level, a computer is a device which can do some form of computation. When computers were first being developed, they were mainly used for mathematical computations; now, we use them for things such as media consumption, communication, and art. But all of these are at the lowest level just computations - arithmetic operations on values in memory. The ability to do arithmetic is at the very core of computing.

In order to do any form of arithmetic, we first need to be able to represent and store numbers. The simplest type of number is the integer, with values such as 10, 15, 0, -12, and so on. As we discussed in The Digital State, computer memory is just a series of bits: ones and zeros. So how can we encode any arbitrary integer into a list of bits?

## Two's Complement Notation

Although this seems like an almost trivial question, it turns out there are a lot of different ways to do this. In grade school, we learn that we can represent integers in different bases. When we write "12", what we mean is "1 10 + 2 1". When we write "153", we mean "1 * 100 + 5 * 10 + 3 * 1". But there's nothing unique about base ten - we can write these same values in base two. "10" becomes "1 * 2 + 0 = 2". Any integer from zero to 2^N - 1 can be represented by a string of ones and zeros of length N.

There are some tricky aspects to this representation, though. How do we write negative one, using a series of bits? It turns out this question has several solutions!

The naive solution is to simply add a bit in front of the number to act as a sign (a "sign bit"). If this bit is a zero, then the number is positive, but if the bit is a one, then the number is negative. This is perfectly valid for some cases, but it makes addition very difficult, because you must introduce special hardware to deal with negative numbers.

The second solution, and the one commonly used in modern hardware, is known as two's complement. Two's complement allows one to encode negative numbers with something similar to a sign bit: the first bit of the number, if it is on, contributes -2^(N-1) to the entire sum. Then, the remaining N-1 bits just form a positive integer between 0 and 2^(N-1) - 1. So, if we have a 4-bit integer, how can we write -3? Well, the first bit has to be negative, and is worth -2^3 (-8) to the sum. Then, the remaining 3 bits should form the number 5, so we set them to 101. -8 + 5 = -3. Therefore, in two's complement notation with a 4-bit number, -3 is encoded as 1101.

Suppose we have two numbers, 3 and -3, encoding both in 4-bit numbers using two's complement notation. These are encoded as 0011 and 1101, respectively. If we add them the way we normally add numbers, with no special considerations, we find that we get 10000 - if we drop everything after the 4th bit, we get 0000. This is the great thing about two's complement notation - by adding two N-bit numbers, and discarding any bits after the Nth bit, we get the correct answer, regardless of whether the numbers are positive or negative.

## Adding Integers

Now that we know how to encode integers, how do we add them? Let's start with a simple circuit, made for adding one-bit integers: Simple adder. Given two bit inputs A and B, add them, and output a carry bit which is on if there was overflow.

This circuit takes two 1-bit integers, adds them, and gives an output. It also outputs a carry bit, indicating whether there was overflow. Constructing this circuit is trivial: the carry bit is only on if both inputs are on, and the output bit is on only if A xor B is true. We can realize this with the following digital gates: Simple adder implemented with boolean functions.

This is the first building block of a multi-bit adder, which is what we ultimately want. We're going to need to propagate the carry through many digits, though. In order to do this, we're going to build a circuit called a full adder, which takes three bits and adds them. By convention, the third bit is called the Carry-In bit, since it's usually the output of a Carry. The schematic for a full adder is represented like this: Full adder circuit element diagram.

Building this is a bit more complicated than the previous 2-bit adder, but not much more. The output will be on if an odd number of inputs are on. The carry bit will be on if more than one of the inputs is on. A good exercise to test whether you really understand this is to try creating a full adder using only AND, OR, and NOT gates. If you're feeling adventurous, try to minimize the number of gates you're using, or implement a full adder directly using MOSFET transistors, as described in the first post, Computing with Transistors.

## Multi-Bit Adders

Full adders are great, but they only operate on one-bit numbers. What we want is to add integers with many bits. It turns out that, as usual, there are many ways to do this, and that these ways are a trade-off in speed and complexity.

The simplest method for adding multi-bit integers is a circuit called a ripple-carry adder. A ripple-carry adder performs the same algorithm we learned as children for adding integers. First, it adds the first two bits of the number. Then, it adds the next two bits of the number, as well as the carry from the previous addition. This is repeated until the entire number has been added; the final carry bit is discarded. We can build this quite easily using several full adder circuits. For example, here is a 4-bit ripple-carry adder: Circuit which adds two four-bit numbers. Note that the first carry input starts out at a constant zero!

Why is this called a ripple-carry adder? The carry-bit starts out as zero, as we can see from the schematic, because for the first digit there is no carry. The adders compute a resulting output digit and a carry, and the carry is propagated to the next adder. In this way, the carry bit ripples through the entire adder chain. The first adder must be finished with its computation before the next adder can begin, and the computation ripples through the circuit sequentially.

What's wrong with a ripple-carry adder? In some sense, nothing. It functions exactly as we want it to. However, it's slow: each adder must wait for all the previous adders before it, so as we increase the number of digits in the inputs, the time it takes for the electrical signals to propagate through the entire circuit increases linearly with the number of digits. Each additional digit adds one more adder, and an equal wait time. There are more advanced adding algorithms which can reduce the time required (and shorten the "criticial path" of the circuit), allowing CPUs to use a higher clock frequency; I won't go into these more advanced adders, but a good one to start with is known as the carry-lookahead adder.

## Verilog

Luckily for us, if we use a higher-level design tool such as Verilog, we might never actually have to deal with the details of two's complement notation and adders. These things are so fundamental that they are built-in to Verilog (in a sense); like in programming languages, Verilog allows you to write the sum of two binary busses using a plus '+' operator, and it implicitly creates an adder circuit in order to compute that sum. Similarly, Verilog assumes by default that your internal representation is two's complement, so you do not have to do anything.

With that said, let's look at a simple computational circuit to compute Fibonacci numbers quickly. This Fibonacci circuit will have a clock signal, a reset signal, an input 6-bit integer, and an output 'ready' signal, and an output 32-bit integer.

module fib(input clock, reset, input [5:0] n, output ready, output [31:0] value)
// Computational circuit
endmodule

In order to use the Fibonacci circuit, we first have to pulse the 'reset' signal. In order to compute Fibonacci numbers, the circuit will use three internal registers to store its state. When the circuit is reset, it sets the initial values on those registers to be the first two Fibonacci numbers, and the index of the Fibonacci number that has been counted so far.

module fib(input clock, reset, input [5:0] n, output ready, output [31:0] value)

reg [31:0] previous, current;
reg [5:0] counter;

always @(posedge reset)
begin
previous <= 32'd0;
current <= 32'd1;
counter <= 32'd1;
end

endmodule

After pulsing the reset signal, we have to continually pulse the clock signal. Once the circuit is done computing, it will set the 'ready' signal to high, and we will know that it is done.

module fib(input clock, reset, input [5:0] n, output reg ready, output [31:0] value)

reg [31:0] previous, current;
reg [5:0] counter;

// Reset the circuit
always @(posedge reset)
begin
previous <= 32'd0;
current <= 32'd1;
counter <= 32'd1;
end

// Compute next Fibonacci number
always @(posedge clock)
begin
// Increment current index
counter <= counter + 1;

// Efficient adders are automatically inferred
current <= current + previous;
previous <= current;

if (counter == n)
ready <= 1;
end

// Read the value of the nth fibonacci number from the internal register
assign value = current;

endmodule

This circuit can compute the $$n$$th Fibonacci number in $$n$$ clock cycles. You can simulate this using a Verilog simulator such as iverilog to test the circuit, or synthesize it onto an FPGA to experiment with it in hardware. The beautiful thing about hardware - especially if you're coming from a software background - is how incredibly fast it can be. In a thousand cycles we can compute the 1000th Fibonacci number: doing this in software, even in assembly language, will take a much longer time. The more specialized your hardware, the faster it can be; this is why we can design circuits to take the Fourier transform of a signal with incredible speed, while doing this on a conventional general-purpose CPU can take quite a while.