An Iterative Function $$Oops$!...$ I Did It Again$[40$ points$]$

Consider the iterative function

f$($x$)=$ a x $+$ b$,$

where a and b are fixed $8-$bit constants, and x is an $8-$bit register that is iteratively

updated according to

xnext $=$ a xcurrent $+$ b$.$

Example Iteration

Suppose a $=2,$ b $=3.$ Start with an initial $8-$bit value x $=5.$ Then the iteration

proceeds as follows $($in decimal just for illustration$)$:

x $=5$

x$=2*5+3=13$

x $=13$

x$=2*13+3=29$

x $=29$

x$=2*29+3=61$

x $=61$

x$=2*61+3=125$

x $=125$

x$=2*125+3=253$

x $=253$

x$=2*253+3=509($but in $8-$bit arithmetic, this wraps around!$)$

so

x$=509-256=253$

Once we reach values beyond the representable range of $8$ bits, the arithmetic wraps

around $($i$.$e$.$ it is computed modulo $256).$ Therefore, repeated iteration can generate

interesting results. In this particular example, the value x $=253$ becomes a fixed point

because further updates do not change x$.$

Problem

You have two $8-$bit constant inputs a $=$ a$0,...,$ a$7$ and b $=$ b$0,...,$ b$7,$ and an $8-$bit

variable register x $=$ x$0,...,$ x$7$ that you load externally. The goal is to iteratively

compute

x $=$ a $$ x $+$ b

on each clock cycle. If and when the circuit reaches a fixed point $($i$.$e$.$ xnext $=$ xcurrent$),$

it should set a Done signal to $1$ and place the final x value on an $8-$bit output Z$.$

In your design, you can use the following $$blocks$$ without specifying their internal

structure: D flip$-$flops, multiplexers, adder circuits, and a comparator circuit $($from

Homework $4).$ Specify everything else.

EE $2301,$ Fall $245$

$($a$)$ Multiplier Units. $[5$ points$]$

$$ Explain how to implement an $8-$bit unsigned multiplier for the product a$$x$.$

You may choose a simple shift$-$and$-$add design or a similar approach.

$($b$)8-$bit