Ripple-carry adder

Figure 3.27 shows a ripple-carry adder formed by stringing together n full-adders. This is the simplest form of parallel adder for adding two n-bit binary numbers. The inputs A₁, A₂, A₃, ..., Aₙ and B₁, B₂, B₃, ..., Bₙ are the two binary numbers to be added (each Aₖ and Bₖ is a 0 or a 1). The circuit generates S₁, S₂, S₃, ..., Sₙ, the n bits of the sum, and C, the carry from the addition. Write a procedure ripple-carry-adder that generates this circuit. The procedure should take as arguments three lists of n wires each -- the Aₖ, the Bₖ, and the Sₖ -- and also another wire C. The major drawback of the ripple-carry adder is the need to wait for the carry signals to propagate. What is the delay needed to obtain the complete output from an n-bit ripple-carry adder, expressed in terms of the delays for and-gates, or-gates, and inverters?


Figure 3.27: A ripple-carry adder for n-bit numbers.

There are no comments yet.

Authentication required

You must log in to post a comment.

(define (make-queue) (cons '() '()))

(define (front-ptr queue) (car queue))
(define (rear-ptr queue) (cdr queue))
(define (set-front-ptr! queue item) (set-car! queue item))
(define (set-rear-ptr! queue item) (set-cdr! queue item))

(define (empty-queue? queue) (null? (front-ptr queue)))

(define (front-queue queue)
  (if (empty-queue? queue)
      (error "FRONT called with an empty queue" queue)
      (car (front-ptr queue))))

(define (delete-queue! queue)
  (cond ((empty-queue? queue)
         (error "DELETE! called with an empty queue" queue))
         (set-front-ptr! queue (cdr (front-ptr queue)))

(define (insert-queue! queue item)
  (let ((new-pair (cons item '())))
    (cond ((empty-queue? queue)
           (set-front-ptr! queue new-pair)
           (set-rear-ptr! queue new-pair)
           (set-cdr! (rear-ptr queue) new-pair)
           (set-rear-ptr! queue new-pair)

(define (get-signal wire)
  (wire 'get-signal))
(define (set-signal! wire new-value)
  ((wire 'set-signal!) new-value))
(define (add-action! wire action-procedure)
  ((wire 'add-action!) action-procedure))

(define (call-each procedures)
  (if (null? procedures)
        ((car procedures))
        (call-each (cdr procedures)))))

(define (make-wire)
  (let ((signal-value 0) (action-procedures '()))
    (define (set-my-signal! new-value)
      (if (not (= signal-value new-value))
          (begin (set! signal-value new-value)
                 (call-each action-procedures))
    (define (accept-action-procedure! proc)
      (set! action-procedures (cons proc action-procedures))
    (define (dispatch m)
      (cond ((eq? m 'get-signal) signal-value)
            ((eq? m 'set-signal!) set-my-signal!)
            ((eq? m 'add-action!) accept-action-procedure!)
            (else (error "Unknown operation -- WIRE" m))))

(define (make-time-segment time queue)
  (cons time queue))
(define (segment-time s) (car s))
(define (segment-queue s) (cdr s))

(define (make-agenda) (list 0))
(define (current-time agenda) (car agenda))
(define (set-current-time! agenda time)
  (set-car! agenda time))
(define (segments agenda) (cdr agenda))
(define (set-segments! agenda segments)
  (set-cdr! agenda segments))
(define (first-segment agenda) (car (segments agenda)))
(define (rest-segments agenda) (cdr (segments agenda)))
(define (empty-agenda? agenda)
  (null? (segments agenda)))

(define the-agenda (make-agenda))
(define inverter-delay 2)
(define and-gate-delay 3)

(define (remove-first-agenda-item! agenda)
  (let ((q (segment-queue (first-segment agenda))))
    (delete-queue! q)
    (if (empty-queue? q)
        (set-segments! agenda (rest-segments agenda)))))

(define (first-agenda-item agenda)
  (if (empty-agenda? agenda)
      (error "Agenda is empty -- FIRST-AGENDA-ITEM")
      (let ((first-seg (first-segment agenda)))
        (set-current-time! agenda (segment-time first-seg))
        (front-queue (segment-queue first-seg)))))

(define (after-delay delay action)
  (add-to-agenda! (+ delay (current-time the-agenda))

(define (add-to-agenda! time action agenda)
  (define (belongs-before? segments)
    (or (null? segments)
        (< time (segment-time (car segments)))))
  (define (make-new-time-segment time action)
    (let ((q (make-queue)))
      (insert-queue! q action)
      (make-time-segment time q)))
  (define (add-to-segments! segments)
    (if (= (segment-time (car segments)) time)
        (insert-queue! (segment-queue (car segments))
        (let ((rest (cdr segments)))
          (if (belongs-before? rest)
               (cons (make-new-time-segment time action)
                     (cdr segments)))
              (add-to-segments! rest)))))
  (let ((segments (segments agenda)))
    (if (belongs-before? segments)
         (cons (make-new-time-segment time action)
        (add-to-segments! segments))))

(define (and-gate a1 a2 output)
  (define (and-action-procedure)
    (let ((new-value
           (logical-and (get-signal a1) (get-signal a2))))
      (after-delay and-gate-delay
                   (lambda ()
                     (set-signal! output new-value)))))
  (add-action! a1 and-action-procedure)
  (add-action! a2 and-action-procedure)

(define (or-gate a b output)
  (let ((c (make-wire))
        (d (make-wire))
        (e (make-wire))
        (f (make-wire))
        (g (make-wire)))
    (and-gate a a c)
    (and-gate b b d)
    (inverter c e)
    (inverter d f)
    (and-gate e f g)
    (inverter g output)))

(define (logical-and x y)
    (if (and (= 1 x) (= 1 y))

(define (inverter input output)
  (define (invert-input)
    (let ((new-value (logical-not (get-signal input))))
      (after-delay inverter-delay
                   (lambda ()
                     (set-signal! output new-value)))))
  (add-action! input invert-input)
(define (logical-not s)
  (cond ((= s 0) 1)
        ((= s 1) 0)
        (else (error "Invalid signal" s))))

(define (propagate)
  (if (empty-agenda? the-agenda)
      (let ((first-item (first-agenda-item the-agenda)))
        (remove-first-agenda-item! the-agenda)

(define (full-adder a b c-in sum c-out)
  (let ((s (make-wire))
        (c1 (make-wire))
        (c2 (make-wire)))
    (half-adder b c-in s c1)
    (half-adder a s sum c2)
    (or-gate c1 c2 c-out)

(define (half-adder a b s c)
  (let ((d (make-wire)) (e (make-wire)))
    (or-gate a b d)
    (and-gate a b c)
    (inverter c e)
    (and-gate d e s)

(define A (make-wire)) 
(define B (make-wire))
(define S (make-wire))
(define C (make-wire))

(ripple-carry-adder (list A) (list B) (list S) C) 
(check-equal? (get-signal C) 0)
(check-equal? (get-signal S) 0)

(set-signal! A 1)
(ripple-carry-adder (list A) (list B) (list S) C) 
(check-equal? (get-signal C) 0)
(check-equal? (get-signal S) 1)

(set-signal! B 1)
(ripple-carry-adder (list A) (list B) (list S) C) 
(check-equal? (get-signal C) 1)
(check-equal? (get-signal S) 0)

(ripple-carry-adder (list A) (list B) (list S) C) 
(check-equal? (get-signal C) 1)
(check-equal? (get-signal S) 1)