# Exercise 1.44 of SICP

Exercise 1.44: The idea of smoothing a function is an important concept in signal processing. If f is a function and dx is some small number, then the smoothed version of f is the function whose value at a point x is the average of f(x - dx), f(x), and f(x + dx). Write a procedure smooth that takes as input a procedure that computes f and returns a procedure that computes the smoothed f. It is sometimes valuable to repeatedly smooth a function (that is, smooth the smoothed function, and so on) to obtained the n-fold smoothed function. Show how to generate the n-fold smoothed function of any given function using smooth and repeated from exercise 1.43.

(define (compose f g)
(lambda (x) (f (g x))))

(define (repeated fn n)
(if (= n 1)
fn
(compose fn (repeated fn (- n 1)))))

(define (smooth f)
(define (average a b c)
(/ (+ a b c) 3))
(let ((dx 0.00001))
(lambda (x)
(average (f (- x dx)) (f x) (f (+ x dx))))))

(define (n-fold-smooth f n)
(if (not (> n 0))
f
(let ((smoothed-n (repeated smooth n)))
(smoothed-n f))))

(define (S x)
(define (>= a b) (not (< a b)))
(cond ((< x 0) 0.0)
((>= x 0) 1.0)))


S is a modified (the argument is not floored) Heaviside step function.

> ((n-fold-smooth S 0) 0)
1
> ((n-fold-smooth S 1) 0)
.6666666666666666
> ((n-fold-smooth S 15) 0)
.5409601581500249