# SICP - Solution: Exercise 1.44

##### October 29, 2018

## 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 obtain then-fold smoothed function. Show how to generate the $n$-fold smoothed function of any given function using`smooth`

and`repeated`

from Exercise 1.43.

## Solution #

By definition, the `smooth`

function is:

```
(define (smooth f)
(lambda (x)
(/ (+ (f (- x dx)) (f x) (f (+ x dx))) 3)))
```

Then we can define `smooth-nth`

:

```
(define (smooth-nth f n)
((repeated smooth n) f))
```