EECS20N: Signals and Systems

Function composition

If f1: X Y and f2: X' Y', and Y is a subset of X', then the composition of f1 and f2 is the function f3: X Y' such that

for all  x Xf3(x) = f2(f1(x))

The function f3 has the same effect as first applying f1, then applying f2, as shown below:

  • The function f3 is written f3 = f2 · f1
  • Note that f1 is applied first, despite being to the right of f2.
  • If the domain and the range of a function are the same, i.e.  f: X → X, then f 2 = f · f, f 3 = f 2 · f,
Recall that a modem transmitter is a function

ModemTransmitter: BitSequencesSounds.

The telephone network is a function

TelephoneNetwork: SoundsSounds.

A modem receiver is a function

ModemReceiver: SoundsBitSequences.

Ideally, a modem design is a function such that

ModemReceiver · TelephoneNetwork · ModemTransmitter  = Identity

where Identity: BitSequencesBitSequences is an identity function, where

for all xBitSequences, Identity(x) = x.

The challenge, of course, is to come up with a design that satisfies this property for a wide range of functions TelephoneNetwork.