x(_{1}n) -> y(_{1}n) |
=> ax(_{1}n) + bx(_{2}n) -> ay(_{1}n) + by(_{2}n) |

x(_{2}n) -> y(_{2}n) | |

The discrete unit impulse function is defined by

(n) = |
1, | n = 0 |

0, | n 0. | |

Let the systems output corresponding to the unit impulse input be *h*(*n*),
the **impulse response** of the system,

1) ** The transfer function** *H*(*z*):

By using the linearity and the time invariance saadaan:

(n+m) -> h(n+m) | |

=> | z(^{-m} n+m) -> z(^{-m} hn+m) |

=> | z(^{n} z^{-n} z^{-m} n+m) -> z(^{n} z^{-n} z^{-m} hn+m) |

=> | z(^{n} n+m) z(^{-n-m} ->
z^{n}hn+m) z^{-n-m} |

=> | z(^{n} -> z^{n}hk) z^{-k} |

=> | H(z) = h(k) z^{-k} |

*H*(*z*) is called the **transfer function** of the system. It is the *Z*-transform of the
impulse response *h*(*n*).

**Example 1: Finding the transfer function**

2) ** Finding the output** *y*(*n*):

X(z) |
= x(n) z(^{-n},
Yz) = y(n) z^{-n} |

z(^{n-1} -> Hz) z^{n-1} | |

=> | X(z) z(^{n-1} -> Hz) X(z) z^{n-1} |

Cauchy formula (from Complex analysis) * => *

x(n) = |
1 | X(z) z -> ^{n-1} dz |
1 | H(z) X(z) z(^{n-1} dz = yn) |

2j |
2j | |||

Thus

y(n) = |
1 | H(z) X(z) z [^{n-1} dz = ^{-1}H(z) X(z)] |

2j | ||

i.e. *y*(*n*)* = h*(*n*)* * x*(*n*), from which we obtain the formula for the output

and

H(z) = |
Y(z) |
. |

X(z) | ||

A discrete system is causal if *h*(*n*) = 0, when *n < *0. Then

If also the input is causal, *x*(*n*) = 0 when *n* < 0, then

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