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3.2. Discrete LTI System(an alternative approach) x(n)   ->   y(n)

Time invariance:

x(n+m)   ->   y(n+m)

Linearity:

 x1(n)  ->   y1(n) => ax1(n) + bx2(n)   ->   ay1(n) + by2(n) x2(n)  ->   y2(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, (n)   ->   h(n).

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 h(n+m) => zn z-n z-m (n+m)   ->   zn z-n z-m h(n+m) => zn  (n+m) z-n-m   ->   zn h(n+m) z-n-m => zn   ->   zn h(k) 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).

2) Finding the output y(n): X(z) = x(n) z-n, Y(z) = y(n) z-n zn-1  ->  H(z) zn-1 => X(z) zn-1  ->  H(z) X(z) zn-1

Cauchy formula (from Complex analysis) =>

 x(n) = 1 X(z) zn-1 dz -> 1 H(z) X(z) zn-1 dz = y(n)  2 j 2 j

Thus

 y(n) = 1 H(z) X(z) zn-1 dz = -1 [H(z) X(z)] 2 j

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

y(n) = h(k) x(n-k)

and

Y(z) = H(z) X(z),

 H(z) = Y(z) . X(z)

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

y(n) = h(k) x(n-k).

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

y(n) = h(k) x(n-k).

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