The discrete time LTI system has the output y(n) described as the convolution of input x(n) and impulse response h(n) assuming that there are no initial conditions.

Therefore, y(n)= x(n) * h(n) …..standard convolution.

The transfer function of a system producing such an output will be

Transfer function=H(z)=(Y(z))/(X(z)) ………………..(1)

This is obvious from above two relations that convolution in time domain is equivalent to multiplication in frequency domain.

When system has unit impulse as input, its output will be called as impulse response,

i.e. when input x(n)=δ(n), output y(n)=h(n)

Substituting these in above equation (1),

Taking z-transform on both the sides,

This proves that, the inverse z-transform of transfer function of LTI system is the impulse response of the system.