Example 2

(4.31)

If impulse area is equal to s, then we consider that we have S-scale impulse of -function which is equal to .

The system reaction to -function is called impulse transient characteristic (impulse response), or weighting function .

Together with transient characteristic it is the most important characteristic of system dynamic properties.

If we have a time shift , then -function will have the following form:

(4.32)

Signals in the form of impulse functions may be considered as derivatives of unit step function

Taking into account an order of a derivative we obtain:

- 1^st order impulse function or -function;

- 2^nd order impulse function, …

To prove this statement we may represent the unit step function as the limit of a certain continuous function:

(4.33)

Within the interval we may obtain:

(4.34)

The derivative of the unit step function:

(4.35)

Thus, we obtain:

- -function = 0 at

- -function = at ;

- if then .

This interpretation is convenient, since it allows to obtain analytical expressions for 2^nd, 3^rd, …, k-th order impulse functions.

Their expressions and graphic forms may be obtained as the limits of derivatives of -functions at according to Eq.(5.8).

You may construct -function block using the following blocks from Matlab6.5:

- Step;

- Derivative (du/dt).

The connection between transient characteristic and weighting function may be also established using two step functions.