Question

A linear, time-invariant system has the following impulse response:

(a) Describe in words the effect of this system.

(b) What is the corresponding polynomial

Short answer

Answer is not given by the question.

Step-by-step solution

Step – 1:

a) consequence of the method:

From the urge response,

ẟ(t) =$\left\{\frac{1}{t_0},t\le t_{0}0,t>t_0-\right\}$ .....1.

 Step – 2:

The response of the system stays constant with both the value of $\frac{1}{t_0}$until it approaches zero t = t₀after extending the (t₀) duration.

As a result, the effect of the system's original signal is defined.

b)

The following is a general representation of a polynomial equation:

Based on the impulsive reaction,

B(x) = b₀ + b₁x + ....... +btx^t ......2.

Extend Equation (1)'s auto - correlation functional as follows::

Whent=0, b₀ = $\frac{1}{t_0}$ .......3.

When t=1, b₁ = $\frac{1}{t_0}$ ……4.

When t=2, b₂ = $\frac{1}{t_0}$ ......5.

When t= t₀, b_t0 = $\frac{1}{t_0}$ .......6.

When t= t₀+ 1, b_t0 = $\frac{1}{t_{0+1}}$ …… (7)

To find the polynomial, change formula (3), argument (4), equation (5), equation (6), then equation (7) in equation (2).

$$\begin{gathered} B\left(x\right)=\frac{1}{t_0}+\frac{1}{t_0}x+\frac{1}{t_0}{x}^2+.....+\frac{1}{t_0}{x}^{t_0} \\ =\frac{1}{t_0}\left(1+x+x^2+....+x^{t_0}\right)\dots \dots \left(8\right) \end{gathered}$$

Therefore, the corresponding polynomial is:

B(x) =$\frac{1}{t_0}$ (1 + x + x^{2 + ......+}x^t0)