Two systems h[n] = A(b1)n u[n] and h2[n] = A(b2)n u[n] are cascaded. If the effective \(H(z)=\frac{4}{4-z^{-2}}\), find a possible value of (A, b1, b2)

Explanation:

Analysis of the Given Problem:

The problem involves cascading two systems with impulse responses h₁[n] = A(b₁)ⁿu[n] and h₂[n] = A(b₂)ⁿu[n]. The overall transfer function of the cascaded system is given as:

H(z) = 4 / (4 - z^-2).

We need to determine the possible values of A, b₁, and b₂ that satisfy the given transfer function. Let us solve step-by-step:

Step 1: Analyze the impulse response and transfer function of individual systems

The impulse response of the first system is:

h₁[n] = A(b₁)ⁿu[n],

where u[n] is the unit step function. The Z-transform of h₁[n] is:

H₁(z) = A / (1 - b₁z^-1) for |z| > |b₁|.

Similarly, the impulse response of the second system is:

h₂[n] = A(b₂)ⁿu[n].

The Z-transform of h₂[n] is:

H₂(z) = A / (1 - b₂z^-1) for |z| > |b₂|.

Step 2: Combine the two systems

When the two systems are cascaded, the overall transfer function is the product of the transfer functions of the individual systems:

H(z) = H₁(z) × H₂(z).

Substituting the expressions for H₁(z) and H₂(z):

H(z) = [A / (1 - b₁z^-1)] × [A / (1 - b₂z^-1)].

H(z) = A² / [(1 - b₁z^-1)(1 - b₂z^-1)].

Step 3: Match the given H(z)

The given H(z) is:

H(z) = 4 / (4 - z^-2).

To match this with the derived expression, rewrite the denominator of the given H(z):

4 - z^-2 = (2 - z^-1)(2 + z^-1).

So, the given H(z) can be expressed as:

H(z) = 4 / [(2 - z^-1)(2 + z^-1)].

Comparing this with the derived H(z), we identify:

b₁ = 1/2, b₂ = -1/2, and A² = 1.

Thus, A = 1 (since A must be positive), b₁ = 1/2, and b₂ = -1/2.

Correct Option: Option 3 (A = 1, b₁ = 1/2, b₂ = -1/2)