Find the inverse z-transform of 1 = (2)X (1– az¯1)? |z|>|a| |
Transfer function
A transfer function (also known as system function or network function) of a system, subsystem, or component is a mathematical function that modifies the output of a system in each possible input. They are widely used in electronics and control systems.
Convolution Integral
Among all the electrical engineering students, this topic of convolution integral is very confusing. It is a mathematical operation of two functions f and g that produce another third type of function (f * g) , and this expresses how the shape of one is modified with the help of the other one. The process of computing it and the result function is known as convolution. After one is reversed and shifted, it is defined as the integral of the product of two functions. After producing the convolution function, the integral is evaluated for all the values of shift. The convolution integral has some similar features with the cross-correlation. The continuous or discrete variables for real-valued functions differ from cross-correlation (f * g) only by either of the two f(x) or g(x) is reflected about the y-axis or not. Therefore, it is a cross-correlation of f(x) and g(-x) or f(-x) and g(x), the cross-correlation operator is the adjoint of the operator of the convolution for complex-valued piecewise functions.
![**Problem Statement:**
Find the inverse z-transform of
\[ X(z) = \frac{1}{(1 - az^{-1})^2} \]
for \(|z| > |a|\).
**Solution Approach:**
To find the inverse z-transform of the given expression, we can utilize properties or known inverse z-transforms and convolution theorems. The inverse z-transform of \(\frac{1}{(1 - az^{-1})}\) is known, and then convolution techniques can be applied due to the power of the denominator. As a result, the convolution yields the required time-domain signal.
**Key Points:**
- The region of convergence \(|z| > |a|\) implies a causal sequence.
- Since the expression involves a power, repeated transformations or convolutions must be done.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fba17cfa5-7b98-4692-8eea-0a66a76af76a%2F41ef42f7-b299-4397-99ce-e6a86a8c19fb%2Fi3bwms_processed.jpeg&w=3840&q=75)
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