Technology
Calculating the Inverse Z-Transform of (e^{z^2}) Using the Taylor Series
Calculating the Inverse Z-Transform of (e^{z^2}) Using the Taylor Series
In this article, we will delve into the method of finding the inverse Z-transform of the function (e^{z^2}) using the Taylor series expansion. The inverse Z-transform is a crucial concept in digital signal processing and control systems, enabling us to convert a Z-transformed function back to its original discrete-time signal. We will follow a step-by-step approach to understand and compute this process.
Step 1: Expand (e^{z^2}) as a Taylor Series
The Taylor series expansion of (e^x) around (x 0) is given by:
[begin{equation}e^x sum_{n0}^{infty} frac{x^n}{n!}end{equation} ]By substituting (z^2) for (x), the expansion for (e^{z^2}) becomes:
[begin{equation}e^{z^2} sum_{n0}^{infty} frac{(z^2)^n}{n!} sum_{n0}^{infty} frac{z^{2n}}{n!}end{equation}Step 2: Identify the Coefficients
The series can be rewritten to identify the coefficients:
[begin{equation}e^{z^2} sum_{k0}^{infty} frac{z^{2k}}{k!}end{equation}In this series, the coefficient of (z^k) is (0) for odd (k) and (frac{1}{k/2!}) for even (k).
Step 3: Write the Series in Terms of the Z-transform
The Z-transform of a discrete-time signal (x[n]) is defined as:
[begin{equation}X(z) sum_{n0}^{infty} x[n] z^{-n}end{equation} ]To find the inverse Z-transform, we need to express (e^{z^2}) in the form of a Z-transform and identify (x[n]).
Step 4: Finding the Inverse Z-transform
From the series expansion, we can see:
[x[n] begin{cases} 0 #39; if n is odd [1ex] frac{1}{n/2!} #39; if n is even end{cases}This allows us to express (x[n]) as follows:
[begin{equation}x[n] begin{cases} 0 #39; if n is odd [1ex] frac{1}{n/2!} #39; if n is even end{cases}end{equation}Final Result
The inverse Z-transform of (e^{z^2}) is:
[x[n] frac{1}{n/2!} quad text{for even } n] $n$ [x[n] 0 quad text{for odd } n]This result gives us the sequence corresponding to the inverse Z-transform of (e^{z^2}). The sequence (x[n]) consists of zeros at odd indices and the factorial over the index at even indices.
Additional Considerations
It's important to note that the Z-transform defined here is the geophysical definition which contains only positive powers of (z). For the bilateral Z-transform, which includes both positive and negative powers of (z), the process would involve some modifications, but the key steps remain similar.