Problem 2. Determine the z-transform for the unit step sequence {uk}={1,1,1,1, …}={1} The z-tran

step 1 So in this question we have to find Z transforms of x transform of UK equals Z UK equal FZ equal

Problem 2. Determine the z-transform for the unit step...

Problem 2. Determine the z-transform for the unit step sequence $\left\{u_{k}\right\}=\{1,1,1,1, \ldots\}=\{1\}$ The z-transform of $\left\{u_{k}\right\}$ is given by: $$ \begin{aligned} Z\left\{u_{k}\right\} &=F(z)=\sum_{k=0}^{\infty} \frac{u_{k}}{z^{k}}=\sum_{k=0}^{\infty} \frac{1}{z^{k}} \\ &=\frac{1}{z^{0}}+\frac{1}{z^{1}}+\frac{1}{z^{2}}+\frac{1}{z^{3}}+\frac{1}{z^{4}}+\ldots \end{aligned} $$ i.e. $\quad \mathbf{Z}\left\{u_{k}\right\}=1+\frac{1}{z}+\frac{1}{z^{2}}+\frac{1}{z^{3}}+\frac{1}{z^{4}}+\ldots$ Using the binomial theorem for $(1+x)^{n}$, the series expansion of $\frac{1}{1-x}$ may be determined: $$ \begin{aligned} \frac{1}{1-x}=&(1-x)^{-1} \\ =& 1+(-1)(-x)+\frac{(-1)(-2)}{2 !}(-x)^{2} \\ &+\frac{(-1)(-2)(-3)}{3 !}(-x)^{3}+\ldots \\ &=1+x+x^{2}+x^{3}+\ldots \text { valid for }|x|<1 \end{aligned} $$ Comparing equations (1) and (2) gives: $\mathrm{F}(\mathrm{z})=\frac{1}{1-\frac{1}{z}}$ provided $\left|\frac{1}{z}\right|<1$ $\frac{1}{1-\frac{1}{z}}=\frac{1}{\frac{z-1}{z}}=\frac{z}{z-1}$ hence, $z\left\{u_{k}\right\}=\frac{z}{z-1}$ provided $|z|>1$

Problem 2. Determine the z-transform for the unit step sequence $\left\{u_{k}\right\}=\{1,1,1,1, \ldots\}=\{1\}$
The z-transform of $\left\{u_{k}\right\}$ is given by:
$$
\begin{aligned}
Z\left\{u_{k}\right\} &=F(z)=\sum_{k=0}^{\infty} \frac{u_{k}}{z^{k}}=\sum_{k=0}^{\infty} \frac{1}{z^{k}} \\
&=\frac{1}{z^{0}}+\frac{1}{z^{1}}+\frac{1}{z^{2}}+\frac{1}{z^{3}}+\frac{1}{z^{4}}+\ldots
\end{aligned}
$$
i.e. $\quad \mathbf{Z}\left\{u_{k}\right\}=1+\frac{1}{z}+\frac{1}{z^{2}}+\frac{1}{z^{3}}+\frac{1}{z^{4}}+\ldots$
Using the binomial theorem for $(1+x)^{n}$, the series expansion of $\frac{1}{1-x}$ may be determined:
$$
\begin{aligned}
\frac{1}{1-x}=&(1-x)^{-1} \\
=& 1+(-1)(-x)+\frac{(-1)(-2)}{2 !}(-x)^{2} \\
&+\frac{(-1)(-2)(-3)}{3 !}(-x)^{3}+\ldots \\
&=1+x+x^{2}+x^{3}+\ldots \text { valid for }|x|<1
\end{aligned}
$$
Comparing equations (1) and (2) gives: $\mathrm{F}(\mathrm{z})=\frac{1}{1-\frac{1}{z}}$ provided $\left|\frac{1}{z}\right|<1$
$\frac{1}{1-\frac{1}{z}}=\frac{1}{\frac{z-1}{z}}=\frac{z}{z-1}$ hence, $z\left\{u_{k}\right\}=\frac{z}{z-1}$ provided $|z|>1$

Key Concepts

Z-Transform

The Z-transform is a fundamental tool in signal processing and discrete-time systems, defined as a series representation of a sequence. It converts a discrete-time signal into a complex frequency domain representation, allowing the analysis of systems and the determination of system behavior such as stability and frequency response.

Unit Step Sequence

The unit step sequence is one of the simplest and most important sequences in discrete-time signal processing. It is defined as being equal to one for all non-negative indices. This sequence is often used as a basic building block in system analysis and to test system responses because its Z-transform represents a sum of powers with coefficients of one.

Geometric Series

The geometric series is a series of the form 1 + x + x^2 + x^3 + ... and plays an important role in the analysis of Z-transforms. In the context of the Z-transform, recognizing the series as a geometric one allows for the summation of the infinite series to a closed-form expression when the series converges (i.e., when the common ratio's magnitude is less than one).

Region of Convergence

The region of convergence (ROC) is a critical concept in the Z-transform framework, specifying the set of values in the complex plane for which the infinite sum defining the transform converges. Determining the ROC is essential for the correct application and interpretation of the Z-transform, as it indicates the conditions under which the transformed expression is valid.

Binomial Theorem

The binomial theorem provides a way to expand expressions of the form (1+x)^n into a power series. Although it is more general, in this context, it underlies the derivation of the series expansion for functions like 1/(1-x) by expressing them as an infinite sum. This series expansion technique is essential in linking the expression obtained from the Z-transform to its closed-form representation.

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