Find the Z tranaform of x(n). Need urgent help. The equation given in the image is:\[ x[n] = -0.5 \delta[n + 1] + \delta[n] \]### Explanation:This equation represents a discrete-time signal $ x[n] $, which is constructed using scaled and shifted impulse functions. - $ \delta[n] $ is the discrete-time unit impulse function, which equals 1 at $ n = 0 $ and 0 elsewhere.- $ \delta[n + 1] $ is the shifted impulse function, which equals 1 at $ n = -1 $ and 0 elsewhere.- The coefficient $-0.5$ scales the amplitude of the impulse at $ n = -1 $.Thus, the signal $ x[n] $ consists of:- A scaled impulse of $-0.5$ at $ n = -1 $.- An impulse of 1 at $ n = 0 $.This is a basic representation used in discrete signal processing to describe signals in terms of impulses.

Answered: × [n] = -0.5 ô [n + 1] + 6 [n] %3D

Introductory Circuit Analysis (13th Edition)

13th Edition

ISBN:9780133923605

Author:Robert L. Boylestad

Publisher:Robert L. Boylestad

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Find the Z tranaform of x(n). Need urgent help.

The equation given in the image is:

\[ x[n] = -0.5 \delta[n + 1] + \delta[n] \]

### Explanation:

This equation represents a discrete-time signal $ x[n] $, which is constructed using scaled and shifted impulse functions.

- $ \delta[n] $ is the discrete-time unit impulse function, which equals 1 at $ n = 0 $ and 0 elsewhere.
- $ \delta[n + 1] $ is the shifted impulse function, which equals 1 at $ n = -1 $ and 0 elsewhere.
- The coefficient $-0.5$ scales the amplitude of the impulse at $ n = -1 $.

Thus, the signal $ x[n] $ consists of:

- A scaled impulse of $-0.5$ at $ n = -1 $.
- An impulse of 1 at $ n = 0 $.

This is a basic representation used in discrete signal processing to describe signals in terms of impulses.

Expert Solution

Step 1

Given:

$x [n] = - 0.5 δ [n + 1] + δ [n]$

Step 2

As,

$Z \{δ [n]\} = 1 . . . (1)$

And

$Z \{δ [n + k]\} = z^{k}$

So,

$\begin{array}{rcl} Z \{δ (n + 1)\} & = & z^{1} \\ = & z . . . . . (2) \end{array}$

Now,

$\begin{array}{rcl} Z \{x [n]\} & = & Z \{- 0.5 δ [n + 1] + δ [n]\} \\ X (z) & = & - 0.5 Z \{δ [n + 1]\} + Z \{δ [n]\} \\ = & - 0.5 z + 1 [From (1) and (2)] \end{array}$

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