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Chapter 22: Problem 2

Solve the difference equation $$ x[k+2]-3 x[k+1]+2 x[k]=\delta[k] $$ subject to the conditions \(x[0]=x[1]=0\).

Short Answer

Expert verified
The solution is \( x[k] = \delta[k] \), so \( x[k] \) is 1 at \( k=0 \) and 0 otherwise.

Step by step solution

01

Understand the Problem

We need to solve a second-order linear difference equation with constant coefficients. The equation is given as \(x[k+2] - 3x[k+1] + 2x[k] = \delta[k]\), where \(\delta[k]\) is the Kronecker delta function, which is 1 when \(k=0\) and 0 otherwise. The initial conditions are \(x[0] = 0\) and \(x[1] = 0\).
02

Write the Homogeneous Equation

First, we solve the homogeneous equation: \(x_h[k+2] - 3x_h[k+1] + 2x_h[k] = 0\). This does not include the \(\delta[k]\) term and represents the equation without the input.
03

Find the Characteristic Equation

The characteristic equation for the homogeneous difference equation is obtained by substituting \(x_h[k] = r^k\): \(r^2 - 3r + 2 = 0\).
04

Solve the Characteristic Equation

We solve \(r^2 - 3r + 2 = 0\): Factor it as \((r - 1)(r - 2) = 0\). Thus, the roots are \(r_1 = 1\) and \(r_2 = 2\).
05

Write the General Solution for the Homogeneous Equation

The general solution for the homogeneous equation is \(x_h[k] = A_1 \, 1^k + A_2 \, 2^k = A_1 + A_2 \, 2^k\).
06

Apply Initial Conditions to Homogeneous Solution

Using the initial conditions \(x[0] = 0\) and \(x[1] = 0\), substitute into the homogeneous solution:1. \(x_h[0] = A_1 + A_2 \cdot 2^0 = 0 \Rightarrow A_1 + A_2 = 0\)2. \(x_h[1] = A_1 \cdot 1 + A_2 \cdot 2 = 0 \Rightarrow A_1 + 2A_2 = 0\).
07

Solve for Constants

Solving the system of equations \(A_1 + A_2 = 0\) and \(A_1 + 2A_2 = 0\), we find that \(A_1 = 0\) and \(A_2 = 0\). Thus, the homogeneous solution based on these initial conditions is zero: \(x_h[k] = 0\).
08

Find Particular Solution for Non-Homogeneous Equation

We need to find a particular solution \(x_p[k]\) to the non-homogeneous equation. For this, consider the structure of \(\delta[k]\). A particular solution could be of the form \(x_p[k] = C \cdot \delta[k]\), so let \(x_p[0] = C\) and \(x_p[k] = 0\) for \(k > 0\).
09

Apply Initial Condition to Particular Solution

Substituting into the non-homogeneous equation at \(k = 0\):\[C - 3 \cdot 0 + 2 \cdot 0 = 1 \Rightarrow C = 1\]. For \(k > 0\): the homogeneous solution \(x_h[k]\) satisfies the equation as \(x_h[k] = 0\).
10

Complete Solution

The complete solution is the sum of the homogeneous and particular solutions. Therefore, for all \(k\), the solution is \(x[k] = x_h[k] + x_p[k] = 0 + 1 \cdot \delta[k] = \delta[k]\).
11

Conclusion

The solution to the given difference equation, subject to the initial conditions, is \(x[k] = \delta[k]\), meaning \(x[k]\) is 1 at \(k=0\) and 0 otherwise.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Second-Order Linear Difference Equation
A second-order linear difference equation is a specific type of difference equation that models relationships involving discrete changes. It is called "second-order" because it involves terms that depend on two preceding values, much like a second derivative in calculus. In this context, the equation given is \(x[k+2] - 3x[k+1] + 2x[k] = \delta[k]\). Here, "second order" refers to the presence of the term \(x[k+2]\).
Key characteristics include:
  • Linearity: Each term is either a constant or multiplied by a variable to the power of one.
  • Constant Coefficients: The multiples (3 and 2 in this case) are constants for all values of \(k\).
Difference equations like these are used in various fields to describe processes that progress in steps, such as population growth models or financial sequences.
Initial Conditions
Initial conditions are the starting values that are usually given alongside a difference equation. They define specific values for the sequence at initial steps. For the equation \(x[k+2] - 3x[k+1] + 2x[k] = \delta[k]\), the initial conditions are given: \(x[0] = 0\) and \(x[1] = 0\). These conditions are crucial as they allow us to determine specific values for any constants in the solution.
Initial conditions help to:
  • Set the Sequence: Provide concrete starting points for calculating future sequence values.
  • Ensure Uniqueness: Ensure that the solution to the difference equation is unique, preventing multiple possible solutions.
  • Align With Reality: In practical applications, these conditions can represent real-world constraints or starting assumptions for a model.
Homogeneous Solution
The homogeneous solution of a difference equation refers to the solution obtained by setting the non-homogeneous part (in this case, \(\delta[k]\)) to zero. This simplifies the equation to \(x_h[k+2] - 3x_h[k+1] + 2x_h[k] = 0\). Solving this helps us understand the underlying structure of the equation apart from any external inputs or perturbations.
Points to consider about homogeneous solutions include:
  • Finding the Characteristic Equation: Typically derived by substituting a trial solution of the form \(r^k\) into the homogeneous equation.
  • Solving for Roots: Solutions to the characteristic equation give us the roots, which help construct the general solution.
  • General Solution Formation: Based on the roots, we form the general solution which in our problem was \(x_h[k] = A_1 + A_2 \cdot 2^k\).
  • Incorporation of Initial Conditions: These fix any constants in the solution, finalizing the homogeneous portion of the full solution.
Kronecker Delta Function
The Kronecker delta function \(\delta[k]\) is a simple yet powerful tool used in discrete mathematics. It is defined as 1 when \(k = 0\) and 0 for all other values of \(k\). In the difference equation given, \(\delta[k]\) acts as a "trigger" that introduces change at a specific point in the sequence without affecting any other points.
This function is used to:
  • Model Impulses: Often used to represent instantaneous events in systems, such as a switch activation or a force applied at a single time step.
  • Simplify Solutions: Allows for a clear incorporation of initial disruptions in the equation solving process.
  • Implementation in Solutions: In this specific equation, \(\delta[k]\) modifies the right-hand side and is critical in determining the particular solution \(x_p[k]\).
The Kronecker delta is essential for converting a well-understood pattern into something that matches the specifics given by input impulses or initial conditions.

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Most popular questions from this chapter

If \(f[k]=4(3)^{k}\) find \(\mathcal{Z}\\{f[k]\\}\). Use the first shift theorem to deduce \(\mathcal{Z}\\{f[k+1]\\}\). Show that \(\mathcal{Z}\\{f[k+1]\\}-3 \mathcal{Z} \mid f[k]\\}=0\)

For each of the following equations (a)-(e): (i) State the order of the equation. (ii) State whether each equation is linear or non-linear. (iii) For each linear equation, state whether it is homogeneous or inhomogeneous. (a) \(n(3 n+x[n])=x[n-1]\) (b) \(\frac{2 z[k-4]}{z[k-3]}=z[k-2\rfloor\) (c) \(y[n-2]+y[n-1]+y[n]=n^{2}\) (d) \(\sqrt{n+x[n]}=x[n-2]+e[n]\), where \(x\) is the dependent variable. (e) \((2 w[n-1]+1)^{2}=w[n-2]+s[n-1]-s[n-2]\), where \(w\) is the dependent variable.

Prove the first and second shift theorems.

Find the inverse \(z\) transforms of the following: (a) \(\frac{4 z}{z-4}\) (b) \(\frac{z^{2}+2 z}{3 z^{2}-4 z-7}\) (c) \(\frac{z+1}{(z-3) z^{2}}\) (d) \(\frac{2 z^{3}+z}{(z-3)^{2}(z-1)}\)

Using the definition of the \(z\) transform, find closed-form expressions for the \(z\) transforms of the following sequences \(f[k\rfloor\) where (a) \(f[0]=0, f[1]=0, f[k]=1\) for \(k \geqslant 2\) (b) \(f[k\rfloor= \begin{cases}0 & k=0,1, \ldots, 5 \\ 4 & k>5\end{cases}\) (c) \(f[k]=3 k, k \geqslant 0\) (d) \(f[k]=\mathrm{e}^{-k}, k=0,1,2, \ldots\) (e) \(f[0]=1, f[1]=2, f[2]=3, f[k]=0, k \geqslant 3\) (f) \(f[0]=3, f[k]=0, k \neq 0\) (g) \(f[k]= \begin{cases}2 & k \geqslant 0 \\ 0 & k<0\end{cases}\)
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