Analysis of Linear Difference Equations and Applications to Signals

This report explores the applications of linear difference equations, particularly in the context of digital signals and discrete-time systems. It begins by introducing linear difference equations and their use in producing numerical solutions for continuous processes modeled by differential equations. The report highlights the application of difference equations in vector spaces dealing with time, specifically in digital signals within electrical and control systems. It discusses discrete-time signals, their representation as sequences, and their relevance in various fields such as biology, physics, economics, and demography. The concept of linear independence in the space of signals is examined, along with the use of the Casorati matrix and Casoratian to determine linear independence. Examples are provided to illustrate how to solve difference equations, find bases for solution spaces, and prove that solutions span the solution set. The report emphasizes the importance of understanding these concepts for analyzing discrete data in scientific and engineering problems.

Applications to
Linear Difference
Equations

Outline
• Introduction
• Applications
• Properties
• Discrete time signals
• Linear Independence in
the space S of Signals
• Examples

Introduction
Difference Equations are often used to produce numerical solutions when
differential equations is used to model a continuous process.
Given scalars a0, …, an, with a0 and an nonzero, and given a signal {zk}, the
equation
a0 yk+n + a1 yk+n-1 + + an-1 yk+1 + an yk = zk for all k (3)⋯
is called a linear difference equation (or linear recurrence relation) of order n. For
simplicity, a0 is often taken equal to 1.

Applications
• Difference Equations are used in a vector space
dealing with time (t) so the application is used
in digital signals that arise in electrical and
control systems: such as stop lights.

Discrete-time signals
• A signal in S is a function defined only on the integers and is
visualized as a sequence of numbers, say, {yk}.
• Figure shows four typical signals whose general terms (.7) k, 1k, (-
1) k, and (-.7) k can be chosen from the pulldown menu.
• Digital signals obviously arise in electrical and control systems
engineering, but discrete-data sequences are also generated in
biology, physics, economics, demography, and many other areas,
wherever a process is measured, or sampled, at discrete time
intervals. When a process begins at a specific time, it is sometimes
convenient to write a signal as a sequence of the form (y0, y1, y2,
…). The terms yk for k < 0 either are assumed to be zero or are
simply omitted.

Linear Independence in the Space S
of Signals
To simplify notation, we consider a set of only three signals in !, say, {uk}, {vk}, and
{wk}. They are linearly independent precisely when the equation
c1 uk + c2 vk + c3 wk = 0 for all k
implies that c1 = c2 = c3 = 0.
The phrase “for all k” means for all integers—positive, negative, and zero.
One could also consider signals that start with k = 0, for example, in which case,
“for all k” would mean for all integers k ≥ 0.

Casorati
Matrix
Suppose , c1, c2, c3 satisfy (1). Then equation (1) holds for any three
consecutive values of k, say, k, k + 1, and k + 2. Thus (1) implies that
c1 uk+1 + c2 vk+1 + c3 wk+1 = 0
and
c1 uk+2 + c2 vk+2 + c3 wk+2 = 0
Hence c1, c2, c3 satisfy
= for all k.
 The coefficient matrix in this system is called the Casorati matrix of the
signals.
 The determinant of the matrix is called the Casoratian of {uk}, {vk}, and {wk}.
 If the Casorati matrix is invertible for at least one value of k, then (2) will
imply that c1 = c2 = c3 = 0, which will prove that the three signals are linearly
independent.

Example 1
Write the following difference equation as a first-order system:
2 – 5 + 6 = 0 for all k.
So for each k,
= .
The difference equation if solved for :
= + 5 + 2 which is
= =
= .
This implies
= A for all k where A = .

Example 2
Find a basis for the solution space of the difference equation. Also prove
that the solution spans the solution set.
Given the equation: – + = 0.
An auxiliary equation can be written as:
– r + = 0 or
9 – 9r +2 =0.
From the quadratic formula, the solution for r:
r = = = and .
With these two solutions: and for the signals implies they are linearly
independent because neither solutions are multiples of the other.
Solution space is two-dimensional by the nth-order
homogeneous linear difference equation (Thm.) with these two signal for a
basis for the solution space by the Basis Theorem.

References

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