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Linear Algebra Lecture 27. Vector Spaces. Applications to Difference Equations. Discrete Time Signal. Let S is the space of discrete-time signals. A signal in S is a function defined only on the integers and is visualized as a sequence of numbers, say, { y k }. ….
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Linear Algebra Lecture 27
Applications to Difference Equations
Discrete Time Signal Let S is the space of 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}. …
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.
Example 1 The crystal clear sounds from a compact disc player are produced from music that has been sampled at the rate of 44,100 times per second. At each measurement, the amplitude of the music signal is recorded as a number, say, yk. …
The original music is composed of many different sounds of varying frequencies, yet the sequence {yk} contains enough information to reproduce all the frequencies in the sound up to about 20,000 cycles per second, higher than the human ear can sense.
Linear Independence in the Space S of Signals
Example 2 Verify that 1k, (-2)k and 3k are linearly independent signals.
Solution …
continued The Casorati matrix is invertible for k = 0. So 1k, (-2)k and 3k are linearly independent.
Definition Given scalars a0, … , an, with a0 and annonzero, and given a signal {zk}, the equation is called a linear difference equation (or linear recurrence relation) of order n. …
continued For simplicity, a0 is often taken equal to 1. If {zk} is the zero sequence, the equation is homogeneous; otherwise, the equation is non-homogeneous.
Note In general, a nonzero signal rk satisfies the homogeneous difference equation if and only if r is a root of the auxiliary equation …
When the auxiliary equation has a complex root, the difference equation has solutions of the form sk coskw and sksinkw, for constants s and w.
Solution Sets of Linear Difference Equations
Given a1, … , an, consider the mapping T: S S that transforms a signal {yk}into a signal {wk} given by It is readily checked that T is a linear transformation. …
This implies that the solution set of the homogeneous equation is the kernel of T (the set of signals that T maps into the zero signal) and hence the solution set is a subspace of S.
Theorem If an 0 and if {zk} is given, the equation yk+n+a1yk+n-1+…+an-1yk+1+anyk=zk, for allk has a unique solution whenever y0,…, yn-1 are specified.
Theorem The set H of all solutions of the nth-order homogeneous linear difference equation yk+n+a1yk+n-1+…+an-1yk+1+anyk=0, for all kis an n-dimensional vector space.
Example 5 Find a basis for the set of all solutions to the difference equation yk+3 – 2yk+2 – 5yk+1 + 6yk = 0for all k
Example 6 Verify that the signalyk = k2satisfies the difference equation yk+2 – 4yk+1 + 3yk = -4k for all k Then find a description of all solutions of this equation.
Reduction to Systems of First Order Equations
A modern way to study a homogeneous nth-order linear difference equation is to replace it by an equivalent system of first order difference equations, …
continued written in the form xk+1 = Axk for k = 0, 1, 2, … Where the vectors xk are in Rn and A is an n x n matrix.
Example 7 Write the following difference equation as a first order system: yk+3 – 2yk+2 – 5yk+1 + 6yk = 0 for all k
Example 8 It can be shown that the signals 2k, 3k , and 3k are solutions of yk+3 – 2yk+2 + 9yk+1 – 18yk = 0 …
Show that these signals form a basis for the set of all solutions of the difference equation.
Linear Algebra Lecture 27
