082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

22.2 Basic definitions 685
an inhomogeneous difference equation could be of the form
y[n] −2y[n −1] = 0.1s[n] +0.2s[n −1] −0.5s[n −2]
where y[n] is the output sequence or dependent variable and s[n] is the input sequence.Notethattheinputsequencecanstillbethoughtofastheindependentvariable
but instead of being expressed analytically in terms ofnit arises as a result of
sampling. The corresponding homogeneous equation is
y[n] −2y[n −1] = 0
22.2.6 Coefficient
Thetermcoefficientreferstothecoefficientofthedependentvariable.InEquation(22.1)
the coefficients are 3nand −1.
Example22.2 (a) State the order ofeach of the following equations (i)--(vii).
(b) State whether each equation islinear or non-linear.
(c) For each linear equation, statewhether itishomogeneous or inhomogeneous.
(i) 2x[n] −3nx[n −1] +x[n −2] +n 2 = 0
1
(ii) (x[n +1] −x[n −1]) =x[n]
3
(iii) z[n +2](2n −z[n −1]) =n +1
7x[n −1]
(iv)
x[n −2] = n +1
n −1
(v) w[n +3]w[n +1] =n 3 −1
(vi) y[n +2] +2y[n +1] = 6s[n +2] −2s[n +1] +s[n]whereyisthedependent
variable
(vii) x[k+3]−2x[k+2]+x[k] =e[k+2]−e[k]wherexisthedependentvariable.
Solution (a) (i) Second order
(ii) Second order
(iii) Third order
(iv) Firstorder
(v) Second order
(vi) Firstorder
(vii) Third order
(b) (i) Linear
(ii) Linear
(iii) Non-linear
(iv) Linear
(v) Non-linear
(vi) Linear
(vii) Linear
(c) Equations (iii) and (v) are non-linear. The linear equations are written in standard
form:
(i) 2x[n] −3nx[n −1] +x[n −2] = −n 2
(ii) x[n +1] −3x[n] −x[n −1] = 0