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

Review exercises 22 721
(e) q[k +3] + √ q[k+2]=q[k]−1
Foreach linearequation state whether itis
homogeneous orinhomogeneous.
2 Given
3(x[n +1]) 2 −2x[n] =n 2 x[0] = 2
findx[1],x[2]andx[3].
3 Rewrite eachequationsothatthe highestargumentof
the dependent variableisasspecified.
(a) 3ny[n +1] −y[n −1] =n 2 ,highest argumentof
the dependent variableisto ben.
(b) z[k +2] + (3 +k/2)z[k] = √ kz[k −1],highest
argumentofthedependentvariableistobek+1.
(c) x[3]x[n] −x[2]x[n −1] = (n +1) 2 ,highest
argumentofthedependentvariableistoben +1.
4 Find f[k]if
F(z) =
z(1 −a)
(z −1)(z −a)
5 Find the inverseztransform of
(a)
(b)
3z(z +2)
(z −2)(z −3) 2
z 2 +3z
3z 2 +2z−5
6 Thesequence δ[k −i]istheKroneckerdeltasequence
shiftediunits to the right. Finditsztransform.
7 Show thatsinak canbe writtenas
Given that
show that
e akj −e −akj
2j
Z{e −ak } =
Z{sinak} =
z
z−e −a
zsina
z 2 −2zcosa+1
Solutions
1 (a) Second order,n,x,linear, inhomogeneous (b) z[k +1] +
(3 + 1 )
2 (k−1) z[k −1]
(b) Second order,k,y,linear, inhomogeneous
= √ k−1z[k−2]
(c) First order,z,y, non-linear
(c) x[3]x[n +1] −x[2]x[n] = (n +2) 2
(d) Second order,n,z,linear, homogeneous
4 u[k] −a k
(e) Third order,k,q,non-linear
5 (a) 12(2 k ) −12(3 k ) +5k3 k
2 1.1547,1.0503,1.4260
u[k] − (−5/3) k /3
(b)
3 (a) 3(n −1)y[n] −y[n −2] = (n −1) 2 2
1
6
z i