Solution of ECE 315 Test 1 F06
1.
A discrete-time system is described by the difference equation
7 y [ n ] − 3 y [ n − 1] + y [ n − 2 ] = 11 .
(a)
The eigenvalues of this difference equation can be expressed in the polar form Ae jθ where A is the
magnitude and θ is the angle or phase. Find the numerical values of A and θ .
A = 0.378
θ = 0.968 (radians)
A = 0.378
θ = -0.968 (radians)
The characteristic equation is 7α 2 − 3α + 1 = 0 . The eigenvalues are the solutions to this equation which
are
3 ± 9 − 28 3 ± j 19
=
= 0.2143 ± j 0.3113 = 0.378 e± j 0.968 .
14
14
(b)
The homogeneous solution approaches zero as n → ∞ . What numerical value does y [ n ] approach as
n→∞?
y [ ∞ ] = 2.2
Since the forcing function is a constant the forced solution is also a constant K. Therefore
7 K − 3K + K = 11 ⇒ K = 2.2 and the final value of y is also 2.2.
2.
A complex number z has five fifth roots, { z1, z2 , z3 , z4 , z5 } and z1 = Ae j 3π / 4 .
(a)
The number z can be expressed in the rectangular form z = x + jy . Find the value of x and y in terms of
the unknown A.
x = 0.707 A 5 ,
(
z = z15 = Ae j 3π / 4
)
5
y = −0.707 A 5
= A 5 e j15π / 4 = A 5 e− jπ / 4 e j16π / 4 = A 5 cos ( −π / 4 ) + jA 5 sin ( −π / 4 ) = 0.707 A 5 − j 0.707 A 5
x = 0.707 A 5 and y = −0.707 A 5
(b)
The other four roots { z2 , z3 , z4 , z5 } can be expressed in the polar form Ae jθ . Find the numerical values of
the angles θ of the other four roots (all in radians).
θ = 3.6128 , 4.8695 , 6.1261 , 7.3827
The spacing between the roots must be 2π / 5 radians. So the angles of the other four roots are
3π / 4 + 2π / 5 = 23π / 20 = 3.6128 or -2.6074 or
3π / 4 + 4π / 5 = 31π / 20 = 4.8695 or -1.4137 or
3π / 4 + 6π / 5 = 39π / 20 = 6.1261 or -0.1571 or
3π / 4 + 8π / 5 = 47π / 20 = 7.3827 or 1.0996 or -5.18
836 or
Solution of ECE 315 Test 1 F06
1.
A discrete-time system is described by the difference equation
8 y [ n ] − 2 y [ n − 1] + y [ n − 2 ] = 11 .
The eigenvalues of this difference equation can be expressed in the polar form Ae jθ where A is the
magnitude and θ is the angle or phase. Find the numerical values of A and θ .
(a)
A = 0.3536
θ = 1.2094 (radians)
A = 0.3536
θ = -1.2094 (radians)
The characteristic equation is 8α 2 − 2α + 1 = 0 . The eigenvalues are the solutions to this equation which
are
2 ± 4 − 32 2 ± j 2 7
=
= 0.125 ± j 0.3307 = 0.3536 e± j1.2094 .
16
16
The homogeneous solution approaches zero as n → ∞ . What numerical value does y [ n ] approach as
(b)
n→∞?
y [ ∞ ] = 1.5714
Since the forcing function is a constant the forced solution is also a constant K. Therefore
8 K − 2 K + K = 11 ⇒ K = 11 / 7 = 1.5714 and the final value of y is also 1.5714.
2.
A complex number z has five fifth roots, { z1, z2 , z3 , z4 , z5 } and z1 = Ae j 3π / 8 .
The number z can be expressed in the rectangular form z = x + jy . Find the value of x and y in terms of
(a)
the unknown A.
x = 0.9239 A 5 ,
(
z = z15 = Ae j 3π / 8
)
5
y = −0.3827 A 5
= A 5 e j15π / 8 = A 5 e− jπ / 8 e j16π / 8 = A 5 cos ( −π / 8 ) + jA 5 sin ( −π / 8 ) = 0.9239 A 5 − j 0.3827 A 5
x = 0.9239 A 5 and y = −0.3827 A 5
(b)
The other four roots { z2 , z3 , z4 , z5 } can be expressed in the polar form Ae jθ . Find the numerical values of
the angles θ of the other four roots (all in radians).
θ = 2.4347 , 3.6914 , 4.948 , 6.2046
The spacing between the roots must be 2π / 5 radians. So the angles of the other four roots are
3π / 8 + 2π / 5 = 31π / 40 = 2.4347 or -3.8485 or
3π / 8 + 4π / 5 = 47π / 40 = 3.6914 or -2.5918 or
3π / 8 + 6π / 5 = 63π / 40 = 4.948 or -1.3352 or
3π / 8 + 8π / 5 = 79π / 40 = 6.2046 or -0.0785 or -6.3
3617 or
Solution of ECE 315 Test 1 F06
1.
A discrete-time system is described by the difference equation
8 y [ n ] − 3 y [ n − 1] + 2 y [ n − 2 ] = 14 .
(a)
The eigenvalues of this difference equation can be expressed in the polar form Ae jθ where A is the
magnitude and θ is the angle or phase. Find the numerical values of A and θ .
A = 0.5
θ = 1.1864 (radians)
A = 0.5
θ = -1.1864 (radians)
The characteristic equation is 8α 2 − 3α + 2 = 0 . The eigenvalues are the solutions to this equation which
are
3 ± 9 − 64 3 ± j 55
=
= 0.1875 ± j 0.4635 = 0.5 e± j1.1864 .
16
16
(b)
The homogeneous solution approaches zero as n → ∞ . What numerical value does y [ n ] approach as
n→∞?
y[∞] = 2
Since the forcing function is a constant the forced solution is also a constant K. Therefore
8 K − 3K + 2 K = 14 ⇒ K = 14 / 7 = 2 and the final value of y is also 2.
2.
A complex number z has five fifth roots, { z1, z2 , z3 , z4 , z5 } and z1 = Ae− j 3π / 8 .
(a)
The number z can be expressed in the rectangular form z = x + jy . Find the value of x and y in terms of
the unknown A.
x = 0.9239 A 5 ,
(
z = z15 = Ae− j 3π / 8
)
5
y = 0.3827 A 5
= A 5 e− j15π / 8 = A 5 e jπ / 8 e− j16π /88 = A 5 cos (π / 8 ) + jA 5 sin (π / 8 ) = 0.9239 A 5 + j 0.3827 A 5
x = 0.9239 A 5 and y = 0.3827 A 5
(b)
The other four roots { z2 , z3 , z4 , z5 } can be expressed in the polar form Ae jθ . Find the numerical values of
the angles θ of the other four roots (all in radians).
θ = 0.0785 , 1.3352 , 2.5918 , 3.8485
The spacing between the roots must be 2π / 5 radians. So the angles of the other four roots are
−3π / 8 + 2π / 5 = π / 40 = 0.0785 or -6.2046 or
−3π / 8 + 4π / 5 = 17π / 40 = 1.3352 or -4.948 or
−3π / 8 + 6π / 5 = 33π / 40 = 2.5918 or -3.6914 or
−3π / 8 + 8π / 5 = 49π / 40 = 3.8485 or -2.4347 or