Solution to ECE 315 Test #6 F02
1.
Which of the periodic CT functions with fundamental period, 2, have
trigonometric CTFS harmonic functions for which X c [ k ] = 0 , for all k ≥ 1.
To meet this requirement, the signal must either be an odd function or the sum of
an odd function and a constant.
2.
A CT function with a fundamental period, T0 , has a CTFS harmonic function,
(
)
(
)
X[ k ] = a δ [ k − ka ] + δ [ k + ka ] + jb δ [ k + kb ] − δ [ k − kb ] .
Using only real-valued functions (no j’s) write a mathematical expression for the CT
function, x( t) , corresponding to this CTFS harmonic function assuming that Tf = T0 .
Using
FS
sin(2πmf 0 t) ←
→
j
(δ[k + m] − δ[k − m])
2
(m an integer)
1
FS
cos(2πmf 0 t) ←
→ (δ [ k − m] + δ [ k + m])
2
(m an integer)
we get x( t) = 2 a cos(2πka f 0 t) + 2b sin(cos 2πkb f 0 t) .
3.
Find the average value, X[0] , of a periodic CT function, x( t) , with fundamental
period, T0 .
1
x( t) dt . The functions graphed were all constant in 4
The average value is
T0 ∫T0
regions, each of which occupied one-fourth of the period. Therefore the average value of
the signal is

1
∫T0 K1dt + ∫T0 K 2 dt + ∫T0 K 3 dt + ∫T0 K 4 dt 
T0  4
4
4
4

or
T
T
T  K + K2 + K3 + K4
1  T0
K1 + K 2 0 + K 3 0 + K 4 0  = 1

4
4
4
4
T0  4
which is simply the average of those four values.