For Questions 1–5
, let
φ
k
(
t
) =
e
j
2
πkt
10
,
for
k
= 0
,
±
1
,
±
2
, . . .
, and for all real
t
. Let
. . . , a
−
1
, a
0
, a
1
, . . .
be complex numbers, and consider the
following representation of
x
as a linear combination of
φ
k
’s:
x
(
t
) =
∞
summationdisplay
k
=
−∞
a
k
φ
k
(
t
)
.
(1)
The objective of Questions 1–5 is to decide whether this representation is possible for a given signal
x
, and if so, to find these numbers
. . . , a
−
1
, a
0
, a
1
, . . .
.
Question 1.
x
(
t
) =
e
j
2
πt
. Select the correct statement:
1.
a
k
=
braceleftbigg
1
,
for
k
= 10
,
0
,
otherwise
.
2. Representation (1) does not exist.
3.
a
k
=
1
2
j
,
for
k
= 1
,

1
2
j
,
for
k
=

1
,
0
,
otherwise
.
4.
a
k
=
k
5.
a
k
=
braceleftBigg
1
5
for
k
= 0
e

jπk
5
πk
sin
(
π
5
k
)
for
k
=
±
1
,
±
2
,
. . . .
6. All the above statements are incorrect.
Solution.
Signals
φ
k
together form a CT Fourier basis with period
T
= 10. Since
x
is periodic with
period 1, then it is also periodic with period 10. Also, it is squareintegrable over any interval of length
10. Hence,
x
can be represented as a linear combination of
φ
k
’s. Now,
x
(
t
) =
e
j
2
πt
=
e
j
2
π
·
10
·
t
10
.
By inspection, we get:
a
k
=
braceleftbigg
1
,
for
k
= 10
,
0
,
otherwise
.
Answer
: 1.
2