Table 1
| Time Domain x[n] |
Frequency Domain X(w) |
Notes |
|
δ
[
n
]
δ
[
n
]
|
1 |
|
| δ[n-M]δ[n-M] |
e-jwMe-jwM |
integer M |
| ∑m=-∞∞δ[n-Mm]∑m=-∞∞δ[n-Mm] |
∑
m
=
-
∞
∞
e
-
j
w
M
m
=
1
M
∑
k
=
-
∞
∞
δ
(
w
2
π
-
k
M
)
∑
m
=
-
∞
∞
e
-
j
w
M
m
=
1
M
∑
k
=
-
∞
∞
δ
(
w
2
π
-
k
M
)
|
integer M |
| e-jane-jan |
2
π
δ
(
w
+
a
)
2
π
δ
(
w
+
a
)
|
real number a |
| u[n]u[n] |
1
1
-
e
-
j
w
+
∑
k
=
-
∞
∞
π
δ
(
w
+
2
π
k
)
1
1
-
e
-
j
w
+
∑
k
=
-
∞
∞
π
δ
(
w
+
2
π
k
)
|
|
| anu(n)anu(n) |
1
1
-
a
e
-
j
w
1
1
-
a
e
-
j
w
|
if |a|<1|a|<1 |
| cos(an)cos(an) |
π
[
δ
(
w
-
a
)
+
δ
(
w
+
a
)
]
π
[
δ
(
w
-
a
)
+
δ
(
w
+
a
)
]
|
real number a |
| W·sinc2(Wn)W·sinc2(Wn) |
t
r
i
(
w
2
π
W
)
t
r
i
(
w
2
π
W
)
|
real number W,
0
<
W
≤
0
.
5
0
<
W
≤
0
.
5
|
| W·sinc[W(n+a)]W·sinc[W(n+a)] |
r
e
c
t
(
w
2
π
W
)
·
e
j
a
w
r
e
c
t
(
w
2
π
W
)
·
e
j
a
w
|
real numbers W,a
0
<
W
≤
1
0
<
W
≤
1
|
| rect[(n-M/2)M]rect[(n-M/2)M] |
sin[w(M+1)/2]sin(w/2)e-jwM/2sin[w(M+1)/2]sin(w/2)e-jwM/2 |
integer M |
|
W
(
n
+
a
)
{
c
o
s
[
π
W
(
n
+
a
)
]
-
s
i
n
c
[
W
(
n
+
a
)
]
}
W
(
n
+
a
)
{
c
o
s
[
π
W
(
n
+
a
)
]
-
s
i
n
c
[
W
(
n
+
a
)
]
}
|
j
w
·
r
e
c
t
(
w
π
W
)
e
j
a
w
j
w
·
r
e
c
t
(
w
π
W
)
e
j
a
w
|
real numbers W,a
0
<
W
≤
1
0
<
W
≤
1
|
|
1
π
n
2
[
(
-
1
)
n
-
1
]
1
π
n
2
[
(
-
1
)
n
-
1
]
|
|
w
|
|
w
|
|
|
|
0
n
=
0
(
-
1
)
n
n
elsewhere
0
n
=
0
(
-
1
)
n
n
elsewhere
|
j
w
j
w
|
differentiator filter |
|
0
n
odd
2
π
n
n
even
0
n
odd
2
π
n
n
even
|
j
w
<
0
0
w
=
0
-
j
w
>
0
j
w
<
0
0
w
=
0
-
j
w
>
0
|
Hilbert Transform |
rect(t) is the rectangle function for arbitrary real-valued tt.
rect(t)
=
0
if
|
t
|
>
1
/
2
1
/
2
if
|
t
|
=
1
/
2
1
if
|
t
|
<
1
/
2
rect(t)
=
0
if
|
t
|
>
1
/
2
1
/
2
if
|
t
|
=
1
/
2
1
if
|
t
|
<
1
/
2
(1)tri(t) is the triangle function for arbitrary real-valued tt.
tri(t)
=
1
+
t
if
-
1
≤
t
≤
0
1
-
t
if
0
<
t
≤
1
0
otherwise
tri(t)
=
1
+
t
if
-
1
≤
t
≤
0
1
-
t
if
0
<
t
≤
1
0
otherwise
"My introduction to signal processing course at Rice University."