This appendix gathers together all of the math facts used
in the text. They are divided into six categories:
Euler's relation
e
±
j
x
=
cos
(
x
)
±
j
sin
(
x
)
Exponential definition of a cosine
cos
(
x
)
=
1
2
e
j
x
+
e
-
j
x
Exponential definition of a sine
sin
(
x
)
=
1
2
j
e
j
x
-
e
-
j
x
Cosine squared
cos
2
(
x
)
=
1
2
1
+
cos
(
2
x
)
Sine squared
sin
2
(
x
)
=
1
2
1
-
cos
(
2
x
)
Sine and Cosine as phase shifts of each other
sin
(
x
)
=
cos
π
2
-
x
=
cos
x
-
π
2
cos
(
x
)
=
sin
π
2
-
x
=
-
sin
x
-
π
2
Sine–cosine product
sin
(
x
)
cos
(
y
)
=
1
2
sin
(
x
-
y
)
+
sin
(
x
+
y
)
Cosine–cosine product
cos
(
x
)
cos
(
y
)
=
1
2
cos
(
x
-
y
)
+
cos
(
x
+
y
)
Sine–sine product
sin
(
x
)
sin
(
y
)
=
1
2
cos
(
x
-
y
)
-
cos
(
x
+
y
)
Odd symmetry of the sine
sin
(
-
x
)
=
-
sin
(
x
)
Even symmetry of the cosine
cos
(
-
x
)
=
cos
(
x
)
Cosine angle sum
cos
(
x
±
y
)
=
cos
(
x
)
cos
(
y
)
∓
sin
(
x
)
sin
(
y
)
Sine angle sum
sin
(
x
±
y
)
=
sin
(
x
)
cos
(
y
)
±
cos
(
x
)
sin
(
y
)
Definition of Fourier transform
W
(
f
)
=
∫
-
∞
∞
w
(
t
)
e
-
j
2
π
f
t
d
t
Definition of Inverse Fourier transform
w
(
t
)
=
∫
-
∞
∞
W
(
f
)
e
j
2
π
f
t
d
f
Fourier transform of a sine
F
{
A
sin
(
2
π
f
0
t
+
Φ
)
}
=
j
A
2
-
e
j
Φ
δ
(
f
-
f
0
)
+
e
-
j
Φ
δ
(
f
+
f
0
)
Fourier transform of a cosine
F
{
A
cos
(
2
π
f
0
t
+
Φ
)
}
=
A
2
e
j
Φ
δ
(
f
-
f
0
)
+
e
-
j
Φ
δ
(
f
+
f
0
)
Fourier transform of impulse
F
{
δ
(
t
)
}
=
1
Fourier transform of rectangular pulse
With
Π
(
t
)
=
1
-
T
/
2
≤
t
≤
T
/
2
0
otherwise
,
F
{
Π
(
t
)
}
=
T
sin
(
π
f
T
)
π
f
T
≡
T
sinc
(
f
T
)
.
Fourier transform of sinc function
F
{
sinc
(
2
W
t
)
}
=
1
2
W
Π
f
2
W
Fourier transform of raised cosine
With
w
(
t
)
=
2
f
0
sin
(
2
π
f
0
t
)
2
π
f
0
t
cos
(
2
π
f
Δ
t
)
1
-
(
4
f
Δ
t
)
2
,
F
{
w
(
t
)
}
=
1
|
f
|
<
f
1
1
2
1
+
cos
π
(
|
f
|
-
f
1
)
2
f
Δ
f
1
<
|
f
|
<
B
0
|
f
|
>
B
,
with the
rolloff factor defined as
β
=
f
Δ
/
f
0 .
Fourier transform of square-root raised cosine (SRRC)
With
w
(
t
) given by
1
T
sin
(
π
(
1
-
β
)
t
/
T
)
+
(
4
β
t
/
T
)
cos
(
π
(
1
+
β
)
t
/
T
)
(
π
t
/
T
)
(
1
-
(
4
β
t
/
T
)
2
)
t
≠
0
,
±
T
4
β
1
T
(
1
-
β
+
(
4
β
/
π
)
)
t
=
0
β
2
T
1
+
2
π
sin
π
4
β
+
1
-
2
π
cos
π
4
β
t
=
±
T
4
β
,
F
{
w
(
t
)
}
=
1
|
f
|
<
f
1
1
2
1
+
cos
π
(
|
f
|
-
f
1
)
2
f
Δ
1
/
2
f
1
<
|
f
|
<
B
0
|
f
|
>
B
.
Fourier transform of periodic impulse sampled signal
With
F
{
w
(
t
)
}
=
W
(
f
)
,
and
w
s
(
t
)
=
w
(
t
)
∑
k
=
-
∞
∞
δ
(
t
-
k
T
s
)
,
F
{
w
s
(
t
)
}
=
1
T
s
∑
n
=
-
∞
∞
W
(
f
-
(
n
/
T
s
)
)
.
Fourier transform of a step
With
w
(
t
)
=
A
t
>
0
0
t
<
0
,
F
{
w
(
t
)
}
=
A
δ
(
f
)
2
+
1
j
2
π
f
.
Fourier transform of ideal
π
/
2 phase shifter
(Hilbert transformer) filterimpulse response
With
w
(
t
)
=
1
π
t
t
>
0
0
t
<
0
,
F
{
w
(
t
)
}
=
-
j
f
>
0
j
f
<
0
.
Linearity property
With
F
{
w
i
(
t
)
}
=
W
i
(
f
) ,
F
{
a
w
1
(
t
)
+
b
w
2
(
t
)
}
=
a
W
1
(
f
)
+
b
W
2
(
f
)
.
Duality property With
F
{
w
(
t
)
}
=
W
(
f
) ,
F
{
W
(
t
)
}
=
w
(
-
f
)
.
Cosine modulation frequency shift property
With
F
{
w
(
t
)
}
=
W
(
f
) ,
F
{
w
(
t
)
cos
(
2
π
f
c
t
+
θ
)
}
=
1
2
e
j
θ
W
(
f
-
f
c
)
+
e
-
j
θ
W
(
f
+
f
c
)
.
Exponential modulation frequency shift property
With
F
{
w
(
t
)
}
=
W
(
f
) ,
F
{
w
(
t
)
e
j
2
π
f
0
t
}
=
W
(
f
-
f
0
)
.
Complex conjugation (symmetry) property If
w
(
t
) is real valued,
W
*
(
f
)
=
W
(
-
f
)
,
where the superscript
* denotes complex conjugation
(i.e.,
(
a
+
j
b
)
*
=
a
-
j
b
) . In particular,
|
W
(
f
)
| is even and
∠
W
(
f
) is odd.
Symmetry property for real signals Suppose
w
(
t
) is real.
If
w
(
t
)
=
w
(
-
t
)
,
then
W
(
f
)
is
real.
If
w
(
t
)
=
-
w
(
-
t
)
,
W
(
f
)
is
purely
imaginary.
Time shift property
With
F
{
w
(
t
)
}
=
W
(
f
) ,
F
{
w
(
t
-
t
0
)
}
=
W
(
f
)
e
-
j
2
π
f
t
0
.
Frequency scale property
With
F
{
w
(
t
)
}
=
W
(
f
) ,
F
{
w
(
a
t
)
}
=
1
a
W
(
f
a
)
.
Differentiation property
With
F
{
w
(
t
)
}
=
W
(
f
) ,
d
w
(
t
)
d
t
=
j
2
π
f
W
(
f
)
.
Convolution
↔ multiplication property
With
F
{
w
i
(
t
)
}
=
W
i
(
f
) ,
F
{
w
1
(
t
)
*
w
2
(
t
)
}
=
W
1
(
f
)
W
2
(
f
)
and
F
{
w
1
(
t
)
w
2
(
t
)
}
=
W
1
(
f
)
*
W
2
(
f
)
,
where the convolution operator “
* ” is defined via
x
(
α
)
*
y
(
α
)
≡
∫
-
∞
∞
x
(
λ
)
y
(
α
-
λ
)
d
λ
.
Parseval's theorem
With
F
{
w
i
(
t
)
}
=
W
i
(
f
) ,
∫
-
∞
∞
w
1
(
t
)
w
2
*
(
t
)
d
t
=
∫
-
∞
∞
W
1
(
f
)
W
2
*
(
f
)
d
f
.
Final value theorem
With
lim
t
→
-
∞
w
(
t
)
=
0 and
w
(
t
) bounded,
lim
t
→
∞
w
(
t
)
=
lim
f
→
0
j
2
π
f
W
(
f
)
,
where
F
{
w
(
t
)
}
=
W
(
f
) .