Digital Signal Processing Reference
In-Depth Information
C.5.2 USING POLAR COORDINATES
The quotient of two complex numbers expressed in polar coordinates is a complex number having a
magnitude equal to the quotient of the two magnitudes, and an angle equal to the difference of the two
angles:
(M
1
θ
1
)/(M
2
θ
2
)
=
(M
1
/M
2
)
(θ
1
−
θ
2
)
Example C.7.
Formulate and compute the ratio 1/j in polar coordinates.
The polar coordinate version of this problem is very straightforward since the magnitude and angle
of each part of the ratio can be stated by inspection:
(
1
0
)/(
1
90
)
=
1
(
−
90
)
=−
j
C.6 POLAR NOTATION USING COSINE AND SINE
Another way of describing a complex number
W
having a magnitude
M
and an angle
θ
is
M(
cos
(θ)
+
j
sin
(θ))
This is true since Re
(W )
is just
M
cos(
θ
), and cos(
θ
), by definition in this case, is Re
(W )/M
;
Im
(W )
is just Msin(
θ
), and sin(
θ
)=Im
(W )/M
,so
M(
Re
(W )/M
+
j
Im
(W )/M)
=
Re
(W )
+
j
Im
(W )
C.7 THE COMPLEX EXPONENTIAL
The following identities are called the Euler identities, and can be demonstrated as true using the Taylor
(infinite series) expansions for
e
jθ
,
cos(θ)
, and
sin(θ)
:
e
jθ
=
cos
(θ)
+
j
sin
(θ)
and
e
−
jθ
cos
(θ)
−
j
sin
(θ)
where
e
is the base of the natural logarithm system, 2.718... Such an expression is referred to as a complex
exponential since the exponent of
e
is complex. This form is very popular and has many interesting traits
and uses.
For example, by adding the two expressions above, it follows that
=
(e
jθ
e
−
jθ
)/
2
cos
(θ)
=
+
and by subtracting it follows that
(e
jθ
e
−
jθ
)/
2
j
sin
(θ)
=
−
A correlation of an input signal
x
[
n
]
with both cosine and sine of the same frequency
k
over the
sequence length
N
can be computed as
