Basic signal processing - Biomedical Engineering Desk Reference - page 80

Solution: Expand each sinusoid into a sum of cosine and
sine, algebraically add the cosines and sines, and recom-
bine them into a single sinusoid. Be sure to convert the
sine into a cosine [recall Eq. 2.4.4 : sin ð u Þ¼ cos ð u t 90
degrees Þ ] before expanding this term.

The beauty of complex numbers and complex vari-
ables is that the real and imaginary parts are orthogonal.
One consequence of orthogonality is that the real and
complex numbers (or variables) can be represented as if
they are plotted on perpendicular axes ( Figure 2.4-3 ).
Orthogonality is discussed in more detail later, but the
importance of orthogonality with respect to complex
numbers is that the real number or variable does not
''interfere'' with the imaginary number or variable and
vice versa. Operations on one component do not affect
the other. This means that a complex number behaves
like two separate numbers rolled into one, and a complex
variable like two variables in one. This feature comes in
particularly handy when sinusoids are involved because
a sinusoid at a given frequency can be uniquely defined by
two variables: its magnitude and phase angle (or equiva-
lently, using Eq. 2.4.8 , its cosine and sine magnitudes,
a and b). It follows that a sinusoid at a given frequency
can be represented by a single complex number.
To find the complex representation, we will use the
identity developed by the Swiss mathematician, Euler
(Leonhard Euler's last name is pronounced ''oiler''. The
use of the symbol e for the basis of the natural loga-
rithmic system is a tribute to his extraordinary mathe-
matical contributions):

4 cos ð 2 t 30 Þ¼a cos ð 2 tÞþb sin ð 2 tÞ

where :
a ¼ C cos ð q Þ¼ 4 cos ð 30 Þ¼ 3 : 5 and
b ¼ C sin ð q Þ¼ 4 sin ð 30 Þ¼ 2
4 cos ð 2 t þ 30 Þ¼ 3 : 5 cos ð 2 tÞ 2 sin ð 2 tÞ

Converting the sine to a cosine then decomposing the
sine into a cosine plus a sine:

3 sin ð 2 t þ 60 Þ¼ 3 cos ð 2 t 30 Þ
¼ 2 : 6 cos ð 2 tÞ 1 : 5 sin ð 2 tÞ

Combining cosine and sine terms algebraically:

4 cos ð 2 t 30 Þþ 3 sin ð 2 t þ 60 Þ
¼ð 3 : 5 þ 2 : 6 Þ cos ð 2 tÞþð 2 1 : 5 Þ sin ð 2 tÞ
¼ 6 : 1 cos ð 2 tÞ 3 : 5 sin ð 2 tÞ
¼ C cos ð 2 t þ q Þ where C ¼

p

e jx ¼ cos x þ j sin x [Eq. 2.4.12]

6 : 1 2 þ 3 : 5 2 and

q ¼ tan 1 3 : 5
6 : 1

This equation links sinusoids and exponentials, providing
a definition of the sine and cosine in terms of complex
exponentials. It also provides a concise representation of
a sinusoid since a complex exponential contains both
a sine and a cosine, although a few extra mathematical
features are required to account for the fact that the
second term is an imaginary sine term. This equation will
prove very useful in two sinusoidally based analysis
techniques: Fourier analysis and phasor analysis.

C ¼ 7 : 0; q ¼ 30 (since b is negative, q is in fourth
quadrant) so q ¼ 30 degrees

xðtÞ¼ 7 : 0 cos ð 2 t 30 Þ

This approach can be extended to any number of sinu-
soids. An example involving three sinusoids is found in
Problem 4.

2.4.1.2 Complex representation

An even more compact representation of a sinusoid is
possible using complex notation. A complex number
combines a real number and an imaginary number. Real
numbers are commonly used, whereas imaginary num-
bers are the product of square roots and are represented

by real numbers multiplied by the

p

1 . In mathematics,

the p is represented by the letter i, whereas engineers

tend to use the letter j , the letter i being reserved for
current. A complex variable simply combines a real and
an imaginary variable: z ¼ x þ jy. Hence, although 5 is
a real number, j 5 is an imaginary number, and 5 þ j 5is
a complex number. We will review here variables and
arithmetic operations.

Figure 2.4-3 A complex number represented as an orthogonal
combination of a real number on the horizontal axis and an
imaginary number on the vertical axis. This graphic representation
is useful for understanding complex numbers and aids in the
interpretation of some arithmetic operations.