Expressing a Sinusoid using a Complex Exponential
Imaginary
x
ωt
A
ϕc
ϕs
A sinusoidal oscillation x(t) can be expressed as the rotation of a vector x(t) in the complex plane. The actual
oscillating quantity x(t) is the projection of x(t) onto the real axis, so that as x rotates around the circle, x
fluctuates sinusoidally.
Â
y0
x0
Real
By using Euler’s formula eiq = cosq + isinq , a complex number that is an angle θ from the real axis can also be
expressed in terms of an exponential with a purely imaginary argument. Multiplying one exponential by
another is equivalent to adding their arguments ( ea eb = ea+b ), so multiplying a complex number by an
exponential with a purely imaginary argument is equivalent to adding to its angle, i.e., rotating it
counterclockwise in the complex plane. Thus, multiplying a complex number by eiw t has the effect of rotating
the number by an angle w t in the counterclockwise direction in the complex plane.
x(t) is thus a vector in the complex plane with constant length A and direction that varies with time. It can be
expressed as x(t) = Âeiw t , where  is a complex number representing the position of x when t = 0, and eiw t
rotates the vector appropriately with time t (i.e, it rotates counterclockwise at speed w ). Â is referred to as
the oscillation’s complex amplitude, since the magnitude of  indicates the amplitude of the oscillation, and
the direction of  indicates the phase of the oscillation. The projection of the complex amplitude onto the real
(
(
)
axis is the initial value, x0 = Re( Â) , and in general the quantity x(t) = Re x(t) = Re Âeiw t
).
The direction of  must be defined relative to some reference, which is another way of saying that the phase
of the oscillation x(t) must be defined relative to some reference oscillation. For example, if the angle of  is
measured counterclockwise from the real axis, it is the phase angle by which the oscillation leads a pure
cosine. Most sources use f to denote phase angle. Here, I use f c for the definition where f is measured
Imaginary
positive counterclockwise from the real axis (leading a cosine), such that the initial condition is
x0 = Re( Â) = Acos(fc ) and in general, x(t) = Re( Âeiwt ) = Acos(w t + fc ) . In this case both f and w t are positive
phase lead
relative to
cosine
phase lag
relative to
cosine
phase lag phase lead
relative to relative to
sine
sine
in the same direction, so the total angle of x (measured counterclockwise from the real axis) is w t + fc .
Real
If the phase is instead defined as a lag relative to a pure cosine, it means that the angle of  is measured
clockwise from the real axis. With f and w t defined as positive in opposite directions, the total angle of x is in
this case their difference, that is, x(t) = Acos(w t - fc ) . Other sources define the phase as the lead or lag relative
to a pure sine. Phase is typically denoted f regardless of the definition used; to avoid ambiguity in the diagram
at top left I label the phase lead relative to a pure sine as f s , which results in x(t) = Asin(w t + fs ) . As the
diagram indicates, fs = fc + p2 , that is, the phase lead relative to a pure sine is
p
2
radians greater than the phase
lead relative to a pure cosine. If f s where instead defined as positive in the clockwise direction (a lag relative
to a pure sine), the result is x(t) = Asin(w t - fs ) . This last definition of phase is used in the textbook by
Ginsberg in section 2.1 (see Figure 2.3 on page 68) but in section 2.2 he uses a phase lag relative to cosine.
The following table shows the relationship between the phase and the angle of the complex amplitude, depending on how the phase is defined.
Phase definition
Lead relative to cosine
x(t) = Acos(w t + phase)
Expression
Lag relative to cosine
x(t) = Acos(w t - phase)
Lead relative to sine
x(t) = Asin(w t + phase)
Lag relative to sine
x(t) = Asin(w t - phase)
Relation to complex amplitude
phase = Â
phase = - Â
( Âi)
= - ( Âi )
phase = Â + p2 =
phase = - Â - p2
Using Two Exponentials
(
) (
)
Taking the real part of a complex number is equivalent to adding the number to its complex conjugate and then dividing by two, since a + bi + a - bi = 2a .
Thus, as alternative to expressing a sinusoid as the real part of one complex exponential, it is possible to express it as the sum of two exponentials that are complex
(
)
conjugates of one another: x(t) = Re Âeiw t = Beiw t + B*e-iw t where B = 12 Â and B* is the complex conjugate of B.
Using Two Sinusoids
Any sinusoid can also be expressed as the sum of a cosine and sine component: x(t) = c1 cos w t + c2 sin w t . In order for this to represent an oscillation with amplitude A, the
Pythagorean sum of c1 and c2 must be A (i.e., A = c12 + c22 ). To find the coefficients of c1 and c2 given a representation of the oscillation as a single sinusoid, one can apply
one of the various “angle sum” trigonometric identities:
For example, Acos(w t + f ) = Acosf cos w t - Asinf sin w t
c1
-c2
or Asin(w t + f ) = Asinf cos w t + Acosf sin w t
c1
c2
The relationship between these coefficients and the complex amplitude is that the coefficient c1 = Re( Â) and the coefficient c2 = - Im( Â) , that is, the coefficients are the
real and imaginary parts not of  but of its complex conjugate Â* .
Conversions
The table on the following page summarizes the formulas necessary to convert between any one representation of an oscillation and any other.
Formulations
Description
Single
exponential
†
Formula
Defining quantities
{
x(t) = Re Âeiw t
}
x(t) = Acos(w t + fc )
of the complex amplitude
Single sine
magnitude A
and phase f c
Two
sinusoids
x(t) = Asin(w t + fs )
magnitude A
and phase f s
iw t
real and imaginary parts
a and b
of the complex amplitudes
x(t) = Be
* -iw t
+B e
x(t) = c1 cosw t + c2 sin w t
if
 = Ae c
( x0 = Acosfc
 = -iAe s
( x0 = Asinfs
and
and
y0 = Asinfc )
y0 = - Acosfs )
A = Â = x02 + y02
Single
sinusoid
††,†††
Two
exponentials
if
real and imagniary parts
x0 and y0
 = x0 + y0i
Single cosine
Conversion from
Single
sinusoid
Single
exponential
fc = fs -
y
(i.e., tanfc = 0 )
x0
( iÂ)
(i.e., tanfs =
x0
)
- y0
Two
sinusoids
 = 2B
 = c1 - c2i
( x0 = 2a
and y0 = 2b )
A = 2 B = 2 a 2 + b2
fc = Â
fs =
Two
exponentials
p
2
fc = B
b
(i.e., tanfc = )
a
fs =
fs = fc + p2
( x0 = c1
and y0 = -c2 )
A = c12 + c22
tanfc =
-c2
c1
tanfs =
c1
c2
( iB)
(i.e., tanfs =
a
)
-b
B = 12 c1 - 12 c2i
B = 12 Â
ifs
B* = 12 c1 + 12 c2i
-ifs
( a = 12 c1
B* = 12 Â*
B = 12 Ae
ifc
B = - 12 iAe
B = a + bi
( a = 12 x0 and
-ifc
and B* = a - bi
B* = 12 Ae
B* = - 12 iAe
b = 12 y0 )
sinusoidal amplitudes
c1 and c2
c1 = x0 = Re( Â)
c1 = Acosfc
c1 = Asinfs
c1 = 2a = B + B*
c2 = - y0 = - Im( Â)
c2 = - Asinfc
c2 = Acosfs
c2 = -2b = i(B - B* )
and b = - 12 c2 )
† It is also possible to define the rotation as time goes by as being clockwise in the complex plane, in which case iw t would be replaced by -iw t (and vice versa).
†† It is also possible to define the phase angle to be positive in the opposite direction, i.e., to express an oscillation as x(t) = Acos(w t - fc ) or x(t) = Asin(w t - fs ) .
In such cases, simply reverse the sign of f when converting to/from other representations.
††† When calculating phase angles via arctangent, be aware that there are multiple solutions that differ by p (= 180° ) , and the signs of the numerator and denominator must be
examined to determine which quadrant the correct answer is in.
-1
p
+1
3p
= 135° . In this case, both -45° and +135° are solutions to tan -1 (-1) .
For instance, if tanf =
then f = - = -45° , but if tanf =
then f =
+1
2
-1
2