Harmonic Phasors and Fourier Series
RA welcome
Various forms of the Fourier series description for periodic signals are based on alternate
ways of writing a cosine signal. Consider
x(t )  a cos( t   )
with amplitude a > 0, frequency  > 0, and radian phase angle  . (The case of negative
amplitude is treated by adding  to  ). Three additional expressions for x(t) are (writing
j  1 )
Trigonometric:
x(t )  a cos( ) cos( t )  a sin( ) sin( t )
Complex exponential:
x(t )  (a / 2)e j e j t  (a / 2)e j e jt
Phasor real part:
x(t )  Re
 a e j( t   ) 
Equivalence of these expressions can be verified by using Euler's formula,
e j  cos   j sin 
and standard trigonometric identities.
RA phasors
We will explore properties of Fourier series using the phasor representation. A phasor
a e j ( t  )
can be viewed as a vector at the origin of the complex plane having length a and, at any
time t, angle ( t   ). The vector rotates counterclockwise with time, since  > 0, and
the projection on the real axis is described by


Re a e j ( t   )  Re a cos( t   )  j a sin( t   )
 a cos( t   )
For graphical representation, projection on the real (horizontal) axis is inconvenient, and
therefore we rotate phasors by /2 radians and project on the vertical axis. This makes
use of the mathematical relationship
Im
 a e j( t   / 2)   Re a e j( t  ) 
Click on the complex plane below to define a vector of length a and initial angle, at t =
0, of ( + /2) radians. Two periods of the corresponding phasor and cosine waveform
will be shown with a convenient frequency .
APPLET
RA phasorsums
In a similar fashion, a sum of cosine waveforms can be represented as the real part of a
phasor sum:

a1 cos(1t  1 )  a2 cos( 2t   2 )  Re a1 e j (1t 1 )  a2 e j ( 2t  2 )

The sum of two vectors in the complex plane can be found by placing the vectors “head
to tail.” Click twice on the complex plane below to define a vector sum, and two periods
of the corresponding two-cosine waveform with radian frequencies of  and 2 will be
displayed.
APPLET
RA fourierseries
A Fourier series is a (possibly infinite) sum of cosine waveforms with the property that
every frequency is a nonnegative integer multiple of a fundamental frequency,  o . We
write this as
x(t ) 


k 0
ak cos( k ot   k )
where o  0 and every ak  0 . Terminology for the various terms is listed below:
 dc-term, or zeroth harmonic term: a0 cos( o )

fundamental frequency term: a1 cos(ot  1)
 k th  harmonic, k  1, term: ak cos(kot   k )
The terms in such an expression are said to be harmonically related. This feature implies
periodicity of x(t), since each term in the sum repeats, at least once, in any time interval
of length To  2 / o .
Simple examples of Fourier series are
w(t) = 4 cos(3t) + 2 cos(6t + /4)
which has fundamental frequency 3, and
y(t) = 2 + 4 cos(3t) + 2 cos(4t + /4)
which has fundamental frequency 1. However,
z(t) = 1 + 4 cos(t) + 2 cos( t)
is not a Fourier series since the frequencies 1 and  cannot be written as integer multiples
of a single fundamental frequency.
The phasor representation for a Fourier series has the form





x(t )  Re   ak e j ( kot  k ) 
 k 0



Of course in practice the infinite sum is truncated to a finite number of terms.
Fourier series can be used to represent essentially any periodic signal. For a specified
signal with period To , the fundamental frequency is given by o  2 / To , and the
calculation of the coefficients ak and  k is relatively straightforward. However, rather
than focus on these calculations, we will explore features of the Fourier series via the
phasor representation. But first, use the applet below to try to generate your favorite
periodic signal, for example a square wave. Each click on the complex plane defines the
amplitude and angle coefficients for an additional harmonic term. Coefficient guesswork
such as this does not easily succeed, and this motivates study of methods for computation
of the coefficients!
APPLET
The applet below presents truncated Fourier series for a triangular wave, a square wave,
and a periodic train of impulses. Explore the effect of using various numbers of terms in
the representation, and you can also compare your attempts above to generate a square
wave with the Fourier series.
APPLET
The first few terms of these Fourier series are given by
 triangular wave:
1
1
x(t )  cos( ot )  cos(3 ot ) 
cos(5 ot ) 
9
25
 square wave:
x(t )  cos(ot   / 2) 

impulse train:
1
1
cos(3ot   / 2)  cos(5 ot   / 2) 
3
5
x(t )  1  cos(ot )  cos(2ot )  cos(3ot ) 
For a periodic signal that is a continuous function of time, such as the triangular wave,
the Fourier series coefficients diminish at least as fast as 1/ k 2 . That is, for some positive
constant M,
M
ak 
, k  1, 2, 3,
k2
You can observe in the applet above that it takes very few terms to achieve a good
approximation to the triangle wave.
RA gibbs
For a function that is continuous except for a finite number of finite jumps in one period,
such as the square wave, the coefficients diminish as 1/k. This raises mathematical issues
of convergence, particularly at points of discontinuity, and gives rise to the Gibbs effect.
The Gibbs effect is the overshoot phenomenon exhibited by the truncated Fourier series
at points of discontinuity. This behavior is apparent with the square wave in the applet.
Notice that as more terms are added to the Fourier series, the overshoot near a
discontinuity decreases only slightly in amplitude, though it decreases significantly in
duration.
A more extreme case is the impulse train, where the signal is a generalized function, and
the Fourier series coefficients remain constant. The mathematical nature of convergence
of the series is far from apparent. However, from the truncated series in the applet, it is
clear that the Gibbs effect is present, and also that some type of convergence might be
possible.
RA window
There are several methods for modifying the coefficients in a truncated Fourier series to
reduce or eliminate Gibbs effect. Essentially these are different ways of weighting
(decreasing) the coefficient values, and are referred to as windowing methods. One simple
method is the Fejer Window, based on Fejer summation of series. In this method, if N
harmonics are included in the truncated Fourier series, then the amplitude of the kth
harmonic is multiplied by (N  k)/N. Thus including the first 5 harmonics (some have
zero amplitude) for the square wave and impulse train yields the expressions


Fejer Window (N = 5) square wave:
4
2
xF (t )  cos(ot   / 2) 
cos(3 ot   / 2)
5
15
Fejer Window (N = 5) impulse train:
xF (t )  1 
4
3
2
1
cos(ot )  cos(2ot )  cos(3ot )  cos(4ot )
5
5
5
5
A second method is the Hamming Window, where the kth harmonic in an N harmonic
series is multiplied by
[0.54  (0.46) cos( k / N )]
For the square wave and impulse train examples, this yields the expressions

Hamming Window (N = 5) square wave:
xH (t )  0.91 cos( ot   / 2) 
0.40
cos(3 ot   / 2)
3
0.08
cos(5 ot   / 2)
5
Hamming Window (N = 5) impulse train:


xH (t )  1  0.91 cos(ot )  0.68 cos(2ot )  0.40 cos(3ot )
 0.17 cos(4ot )  0.08 cos(5ot )
Notice that both windows apply a unity multiplier to the dc-term (k = 0), and that as N
increases the multiplier on the kth-term tends to unity. Therefore the limit function of the
series is unchanged. The success of these two approaches in eliminating the Gibbs effect
can be explored in the applet below. Also, there are a number of other windows that have
been devised by choosing the weights according to various criteria.
APPLET