Staffordshire University
Faculty of Computing, Engineering and Technology
Signal Processing
November 2005
Page 1
FOURIER SERIES
Contents
FOURIER SERIES (F.S.)
FOURIER SERIES FOR A UNIPOLAR RECTANGULAR PULSE TRAIN
FOURIER SERIES (F.S.)
Review
Rules of Thumb
Exercise
COMPLEX FOURIER SERIES
Exercise
FURTHER EXERCISES
Staffordshire University
Faculty of Computing, Engineering and Technology
November 2005
Signal Processing
Page 2
FOURIER SERIES (F.S.)
Fourier’s Theorem states that any periodic function of time, f(t), (i.e. a periodic signal)
can be expressed as a Fourier Series consisting of:
a)
A DC component – the average value of f(t).
b)
A component at a fundamental frequency and harmonically related
components, collectively the AC components.
ie
f(t) = DC + AC components.
The Fourier Series for a periodic signal may be expressed by:

f ( t )  a 0   {an cos nt  bn sin nt}
n 1
DC or average
component
AC components
Fundamental frequency (n=1)
at ω rads per second.
a0, an and bn are coefficients given by:

a0 
an 
1
T
2
T
T
2
 f (t )dt ,
 f (t ) cos ntdt ,


2
bn 
T
which gives the average value or DC component.
T
2
T

2

T
2
T
2
 f (t ) sin ntdt .

T
2
These give the amplitudes of the AC components
Staffordshire University
Faculty of Computing, Engineering and Technology
Signal Processing
November 2005
Page 3
NOTE
a) The function must be periodic, i.e. f(t) = f(t+T). Periodic time = T. Frequency f =
1
Hz.
T
b)
If f(t) = f(-t) the function is EVEN and only cosine terms (and a0) will be
present in the F.S.
time
c)
If f(t) = -f(-t) the function is ODD and only sine terms (and a0) will be present
in the F.S.
time
d) The coefficients an and bn are the amplitudes of the sinusoidal components.
For example, in general,
an cos nωt
The component at the lowest frequency (excluding the DC component) is when n = 1,
i.e. a1 cos ωt – this is called the fundamental or first harmonic. The component for n = 2 is
called the second harmonic, n = 3 is the third harmonic and so on.
Staffordshire University
Faculty of Computing, Engineering and Technology
November 2005
Signal Processing
Page 4
FOURIER SERIES FOR A UNIPOLAR RECTANGULAR PULSE TRAIN
Consider the rectangular pulse train below.
Pulse width τ
E
time
0

T
2
T
2
t=0
Pulse width τ seconds, periodic time T seconds, amplitude E volts (unipolar).
As shown, the function is chosen to be even, ie f(t) = f(-t) so that a DC term and cosine terms
only will be present in the F.S.
We define
f(t) = E,


t

2
2
f(t) = 0, ‘elsewhere’
And
As noted, the Fourier Series for a periodic signal may be expressed by:

f ( t )  a 0   {an cos nt  bn sin nt}
n 1
DC or average
component
AC components
Fundamental frequency (n=1)
at ω rads per second.
where a0, an and bn are coefficients given by:

a0 
1
T
T
2

T

2

f (t )dt , a n 
2
T
T
2


f (t ) cos ntdt , bn 
T

2

Applying to find
1
a0 
T
 f (t ) sin ntdt .

T
2

2
 Edt ,

2
T
T
2
2

E
a 0  t  2
T
2
Staffordshire University
Faculty of Computing, Engineering and Technology
November 2005
Signal Processing
Page 5
E
a0 
Note this is a standard result..
T

The an coefficients are given by a n 
2
T

an 
an 
2
T
T
2
 f (t ) cos ntdt

T
2


2
 E cos ntdt 


2 E  sin nt  2
T  n  
2
2
2E
nT
 n
 n
sin 2  sin   2





Since sin(-A) = -sinA
an 
4E
 n 
sin 

nT
 2 
In this case it may be show that bn = 0 (because the choice of t = 0 gives an even function).
E
a0 
,
T
Hence:
Simplifying, by noting  
an 
4E
 n 
sin 
 and
nT
 2 
bn  0
2
and substituting back into the F.S. equation:
T
 2 
 n   2E
4E
4E
 n 
 n 
an 
sin 
sin  T  
sin 


nT  2  n 2 T  2  n
 T 


T



f (t )  a 0   an cos nt
n 1
f (t ) 
 2 E  n 
E

sin 
 cos nt
T n1 n
 T 
Fourier Series for a unipolar
pulse train.
Staffordshire University
Faculty of Computing, Engineering and Technology
Signal Processing
November 2005
Page 6
But NOTE, it is more usual to convert this to a ‘Sinc function’.
ie Sinc(X) =
sin X
X
 n 
sin 

4E
4E
2   n 
 n 

an 
sin 
.


nT  2  nT  n   2 


 2 
n
Note the ‘trick’, i.e multiply by
This reduces to
Hence
an 
n
2
2
2 E
 n 
Sinc 

T
 2 
f (t ) 
E  2 E
 n 

Sinc 
 cos nt
T n1 T
2


This is an important result, the F.S. for a periodic pulse train and gives a spectrum of the form
shown below:
‘Sinc’ envelope
frequency
Staffordshire University
Faculty of Computing, Engineering and Technology
November 2005
Signal Processing
FOURIER SERIES (F.S.)
We have discussed that:

f (t )  a0   an cos nt
n1
f (t ) 
Page 7
Review
General Fourier Series for an Even function
 2 E  n 
E

sin 
 cos nt
T n1 n
 T 
Fourier Series for a unipolar pulse train.
f (t ) 
E  2 E
 n 

Sinc 
 cos nt
T n1 T
 2 
This is a ‘Sinc function’, ie Sinc(X) =
sin X
.
X
This gives an ‘envelope’ for the amplitudes of the harmonics.
The Sinc function, in conjunction with
2 E
, gives the amplitudes of the harmonics.
T
Note that Sinc(0) =1. (As an exercise, justify this statement).
The amplitudes of the harmonic components are given by:
an 
2 E
 n 
Sinc 

T
 2 
To calculate, it is usually easier to use the form
an 
2 E
 n 
sin 

T
 T 
The harmonics occur at frequencies nω radians per second.
We normally prefer to think of frequency in Hertz, and since ω = 2πf, we can consider
harmonics at frequencies nf Hz.
The periodic time, T, and frequency are related by f =
1
Hz.
T
Staffordshire University
Faculty of Computing, Engineering and Technology
November 2005
Signal Processing
Page 8
‘Rules of Thumb’
The following ‘rules of thumb’ may be deduced for a pulse train, illustrated in the waveform
below.
τ
E
volts
T
1
1 2 3
Hz , ie , , etc. OR f, 2f, 3f, etc.
T
T T T

Harmonics occur at intervals of f =

Nulls occur at intervals of

If

For example if T = 10 ms and τ = 2.5 ms, then

As τ is reduced, ie the pulse gets narrower, the first and subsequent nulls move to a
higher frequency.

As T increases, ie the pulse frequency gets lower, the first harmonic moves to a lower
frequency and the spacing between the harmonics reduces, ie they move closer together.
T

1

1 2 3
Hz , ie , , , etc
  
= x is integer, then nulls occur every xth harmonic.
T
= 4 and there will be nulls at the 4th

harmonic, the 8th harmonic, the 12th harmonic and so on at every 4th harmonic.
Staffordshire University
Faculty of Computing, Engineering and Technology
Signal Processing
November 2005
Page 9
Exercise
Q1.
Label the axes and draw the pulse waveform corresponding to the spectrum below.
frequency
4 kHz
Q2.
What pulse characteristic would give this spectrum?
frequency
1 kHz
Q3. Suppose a triac firing circuit produces a narrow pulse, 1 nanosecond pulse width, with a
repetition rate of 50 pulses per second.
a) What is the frequency spacing between the harmonics?
b) At what frequency is the first null in the spectrum?
c) Why might this be a nuisance for radio reception?
Staffordshire University
Faculty of Computing, Engineering and Technology
November 2005
Signal Processing
Page 10
COMPLEX FOURIER SERIES
Up until now we have been considering trigonometric Fourier Series.
An alternative way of expressing f(t) is in terms of complex quantities, using the
relationships:
cos nt 
e
jnt
e
2
 jnt
and sin nt 
e
jnt
e
2j
 jnt

f (t )  a 0   {a n cos nt  bn sin nt} , then this
Since the ‘trig’ form of F.S. is:
n 1
may be written in the complex form:
f (t )  a0 
   e jnt  e jnt
 an 
2
n  1 

 jnt  e jnt
b e
 n
2j






The complex F.S. may be written as:
f (t ) 

 Cn e
n  
When n = 0, C0 ej0 = C0 is the average value.
jnt
n = ± 1, n = ± 2, n = ± 3 etc represent pairs of harmonics.
Where:
Cn 
Cn 
1
an  jbn 
2
1
T
T
2
 f (t )e
where an and bn are coefficients in the ‘trig’ F.S.
 jnt
dt
T

2
Note when n = 0,
e
j0
 1, and:
These are general for any periodic function.
CO 
1
T
T
2
 f (t )dt  ao in the ‘trig’ F.S.

T
2
Staffordshire University
Faculty of Computing, Engineering and Technology
November 2005
Signal Processing
Page 11
In particular, for a periodic unipolar pulse waveform, we have:
an 
2 E  n 
sin 

n
 T 
an 
OR
2 E
 n 
Sinc 

T
 2 
with bn = 0
Hence
Cn 
1
E
 n  E
 n 
an 
sin 
Sinc 


2
n
 T  T
 2 
f (t ) 

 n  jnt
e
T 
E
sin 

n


n  
Alternative forms of Complex
Fourier series for a unipolar
pulse.
f (t ) 

E

n  T
 n  jnt
Sinc
e
2


Example
Express the equation below (for a periodic pulse train) in complex form.
f (t ) 
E  2 E
 n 

Sinc 
 cos nt
T n1 T
2


NOTE, we change the ‘cos’ term, We DON’T change the Sinc term.
Since:
cos nt 
e
jnt
e
2
 jnt
, we may write;
jnt
 jnt
E  2 E
e
 n  e
f (t ) 

Sinc 

T n1 T
2
 2 
Staffordshire University
Faculty of Computing, Engineering and Technology
November 2005
Signal Processing
Page 12
By changing the sign of the ‘-n’ and summing from -∞ to -1, this may be written as:
f (t ) 
 E
E
 n

Sinc 
T
 2
n1 T

1
We the have

n  
1
E
 jnt
 n
e

Sinc 



 2
n T
and

n 1

.
We want

n  
We need to include the term for n = 0 and may show that for n = 0, the term
Consider
 jnt
e

E
results.
T
E
 n  jnt
Sinc 
when n = 0.
e
T
 2 
Sinc(0) = 1 and ej0 = 1, ie
Hence we may write:
E
E
 n  jnt
Sinc 
=
when n = 0.
e
T
T
 2 
f (t ) 

E

n  T
 n  jnt
Sinc
e
2


Comments
1) Fourier Series apply only to periodic functions.
2) Two main forms of F.S., Trig’ F.S. and ‘Complex’ F.S. which are equivalent.
3) Either form may be represented on an Argand diagram, and as a single-sided or two-sided
(bilateral) spectrum.
4) The F.S. for a periodic function effectively allows a time-domain signal (waveform) to be
represented in the frequency domain, (spectrum).
Staffordshire University
Faculty of Computing, Engineering and Technology
November 2005
Signal Processing
Page 13
Exercise
Q1.
A pulse waveform has a ratio of
T

= 5. Sketch the spectrum up to the second null
using the ‘rules of thumb’.
Q2.
A pulse has a periodic time of T = 4 ms and a pulse width τ = 1 ms. Sketch, but do not
calculate in detail, the single-sided and two-sided spectrum up to the second null,
showing frequencies in Hz.
Q3.
With T = 4 ms and τ = 1 ms as in Q2, now calculate, tabulate and sketch the singlesided and two-sided spectrum.
Q4.
Convert the ‘trig’ FS to complex by using the substitution :
cos nt 
Q5.
e
jnt
e
2
 jnt
A signal, v(t) = VcosωCt is multiplied by a pulse signal, p(t) as illustrated in
figure Q5(a) below.
VcosωCt
S(t)
Figure Q5(a)
Pulses, p(t)
The pulse train, illustrated in figure Q5(b) is given by:
p(t) =

TS

2
TS

 n s 
 cosn st
2 
 Sinc
n 1
‘1’

Figure Q5(b)
‘0’
(a)
T
Derive an expression for S(t)
(b)
If v(t) is a 10 kHz signal, and p(t) has T = 4 ms and τ = 1 ms as in Q2,
calculate, tabulate and sketch the single-sided for S(t).
(c)
What bandwidth does a band pass filter require if it is to pass components
between the first nulls in the spectrum of S(t)?
Staffordshire University
Faculty of Computing, Engineering and Technology
November 2005
Signal Processing
Page 14
FURTHER EXERCISES
Q1
A unipolar periodic pulse signal, f(t), may be expressed by a Fourier Series of
the form:
f (t ) 
E 2 E

T
T

 n 
 cosnt
2 
 Sinc
n 1
Equation 3.1
where E = pulse amplitude, volts
 = pulse width, seconds
T = periodic time, seconds
(a)
If T = 3 sec and  = 1 sec, calculate and tabulate component
amplitudes and frequencies up to the second null and sketch the singlesided spectrum.
[8 Marks]
(b)
(i)
Show that equation 3.1 may be written in the form:
f (t ) 

E
n 
Sinc
 exp jnt 
 2 
T
n  

Equation 3.2
[4 Marks]
(ii)
Sketch the bilateral spectrum for f(t).
recalculate values).
(You do not need to
[4 Marks]
(c)
The first pair of components (i.e. n = +1 and n=-1) in equation 3.2 is
given by:
E

E
 
Sinc  exp jt  
Sinc
 exp  jt 
 2
 2 
T
T
Sketch clearly, an Argand diagram representation for this pair, at a time
t = 1 sec.
[4 Marks]
Staffordshire University
Faculty of Computing, Engineering and Technology
November 2005
Signal Processing
Q2.
a)
Page 15
Discuss briefly the terms:
i)
Continous-time and discrete-time signals.
ii)
Deterministic and non-deterministic signals.
iii)
Periodic and non-periodic signals.
[3 Marks]
b)
The equation below represents a general periodic signal.

v ( t )   An cos nt
....... Equation 2.1
n 1
Assume that An 
c)
kV
where k is a constant.
n
i)
Express the nth component in exponential form and sketch this
component on an Argand diagram.
[3 Marks]
ii)
Sketch the single-sided spectrum for the signal v(t), showing
components up to the 5th harmonic.
[2 Marks]
iii)
Express Equation 2.1 in exponential form and sketch the twosided spectrum showing components up to the 5th harmonic.
[4 Marks]
i)
Re-write Equation 2.1 to express the normalized average power
of the signal.
[4 Marks]
ii)
If V = 2 volts peak and k = 1,determine the total average power in
the first five harmonics if the load resistance is RL = 100 ohms.
[4 Marks]
Staffordshire University
Faculty of Computing, Engineering and Technology
November 2005
Signal Processing
Q3.
Page 16
A unipolar periodic pulse signal, f(t), may be expressed by a Fourier Series of
the form:
f (t ) 
E 2 E 
n 

Sinc
 Cos nt

 2 
T
T n 1
Equation 3.1
where E = pulse amplitude, volts
 = pulse width, seconds
T = periodic time, seconds
2

T
a)
If the signal is a square wave with T = 1 millisecond and  = 0.5
millisecond,
i)
ii)
iii)
b)
i)
Give a suitable equation for f(t), developed from equation 3.1.
[4 Marks]
Calculate and tabulate component amplitudes and frequencies up
to the 5th harmonic.
[4 Marks]
Sketch the single-sided spectrum.
[2 Marks]
Show that equation 3.1 may be written in the form:
f (t ) 

E
n 
Sinc
 exp jnt 
 2 
T
n  

Equation 3.2
[6 Marks]
(ii)
Sketch the bilateral spectrum for f(t). Explain your method. (You
do not need to recalculate values).
[4 Marks]
Staffordshire University
Faculty of Computing, Engineering and Technology
November 2005
Signal Processing
Page 17
Q4. The Fourier Series for a periodic signal may be expressed by:

f ( t )  a 0   {an cos nt  bn sin nt}
n 1

where, a 0 
a)
1
T
T
2

 f (t )dt ,
T

2
an 
2
T
T
2
 f (t )cos ntdt ,

bn 
T

2
2
T
T
2
 f (t )sin ntdt

T
2
Show that a periodic pulse signal, f(t) may be expressed by a Fourier
Series of the form:
f (t ) 
E 2 E 
n 

Sinc
 Cos nt

 2 
T
T n 1
Equation 4.1
where
E = pulse amplitude, volts
 = pulse width, seconds
T = periodic time, seconds
2

T
[6 Marks]
b)
If T = 1 sec and  = 1/3 sec, calculate and tabulate component
amplitudes and frequencies up to the second null and sketch the singlesided spectrum.
[6 Marks]
c)
i)
Show that equation 4.1 may be written in the form:
f (t ) 

E
n 
Sinc
 exp jnt 
 2 
T
n  

Equation 4.2
[4 Marks]
ii)
Sketch the bilateral spectrum for f(t). (You do not need to
recalculate values). Explain your answer.
[4 Marks]
Staffordshire University
Faculty of Computing, Engineering and Technology
November 2005
Signal Processing
Q5.
a)
Page 18
Discuss briefly the terms:
i)
Continous-time and discrete-time signals.
ii)
Deterministic and non-deterministic signals.
iii)
Periodic and non-periodic signals.
[3 Marks]
b)
The equation below represents a general periodic signal.
v(t ) 
n  

n  
c)
E
n
Sinc(
) exp( jnt )
T
2
....... Equation 5.1
i)
If  = 2 104 radians per second, sketch the first harmonic on an
Argand diagram at time t = 25 s.
[3 Marks]
ii)
If  
i)
Re-write Equation 5.1 to express the normalized average power of the
signal.
[5 Marks]
ii)
If  
T
, (square wave), sketch the two-sided spectrum for the signal
2
v(t), showing amplitudes and frequencies of components up to the 5th
harmonic.
[5 Marks]
T
, and E = 2 volts, determine the normalized average power in
2
the dc and first five harmonics.
[4 Marks]
Staffordshire University
Faculty of Computing, Engineering and Technology
November 2005
Signal Processing
Page 19
Q6. A unipolar periodic pulse signal, f(t), may be expressed by a Fourier Series of
the form:
f (t ) 
E 2 E

T
T

 n 
 cos nt
2 
 Sinc
n 1
Equation 6.1
where E = pulse amplitude, volts
 = pulse width, seconds
T = periodic time, seconds
a)
If T = 3 sec and  = 1 sec, calculate and tabulate component
amplitudes and frequencies up to the second null and sketch the singlesided spectrum.
[8 Marks]
b)
(i)
Show that equation 6.1 may be written in the form:
f (t ) 

E
n 
Sinc
 exp jnt 

2 
n   T

Equation 6.2
[4 Marks]
(ii)
Sketch the bilateral spectrum for f(t).
recalculate values).
(You do not need to
[4 Marks]
c)
If the pulse is passed through an ideal narrow band filter, with a voltage
gain equal to 1, which removes all components (including the DC
component) except the fundamental, determine the amplitude and
frequency of the waveform at the filter output.
[4 Marks]
Staffordshire University
Faculty of Computing, Engineering and Technology
November 2005
Signal Processing
Q7.
a)
Page 20
Give an equation for the average voltage of the signal given by:
v( t ) 
E 2 E

T
T

 n 
 Cos nt
2 
 Sinc
n 1
Equation 7.1
Explain your answers clearly.
b)
[4 Marks]
Show that Equation 7.1 may be written in the complex form:
v( t ) 
E
 n 
Sinc
 exp jnt 
 2 
n  T


Equation 7.2
[4 Marks]
c)
The above equations describe a square wave with amplitude E, if  equals T
and  
2
.
T
2
If a square wave is bandlimited by a low-pass filter with a cut-off frequency
2 Hz , determine the distortion D, defined as:
T
PT  PBL
PT
where: PT is the total power in the signal,
PBL is the power in the bandlimited signal.
D
d)
If T
[6 Marks]
  4 , tabulate and plot the single-sided spectrum for components up to
the first null. Express values in terms on E,  and T as appropriate.
[6 Marks]