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Signal ProcessingMike Doggett

Staffordshire University

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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 DC component – the average value of f(t).

A component at a fundamental frequency and harmonically related components, collectively the AC components.

ie f(t) = DC + AC components.

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The Fourier Series for a periodic signal may be expressed by:

f t a a n t b n tn n

n

( ) { cos sin }

0

1

DC or average componentAC components

Fundamental frequency (n=1) at ω rads per second.

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a0, an and bn are coefficients given by:

aT

f t dtT

T

0

2

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( )

2

2

cos)(2

T

T

tdtntfTna

tdtntfTnb

T

T

2

2

sin)(2

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NOTE The function must be periodic, i.e. f(t) = f(t+T).

Periodic time = T. Frequency f = Hz. If f(t) = f(-t) the function is EVEN and only cosine

terms (and a0) will be present in the F.S.

time

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If f(t) = -f(-t) the function is ODD and only sine terms (and a0) will be present in the F.S.

The coefficients an and bn are the amplitudes of the sinusoidal components.

For example, in general, an cos nωt

time

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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.

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FOURIER SERIES FOR A UNIPOLAR RECTANGULAR PULSE TRAIN

Consider the rectangular pulse train below.

time

t=0 2

T

2

T

Pulse width τ

E

0

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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,

And f(t) = 0, ‘elsewhere’

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t

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As noted, the Fourier Series for a periodic signal may be expressed by:

Applying to find

f t a a n t b n tn n

n

( ) { cos sin }

0

1

2

2

10

EdtT

a

2

20

tT

Ea

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The an coefficients are given by

T

Ea

0

2

2

cos)(2

T

T

tdtntfTna

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Since sin(-A) = -sinA

2

2

2

2

sin2cos

2

n

tn

T

EtdtnE

Tna

2sin

2sin

2

nn

Tn

Ena

2sin

4

n

Tn

Ena

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In this case it may be show that bn = 0 (because the choice of t = 0 gives an even function).

Hence:

and

T

Ea

0

2sin

4

n

Tn

Ena

0nb

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Simplifying, by noting

substituting back into the F.S. equation:

T

2

T

n

n

ETn

TT

n

En

Tn

Ena

sin

2

2

2

sin24

2sin

4

1

0 cos)(n

n tnaatf

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Fourier Series for a unipolar pulse train.

But NOTE, it is more usual to convert this to a ‘Sinc function’.

ie Sinc(X) =

tnn T

n

n

E

T

Etf

cos1

sin2

)(

X

Xsin

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Note the ‘trick’, i.e multiply by

This reduces to

2.

2

2sin

4

2sin

4

n

n

n

Tn

En

Tn

Ena

2

2

n

n

2

2 nSinc

T

Ena

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Hence

This is an important result, the F.S. for a periodic pulse train and gives a spectrum of the form shown below:

tnn

nSinc

T

E

T

Etf

cos1 2

2)(

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‘Sinc’ envelope

frequency

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FOURIER SERIES (F.S.) Review We have discussed that the general FS for an

Even function is:

Fourier Series for a unipolar pulse train.

1cos)( 0

ntnnaatf

tnn T

n

n

E

T

Etf

cos1

sin2

)(

tnn

nSinc

T

E

T

Etf

cos1 2

2)(

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The Sinc function gives an ‘envelope’ for the amplitudes of the harmonics.

The Sinc function, in conjunction with

gives the amplitudes of the harmonics.

Note that Sinc(0) =1. (As an exercise, justify this statement).

T

E2

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The amplitudes of the harmonic components are given by:

To calculate, it is usually easier to use the form

2

2 nSinc

T

Ena

T

n

T

Ena

sin

2

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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 = Hz.T

1

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‘Rules of Thumb’

The following ‘rules of thumb’ may be deduced for a pulse train, illustrated in the waveform below.

E volts

τ

T

E volts

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Harmonics occur at intervals of f =

OR f, 2f, 3f, etc.

Nulls occur at intervals of

If = x is integer, then nulls occur every xth

harmonic.

.3

,2

,1

,1

etcTTT

ieHzT

etcieHz ,3

,2

,1

,1

T

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For example if T = 10 ms and τ = 2.5 ms, then

= 4 and there will be nulls at the 4th

harmonic, the 8th harmonic, the 12th harmonic and so on at every 4th harmonic.

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

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Exercise Q1. Label the axes and draw the pulse waveform corresponding to the spectrum below.

frequency

4 kHz

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Q2. What pulse characteristic would give this spectrum?

frequency

1 kHz

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Q3. Suppose a triac firing circuit produces a narrow

pulse, with 1 nanosecond pulse width, and a repetition rate of 50 pulses per second.

What is the frequency spacing between the harmonics?

At what frequency is the first null in the spectrum?

Why might this be a nuisance for radio reception?

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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:

2cos

tjnetjnetn

j

tjnetjnetn

2sin

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Since the ‘trig’ form of F.S. is:

, then this may be written in the complex form:

1

}sincos{0)(n

tnnbtnnaatf

1220)(

nj

tjnetjnenb

tjnetjnenaatf

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The complex F.S. may be written as:

where:

n

tjnenCtf )(

njbnanC 2

1

dttjnetfTnC

T

T

2

2

)(1

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When n = 0, C0 ej0 = C0 is the average value.

n = ± 1, n = ± 2, n = ± 3 etc represent pairs of harmonics.

These are general for any periodic function.

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In particular, for a periodic unipolar pulse waveform, we have:

OR

T

n

n

Ena

sin2

2

2 nSinc

T

Ena

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Hence

Alternative forms of complex F.S. for pulse train:

2sin

2

1

nSinc

T

E

T

n

n

EnanC

tjnen T

n

n

Etf

sin)(

tjnen

Sincn T

Etf

2)(

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Example

Express the equation below (for a periodic pulse train) in complex form.

NOTE, we change the ‘cos’ term, We DON’T change the Sinc term.

tnn

nSinc

T

E

T

Etf

cos1 2

2)(

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Since:

By changing the sign of the ‘-n’ and summing from -∞ to -1, this may be written as:

2cos

tjnetjnetn

21 2

2)(

tjnetjne

n

nSinc

T

E

T

Etf

tjnen

Sincn T

Etjnen

nSinc

T

E

T

Etf

21 2)(

1

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We the have and

We want

We need to include the term for n = 0 and may

show that for n = 0, the term results.

1

n

1n

n

T

E

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Consider when n = 0

Sinc(0) = 1 and ej0 = 1,

ie = when n = 0

Hence we may write:

tjnen

SincT

E

2

tjnen

SincT

E

2 T

E

tjnen

Sincn T

Etf

2)(

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Comments Fourier Series apply only to periodic functions.

Two main forms of F.S., Trig’ F.S. and ‘Complex’ F.S. which are equivalent.

Either form may be represented on an Argand diagram, and as a single-sided or two-sided (bilateral) spectrum.

The F.S. for a periodic function effectively allows a time-domain signal (waveform) to be represented in the frequency domain, (spectrum).

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Exercise Q1. A pulse waveform has a ratio of

= 5.

Sketch the spectrum up to the second null using the ‘rules of thumb’.

T

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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.

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Q3.

With T = 4 ms and τ = 1 ms as in Q2, now calculate, tabulate and sketch the single-sided and two-sided spectrum.

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Q4.

Convert the ‘trig’ FS to complex by using the substitution :

2cos

tjnetjnetn