# Mike Doggett Staffordshire University

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Mike Doggett Staffordshire University
Signal Processing Mike Doggett Staffordshire University

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.

Fundamental frequency (n=1) DC or average component
The Fourier Series for a periodic signal may be expressed by: AC components Fundamental frequency (n=1) at ω rads per second. DC or average component

a0, an and bn are coefficients given by:

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.

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

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.

FOURIER SERIES FOR A UNIPOLAR RECTANGULAR PULSE TRAIN
Consider the rectangular pulse train below.

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’

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

The an coefficients are given by

Since sin(-A) = -sinA

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

Simplifying, by noting substituting back into the F.S. equation:

Fourier Series for a unipolar pulse train.
But NOTE, it is more usual to convert this to a ‘Sinc function’. ie Sinc(X) =

Note the ‘trick’, i.e multiply by
This reduces to

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

FOURIER SERIES (F.S.) Review
We have discussed that the general FS for an Even function is: Fourier Series for a unipolar pulse train.

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).

The amplitudes of the harmonic components are given by:
To calculate, it is usually easier to use the form

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.

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

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.

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.

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

Q2. What pulse characteristic would give this spectrum?

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?

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:

Since the ‘trig’ form of F.S. is:
, then this may be written in the complex form:

The complex F.S. may be written as:
where:

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.

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

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

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.

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

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.

Consider when n = 0 Sinc(0) = 1 and ej0 = 1, ie = when n = 0 Hence we may write:

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).

Exercise Q1. A pulse waveform has a ratio of = 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 single-sided and two-sided spectrum.

Q4. Convert the ‘trig’ FS to complex by using the substitution :