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1 Signal Processing Mike Doggett Staffordshire University

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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: n A DC component – the average value of f(t). n A component at a fundamental frequency and harmonically related components, collectively the AC components. n ie f(t) = DC + AC components.

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

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4 n a 0, a n and b n are coefficients given by:

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5 n NOTE n The function must be periodic, i.e. f(t) = f(t+T). Periodic time = T. Frequency f = Hz. n If f(t) = f(-t) the function is EVEN and only cosine terms (and a0) will be present in the F.S.

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6 n If f(t) = -f(-t) the function is ODD and only sine terms (and a 0 ) will be present in the F.S. n The coefficients a n and b n are the amplitudes of the sinusoidal components. n For example, in general,a n cos nωt

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7 n The component at the lowest frequency (excluding the DC component) is when n = 1, i.e. a 1 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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8 FOURIER SERIES FOR A UNIPOLAR RECTANGULAR PULSE TRAIN n Consider the rectangular pulse train below.

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9 n Pulse width τ seconds, periodic time T seconds, amplitude E volts (unipolar). n 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. n We define f(t) = E, n And f(t) = 0, ‘elsewhere’

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

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11 n The a n coefficients are given by

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12 n Since sin(-A) = -sinA

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

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

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

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

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

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

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20 n The Sinc function gives an ‘envelope’ for the amplitudes of the harmonics. n The Sinc function, in conjunction with gives the amplitudes of the harmonics. Note that Sinc(0) =1. (As an exercise, justify this statement).

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

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22 n The harmonics occur at frequencies nω radians per second. n We normally prefer to think of frequency in Hertz, and since ω = 2πf, we can consider harmonics at frequencies nf Hz. n The periodic time, T, and frequency are related by f = Hz.

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

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24 n Harmonics occur at intervals of f = OR f, 2f, 3f, etc. n Nulls occur at intervals of n If = x is integer, then nulls occur every xth harmonic.

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25 n 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.

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26 ExerciseQ1.Label the axes and draw the pulse waveform corresponding to the spectrum below.

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27 Q2.What pulse characteristic would give this spectrum?

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28 n Q3. n Suppose a triac firing circuit produces a narrow pulse, with 1 nanosecond pulse width, and a repetition rate of 50 pulses per second. n What is the frequency spacing between the harmonics? n At what frequency is the first null in the spectrum? n Why might this be a nuisance for radio reception?

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29 COMPLEX FOURIER SERIES n Up until now we have been considering trigonometric Fourier Series. n An alternative way of expressing f(t) is in terms of complex quantities, using the relationships:

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

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

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32 n When n = 0, C 0 e j0 = C 0 is the average value. n n = ± 1, n = ± 2, n = ± 3 etc represent pairs of harmonics. n These are general for any periodic function.

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

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

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35 Example n Express the equation below (for a periodic pulse train) in complex form. n NOTE, we change the ‘cos’ term, We DON’T change the Sinc term.

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

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37 n We the have and n We want n We need to include the term for n = 0 and may show that for n = 0, the term results.

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38 n Considerwhen n = 0 n Sinc(0) = 1 and e j0 = 1, n ie = when n = 0 n Hence we may write:

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39 n Comments n Fourier Series apply only to periodic functions. n Two main forms of F.S., Trig’ F.S. and ‘Complex’ F.S. which are equivalent. n Either form may be represented on an Argand diagram, and as a single-sided or two-sided (bilateral) spectrum. n 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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40 Exercise n Q1. n A pulse waveform has a ratio of = 5. Sketch the spectrum up to the second null using the ‘rules of thumb’.

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41 n Q2. n A pulse has a periodic time of T = 4 ms and a pulse width τ = 1 ms. n 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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42 n Q3. n 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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43 n Q4. n Convert the ‘trig’ FS to complex by using the substitution :

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