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1 Signal Processing Mike Doggett Staffordshire University.

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1 1 Signal Processing Mike Doggett Staffordshire University

2 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.

3 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.

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

5 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.

6 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

7 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.

8 8 FOURIER SERIES FOR A UNIPOLAR RECTANGULAR PULSE TRAIN n Consider the rectangular pulse train below.

9 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’

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

11 11 n The a n coefficients are given by

12 12 n Since sin(-A) = -sinA

13 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

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

15 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) =

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

17 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:

18 18

19 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.

20 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).

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

22 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.

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

24 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.

25 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.

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

27 27 Q2.What pulse characteristic would give this spectrum?

28 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?

29 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:

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

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

32 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.

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

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

35 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.

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

37 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.

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

39 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).

40 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’.

41 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.

42 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.

43 43 n Q4. n Convert the ‘trig’ FS to complex by using the substitution :


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