Presentation on theme: "Mike Doggett Staffordshire University"— Presentation transcript:
1Mike Doggett Staffordshire University Signal ProcessingMike DoggettStaffordshire University
2FOURIER 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.
3Fundamental frequency (n=1) DC or average component The Fourier Series for a periodic signal may be expressed by:AC componentsFundamental frequency (n=1)at ω rads per second.DC or average component
5NOTEThe 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.
6If 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
7The component at the lowest frequency (excluding the DC component) is when n = 1, i.e. a1 cos ωtThis 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.
8FOURIER SERIES FOR A UNIPOLAR RECTANGULAR PULSE TRAIN Consider the rectangular pulse train below.
9Pulse 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’
10As noted, the Fourier Series for a periodic signal may be expressed by: Applying to find
19FOURIER SERIES (F.S.) Review We have discussed that the general FS for an Even function is:Fourier Series for a unipolar pulse train.
20The Sinc function gives an ‘envelope’ for the amplitudes of the harmonics. The Sinc function, in conjunction withgives the amplitudes of the harmonics.Note that Sinc(0) =1. (As an exercise, justify this statement).
21The amplitudes of the harmonic components are given by: To calculate, it is usually easier to use the form
22The 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 relatedby f = Hz.
23‘Rules of Thumb’The following ‘rules of thumb’ may be deduced for a pulse train, illustrated in the waveform below.E volts
24Harmonics occur at intervals of f = OR f, 2f, 3f, etc.Nulls occur at intervals ofIf = x is integer, then nulls occur every xthharmonic.
25For example if T = 10 ms and τ = 2.5 ms, then = 4 and there will be nulls at the 4thharmonic, 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.
26Exercise Q1. Label the axes and draw the pulse waveform corresponding to the spectrum below.
27Q2. What pulse characteristic would give this spectrum?
28Q3.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?
29COMPLEX 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:
30Since the ‘trig’ form of F.S. is: , then this may be written in the complex form:
32When 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.
33In particular, for a periodic unipolar pulse waveform, we have:
34HenceAlternative forms of complex F.S. for pulse train:
35ExampleExpress the equation below (for a periodic pulse train) in complex form.NOTE, we change the ‘cos’ term, We DON’T change the Sinc term.
36Since:By changing the sign of the ‘-n’ and summing from -∞ to -1, this may be written as:
37We the have andWe wantWe need to include the term for n = 0 and mayshow that for n = 0, the term results.
38Consider when n = 0Sinc(0) = 1 and ej0 = 1,ie = when n = 0Hence we may write:
39CommentsFourier 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).
40Exercise Q1. A pulse waveform has a ratio of = 5. Sketch the spectrum up to the second null using the ‘rules of thumb’.
41Q2.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.
42Q3.With T = 4 ms and τ = 1 ms as in Q2, now calculate, tabulate and sketch the single-sided and two-sided spectrum.
43Q4.Convert the ‘trig’ FS to complex by using the substitution :