An Introduction to the Analysis and Processing of Signals by Paul A. Lynn (auth.)

By Paul A. Lynn (auth.)

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Extra resources for An Introduction to the Analysis and Processing of Signals

Example text

8. Show that the Fourier series for the 'half-wave rectified' sine wave shown in the figure (that is, a sine wave with alternate half-cycles eliminated) is given by 1 1 2 2 [(t) = 1t +- cos wIt + - cos 2w)t - cos 4w)t + ... 2 37t 157t 2(- 1)) +n/2 + (2 cos nw)t + ... (n even) 7t n - I) 9. A wave [)(t) = 0·5 sin w) t is added to another wave h(t) = cos w) t. Express their sum in the form of a single waveform of sinusoidal form with an associated phase angle. Show that your result is reasonable by tabulating values of f) (t) and [2(t) and summing them graphically.

F(t)}. e-st . dt o J o f(t). e -(s+a)t . dt o ~d t which may be integrated by parts to give G4 (s) = [r(t). e -stI + s I f(t). e -st. G(s)-f(O) where f(O) represents the value of f(t) immediately prior to t = O. Since we are generally concerned with functions which are zero before t = 0, the term f(O) may normally be disregarded, giving G 4 (s) = s. G(s) However, for generality, we should retain the term f(O). The reasons for this will become clearer in chapter 7 when we deal with the response of linear systems which have energy stored in them prior to t = O.

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