By Paul A. Lynn (auth.)
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Additional info for An Introduction to the Analysis and Processing of Signals
Its Laplace transform is given by G(s) = 1=0 f f(t). dt 2J 00 C(s) = o o :( s; 1[ = -2' - ( =2j 1 . ) exp [-(s - jwo)t] + ( 1) JWo (s - jwo) - (s + jwo) w: + =(s + jwo)(s - 1 . 11. Of course, the variable s = (a + jw) can be real, imaginary, or complex, and if we make it imaginary (s = jw) we are in effect considering the signal to be composed of sinusoidal waves, as in the Fourier transform. 13. 13 w- -450 -----------------_ -90· - - - - -- - - - - - - -- - -- - - - - -- - - - - - - - - - - - - (a) Magnitude /unction and (b) phase function of the spectrum of the waveform offigure 3.
The waveform is given by f(t) = WI t/2 in the interval -1t/WI < t <1t/WI; replacing WIt by x for convenience, and changing the limits to x = ±1t, we have and which may be integrated by parts to give I 2-1t [sin nx xcos nx]1t --2- - n n -1t 1 = - 2 (sin n1t - n1t cos n1t) 1tn 21 PERIODIC SIGNALS Ifn is an odd integer, sin me = 0 and cos n7t = -I, givingBn integer, sin n7t = 0 and cos n7t = I, giving Bn = -lin. Thus = lin; ifn is an even B4 = -i, ... 2 [(t) = sin wit -! sin 2w}t +! sin 3w 1 t -! sin 4w}t +....
2 5m- (b) (a) A triangular wave, and (b) the TfIIlgnitudes of its real exponential Fourier coefficients exhibits half-wave symmetry, the coefficients am will be zero when m is even, and it is sufficient to integrate over the interval x = 0 t01t. Over this interval, the wave is described by f(x) = (~ - 1), o