By Paul A. Lynn (auth.)
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Extra info for An Introduction to the Analysis and Processing of Signals
Now when an odd function is integrated over an interval symmetrical about t = 0, the result is always zero. Hence all the B coefficients are zero and we are left with a series containing only cosines. By similar arguments, if f(t) is odd the A coefficients must be zero and we are left with a sine series. It is indeed intuitively clear that an even function can only be built up from a number of other functions which are themselves even, and vice versa. 4, which is odd. In this case, all An will be zero, except possibly A o.
Sin 4wlt + ... Tabulate and plot the first four components of the series and add them, and hence show that it is likely that the infmite series will converge to the given waveform. 5. The diagram shows a function in the interval between x =0 and x =rr:/2. +------------------~----~---4---------4--- Tr x-J!.. l!.. -Tr 4 2 29 PERIODIC SIGNALS Complete the function over the full period from x = -7t to X = 7t in such a way that: (i) the Fourier series contains only cosine terms, (ii) the Fourier series contains only sine terms, (iii) the function displays half-wave symmetry.
S - Z2)(S - Z3) .. )(s - P2)(S - P3) . . where A is a constant, then the values s = Z .. , represent poles. It is indeed always possible to write a Laplace transform in the above way, and therefore to define the signal in terms of a set of poles and zeros. It is found 17 that poles are either real, or they occur in complex conjugate pairs, and the same is true of zeros. Thus a pole at s = 0: + j~ is matched by one at s = 0: - j~, and this must always be the case when the corresponding signal is a real function of time.