Evaluate the Fourier series representation of the periodic function 2.1.4 The periodic function of Fig. P2.1.1(c) is expressed in a trigonometric Fourier series. Find the error if only three terms of the Fourier series are used. Repeat for four terms and five terms. 2.1.5 Find the Fourier series expansion of a gated sinusoid, as shown in Fig. P2.I.5. 2.1.6 The triangular waveform shown in Fig. P2.I.1(b) forms the input to a diode circuit, as shown in Fig. P2.I.6. Assume that the diode is ideal. Find the Fourier series expansion of the current i(t). 2.1.7 In Fig. P2.1.7, find a Fourier series expansion of s(t) that applies for —r12
2.1.8 Find the complex Fourier series representation or Au = r2 that applies in the interval 0
23.2 Evaluate the Fourier transform of the following functions of time: 2.5.1 Convolve elm -0 with c'd/./(0. These two functions are shown in Fig. P2.5.I. 2.5.2 onvolve together the two functions shown in Fig. P2.5.2, using the convolution integral. Repeat using graphical techniques. 25.4 Evaluate the following integral using Parseval's theorem: 2.5.5 Show that if S( f ) = 0 for Ifl >fin, then
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provided that at2ir >fin. th…..)6 Evaluate the following integrals: 2.6.1 (a) Write the convolution of s(t) with U(t) in integral forn this as the integral of s(t). (b) What is the transform of s(t)*U(t)? Solve this using the 2.6.2 Any arbitrary function can be expressed as the sum of an evi (a) Show that se(t) is an even function and that so(t) is an odi (b) Show that s(t) = s41) + so(t). (c) Find Mt) and so(t) for At) = U(t), a unit step function. (d) Find Mt) and so(t) for s(t) = cos2Ont 2.6.3 Given a function s(t) that is zero for negative t, find a rela Can this result be used to find a relationship between the Fourier transform of s(t)?
. See whether you can identify
onvolution theorem. n and odd function; that is, I function.
tionship between Mt) and so(t). real and imaginary parts of the
SEvaluate the Fourier transform of cos5in, starting with the Fourier transform of eosin and using the time-scaling property. 2.6.5 Given that the Fourier transform of s(t) is 5(f): (a) What is the Fourier transform of ds/dt in terms of S(f)? (b) What is the Fourier transform of in terms of S( J.)? 2.6.6 Use the time shift property to find the Fourier transform of From this result, find the Fourier transform of ds/dt. 2.6.7 A signal s(t) is put through a gate and truncated in time as shown in Fig. P2.6.7. The gate is closed for I