EE160. Spring 2003. San Jose State UniversityHOMEWORK # 1 Due: Thursday 2/20/031. (Textbook, Problem 2.3) Show that for a real and periodic signal x(t), we havexe(t)=a02+∞n=1ancos2πnT0t ,xo(t)=∞n=1bnsin2πnT0t ,where xe(t)andxo(t) are the even and odd parts of x(t), defined asxe(t)=x(t)+x(−t)2,xo(t)=x(t) − x(−t)2.2. Determine the Fourier series expansion of the sawtooth waveform, shown belowT2T3T-T-2T-3T-11x(t)t3. (Textbook, problem 2.23) By computing the Fourier series coefficients for the periodicsignal∞n=−∞δ(t − nTs), show that∞n=−∞δ(t − nTs)=1Ts∞n=−∞ejn2πtTs.Using this result, show that for any signal x(t)andanyperiodTs, the following identityholds∞n=−∞x(t − nTs)=1Ts∞n=−∞XnTs ejn2πtTs.From this, conclude the following relation, known as Poisson’s sum formula:∞n=−∞x(nTs)=1Ts∞n=−∞XnTs .4. (Textbook, problem 2.36) Let the signal x(t)=sinc(1000t) be sampled with a sam-pling frequency of 2000 samples per second. Determine the most general class ofreconstruction filters for the perfect reconstruction of x(t) from its samples.5. (Textbook, problem 2.37) The lowpass signal x(t) with a bandwidth of W is sampledat intervals of Tsseconds, and the signalxp(t)=∞n=−∞x(nTs)p(t − nTs)is generated, where p(t) is an arbitrary pulse (not necessarily limited to the interval[0,Ts]).(a) Find the Fourier transform of xp(t).(b) Find the conditions for perfect reconstruction of x(t)fromxp(t).(c) Determine the required reconstruction
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