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1 EE2353: Continuous-Time Signals and Systems: HOMEWORK #4 Fall 2015, SOLUTIONS 3.11 d) only Solution: Each impulse response has two terms which means we need to perform four convolutions, however the “cross-terms” can be combined due to the commutative property. Also, we can add shifts since they are equivalent to convolutions with impulses, in general: x(t-t1)*h(t-t2) = x(t)*(t-t1)*h(t)*(t-t2)= x(t)*h(t)*(t-t1-t2) )= y(t)*(t-t1-t2)=y(t-(t1+t2)) Also convolution of a step with a step gives a ramp. 3.13 Here you may want to use well known results of convolutions: anything convolved with an impulse; steps convolved with steps; steps with ramps, etc. Try to avoid actually performing any convolutions by recognizing well known results. Solution: Call mi(t) the output of each LTI with impulse response hi(t). 2 3.16 a)-c) Solution: Here, h(t) is a left-sided signal (LSS) which decays on the left side 3.18 c), e) 3 Solution: for c) to prove stability: the decaying exponential et for t<0 is a bigger integrand than the smaller product integrand |et sin(-5t)| . Since it converges, the product integrand must converge (smaller<bigger< ). For e) the signal is clearly left-sided so cannot be causal. The integral is of a growing exponential for t<0. 3.19 Your answer to a) can be checked by looking at the input-output relationship below and making it look like a convolution with h(t). x(t) = u(t+1) Solution: One approach is to match the input/output relationship to a convolution integral. It should give the same h(t) as found here by letting x(t)=(t). 3.22 a), b) and C C-Also, determine the step response of this LTI system and plot it. Solution: b) Stable since |h(t)|=1 for 0<t<2 so its integral is 2<. C) The running integral is clearly increasing linearly from 0 to 1 for 0<t<1 then decreases linearly from 1 to 0 for 1<t<2, so. otherwise0)(; 21;2 10; )( ts tt tt ts 4.2 a), b) for (i), (ii) and (iv) In all cases use the largest possible fundamental frequency 4 Solution:(i) Clearly all frequencies are multiples of o=4 thus C0=7 4.4 c) Use Trig. Identities from Appendix A Solution: Using orthogonality of complex sinusoids would be simplest way (4 terms all integrating to 0) but this is another doable way.
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