Exercicios resolvidos HW04V3withProbs Fall2015 EE2353

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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.