Download EE422G Homework #1 Solutions: Linear Systems, Fourier Transform, and Laplace Transform and more Assignments Signals and Systems in PDF only on Docsity! EE422G Homework #1 Solution 1. (4 pts) Determine which of the following systems are linear. Explain your answer: (a) dy(t)dt + 3y(t) = 4x(t) Solution: Yes, it is linear. The reason is that given two pairs of input-output signals x1(t), y1(t) and x2(t), y2(t) that satisfy the differential equation, it is clear that ax1(t)+bx2(t) and ay1(t)+by2(t) are also a solution to the differential equation as shown below: d dt (ay1(t) + by2(t)) + 3(ay1(t) + by2(t)) = a [ d dt y1(t) + 3y1(t) ] + a [ d dt y1(t) + 3y1(t) ] = ax1(t) + bx + 2(t) (b) The impulse response: h(t) = δ(t + 2) Solution: Since the system has an impulse response, it is definitely linear. (c) y(t) = x2(t) + 2dx(t)dt Solution: It is non-linear due to the second degree term – when using a similar test as part a), you will notice that the second degree term creates a cross term x1(t)x2(t) that is not present in the differential equation. (d) Solution: The system is non-linear. This is a little bit tricky. As we know, for all lin- ear systems, we can relate the input and output Fourier Transforms using the following relationship: Y (jω) = X(jω)H(jω) where H(jω) is the Fourier Transform of the impulse response. Using this rela- tionship, if X(jω) is zero at a particular frequency, Y (jω) must also be zero no matter what H(jω). This is clearly not the case here as X(jω) is zero except at ω0 and −ω0. 2. (2 pts) Give the definition of Fourier transform and Fourier series. Does every signal have a Fourier transform? If yes, prove it. If no, give a counter-example. Solution: The Fourier transform of a sequence x[n] is defined as: X(ejω) = ∞∑ n=−∞ x[n]e−jωn The discrete Fourier series analysis-synthesis pair is expressed as: Analysis equation: X̃[k] = N−1∑ n=0 x̃[n]e −j2πkn N 1 Synthesis equation: x̃[k] = 1 N N−1∑ n=0 X̃[n]e j2πkn N Fourier Transform does not exist for signals whose Laplace Transforms have poles on the open right half plane. An example of such signal is x(t) = etu(t). 3. (2 pts) (THIS PROBLEM WILL NOT BE GRADED) Get the Laplace transform for the following transform and the corresponding region of convergence (ROC): (a) x(t) = sin(ωt)u(t) Solution: sin(ωt) = 1 2j ( ejωt − e−jωt) e−atu(t) ⇔ 1 s + a By the linearity property, the sum of these terms corresponds to the sum of their Laplace transforms. Therefore, L [sin(ωt)u(t)] = 1 2j [ 1 s− jω − 1 s + jω ] = ω s2 + ω2 for σ > 0 (b) x(t) = δ(t− t0), t0 > 0 Solution: X(s) = ∫ ∞ −∞ δ(t− t0)e−stdt = e−st0 for all s 4. (2 pts) Use Matlab to plot y versus t with t = 0 : 0.01 : 5 and y defined as: y = 3e−4t cos(5t)− 2e−3t sin(2t) + t 2 t + 1 Solution: t=0:0.01:5; y=3*exp(-4*t).*cos(5.*t)-2*exp(-3*t).*sin(2*t)+t.^2./(t+1); plot(t,y) xlabel(’t’) ylabel(’y’) 2
