EE 3054: Signals, Systems, and Transforms Summer It is observed of some continuous-time LTI system that the input signal.

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1 EE 34: Signals, Systems, and Transforms Summer 7 Test No notes, closed book. Show your work. Simplify your answers. 3. It is observed of some continuous-time LTI system that the input signal = 3 u(t) produces the output signal y(t) = 4 u(t) + cos(πt) u(t). (a) What can be immediately concluded about the pole positions of the LTI system? (b) What is the dc gain of the system? (c) Can you make any conclusion about the stability of the system? (d) Find the impulse response h(t) of the system. 4. A causal continuous-time LTI system is described by the equation y (t) + y (t) + y(t) = where x is the input signal, and y is the output signal.. A continuous-time system is described by the following rule y(t) = sin(πt) + cos(πt) x(t ) where x is the input signal, and y is the output signal. Classify the system as: (a) linear/nonlinear (b) time-invariant/time-varying (c) stable/unstable. Sketch the convolution of the following two signals. f(t) g(t) t t (a) Find the impulse response of the system. (b) Accurately sketch the pole-zero diagram. (c) What is the dc gain of the system? (d) Classify the system as either stable or unstable. (e) Write down the form of the step response of the system, as far as it can be determined without actually calculating the resides. (You do not need to complete the partial fraction expansion).. In this problem you are to design a simple causal continuous-time LTI system with the following properties: The system should remove the dc component of the input signal. The system should have a pole at s =. The frequency response H f (ω) should approach. as ω goes to infinity. For the system you design: (a) Find the differential equation to implement the system. (b) Find and sketch the impulse response of the system. (c) Find the frequency response H f (ω) and roughly sketch H f (ω).

2 6. Consider a continuous-time LTI system with the impulse response h(t) = e 3πt u(t). (a) Find the the frequency response H f (ω). (b) Roughly sketch H f (ω) and H f (ω). (c) Find the output signal y(t) produced by the input signal = + cos(3πt). (d) For the input signal = u(t) + cos(3πt) u(t) find the output signal y(t) as far as can be determined without actually performing partial fraction expansion. 4π π π 4π then accurately sketch the spectrum Y f (ω).. A continuous-time LTI system has the impulse response h(t) = 3 sinc(3t) sinc(t). The input signal has the spectrum shown, ω 7. Consider a continuous-time LTI system with the impulse response h(t) = 3 sinc(3 t). (a) Accurately sketch the frequency response H f (ω). (b) Find the output signal y(t) produced by the input signal = + cos(πt) + 7 cos(4πt). (c) Consider a second continuous-time LTI system with impulse response g(t) = h(t ) where h(t) is as above. For this second system, find the output signal y(t) produced by the input signal π (.π) (.π) (π) (π) 4π 3π π π π π 3π 4π Find the output signal y(t). (Hint: first find H f (ω).) ω = + cos(πt) + 7 cos(4πt). 8. Find the Fourier transform of the signal = cos πt + π In amplitude modulation (AM) the signal to be transmitted is multiplied by cos(ω o t). Usually, a constant is added before the multiplication by cosine. If y(t) is given by y(t) = ( + ) cos(4πt) and the spectrum is as shown,. The first six seconds of the impulse responses of eight causal continuoustime systems are illustrated below, along with the pole/zero diagram of each system. But they are out of order. Match the figures with each other by completing the table (copy the table into your answer book). Impulse Response Pole/ero Diagram

3 IMPULSE RESPONSE IMPULSE RESPONSE POLE/ERO DIAGRAM POLE/ERO DIAGRAM IMPULSE RESPONSE 3 IMPULSE RESPONSE 4 POLE/ERO DIAGRAM 3 POLE/ERO DIAGRAM IMPULSE RESPONSE IMPULSE RESPONSE 6 POLE/ERO DIAGRAM POLE/ERO DIAGRAM IMPULSE RESPONSE 7 IMPULSE RESPONSE 8 POLE/ERO DIAGRAM 7 POLE/ERO DIAGRAM TIME (SEC) TIME (SEC)

4 . The frequency responses of eight causal continuous-time systems are illustrated below, along with the pole/zero diagram of each system. But they are out of order. Match the figures with each other by completing a table. POLE ERO DIAGRAM POLE ERO DIAGRAM FREQUENCY RESPONSE FREQUENCY RESPONSE POLE ERO DIAGRAM 3 POLE ERO DIAGRAM FREQUENCY RESPONSE 3 FREQUENCY RESPONSE 4 POLE ERO DIAGRAM POLE ERO DIAGRAM FREQUENCY RESPONSE FREQUENCY RESPONSE 6 POLE ERO DIAGRAM 7 POLE ERO DIAGRAM FREQUENCY RESPONSE 7 FREQUENCY RESPONSE

5 useful formulas name Euler s formula formula e jθ = cos(θ) + j sin(θ)... for cosine cos(θ) = ejθ + e jθ... for sine sin(θ) = ejθ e jθ sinc function sinc(θ) := j sin(π θ) π θ selected Laplace transform pairs X(s) ROC e st dt (def.) δ(t) all s u(t) e a t u(t) cos(ω ot) u(t) s s + a s s + ω o Re(s) > Re(s) > a Re(s) > formulas for continuous-time LTI signals and systems name formula area under impulse δ(t) dt = multiplication by impulse f(t) δ(t) = f() δ(t)... by shifted impulse f(t) δ(t t o) = f(t o) δ(t t o) convolution f(t) g(t) = f(τ) g(t τ) dτ sin(ω ot) u(t) e a t cos(ω ot) u(t) e a t sin(ω ot) u(t) ω o s + ωo s + a (s + a) + ωo ω o (s + a) + ω o Note: a is assumed real. Laplace transform properties Re(s) > Re(s) > a Re(s) > a... with an impulse f(t) δ(t) = f(t)... with a shifted impulse f(t) δ(t t o) = f(t t o) transfer function H(s) = h(t) e st dt frequency response H f (ω) = h(t) e jωt dt... their connection H f (ω) = H(jω) provided jω-axis ROC a + b g(t) g(t) d dt x(t t o) X(s) a X(s) + b G(s) X(s) G(s) s X(s) e s to X(s)

6 selected Fourier transform pairs π e jωt dω δ(t) e jωt dt π δ(ω) u(t) π δ(ω) + jω e jωot π δ(ω ω o) (def.) cos(ω o t) π δ(ω + ω o) + π δ(ω ω o) sin(ω o t) j π δ(ω + ω o) j π δ(ω ω o) ω o π sinc ωo π t symmetric pulse width T, height ideal LPF cut-off frequency ω o «T ω sin ω Fourier transform properties a + b g(t) x(a t) g(t) g(t) x(t t o) a + b G f (ω) ω a X a G f (ω) π Xf (ω) G f (ω) e jtoω X(ω) e jωot X(ω ω o) cos(ω ot). X(ω + ω o) +. X(ω ω o) sin(ω ot) j. X(ω + ω o) j. X(ω ω o) d dt j ω impulse train period T, height impulse train period, height ω o = π T