ece 3500 hw 7 signals and systems university of utah ece.utah.edu

Full Name: Lab Section:

ECE 3500 (Fall 2015) – HW #7 Due Date: Oct. 29, 2015

Homework learning objectives: • Apply the Laplace and inverse Laplace transforms • Apply and analyze amplitude modulation

Question #1: (2 pts) How many hours did you spend on this homework?

Question #2: (12 pts) For the following impulse responses h(t): (a) Compute the corresponding transfer function H(s) in the Laplace domain, or “s” domain

(b) Sketch the pole-zero plot for the system.

(c) Determine the region of convergence for H(s).

Use the Laplace transform pairs and properties tables.

(a) h(t) = e−3tu(t)

(b) h(t) = e−4tu(t) + e−2tu(t)

(c) h(t) = e−tu(t) + e7tu(−t) (d) h(t) = et cos(3t)u(−t) (e) h(t) = (t− 1)e−2(t−1)u(t− 1) (f) h(t) =

( e−2tu(t)

) ∗ ( e−4tu(t)

) ∗ ( e−2t cos(5t)u(t)

)

Question #3: (12 pts) For the following transfer functions H(s = σ + jω) and region of con- vergence, compute the impulse response h(t). Use the Laplace transform tables and partial fraction decomposition.

(a) H(s) = 1

s+ 3 , ROC: σ ≥ −3

(b) H(s) = s+ 2

s2 + 4 , ROC: σ ≥ 0

(c) H(s) = s− 1

(s+ 2)(s+ 3) , ROC: σ ≤ −3

(d) H(s) = s− 1

(s+ 2)(s+ 3) , ROC: −3 ≤ σ ≤ −2

(e) H(s) = 24

s(s+ 2)(s− 4) , ROC: σ ≥ 4

(f) H(s) = 1

s2(s− 1) , ROC: σ ≥ 1

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Question #4: (12 pts) [Problem 4.7-6 from the text book, modified] Answer the following for the baseband (i.e., low frequency near 0 Hz) signal x(t) = cos(100t) + 2 cos(300t).

(a) Sketch the Fourier transform of x(t).

(b) Find and sketch the Fourier transform of the double sideband (DSB) signal xDSB−SC(t) = 2x(t) cos(1000t).

(c) From the result in (b), use an ideal filter to suppress the lower sideband (LSB) spectrum (the lower half frequencies in each band) to obtain the upper sideband (USB) spectrum. Sketch the result, XUSB(ω).

(d) Now sketch the Fourier transform of xUSB(t) cos(1000t), where xUSB(t) is the inverse Fourier transform of XUSB(ω).

(e) From the result in (b), use an ideal filter to suppress suppress the upper sideband (USB) spectrum (the upper half frequencies in each band) to obtain the lower sideband (LSB) spectrum. Sketch the result, XLSB(ω).

(f) Now sketch the Fourier transform of xLSB(t) cos(1000t), where xLSB(t) is the inverse Fourier transform of XLSB(ω).

Question #5: (8 pts) Quadrature-amplitude modulation (QAM) is a bandwidth conservation tech- nique in which multiple messages occupy the same transmission band. Consider two message signals x1(t) and x2(t) (with Fourier Transforms X1(ω) and X2(ω)), both with bandwidths ωb (the width of the signal in the frequency domain), and a carrier frequency ωc. A QAM signal xQAM (t) is defined by

xQAM (t) = x1(t) cos(ωct) + x2(t) sin(ωct)

(a) Find the Fourier transform of xQAM (t), i.e., XQAM (ω), in terms of X1(ω) and X2(ω).

(b) Show that we can recover x1(t) from xQAM (t) by low-pass filtering y(t) = xQAM (t) cos(ωct).

(c) For the result in (b), determine the cutoff frequency for the low-pass filter to safely recover x1(t).

(d) Devise a procedure to recover x2(t) from xQAM (t).

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