exercise_02

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Communication and
Detection Theory
Spring Semester 2018
Prof. Dr. A. Lapidoth
Signal and Information
Processing Laboratory
Institut für Signal- und
Informationsverarbeitung
Exercise 2 of Februar 27, 2018
http://www.isi.ee.ethz.ch/teaching/courses/cdt
Problem 1 Reflection of Passband Signal
Let xPB and yPB be real integrable passband signals that are bandlimited to W Hz around the
carrier frequency fc. Let xBB and yBB be their baseband representations.
(i) Express the baseband representation of ~xPB in terms of xBB.
(ii) Express 〈xPB, ~yPB〉 in terms of xBB and yBB.
Problem 2 Symmetries of the FT
Let x : R→ C be integrable, and let xˆ be its FT.
(i) Show that if x is a real signal, then xˆ is conjugate-symmetric, i.e., xˆ(−f) = xˆ∗(f), for every
f ∈ R.
(ii) Show that if x is purely imaginary (i.e., takes on only purely imaginary values), then xˆ is
conjugate-antisymmetric, i.e., xˆ(−f) = −xˆ∗(f), for every f ∈ R.
(iii) Show that xˆ can be written uniquely as the sum of a conjugate-symmetric function gcs and
a conjugate-antisymmetric function gcas. Express gcs & gcas in terms of xˆ.
Problem 3 Phase Shift
Let x be a real integrable signal that is bandlimited to W Hz. Let fc be larger than W.
(i) Express the baseband representation of the real passband signal
zPB(t) = x(t) sin(2pifct+ φ), t ∈ R
in terms of x(·) and φ.
(ii) Compute the Fourier Transform of zPB.
c© Amos Lapidoth, 2018 1
Problem 4 Purely Real and Purely Imaginary
Baseband Representations
Let xPB be a real integrable passband signal that is bandlimited to W Hz around the carrier
frequency fc, and let xBB be its baseband representation.
(i) Show that xBB is real if, and only if, xˆPB satisfies
xˆPB(fc − δ) = xˆ
∗
PB(fc + δ), |δ| ≤
W
2
.
(ii) Show that xBB is imaginary if, and only if,
xˆPB(fc − δ) = −xˆ
∗
PB(fc + δ), |δ| ≤
W
2
.
Problem 5 Symmetry around the Carrier Frequency
Let xPB be a real integrable passband signal that is bandlimited to W Hz around the carrier
frequency fc.
(i) Show that xPB can be written in the form
xPB(t) = w(t) cos(2pifct)
where w(·) is a real integrable signal that is bandlimited to W/2 Hz if, and only if,
xˆPB(fc + δ) = xˆ
∗
PB(fc − δ), |δ| ≤
W
2
.
(ii) Show that xPB can be written in the form
xPB(t) = w(t) sin(2pifct), t ∈ R
for w(·) as above if, and only if,
xˆPB(fc + δ) = −xˆ
∗
PB(fc − δ), |δ| ≤
W
2
.
Problem 6 Table
Be sure you can justify and derive all the entries in Table 7.1.
c© Amos Lapidoth, 2018 2