exercise_01

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Communication and
Detection Theory
Spring Semester 2019
Prof. Dr. A. Lapidoth
Signal and Information
Processing Laboratory
Institut für Signal- und
Informationsverarbeitung
Exercise 1 of February 19, 2019
http://www.isi.ee.ethz.ch/teaching/courses/cdt
Problem 1 Manipulating Inner Products
Show that if u, v, and w are energy-limited complex signals, then
〈u+ v, 3u+ v + iw〉 = 3 ‖u‖2
2
+ ‖v‖2
2
+ 〈u,v〉+ 3 〈u,v〉∗ − i 〈u,w〉 − i 〈v,w〉 .
Problem 2 Orthogonality to All Signals
Let u be an energy-limited signal. Show that
(
u ≡ 0
)
⇐⇒
(
〈u,v〉 = 0, v ∈ L2
)
.
Problem 3 Finite-Energy Signals
Let x be an energy-limited signal.
(i) Show that, for every t0 ∈ R, the signal t 7→ x(t− t0) must also be energy-limited.
(ii) Show that the reflection of x is also energy-limited. I.e., show that the signal ~x that maps t
to x(−t) is energy-limited.
(iii) How are the energies in t 7→ x(t), t 7→ x(t− t0), and t 7→ x(−t) related?
Problem 4 Inner Products of Mirror Images
Express the inner product 〈~x, ~y〉 in terms of the inner product 〈x,y〉.
Problem 5 Truncated Polynomials
Consider the signals u : t 7→ (t + 2) I{0 ≤ t ≤ 1} and v : t 7→ (t2 − 2t − 3) I{0 ≤ t ≤ 1}. Compute
the energies ‖u‖2
2
and ‖v‖2
2
as well as the inner product 〈u,v〉.
c© Amos Lapidoth, 2019 1