ECE 8700 Communication System Engineering, Spring 2009
Homework Set # 1
Suggested Problems from the Text
2.1,2.2,2.7,2.9 (signal & system theory for digital communications);
2.3,2.6,2.8,2.11,2.12,2.13 (signal space representation)
Homework # 1 (Due Wed., Jan. 21 before class): (Do all. Submit problems 2, 3, 4,
7, 8.)
1. Problem 2.2 of the Course Text.
2. Problem 2.9 of the Course Text.
3. Consider the set of signals {xk(t) = sinc(t−k); k= 0,±1,±2,···}. Show that they
form an orthonormal set. (Hints: the sinc function is define on p. 17 of the Course
Text. Use the Fourier transform representations of the xk(t) when evaluating their
inner products. Use Table 2.0-2 on p. 19 of the Course Text and the delay property
in Table 2.0-1 for xk(t) Fourier transforms. Use the following fact from generalized
functions,
Z∞
−∞
ej2πf t dt =δ(f) (1)
where δ(f) is the continuous impulse function.)
4. Low rank representation of vectors:
(a) In Lectures 1, after Eq (40), it is noted that the coefficients sk=vH
kvminimize
the Euclidean norm (i.e. the energy) of the error vector
e=v−ˆv=v−
m
X
k=1
skvk=v−V s (2)
of the the low rank orthonormal expansion of n-dimensional vector vwith respect
to the orthonormal vectors vk;k= 1,2,···, m (where m < n). Prove this by
taking the derivatives of ||e||2with respect to the sk;k= 1,2,···, m and setting
them equal to zero. To simplify this, assume all values are real-valued.
(b) Given the optimum s′
ks, and starting with Eq (40) of Lecture 1, prove Eq (41).
5. Problem 2.3 of the Course Text.
6. Consider orthonormal φ1(t) and φ2(t), and three symbols s1(t) = φ1(t) + φ2(t),
s2(t) = −φ1(t)+φ2(t) and s3(t) = −φ2(t).Determine the signal space diagram of these
three symbols, and the minimum Euclidean distance D(e)
min between these symbols.
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