ECE 614 – Principles of Digital Communications
Homework 2
Assigned on: 02/09/2016
Due by: 02/18/2016
2.1
Two subspaces S1 and S 2 of a Hilbert space are said to be orthogonal if every vector in S1
is orthogonal to every vector in S 2 . The sum of the two subspaces S1  S 2 is the subspace
consisting of vectors that can be expressed as the sum of a vector in S1 and a vector in S 2 .
Given two orthogonal subspaces S1 and S 2 of a Hilbert space H and an arbitrary vector
x  H , show that the projection x̂ of x onto S1  S 2 can be expressed uniquely as
xˆ  xˆ 1  xˆ 2
where x̂1 is the projection of x onto S1 and x̂ 2 is the projection of x onto S 2 .
2.2
Let 1 t , 2 t , 3 t  be an orthonormal basis for a subspace S. Let us define three new
waveforms  f1 t , f 2 t , f 3 t  according to
f1 t   A111 t   A122 t   A133 t 
f 2 t   A211 t   A222 t   A233 t 
f 3 t   A311 t   A322 t   A333 t 
Under what conditions will  f1 t , f 2 t , f 3 t  also be an orthonormal basis for S?
2.3
Consider the signals r t  , s1 t  and s2 t  sketched below:
1
r(t)
2 t 1  t 
2
1
3
t
1
s1(t)
2
1
3
t
-1
1
s2(t)
2
1
3
t
Find r̂ t  , the projection of r t  onto the space spanned by s1 t  and s2 t  . Carefully
sketch r̂ t  .
2.4
Let S  st  | st  T   st , for all t be the subspace of complex-valued periodic signals
with fundamental period T. Show that the complex sinuisiod signal set
 

k 
exp  j 2 t , k  Z  (where Z is the integer set) forms an orthonormal basis of S under
T 
 

the following inner product definition.
xt , y t  
1
T
T
2
 xt y t dt
*

T
2
Given st   S , express st  as linear combination of the basis signals.
2.5
Suppose you have a set of real-valued waveforms s1 t , s 2 t ,, s N t , and you want to
find a basis for the span of their complex envelopes. The obvious approach would be to first
downconvert each of the waveforms, and then apply the Gram-Schmidt procedure to the set
of complex envelopes. Will we get the same answer if we first apply Gram-Schmidt, and
then downconvert? Justify your answer