Communication Technology Laboratory
Prof. Dr. H. Bölcskei
Sternwartstrasse 7
CH-8092 Zürich
Harmonic Analysis: Theory and Applications in Advanced
Signal Processing
Spring semester 2017
Homework 1 - Solutions available on March 16, 2017
Problem 1
Overcomplete expansion in R2
Consider the following example discussed in class. For the vectors
" #
" #
1
e1 =
,
0
"
0
e2 =
,
1
1
e3 = e1 − e2 =
−1
#
we found that any vector x ∈ R2 can be represented according to
x = hx, ẽ1 i e1 + hx, ẽ2 i e2 + hx, ẽ3 i e3
where
ẽ2 = −e3 ,
ẽ1 = 2e1 ,
ẽ3 = −e1 .
a) Find another set of vectors ẽ01 , ẽ02 , ẽ03 , neither of which is collinear to neither of the vectors
ẽ1 , ẽ2 , ẽ3 and such that any vector x ∈ R2 can be represented as
x = x, ẽ01 e1 + x, ẽ02 e2 + x, ẽ03 e3 .
Hint: Look for another right-inverse of the matrix
"
#
1 0 1
.
0 1 −1
b) Now consider the following example discussed in class. For the vectors
" #
1
e1 =
,
0
√ #
1/√2
,
e2 =
1/ 2
"
"
#
1
ẽ1 =
,
−1
any vector x ∈ R2 can be represented as
x = hx, e1 i ẽ1 + hx, e2 i ẽ2 .
Show that x can also be written as
x = hx, ẽ1 i e1 + hx, ẽ2 i e2 .
"
#
0
ẽ2 = √
2
Is it possible to find two vectors e01 , e02 , neither of which is collinear to neither of the vectors
e1 , e2 such that
x = x, e01 ẽ1 + x, e02 ẽ2 ?
If the answer is “yes”, find these vectors. If the answer is “no”, explain why this is not
possible.
Problem 2
Equality in the Cauchy-Schwarz inequality
Prove that if the elements x and y of a complex Hilbert space H satisfy |hx, yi| = kxkkyk and
y 6= 0, then x = cy for some c ∈ C.
Hint: Assume kxk = kyk = 1 and hx, yi = 1. Then x − y and x are orthogonal, while
x = x − y + y. Therefore, kxk2 = kx − yk2 + kyk2 .
Problem 3
Parallelogram law
a) Let (X , h·, ·i) be an inner product space and k · k the norm induced by h·, ·i. Show that for
all x, y ∈ X , the following holds
kx − yk2 + kx + yk2 = 2kxk2 + 2kyk2 .
(1)
b) Let (X , k · k) be a normed space. Show that
p if (1) holds for all x, y ∈ X , then there exists
an inner product h·, ·i such that kxk = hx, xi for all x ∈ X . For simplicity, you may
assume that X is a real normed space.
Use the last statement to show that `2 (Z) is a Hilbert space. What about `1 (Z)?
Recall: `p (Z), p ∈ [1, ∞), is the space
`p (Z) ,


{uk }k∈Z :

+∞
X


|uk |p < ∞

k=−∞
equipped with the norm

kuk`p (Z) , 
+∞
X
1/p
|uk |p 
.
k=−∞
Problem 4
Projection on closed subspaces
Let H be a Hilbert space, x ∈ H, and S a closed subspace of H. We proved that there exists an
y ∈ S such that
kx − yk = min kx − zk.
z∈S
a) Use the parallelogram law to show that y is the unique element in S fulfilling (2).
b) Show that y is the unique element in S such that (x − y) ∈ S ⊥ .
(2)
Problem 5
Unconditional convergence
Let H = l2 (N) and define
xk = (0, . . . , 0, 1/k, 0, . . . , 0, . . . ),
k ∈ N,
where the only non-zero entry 1/k of the sequence xk is at position k ∈ N. Does the sum
∞
X
xk
k=1
converge unconditionally?
Problem 6
Discrete Fourier Transform (DFT) as a signal expansion
The DFT of an N -point signal f (n), n = 0, 1, . . . , N − 1, is defined as
−1
k
1 NX
fb(k) = √
f (n)e−i2π N n
N n=0
Find the corresponding inverse transform and show that the DFT can be interpreted as a signal
expansion in CN .