Exercise involving DFT The fourier matrix is a transformation matrix where each component is define

groupweird40 2022-05-23 Answered
Exercise involving DFT
The fourier matrix is a transformation matrix where each component is defined as F a b = ω a b where ω = e 2 π i / n . The indices of the matrix range from 0 to n 1 (i.e. a , b { 0 , . . . , n 1 })
As such we can write the Fourier transform of a complex vector v as v ^ = F v, which means that
v ^ f = a { 0 , . . . , n 1 } ω a f v a
Assume that n is a power of 2. I need to prove that for all odd c { 0 , . . . , n 1 }, every d { 0 , . . . , n 1 } and every complex vectors v, if w b = v c b + d , then for all f { 0 , . . . , n 1 } it is the case that:
w ^ c f = ω f d a v ^ f
I was able to prove it for n = 2 and n = 4, so I tried an inductive approach. This doesn't seem to be the best way to go and I am stuck at the inductive step and I don't think I can go any further which indicates that this isn't the right approach.
Note that I am not looking for a full solution, just looking for a hint.
You can still ask an expert for help

Want to know more about Discrete math?

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

Crevani9a
Answered 2022-05-24 Author has 8 answers
Step 1
You need to use the fact that n is a power of 2 . I believe a key observation is that because c is odd, it is a unit in the ring of integers mod   n = 2 r   (for some positive integer r ), and so, as k ranges from 0 to   2 r 1   in the sum
ω ^ c f = k = 0 2 r 1 e 2 π i k c f 2 r v c k + d ( mod 2 r )   ,
the residues of ck mod   2 r   will range over the same set of values, but in a different order. For each   k { 0 , 1 , , there is a (unique) integer   u k   with   0 such that   c k = u k + z k 2 r   for some integer   z k  , and u will be a permutation on the set of integers   { 0 , 1 , , 2 r 1 }  .
Step 2
We then have
k = 0 2 r 1 e 2 π i k c f 2 r v c k + d ( mod 2 r ) = k = 0 2 r 1 e 2 π i ( u k + z k 2 r ) f 2 r v u k + z k 2 r + d ( mod 2 r )   , = k = 0 2 r 1 e 2 π i u k f 2 r v u k + d ( mod 2 r )   , = t = 0 2 r 1 e 2 π i t f 2 r v t + d ( mod 2 r )   , the last expression being obtained by rearranging the order of summation.
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2020-11-09
Use proof by Contradiction to prove that the sum of an irrational number and a rational number is irrational.
asked 2021-08-18
Discrete Mathematics Basics
1) Determine whether the relation R on the set of all Web pages is reflexive, symmetric, antisymmetric, and/or transitive, where (a,b)R if and only if
I) everyone who has visited Web page a has also visited Web page b.
II) there are no common links found on both Web page a and Web page b.
III) there is at least one common link on Web page a and Web page b.
asked 2021-07-28

Let A, B, and C be sets. Show that (AB)C=(AC)(BC)
image

asked 2021-08-02
Suppose that A is the set of sophomores at your school and B is the set of students in discrete mathematics at your school. Express each of these sets in terms of A and B.
a) the set of sophomores taking discrete mathematics in your school
b) the set of sophomores at your school who are not taking discrete mathematics
c) the set of students at your school who either are sophomores or are taking discrete mathematics
Use these symbols:
asked 2021-02-02
List all the steps used to search for 9 in the sequence 1,3,4,5,6,8,9,11 using linear search
asked 2021-08-22
Let A2 be the set of all multiples of 2 except for 2. Let A3 be the set of all multiples of 3 except for 3. And so on, so that An is the set of all multiples of n except for n, for any n2. Describe (in words) the set A2A3A4
asked 2021-08-15
Let n8. Give a bigΘ estimate for the number of circuits of length 8 in Kn. Θ(n)Θ(n8)Θ(8)Θ(8n)Θ(8n)

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question