Find the 5-point DFT of the sequence:

Good day.

I'm trying to code the 5-point DFT of the following sequence:

u = {sinj} between the limits 4 & j=4

The answer I'm looking for is:

(1/2i) 1-e^5i/1 - e^i - (1/2i) 1-e^-5i/1 - e^-i

This is the code I have thus far:

syms u

u = sin(j);

u = sin(j) * exp((-2*pi*i*j*k)/5);

symsum(u, 0, 5)

Giving me the result:

sin(1) + sin(exp(-2*i*j*k)) + sin(exp(-(2*i*j*k)/5)) + sin(exp(-(4*i*j*k)/5)) + sin(exp(-(6*i*j*k)/5)) + sin(exp(-(8*i*j*k)/5))

Not sure where I'm making my error or if my code is just completely wrong.

Thank you in advance.

Regards

Ruwan

[Paul]

Presumably the sequence is x[j] = sin(j) for j = 0 to 4?

By definition the 5-point DFT of x[j] is a 5-point sequence. But the question says the answer is supposed to be a single value. Is that value supposed to be just one point in the DFT sequence?

The symsum is going over 0-5, which is 6 points.

Can you clarify your question?

[Ruwan Badenhorst]

Hi Paul.

Thank you for your feedback.

The question only states: Find the 5-point DFT of the sequence.

I have attached a file showing the question and the workings.

Apologies if this does not answer your question.

Regards

Answer 1

[Paul]

The code posted has a few issues.

The second line is:

>> u = sin(j)

u =

   0.0000e+00 + 1.1752e+00i

>> whos u
  Name      Size            Bytes  Class     Attributes

  u         1x1                16  double    complex   

Unless otherwise defined, Matlab treats j as sqrt(-1). So if you haven't declared j as a sym object (or assigned it some other value), the variable u is becomes a double with value u = sin(sqrt(-1)).

The summation in this line:

symsum(u, 0, 5)

should be from 0 to 4, which is five points. Also, it's safer to be unambiguous about the variable to sum over. Also, the code is using i, which also is equal to sqrt(-1) unless otherwise defined, but it's better to use 1i, which is unambiguous. So the corrected code should be

>> syms j k real
>> U(k) = symsum(sin(j)*exp(-2*pi*1i*j*k/5),j,0,4); % five point DFT of sin(j)
>> U(k)
 
ans =
 
exp(-(pi*k*2i)/5)*sin(1) + exp(-(pi*k*4i)/5)*sin(2) + exp(-(pi*k*6i)/5)*sin(3) + exp(-(pi*k*8i)/5)*sin(4)

Verify this answer:

>> vpa(norm((U(0:4))-fft(sin(0:4))))
 
ans =
 
0.0000000000000001690981417154962344172159894936

It may be possible to simplifty the form of U(k).

[Ruwan Badenhorst]

Thank you Paul.

Much appreciated. I see where I may my error now.

Regards

Ruwan