Trying to prove that:

$$ X^{*}[k] = \sum^{N-1}_{n=0} x^{*}\left((N-n)\right)_N\ W_N^{nk} $$

Where: $$((x))_N \text{ = x modulus N}$$

$$W_{N}^{nk} = e^{-j\ 2\pi / N}$$

So I start out with definition for DFT:

$$ X[k]=\sum^{N-1}_{n=0} x[n]\ W_{N}^{nk} $$

Then I conjugate both sides of equation.

$$ X^{*}[k]=\left[\sum^{N-1}_{n=0} x[n]\ W_{N}^{nk}\right]^{*} $$

$$ X^{*}[k]=\sum^{N-1}_{n=0} x^{*}[n]\ W_{N}^{-nk} $$

then somehow the -n changes to N-n which i'm guessing is this:

$$((-n))_N = N - n$$

I don't really understand it... but, seems its equivalent if you do this inside of an exponent… to just randomly take a modulus of a negative signed number and replace it with its modulus equivalent?

$$ X^{*}[k]=\sum^{N-1}_{n=0} x^{*}[n]\ W_{N}^{(N-n)k} $$

Then, I get stuck on the proof because somehow $x^{*}[n]$ changes to $x^*((N-n))$,

and $W_{N}^{(N-n)k}$ somehow changes to $W_{N}^{nk}$.

and thus, i'm unable to obtain final result of:

$$ X^{*}[k] = \sum^{N-1}_{n=0} x^{*}((N-n))_N\ W_N^{nk} $$

still wondering what the rules are to apply to get there?

  • 1
    $\begingroup$ to deal with the $\mathrm{mod}$ operator, just periodically extend $x[n]$ so that $$ x[n+N] = x[n] \qquad \forall n \in \mathbb{Z} $$ then just get rid of all that $\mathrm{mod}$ crap. $\endgroup$ – robert bristow-johnson Jan 25 '19 at 3:27
  • $\begingroup$ is it also true that x[N - n] = x[n]? and x[-n] = X[N-n]? $\endgroup$ – MrCasuality Jan 25 '19 at 3:51
  • $\begingroup$ if you are curious where i'm at now... its Shaum's DSP outline, 2nd edition, page 262, problem 6.11(b). $\endgroup$ – MrCasuality Jan 25 '19 at 3:53
  • $\begingroup$ it is true that $x[N-n]=x[-n]$ but it is not true in general that $x[-n]=x[n]$. it could be the case, but that is a special case with even symmetry. $\endgroup$ – robert bristow-johnson Jan 25 '19 at 21:14

So you want to show $X^*[k]$ ?

Actually it follows very simply from the fundamental properties

  • $x[n] \longleftrightarrow X[k]$
  • $x[-n] \longleftrightarrow X[-k]$
  • $x[n]^* \longleftrightarrow X^*[-k]$

and by combining the last two you have:

  • $x[-n]^* \longleftrightarrow X^*[k]$

which shows that $X^*[k]$ is given by the forward DFT of the signal $x[-n]^*$.

Note that, due to periodicity of DFT sequences, the negative index can be repaced by $-n = N-n$, and hence $x[-n]^* = x[N-n]^*$ too.

Direct derivation is as follows:

$$ X[k] = \sum_{n=0}^{N-1} x[n] W_N^{nk} $$

where $W_N= e^{-j \frac{2\pi}{N} }$.

Take conjugate of both sides: $$ X^*[k] = (\sum_{n=0}^{N-1} x[n] W_N^{nk})^* $$

$$ X^*[k] = \sum_{n=0}^{N-1} x^*[n] W_N^{-nk} $$

replace $n$ with $-n$ so that forward DFT appaers:

$$ X^*[k] = \sum_{n=0}^{N-1} x^*[-n] W_N^{nk} $$

which again shows that $X^*[k]$ is given by DFT of $x[-n]^*$. Note again that periodicity of $x[n]$ is utilized to rearrange the limits.

  • $\begingroup$ Thanks, that answers it... then the last step is to replace x*[-n] with x*[N-n] so that index is within 0 to N-1 range... not sure why they also add modulus to x*[N-n] as well since range of sum is 0 to N-1..., but, i think that's not really needed.. $\endgroup$ – MrCasuality Jan 25 '19 at 12:51
  • $\begingroup$ I think x[n] is implicitly modulus when dealing with DFT X[k]... so you really don't need to write it explicitly... $\endgroup$ – MrCasuality Jan 25 '19 at 12:59
  • $\begingroup$ Every (repeat every) expression involving DFT sequences can be written using modulus operator. And indeed it's implicitly implied as well. Note that for $ 0 \leq n \leq N-1$ , we have $\text{mod}(n,N) = n$ and $\text{mod}(-n,N) = N-n$.. $\endgroup$ – Fat32 Jan 25 '19 at 20:07
  • $\begingroup$ just a notational curiosity? why do you put the asterisk in different places for $X^*[\cdot]$ and $x[\cdot]^*$? $\endgroup$ – robert bristow-johnson Jan 25 '19 at 21:16
  • $\begingroup$ oh, and i think that there is one pesky exception. for $1 \le n < N$ then $$ \mod(-n,N) = N-n $$ but not for $n=0$. $\endgroup$ – robert bristow-johnson Jan 25 '19 at 21:18

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