Plot spectrum of $X(-k)$

Question

Assume FFT of $x[n]$ is $X(k)$ and I would like to find $X(-k)$. I have implemented two techniques:

1. I am calculating the X(k) = fftshift(fft(x)) and then find X(-k) to be fliplr(X(k)) ;
2. I write the for loop to correspond each symbol of in $X(k)$ to $X(N-K+1)$ where $N$ is the number of points used for FFT.

I get kind of similar answer but in the second approach I am mapping $X(N)$ to $X(1)$ which is DC, which should not be the case. Therefore I am suspecting I am doing something wrong.

Can anyone tell me what is the correct way to implement $X(-k)$ when you have the $X(k)$ ?

Answer 1

FFT (DFT) is defined as

$$\mathsf{DFT}\{x[n]\}=X[k]=\sum_{n=0}^{N-1}x[n]e^{-jk2\pi n/N}$$

Hence, $X[-k]$ is $$X[-k]=\sum_{n=0}^{N-1}x[n]e^{jk2\pi n/N}$$

By investigating the formula of inverse DFT (and a quick change of variable $n\leftrightarrow k$ for more clarity),

$$\mathsf{IDFT}\{X[k]\}=x[n]=\frac1N\sum_{k=0}^{N-1}X[k]e^{jk2\pi n/N}\Rightarrow x[k]=\frac1N\sum_{n=0}^{N-1}X[n]e^{jk2\pi n/N}$$

We can see that $$X[-k]=N\cdot\mathsf{IDFT}\{x[n]\}$$ So instead of calculating the FFT and then converting the result, you can directly calculate $X[-k]$ by taking the IFFT.