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I know that a function $f_a\in \text{L}^2(\mathrm{R})$ is an analytic signal if its Fourier transform is 0 for negative frequencies, i.e.

$\hat{f_a}(\omega)=0$ if $\omega<0$.

We can characterize an analytic function by its real part $f = \text{Re}[f_a]$, and the Fourier transform of this function is

$\hat{f_a}(\omega)=\left\{\begin{matrix} 2\hat{f}(\omega) & \text{if }\omega\geq0 \\ 0 & \text{if }\omega<0 \end{matrix}\right.$

Now if we have a discrete signal $f[n]$ of size $N$, we can also compute $f_a[n]$ by setting the negative frequency components of its discrete Fourier transform to 0.

Now why is it that if we want $\text{Re}[f_a]=f$, then we must have:

$\hat{f_a}[k]=\left\{\begin{matrix} \hat{f}[k] && \text{if }k=0,N/2\\ 2\hat{f}[k] && \text{if }0<k<N/2\\ 0 && \text{if } N/2<k<N \end{matrix}\right.$

?

Thanks in advance! :)

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  • $\begingroup$ I thought a function was analytic on the line or plane if it can be approximated by a convergent series. $\endgroup$ – Tom Kealy Sep 4 '13 at 8:27
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    $\begingroup$ @Tom Kealy: The OP confuses analytic functions and analytic signals. $\endgroup$ – chirlu Sep 4 '13 at 10:26
  • $\begingroup$ @chirlu I mentioned that in my answer below. $\endgroup$ – Tom Kealy Sep 4 '13 at 10:54
  • $\begingroup$ Your characterization of an analytic signal in the continuous-frequency domain is incorrect. See Wikipedia's definition. $\endgroup$ – Peter K. Sep 4 '13 at 12:01
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Analytic signals to get a but more tricky in the discrete domain because of the periodicity in the time domain and the circular nature of the convolution. This article treats that fairly well http://www.andrewduncan.ws/air/

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It's not clear what you're asking, but I think you want to know why the DFT of a signal has extra conditions for it to real.

For the continuous case: the 2 is there because we need a function to be symmetric to have a real Fourier transform.

In the discrete case, the DFT obey the symmtery:

$$ X_{N-k} = X_{-k} = X_k^* $$ i.e. conjugate symmetry (half the values are just complex conjugates of the other half). Thus $X_0$ and $X_{N/2}$ are real numbers whilst the remainder of the DFT is specified by $N/2-1$ complex numbers (hence the 2 in the second line).

Also, the answers to this question.

offer counterexamples to your first statement - that analytic functions have vanishing negative frequencies.

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