What qualities of $h[n]$ are necessary for:

$$ H(e^{j\omega}) = DTFT\{h_{even}[n]\} + j\ DTFT\{h_{odd}[n]\} $$

Do all real / causal h[n] have the property that:

$$ H(e^{j\omega}) = DTFT\{h_{even}[n]\} + j\ DTFT\{h_{odd}[n]\} $$


$$ h_{even}[n] = \frac{1}{2}(h[n] + h[-n]) $$

$$ h_{odd}[n] = \frac{1}{2}(h[n] - h[-n]) $$


The DTFT relationships



$$x_{odd}[n]=\frac12\left(x[n]-x^*[-n]\right)\Longleftrightarrow j\,\textrm{Im}\left\{X(e^{j\omega})\right\}$$

hold for any sequence $x[n]$ for which the DTFT exists. There is no assumption about $x[n]$ being real-valued or causal (note the complex conjugation $^*$ in the definition of even and odd signals). If $x[n]$ is real-valued you can leave out the conjugation.

Note that the DTFT of the odd part $x_{odd}[n]$ equals $j$ times the imaginary part of the DTFT $X(e^{j\omega})$, so you have


(without a $j$ on the right-hand side).

  • $\begingroup$ thanks, makes sense now. any suggestion for title? $\endgroup$ Jan 12 '19 at 15:55
  • $\begingroup$ @MrCasuality: If your question has been answered you can accept this answer by clicking on the green check mark to its left, thanks. $\endgroup$
    – Matt L.
    Jan 12 '19 at 17:07

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