DTFT Properties for Real and Odd Signals

We need to determine the characteristics (real/imaginary, even/odd) of the Discrete Time Fourier Transform (DTFT), $X(e^{j\omega})$, for a signal $x[n]$ that is both real and odd.

Signal Properties and DTFT

Key properties relating a signal $x[n]$ and its DTFT $X(e^{j\omega})$ are:

• If $x[n]$ is real, then $X(e^{j\omega}) = X^*(e^{-j\omega})$.
• If $x[n]$ is odd, then $x[n] = -x[-n]$.

Deriving $X(e^{j\omega})$ Properties

Let's combine these properties:

1. Real Signal Property: Since $x[n]$ is real, we have:

   $X(e^{j\omega}) = X^*(e^{-j\omega})$

2. Odd Signal Property: For an odd signal, the DTFT satisfies $X(e^{-j\omega}) = -X(e^{j\omega})$.
3. Combining Properties: Substitute the odd signal property into the real signal property:

   $X(e^{j\omega}) = (X(e^{-j\omega}))^* = (-X(e^{j\omega}))^*$

   Let $X(e^{j\omega}) = A + jB$, where $A$ is the real part and $B$ is the imaginary part.

   Then $X^*(e^{j\omega}) = A - jB$. The equation becomes:

   $A + jB = -(A - jB)$

   $A + jB = -A + jB$

   Equating the real parts gives $A = -A$, which implies $A = 0$.

   This means $X(e^{j\omega})$ has no real part, so it must be purely imaginary ($X(e^{j\omega}) = jB$).

4. Odd Function Property: We already established from the odd signal property that $X(e^{-j\omega}) = -X(e^{j\omega})$. This is the definition of an odd function.

Conclusion

Combining the results, the DTFT $X(e^{j\omega})$ of a real and odd signal $x[n]$ is both purely imaginary and an odd function.