# When is a system's frequency response symmetric?

When is the frequency response of an LTI system symmetric, i.e. which property does it need to satisfy? Does it have to do with the system being causal (I believe this means that h(n) = 0 for n < 0)? If so, why?

Thanks!

## 1 Answer

This property applies to any system whose impulse response is real-valued (as opposed to complex-valued).

For a real-valued function of time $f(x)$, the Fourier transform $F(\omega)$ has the property

$$F(\omega)=F^\star{}(-\omega),$$

where ${}^\star$ indicates the complex conjugate. Since $\|z^\star\|=\|z\|$, the magnitude spectrum of a real-valued function is symmetric.

• It should be $F(\omega)=F^*(-\omega)$ (conjugate symmetry). – Matt L. Jan 11 '15 at 21:13
• @MattL., good catch. Nobody noticed that for 2 months. – The Photon Jan 11 '15 at 21:19