# Is complex-conjugation an LTI operation?

Started by November 24, 2009
```Good morning all,

This conundrum may be due to it being too early in the day for me to
think clearly, so please tell me where I've gone wrong!

It would appear to me that complex conjugation would satisfy the
properties of linearity and time-invariance, and therefore could be
categorised as an LTI operation.  However, clearly it creates
frequency content that was not present in the original signal
(specifically, it reflects the spectrum), which would go against what
I have always held to be a characteristic of LTI systems.

What is wrong here?

--
Oli
```
```Oli Charlesworth <catch@olifilth.co.uk> wrote:

> This conundrum may be due to it being too early in the day for me to
> think clearly, so please tell me where I've gone wrong!

> It would appear to me that complex conjugation would satisfy the
> properties of linearity and time-invariance, and therefore could be
> categorised as an LTI operation.

I don't think so.  You can't get the complex conjugate by
multiplication.

It is time invariant, though.

-- glen

```
```On Nov 24, 3:57&#4294967295;am, Oli Charlesworth <ca...@olifilth.co.uk> wrote:

>
> It would appear to me that complex conjugation would satisfy the
> properties of linearity and time-invariance, and therefore could be
> categorised as an LTI operation. &#4294967295;

Complex conjugation is not a linear operation over
the complex field.  For linearity to hold, it must be
that L(cX) = cL(X) for all complex numbers c, which
is not true.  Complex conjugation *is* a linear operation
if we construe complex numbers as constituting a
two-dimensional space over the real field. The mapping
(a,b) --> (a, -b) is a linear map from R^2 to R^2, while
the mapping X --> X* is not a linear map from C to C.

Hope this helps

Dilip Sarwate

```