# Why do we need to conjugate complex signals in autocorrelation and cross correlation

Why is it necessary to conjugate $f(t)$ while performing auto correlation or cross correlation with respect to $g(t)$, if $f(t)$ and $g(t)$ are complex signals?

• Can I please ask what do you think about this? Where exactly is the difficulty? Have you handled any complex $f(t)$ so far? Did you notice anything "strange" about them, compared to real $f(t)$s? – A_A Dec 8 '17 at 9:06
• maybe look up the meaning of inner product in what we normally call a Hilbert space. an example: $$\langle x, y \rangle = \sum_{n} x_n \overline{y_n}$$ then they define the norm of the vector $x$ as the square root of the inner product of $x$ with itself. – robert bristow-johnson Dec 8 '17 at 18:28

$$\underline{Prologue :}$$

Let me ask you another question, How will you compare two complex numbers $$U$$ (a+jb) and $$V$$ (c+jd)? By comparing magnitude? Subtract them and take real part? Multiply them and compare?

Since any complex number involves two entities ( one for magnitude |$$z$$| and other for argument $$\theta$$ ) any comparison involves comparison of those two entities. Suppose we represent $$U$$ and $$V$$ in a complex argument plane as $$M e^{j \theta}$$ and $$N e^{j \phi }$$ if we multiply both of them together, we get

$$U$$ x $$V = M e^{j \theta} * N e^{j \phi } = MN e^{j ( \theta + \phi )}$$

If we seek maximum value out of the above multiplication then

$$|U$$x$$V|_{max} = |MN|_{max} * |e^{j ( \theta + \phi )}|_{max}$$

Suppose our comparison is only about the relative location of $$V$$ with respect to $$U$$ in the complex argument plane, for the time being, forget about the magnitudes $$M$$ and $$N$$.

maximum value of the second term $$\;e ^{j ( \theta + \phi )}$$ occurs at 1.

for that to happen
$$\theta+\phi = 2\pi k \;\;\;\;$$ where $$\:k\:\epsilon\;( 0,1,2...$$

if $$k = 0$$,

$$\theta + \phi = 0$$

$$\phi = - \theta$$

ie, when two complex numbers with bounded magnitudes get multiplied, their maximum value occurs, when the second number ($$V$$) is in the negative argument direction of the first number ($$U$$), ie in the direction of its conjugate ($$U^*$$).

ie maximum of $$U \times V$$ occurs when $$V$$ points in the direction of $$U^*$$ { the orange dashed line in the picture }

$$\textit{-or conversely-}$$

If $$V$$ is getting multiplied by $$U^*$$ then their maximum occurs when $$V \textit{points in the direction of } U$$

In another words $$U^* \times V$$ gives nearness of $$V$$ w.r.t $$U$$ in terms of angle ( complex argument ) and it will decrease as a function of $$cos \:\alpha$$, where $$\alpha$$ is the angle between $$U$$ and $$V$$.

$$\underline{Epilogue:}$$

Considering two signals $$f$$ and $$g$$, correlation is given as

$$\hspace{1cm} (f \otimes g)(\tau) = \int_{-\infty}^\infty f^*(t) g(t + \tau) dt$$

At any instant '$$t$$', a signal is just a point in the complex plane. So at any moment $$f$$ and $$g$$ are just two points in the complex plane. Then the job of comparison of two signal becomes a mere comparison of two complex points. With this logic, to compare two signals with a given lag '$$\tau$$' between them, just multiply point by point in '$$t$$' by taking conjugate of the other signal. If you get a big number that means both reference and compared signals are just looking in the same direction in the complex plane. When integrated on the whole '$$t$$', the value of the integration shows the how much similarity is between signal $$f$$ and signal $$g$$. that is which is known as the cross-correlation between two signals.

{ I didn't talk about magnitudes $$M$$ and $$N$$, right? The method in which everything is put in a single benchmark by avoiding magnitudes $$M$$ and $$N$$ is known as Normalized cross-correlation }

• Thanks Abhilash for such a detailed explanation and intuition.. please keep contributing and sharing.. :).. – Vinayak Killedar Jan 27 '19 at 20:28

HINT:

What is the physical meaning of the autocorrelation evaluated at lag $\tau=0$?