# What is the significance of a spike like autocorrelation function

I have some questions and doubts to which the answers are difficult to extract from text books. Hence I have posted this question here. Help would be extremely beneficial. Thank you.

(1) The normalized ACF of white noise is similar to delta impulse function. What is the purpose and benefit of this spike nature in signal processing and statistics?

(2) If the length of the spike for white noise is higher in comaprison to the length of the spike for a random process then is there a significance and what soes this convey? Does the length of the spike have a special name and a role in applications to signal processing?

(3) A delta function is not a random process. Then why is the ACF of white noise, which is closer to the delta function preferred?

• question 2 is puzzling because your terms height and length of spike could use some clarification. One thinks of spikes as”skinny” and “pointy” which would imply narrow width compared to height as intrinsic – user28715 Oct 2 '17 at 2:48
• @StanleyPawlukiewicz: by height and length of spike I mean the length in Y axis. Indeed spikes is the correct term. Therefore, is there a significance to the value of spike on Y axis? – SKM Oct 3 '17 at 3:13

Consider a linear time-invariant system with unknown impulse response $h(t)$. If the input is a zero-mean wide-sense-stationary random process $\{X(t)\}$, then, with $\{Y(t)\}$ denoting the output process, we have that $$R_{Y,X}(\tau) = E[Y(t)X(t+\tau)] = h\star R_x\big|_\tau.$$ So. if $\{X(t)\}$ is a white noise process wth autocorrelation function $K\delta(t)$, then $R_{Y,X}(\tau) = Kh(t)$. Why is this at all important? Well, most processes are assumed to be ergodic, and so $R_{Y,X}(\tau)$ can be estimated by cross-correlating a sample path of the output with a sample path of the input with a delay of $\tau$. Thus, the spikiness of the autocorrelation function has a role in system identification of unknown systems.