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This question already has an answer here:

I am new to dsp, and I'd really appricate it if someone please help me with this problem. I guess it's a basic simple question, but I cannot get it write. Here is the question:

The variance of a random signal generated using MATLAB command randn is approximately 1. We need a random signal with variance 0.01. With which factor you need to scale the signal?

With amplitude? Like we need to multiple the amplitude of the signal by 0.1?

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marked as duplicate by Matt L., MBaz, A_A, Stanley Pawlukiewicz, lennon310 Nov 12 '18 at 4:56

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

  • $\begingroup$ Assuming you want a normal independent RV sqrt(var)*randn. $\endgroup$ – Stanley Pawlukiewicz Nov 11 '18 at 17:35
  • $\begingroup$ I need to scale the signal with this sqrt(var)*randn? Why? $\endgroup$ – Niousha Nov 11 '18 at 18:48
  • $\begingroup$ Because it answered the question you asked. Google "standardizing a random variable" and work backwards $\endgroup$ – Stanley Pawlukiewicz Nov 11 '18 at 21:53
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In a practical setting to adjust the variance (thereof the power) of a random process, you could use the following to get what you want.

Let the variance of a given RV $X$ be $$\text{Var}\{X\} = \sigma_X^2 $$

Then the following transform $$ Y = K X $$ ($K$ being a scalar) will define a RV $Y$ with a variance given by

$$\text{Var}\{Y\} = \text{Var}\{ K X\} = K^2 \text{Var}\{ X\} = K^2 \sigma_X^2 = \sigma_Y^2$$

So, given a variance of $\sigma_X^2$ and a desired variance of $\sigma_Y^2$, you shall compute the necessary gain $K$ as

$$ K = \frac{ \sigma_Y }{\sigma_X} $$

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standardizing a Normal random variable $x$ $$ \frac{x -m}{\sigma} \sim \mathcal{N}(0,1) $$ so if $y$ is a random variable that is Normal with mean zero, with standard deviation $1$, $$ \sqrt{\sigma^2}y \sim \mathcal{N}(0, \sigma^2) $$ This standardization actually works for more than just Normal Distributions.

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