White noise & 1/f noise after a system h(t)

In summary, the conversation discusses finding the variance of noise after passing through a system with an impulse response h(t). It is mentioned that X is the input noise and Y is the output noise after passing through the system. The formula for calculating the variance of Y is given, where Rxx is the autocorrelation function of noise X and h(u) and h(v) are the impulse response of the system. Two scenarios are presented - one where Rxx models a white noise and the other where it models a 1/f noise. In both cases, the formula for calculating the variance of Y is given. However, the person admits not understanding the math behind statistics well and requests help.
  • #1
iVenky
212
12
Hi,

I am trying to solve this math equation on finding the variance of a noise after passing through a system whose impulse response is h(t)
X is the input noise of the system and Y is the output noise after system h(t)
if let's say variance of noise Y is
σy2=∫∫Rxx(u,v)h(u)h(v)dudv

where integration limits are from -∞ to +∞. Rxx is the autocorrelation function of noise X. Can you show that if Rxx (τ)=σx2 δ(τ) (models a white noise), then

σy2=σx2∫h2(u)du (integration limits are from -∞ to +∞)

and if Rxx (τ)=σx2 (models a 1/f noise), then

σy2=σx2(∫h(u)du)2 (integration limits are from -∞ to +∞)

I don't understand the math behind statistics that well
Thanks
 
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  • #2
This sounds like a homework problem. You need to post it in the appropriate Homework forum using the template to show your attempt at a solution.
 

1. What is white noise?

White noise is a type of random signal that has a constant power spectral density, meaning it has an equal amount of energy at all frequencies. It is often described as sounding like static or the sound of a waterfall.

2. What is 1/f noise?

1/f noise, also known as pink noise, is a type of random signal that has a power spectral density inversely proportional to the frequency. This means that lower frequencies have more energy than higher frequencies. It is often described as sounding like a heartbeat or the sound of rustling leaves.

3. How does a system affect white noise and 1/f noise?

When a system is applied to white noise or 1/f noise, it can change the characteristics of the noise. For example, a low-pass filter can decrease the high-frequency components of white noise, making it sound more like 1/f noise. Similarly, a high-pass filter can decrease the low-frequency components of 1/f noise, making it sound more like white noise.

4. What is the significance of h(t) in relation to white noise and 1/f noise?

h(t) represents the impulse response of a system. It is used to determine how the system will affect the input signal, such as white noise or 1/f noise. By analyzing h(t), scientists can better understand how a system will modify or filter the noise signals.

5. How is white noise and 1/f noise used in scientific research?

White noise and 1/f noise are commonly used in scientific research to study the behavior of complex systems. They can provide insight into the dynamics of systems and help identify patterns or anomalies. Additionally, they are used in experiments to test the effectiveness of various signal processing techniques and algorithms.

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