Understanding White Gaussian Noise: Probability of Inclusion in [0.25A, 1.15A]

[nightworrier]

Homework Statement

Consider the random process X(t)=A+N(t), where A is a constant and N(t) is a white Gaussian
noise with spectral density equal to 1. The process X(t) is filtered with a system with impulse
response h(t)=u(t)exp(‐t/T).
Compute the probability that X(t), once filtered, is included in the interval [0.25A, 1.15A].

I have problems about white gaussian noise. I actually don't know the concept. I appreciate if you help me to figure it out this.

 

[poor mystic]

Hi Nightworrier
Many here will know this much better than i do, but I can say that Gaussian white noise is noise which is present at all frequencies at equal amplitude.
Essentailly if Gaussian white noise is filtered by some process X(t) you'll get an output from X(t) that looks a lot like the frequency-wise transfer function.
Hope that might be of value!

I remember a wonderful book by F.R. Connor on Noise which may be available at your library.

 

[nightworrier]

I am actually not familiar with that. I am a mechanical engineer and far from the idea of signals just trying to learn by myself. Could you tell the solution of this problem ? I want to understand the concept.

 

[poor mystic]

Dear Nightworrier
I apologise that I am unable to solve this problem, which lies beyond my skill. The reference I mentioned is very good, but it has been 30 years since I even looked at this kind of work.
I make this apology in the hope that some kind communications engnner may see that you still do not have the help that you need.

 

[DeeNos]

I can provide a response to your question about white Gaussian noise. White Gaussian noise is a type of random process where the amplitude of the noise is independent of frequency and follows a Gaussian distribution. This means that the noise has equal power at all frequencies and is characterized by a bell-shaped curve.

In the context of your question, the random process X(t)=A+N(t) represents a signal with a constant amplitude A, which is being corrupted by white Gaussian noise N(t). The spectral density of the noise is equal to 1, which indicates that the noise has equal power at all frequencies. This type of noise is commonly found in many natural and man-made systems, such as electronic circuits and communication systems.

The process X(t) is then filtered with a system with impulse response h(t)=u(t)exp(‐t/T). This means that the signal is being passed through a filter that has a step function (u(t)) and an exponential decay (exp(-t/T)). The step function acts as a low-pass filter, allowing only low-frequency components of the signal to pass through, while the exponential decay acts as a high-pass filter, allowing only high-frequency components to pass through.

To compute the probability that X(t), once filtered, is included in the interval [0.25A, 1.15A], we need to consider the characteristics of the filtered signal. Since the filter only allows certain frequencies to pass through, the amplitude of the filtered signal will be dependent on the amplitude of the original signal A and the frequency components of the noise. As the noise has equal power at all frequencies, the filtered signal will also have equal power at all frequencies. This means that the amplitude of the filtered signal will also follow a Gaussian distribution.

To compute the probability, we can use the properties of the Gaussian distribution. The interval [0.25A, 1.15A] represents a range of values that the filtered signal can take. To find the probability, we need to calculate the area under the Gaussian curve within this interval. This can be done using a mathematical formula or by using a graphing calculator.

In summary, understanding white Gaussian noise is important in signal processing and communication systems. It is a type of noise that has equal power at all frequencies and follows a Gaussian distribution. By considering the characteristics of the filtered signal, we can compute the probability of the signal falling within a certain range of values.