purplebird said:given that x has an exponential density function ie p(x) = exp (-x) and x(n) & x(m) are statistically independent.
Now y(n) = x(n-1)+x(n)
what is the pdf (probability density function) of y(n)
A probability density function (PDF) is a mathematical function that describes the relative likelihood of a random variable taking on a specific value. In the context of digital filters, the PDF represents the distribution of the values that the filter outputs in response to a given input signal. It is often used to analyze the performance and behavior of digital filters in various applications.
The PDF of a digital filter is determined by the design and characteristics of the filter, such as its transfer function, order, and coefficients. It can also be influenced by the input signal and any noise or disturbances present in the system. The PDF can be calculated analytically or through simulations and experiments.
The PDF provides valuable information about the behavior and performance of a digital filter. It can reveal important characteristics, such as the filter's frequency response, passband and stopband ripple, and attenuation. The PDF can also be used to evaluate the filter's stability, linearity, and other properties.
The shape of the PDF can have a significant impact on the performance of a digital filter. For example, a narrow and tall PDF indicates a smaller spread of output values, which may result in a more precise and accurate response. On the other hand, a wider and flatter PDF may indicate a larger spread of output values, which can lead to more errors and distortions in the filter's output.
Yes, the PDF can be used to optimize or improve the performance of a digital filter. By analyzing the PDF, engineers can identify areas where the filter may be underperforming or exhibiting unwanted behaviors. This information can be used to make design modifications or adjustments to the filter's parameters, ultimately leading to a better performing filter.