The following autocorrelation sequence is given, rs s(m)=(0.8^|m|)/(0.36). Describe a source-sys

The following autocorrelation sequence is given, rs...

Key Concepts

White Noise

White noise refers to a sequence of uncorrelated random variables that typically have zero mean and constant variance. In many signal generation mechanisms, white noise acts as a driving force. By inputting white noise into a properly configured linear filter or difference equation, one can generate a signal with a specific autocorrelation structure, as white noise contains all frequency components uniformly.

Linear Filtering and Difference Equation

Linear filtering involves processing an input signal, such as white noise, through a linear system whose impulse response shapes the statistical properties of the output. By designing a filter (often represented by a difference equation) with appropriate feedback coefficients, the memory characteristics of the output signal can be controlled to match a given autocorrelation function. Specifically, a first?order linear difference equation can generate a process whose autocorrelation decays exponentially, mirroring the behavior of an AR(1) process.

Autocorrelation Function

The autocorrelation function measures how a random process correlates with itself at different time lags. It is a key statistical tool used to understand the memory or time-dependence in a process. A decaying autocorrelation function, such as the one given, indicates that values closer in time are more strongly correlated than those that are further apart, often signaling an underlying first?order Markov (or autoregressive) behavior in the process.

Autoregressive (AR) Process

An autoregressive process is a type of stochastic process in which current values are expressed as a linear combination of past values plus a random disturbance. In an AR(1) process, the current output is directly related to the immediately preceding output. The exponential decay in the autocorrelation function is a hallmark of AR(1) processes, where the coefficient in the difference equation determines the rate of decay of correlation over time.

Transcript

00:01 Okay, let's do this like a frequency response or both loudly designed for a unity feedback system.

00:08 So we have g of s equals k times s plus 7 over s times s plus 5 times s plus 15.

00:18 The specs, the percent os is 15 percent, t sub s is 0 .1 seconds, and we're assuming 2 percent setting time rule and k sub v is 1 ,000.

00:29 So from the percent overshoot, m -s -p is e to the negative zeta pi over root 1 minus zeta squared, which is 0 .15.

00:50 So that means zeta is about 0 .517.

01:00 From settling time, omega sub -n with the 2 % rule t -s is about 4 over.

01:08 Zeta omega sub n so omega n is four over zeta t sub s which is four over 0 .517 times 0 .1 and that's about 77 .4 radiance per second.

01:26 The damping ratio is going to lead to the desired phase margin and a common mapping is going to give p ms about 53 degrees that's for a zeta of about 0 .52.

01:36 We'll design a little higher to budget for a lag phase loss.

01:40 So we'll use a target of 58 degrees.

01:47 And then you pick a target gain crossover frequency and a typical choice to put in would be about 0 .125 times omega sub n.

01:57 And so that would be 1 .25 times 77 .4, which is about 97.

02:03 Let's just call 100 radiance per second.

02:07 Then you check the plant at 100 radiance per second without compensation.

02:11 So we let g0 of s be s plus 7 over s times s plus 5 times s plus 15 with no k.

02:20 And at an omega of 100, then the angle is about negative 172 degrees, negative 172 .6 degrees.

02:35 So the current phase margin would be 180 minus 172 .6, which is about 72 .6 degrees.

02:47 7 .4 degrees.

02:50 Now that's way too small, so we need a lead compensator to add phase...

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