Lecture notes on probability theory and random processes. As indicated by the table of contents, the notes cover traditional, introductory. Hf x t yt the crosscorrelation of input and output and the autocorrelation of the output can be com puted via application of the lti lter as well. If one scans all possible outcomes of the underlying random experiment, we shall get an ensemble of signals. Noise source noise can often be modeled as a gaussian. Prerequisites for lti systems laplace transform topics discussed. By the principle of superposition, the response yn of a discretetime lti system is the sum. When a continuous time random process x t is applied on this system, the output response is also a continuous time random process y t. For the moment we show the outcome e of the underlying random experiment. Since the output is a ct signal with uncertainty described prob abilistically, the output is a ct random process. A random process is a family of random variables indexed by a parameter, where is called the i ndex set. C h a p t e r 9 random processes introduction much of your background in signals and systems is assumed to have focused on the e.
Statistics of the wss processes passed through lti systems. Power spectral density continuoustime random processes if r x. Then the input xt and output yt are jointly wss with. R, be a wss process input to a stable lti system with real impulse response ht and transfer function hf.
Unit55 linear systems response to random inputs consider a continuous lti system with impulse response h t. Random process through lti systems fourier transform view a time domain signal in the frequency domain. Lti systems on signals modeled as the outcome of probabilistic experiments, i. Lecture 5 time invariance we will work with timeinvariant or shiftinvariant systems. Transmission of wss random process through lti system duration. Stationary random process an overview sciencedirect topics. You may be surprised to learn that a random variable does not vary. What can we say about y when we have a statistical description of x and a description of the system. When a wssinput process is applied to an lti system with impulse response. Digital signal processing ztransforms and lti systems. End of chapter problems probability, statistics and random. Linear timeinvariant theory, commonly known as lti system theory, investigates the response of a linear and timeinvariant system to an arbitrary input signal.
Power spectral density discretetime random processes if r x m is the autocorrelation function of xn then its power spectral density is s x ej. Note that l does not need to exhibit random behavior for y to be random. Let yt,elxt,e be the output of a linear system when xt,e is the input. Passing a random process through an lti system xt lti system yt consider a linear timeinvariant lti system ht which has random processes xt and yt as input and output yt z 1 1 h. For each sample path input, the output is a deterministic signal. For a time domain signal xt, define the fourier transform x fxt xtedt j2 ft f and the inverse fourier transform x txfxfedf12 jft f examples. Notes for signals and systems electrical and computer.
Lti system models for random signals ar, ma and arma models. Digital signal processing ztransforms and lti systems d. Clearly, yt,e is an ensemble of functions selected by e, and is a random process. Random processes 5 for such a lti system, if ut is a stationary and ergodic random process then yt is also stationary and ergodic.
Spectrum given that a random process that is a stationary and ergodic with an expected value of zero and auto. Proposition if xt passes through an lti system to yield yt, then the mean. In the model given below, the random signal \xn\ is observed. Random process can be continuous or discrete real random process also called stochastic process example. Linear time invariant systems imperial college london. Transmission of a random process through a system, effects on acf, effects on psd. The mean of the output rp is equal to the result of passing the input mean through the lti system. Trajectories of these systems are commonly measured and tracked as they move through time e. Filtering random processes let xt,e be a random process.
Given the observed signal \xn\, the goal here is to find a model that best describes the spectral properties of \xn\ under the following assumptions. A linear timeinvariant lti system can be represented by its impulse response figure 10. When the input is wss and the system is time invariant the output is also wss. In the world of signals and systems modeling, analysis, and implementation, both discretetime and continuoustime signals are a reality. Prerequisites for lti systems laplace transform youtube. Jul 20, 2018 transmission of wss random process through lti system duration. What is the output of a ct lti system when the input is a ct random process. Stationary and ergodic random processes given the random process yz,t we assume that the expected value of the random process is zero, however this is not always the case. Linear system with random process input lti system with. Linear timeinvariant lti systems with random inputs. Deepa kundur university of torontofrequency domain analysis of lti systems25 39 chapter 5. May 27, 2012 the properties of the lti system its frequency response or its impulse response affects the power spectrum and the autocorrelation of the process the lti system is working on.
For an lti system with impulse response hn, we have. Statistics of random processes passed through an lti system. The autocorrelation function and the rate of change. May 28, 2012 statistics of the wss processes passed through lti systems. Once you understand that concept, the notion of a random variable should become transparent see chapters 4 5. Suppose a widesense stationary random process xt is applied to a linear timeinvariant lti system whose impulse response is ht and frequency response is hf, the system output is then a widesense stationary random process yt. The autocorrelation function can be found for a process that is not wss and then specialized to the wss case without doing much additional work. Random processes in linear systems linear system with random process input lti system with wss process input process linear estimation in. If the expected value equals some constant x o we can adjust the random process such that the expected value is indeed zero. Random processnoise as an input to a low pass filter and calculation of output noise power. This random process is stationary and ergodic with an expected value of zero. Assume that the system is always causal and stable. Stationary random processes linear estimation the random.
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