Applied Digital Signal Processing Theory and Practice

In this chapter we are concerned with probability models for the mathematical description of random signals. We start with the fundamental concepts of random experiment, random variable, and statistical regularity and we show how they lead into the concepts of probability, probability distributions, and averages, and the development of probabilistic models for random signals. Then, we introduce the concept of stationary random process as a model for random signals, and we explain how to characterize the average behavior of such processes using the autocorrelation sequence (time-domain) and the power spectral density (frequency-domain). Finally, we discuss the effect of LTI systems on the autocorrelation and power spectral density of stationary random processes.

Study objectives

After studying this chapter you should be able to:

• Understand the concepts of randomness, random experiment, statistical variability, statistical regularity, random variable, probability distributions, and statistical averages like mean and variance.

• Understand the concept of correlation between two random variables, its measurement by quantities like covariance and correlation coefficient, and the meaning of covariance in the context of estimating the value of one random variable using a linear function of the value of another random variable.

• Understand the concept of a random process and the characterization of its average behavior by the autocorrelation sequence (time-domain) and power spectral density (frequency-domain), develop an insight into the processing of stationary processes by LTI systems, and be able to compute mean, autocorrelation, and power spectral density of the output sequence from that of the input sequence and the impulse response.

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