This prelim covers deterministic and stochastic signal processing concepts and mathematical tools. The recommended preparatory courses are EECS 123 (deterministic signal processing), EECS 126, and EECS 225A (stochastic signal processing). You are responsible for the following topics regardless of whether or not they are covered in a course

The deterministic signal processing component includes Laplace transform, Z transforms, poles and zeros, frequency response, difference equations, differential equations, LTI systems, sampling theorems, up-sampling, down-sampling, FFT algorithms and their variations, FIR and IIR filter design. The emphasis is on discrete-time, including multi-rate systems, but continuous-time systems are also considered in light of their interface to discrete-time systems through sampling and reconstruction. In this context, the relationship between the "four Fourier transforms": Fourier transform, Fourier series, DTFT, DFT is important. Digital filter realizations and design are covered, including round off errors, and coefficient quantization effects.

Basic probability: sample spaces, events, and probability functions. Independence and conditioning on events using Bayes rule; distributions, expectation, conditional expectation, multiple random variables, transformation of random variables.

Discrete random variables: Uniform, Bernoulli, Geometric, probability mass functions, conditioning on random variables. Counting arguments.

Summary statistics: Expectations, variances, moment generating functions

Continuous random variables: Uniform, Exponential, Gaussian, probability density functions, jointly continuous random variables, conditioning on continuous or discrete random variables.

Laws of large numbers and bounding: Markov Inequality, Chebychev Inequality, Chernoff Bounding, Weak law of large numbers, Central Limit Theorem

Estimation: Very basic MMSE and LLSE estimation

Stochastic Processes: Bernoulli processes and basic Poisson processes. Finite state Markov chains, stationary distributions, and recurrent classes. Wide sense stationary random processes, power spectra, modulation, LTI filtering of random processes.

Hypothesis testing, MAP rule, minimum mean-square estimation, linear least square estimation, maximum likelihood estimation, confidence intervals.

Spectral factorization, the innovations process model, and whitening filters for discrete-time random processes are covered. Wiener filtering problems are included i.e. unconstrained, causal, and FIR, together with applications to echo cancellation, noise cancellation, linear equalization, and linear prediction. The normal equations and the orthogonality principle form the basis for much of this. Students should understand as well how to apply these techniques to AR modeling and AR spectral estimation. Adaptive filters for solving Wiener filtering problems are also covered (the LMS algorithm is sufficient). Eigen decomposition and singular value decomposition are also covered at an introductory level, particularly with regard to their connection to the above topics.

- Oppenheim & Shafer,
*Discrete-Time Signal Processing*, 2nd edition, Prentice-Hall, 1999. Chapters 1-10. - Simon Haykin,
*Adaptive Filter Theory*, 3rd edition, Prentice-Hall, 1996. Chapters 1-6. - Alberto Leon-Garcia,
*Probability and Random Processes in Electrical Engineering*, 2nd edition, Addison Wesley, 1994. Chapters 1-5, 8.

**NOTE: ** The following courses may not be used to fulfill the prelim breadth requirement if
you take the DSP prelim: EE 120, 123, 126, 225AB, 226, 290T, and CS 280.

ruthg@eecs.berkeley.edu 03/18/09