Review of Signals and Systems: Discrete time complex exponentials and other basic signals—scaling of the independent axis and differences from its continuous-time counterpart—system properties (linearity, time-invariance, memory, causality, BIBO stability)—LTI systems described by linear constant coefficient difference equations (LCCDE)—autocorrelation.

Discrete-Time Fourier Transform (DTFT): Complex exponentials as eigensignals of LTI systems—DTFT definition—inversion formula—properties—relationship to continuous-time Fourier series (CTFS).

Z-Transform: Generalized complex exponentials as eigensignals of LTI systems—z-transform definition—region of convergence (RoC)—properties of RoC—properties of the z-transform—inverse z-transform methods (partial fraction expansion, power series method, contour integral approach)—pole-zero plots—time-domain responses of simple pole-zero plots—RoC implications of causality and stability.

Frequency Domain Analysis of LTI Systems: Frequency response of systems with rational transfer function—definitions of magnitude and phase response—geometric method of frequency response evaluation from pole-zero plot—frequency response of single complex zero/pole—frequency response of simple configurations (second order resonator, notch filter, averaging filter, comb filter, allpass systems)—phase response—definition of principal phase—zero-phase response—group delay—phase response of single complex zero/pole—extension to higher order systems—effect of a unit circle zero on the phase response—zero-phase response representation of systems with rational transfer function—minimum phase and allpass systems—constant group delay and its consequences—generalized linear phase—conditions that have to be met for a filter to have generalized linear phase—four types of linear phase FIR filters—on the zero locations of a linear phase FIR filter—constrained zeros at z = 1 and at z = -1 and their implications on choice of filters Type I through Type IV when designing filters—frequency response expressions for Type I through Type IV filters.

Sampling: Impulse train sampling—relationship between impulse trained sampled continuous-time signal spectrum and the DTFT of its discrete-time counterpart—scaling of the frequency axis—relationship between true frequency and digital frequency—reconstruction through sinc interpolation—aliasing—effect of sampling at a discontinuous point—relationship between analog and digital sinc—effects of oversampling—discrete-time processing of continuous-time signals—non-integer delay—up-sampling and down-sampling—introduction to sample-rate alteration.

Discrete Fourier Transform (DFT): Definition of the DFT and inverse DFT—relationship to discrete-time Fourier series—matrix representation—DFT as the samples of the DTFT and the implied periodicity of the time-domain signal—recovering the DTFT from the DFT—circular shift of signal and the "index mod N" concept—properties of the DFT—circular convolution and its relationship with linear convolution—sectioned convolution methods: overlap add and overlap save—effect of zero padding—introduction to the estimation of frequencies of sinusoids—windowing and spectral leakage—introduction to the Fast Fourier Transform (FFT) algorithm—decimation-in-time and decimation-in-frequency algorithms.