Categorization of Stochastic Processes Discrete time; discrete variable Discrete time; continuous variable Continuous time; discrete variable Continuous time; continuous variable Options, Futures, and Other Derivatives,5 th edition2002 by John.hull

Transform-Domain Representation of Discrete-Time Signals Three useful representations of discrete-time sequences in the transform domain: Discrete-time- Fourier Transform(DTFT) YDiscrete Fourier Transform(DFT) z- -Transform

◆8.0 Introduction ◆8.1 Representation of Periodic Sequence: the Discrete Fourier Series (DFS) ◆8.2 Properties of the DFS ◆8.3 The Fourier Transform of Periodic Signal ◆8.4 Sampling the Fourier Transform ◆8.5 Fourier Representation of Finite-Duration Sequence: the Discrete Fourier Transform(DFT) ◆8.6 Properties of the DFT ◆8.7 Linear Convolution using the DFT ◆8.8 the discrete cosine transform (DCT)

◆8.0 Introduction ◆8.1 Representation of Periodic Sequence: the Discrete Fourier Series (DFS) ◆8.2 Properties of the DFS ◆8.3 The Fourier Transform of Periodic Signal ◆8.4 Sampling the Fourier Transform(DTFT) ◆8.5 Fourier Representation of Finite-Duration Sequence: the Discrete Fourier Transform(DFT) ◆8.6 Properties of the DFT ◆8.7 Linear Convolution using the DFT ◆8.8 the discrete cosine transform (DCT)

l Chapter 6- Learning objectives Distinguish between discrete and continuous randomⅴ ariables Differentiate between the binomial and the poisson discrete probability distributions and their applications Construct a probability distribution for a discrete random variable, determine its mean and variance, and specify the

§3.1 Discrete-Time Fourier Transform §3.2 DTFT Properties §3.3 Discrete Fourier Transform (DFT) §3.5 Circular Shift of a Sequence §3.6 Circular Convolution §3.7 z-transform §3.8 z-Transform Properties §3.9 Discrete Fourier Transform Computation

Discrete-Time Systems A discrete-time system processes a given input sequence x[] to generates an output sequencey[n] with more desirable properties In most applications, the discrete-time system is a single-input, single-output system:

◼ Introduction ◼ Discrete-time signals ◼ Discrete-time systems ◼ Time-domain characterization of LTI discrete-time systems ◼ Digitalization of Analog Digital

Stability Condition of a Discrete-Time LTI System · BIBO Stability Condition-A- discrete--time LTI system is BIBO stable if the output sequence {y[n]} remains bounded for any bounded input sequence{x[n]} A discrete-time LTI system is BIBO stable if and only if its impulse response sequence {h[n]} is absolutely summable

Discrete-Time Systems A discrete-time system processes a given input sequence x[] to generates an output sequencey[n] with more desirable properties In most applications, the discrete-time system is a single-input, single-output system: