Review of DSP

Signal and Systems: Signal are represented mathematically as functions of one or more independent variables. Digital signal processing deals with the transformation of signal that are discrete in both amplitude and time. Discrete time signal are represented mathematically as sequence of numbers.

Signals and Systems: A discrete time system is defined mathematically as a transformation or operator. y[n] = T{ x[n] } T{.} x [n] y [n]

Linear Systems: The class of linear systems is defined by the principle of superposition. And Where a is the arbitrary constant. The first property is called the additivity property and the second is called the homogeneity or scaling property.

Linear Systems: These two property can be combined into the principle of superposition, H Linear System H H

Time-Invariant Systems: A Time-Invariant system is a system for witch a time shift or delay of the input sequence cause a corresponding shift in the output sequence. H H

LTI Systems: A particular important class of systems consists of those that are linear and time invariant. LTI systems can be completely characterized by their impulse response. From principle of superposition: Property of TI:

LTI Systems (Convolution): Above equation commonly called convolution sum and represented by the notation

Convolution properties: Commutativity: Associativity: Distributivity: Time reversal:

…Convolution properties: If two systems are cascaded, The overall impulse response of the combined system is the convolution of the individual IR: The overall IR is independent of the order: H1 H2 H2 H1

Duration of IR: Infinite-duration impulse-response (IIR). Finite-duration impulse-response (FIR) In this case the IR can be read from the right-hand side of:

Transforms: Transforms are a powerful tool for simplifying the analysis of signals and of linear systems. Interesting transforms for us: Linearity applies: Convolution is replaced by simpler operation:

…Transforms: Most commonly transforms that are used in communications engineering are: Laplace transforms (Continuous in Time & Frequency) Continuous Fourier transforms (Continuous in Time) Discrete Fourier transforms (Discrete in Time) Z transforms (Discrete in Time)

The Z Transform: Definition Equations: Direct Z transform The Region Of Convergence (ROC) plays an essential role.

The Z Transform (Elementary functions): Elementary functions and their Z-transforms: Unit impulse: Delayed unit impulse:

The Z Transform (…Elementary functions): Unit Step: Exponential:

Z Transform (Cont’d) Important Z Transforms |z| > |a| Region Of Convergence (ROC) Whole Page Whole Page Unit Circle |z| > |a|

The Z Transform (Elementary properties): Elementary properties of the Z transforms: Linearity: Convolution: if ,Then

The Z Transform (…Elementary properties): Shifting: Differences: Forward differences of a function, Backward differences of a function,

The Z Transform (…Region Of Convergence for Z transform): Since the shifting theorem

The Z Transform (Region Of Convergence for Z transform): The ROC is a ring or disk in the z-plane centered at the origin :i.e., The Fourier transform of x[n] converges at absolutely if and only if the ROC of the z-transform of x[n] includes the unit circle. The ROC can not contain any poles.

The Z Transform (…Region Of Convergence for Z transform): If x[n] is a finite-duration sequence, then the ROC is the entire z-plane, except possibly or . If x[n] is a right-sided sequence, the ROC extends outward from the outermost finite pole in to . The ROC must be a connected region.

The Z Transform (…Region Of Convergence for Z transform): A two-sided sequence is an infinite-duration sequence that is neither right sided nor left sided. If x[n] is a two-sided sequence, the ROC will consist of a ring in the z-plane, bounded on the interior and exterior by a pole and not containing any poles. If x[n] is a left-sided sequence, the ROC extends in ward from the innermost nonzero pole in to .

The Z Transform (Application to LTI systems): We have seen that By the convolution property of the Z transform Where H(z) is the transfer function of system. Stability A system is stable if a bounded input produced a bounded output, and a LTI system is stable if:

Fourier Transform Time Transform Type Frequency Continuous Continuous Discrete Continuous Discrete Time Continuous FFT Continuous Discrete Fourier Series Discrete Discrete Discrete Time Discrete FFT

The Discrete Fourier Transform (DFT) It is customary to use the Then the direct form is:

The Discrete Fourier Transform (DFT) With the same notation the inverse DFT is

The DFT (Elementary functions): Elementary functions and their DFT: Unit impulse: Shifted unit impulse:

The DFT (…Elementary functions): Constant: Complex exponential:

The DFT (…Elementary functions): Cosine function:

The DFT (Elementary properties): Elementary properties of the DFT: Symmetry: If ,Then Linearity: if and

The DFT (…Elementary properties): Shifting: because of the cyclic nature of DFT domains, shifting becomes a rotation. if ,Then Time reversal:

The DFT (…Elementary properties): Cyclic convolution: convolution is a shift, multiply and add operation. Since all shifts in the DFT are circular, convolution is defined with this circularity included.