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Unformatted text preview: MODULE 4 BASIC DISCRETETIME CONCEPTS Discrete Linear Systems Discrete Convolution Stability Difference Equations Page 4.1 INDEX Basic DiscreteTime Concepts Unit Impulse Function Sifting Property of Unit Impulse Unit Step Function Discrete Linear Time Invariant (LTI) Systems Impulse Response of Linear System Response of LTI System to General Input Convolution Summation Causality RealTime Processing and Realizability Stability Frequency Response Difference Equations Homogeneous Solution of LCCDE Solving the Characteristic Equation On the Roots of the Characteristic Equation Particular Solution of LCCDE Method of Undetermined Coefficients Example  Finding Total Solution of a LCCDE Comments on the Form of the LCCDE Interesting Application of Difference Equations MAIN INDEX Page 4.2 4. BASIC DISCRETETIME CONCEPTS index READ : Sections 2.0  2.6 of Oppenheim & Schafer. Work as many related problems as possible. Now we return to the study of discretetime signals. These may or may not be sampled continuoustime signals. For now, interpret discretetime signals simply as indexed sequences of numbers. The index n is discrete time, such as the clock instants of a digital processor. We will explore the relationship between continuous signals and their sampled versions, in detail, later. Page 4.3 Unit Impulse Function index The simplest discretetime function is the unit impulse function : = = ; ; 1 ) ( n n n or = = k n k n k n ; ; 1 ) ( n 0 1 2 3 412 1 k k +1 ( n k ) Also called the Kronecker delta function . The Kronecker function plays a similar role as the singularity ( t ) (Dirac delta function) in continuous theory. But complications do not arise  no need for interpretation as a limit or generalized function. However, the key integral properties (sifting, etc.) of the Dirac function occur as summation properties of the discrete impulse. Page 4.4 Sifting Property of Unit Impulse index The following holds for any discretetime signal x ( n ): x ( n ) =  = m x ( nm ) ( m ) =  = m x ( m ) ( nm ) This simple equation has several interpretations. (a) As a generalized function : ( n ) assigns a number to each function that it is placed in a summation with. (b) As a representation : Any signal x ( n ) can be expressed as a sum of weighted, shifted unit impulses. Example: n 1 2 3 412 x ( n )34 1 1 1 2 2 3 3 x ( n ) = ( n +4) + 3 ( n +2) + 2 ( n +1) + ( n ) + 2 ( n1) + ( n2) + 3 ( n4) (c) As discretetime convolution between x ( n ) and ( n ) (later). Page 4.5 Unit Step Function index Another important signal is the unit step function : < = ; ; 1 ) ( n n n u or < = k n k n k n u ; ; 1 ) ( A shifted unit step function: n 0 112 1 k u ( n k ) k +1 k +2 k +3 k +4 Of course the unit step can be written as a sum of unit impulses:...
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 Spring '04
 AlanC.Bovik
 Digital Signal Processing, Signal Processing

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