MODULE-4 - MODULE 4 BASIC DISCRETE-TIME CONCEPTS Discrete...

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Unformatted text preview: MODULE 4 BASIC DISCRETE-TIME CONCEPTS Discrete Linear Systems Discrete Convolution Stability Difference Equations Page 4.1 INDEX Basic Discrete-Time 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 Real-Time 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 DISCRETE-TIME 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 discrete-time signals. These may or may not be sampled continuous-time signals. For now, interpret discrete-time 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 discrete-time function is the unit impulse function : = = ; ; 1 ) ( n n n or = =- k n k n k n ; ; 1 ) ( n 0 1 2 3 4-1-2 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 discrete-time signal x ( n ): x ( n ) = - = m x ( n-m ) ( m ) = - = m x ( m ) ( n-m ) 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 4-1-2 x ( n )-3-4 1 1 1 2 2 3 3 x ( n ) = ( n +4) + 3 ( n +2) + 2 ( n +1) + ( n ) + 2 ( n-1) + ( n-2) + 3 ( n-4) (c) As discrete-time 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 1-1-2 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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MODULE-4 - MODULE 4 BASIC DISCRETE-TIME CONCEPTS Discrete...

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