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# 3 - Lecture 3 System Representation Transfer Functions...

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1 Lecture 3: System Representation • Transfer Functions • Graphical Representation • State Space Representation • Reading: Chap. 2.7-2.10 Linear Difference Equation Representation e ( k ) y ( k ) • Input e ( k ), and output y ( k ), k =0,1,2,…. • LTI system given by a linear difference equation Typically assume zero initial conditions: ¡ a n ¡ 1 y ( k ¡ 1) ¡¢¢¢¡ a 0 y ( k ¡ n ) y ( k ) = b n e ( k ) + b n ¡ 1 e ( k ¡ 1) + ¢¢¢ + b 0 e ( k ¡ n ) y ( ¡ 1) = y ( ¡ 2) = ::: = y ( ¡ n ) = 0 and e ( ¡ 1) = e ( ¡ 2) = = e ( ¡ n ) = 0

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2 Transfer Function Take the z -transform of the linear difference equation to obtain where is called the transfer function of the discrete-time LTI system Question: given the transfer function , what is the corresponding linear difference equation representation? Y ( z ) = b n E ( z ) + b n ¡ 1 z ¡ 1 E ( z ) + ¢¢¢ + b 0 z ¡ n E ( z ) ¡ a n ¡ 1 z ¡ 1 Y ( z ) ¡¢¢¢¡ a 0 z ¡ n Y ( z ) ) Y ( z ) = b n + b n ¡ 1 z ¡ 1 + ¢¢¢ + b 0 z ¡ n 1+ a n ¡ 1 z ¡ 1 + ¢¢¢ + a 0 z ¡ n E ( z ) = G ( z ) E ( z ) G ( z ) = b n + b n ¡ 1 z ¡ 1 + ¢¢¢ + b 0 z ¡ n 1+ a n ¡ 1 z ¡ 1 + ¢¢¢ + a 0 z ¡ n E ( z ) Y ( z ) G ( z ) G ( z ) = z ¡ 1 ( z ¡ 2) 2 Time-Delay Element E ( z ) Y ( z ) = z ¡ 1 E ( z ) z ¡ 1 Consider a simple LTI discrete-time system whose output y ( k ) is obtained from the input e ( k ) by a delay of one time step: If the input e ( k ) is obtained by sampling a continuous-time signal at sampling period T , i.e., e ( k )= e ( kT ), then the above operation is a time delay element by time T : The transfer function of the time-delay element is Easily implemented by hardware y ( k ) = e ( k ¡ 1) e ( k ) y ( kT ) = e (( k ¡ 1) T ) e ( kT ) T T
3 Series Connection of Time Delay Elements More complicated connection y ( k ) e ( k ) T T y ( k ) e ( k ) T T + + Simulation Diagram Simulation diagram is a graphical representation of systems consisting of basic elements of operations: Time-delay elements Summation Multiplication by constant Example: can be represented by a simulation diagram: y ( k ) = 2 e ( k ) ¡ e ( k ¡ 1) ¡ y ( k ¡ 1) T T e ( k ) e ( k ¡ 1) ¡ ¡ + y ( k ) y ( k ¡ 1) 2

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4 Example y ( k ) ¡ 4 y ( k ¡ 1) +3 y ( k ¡ 2) = e ( k ¡ 1) ¡ 2 e ( k ¡ 2) Simulation diagram: Simulation Diagram for General Linear Difference Equation has simulation diagram: ¡ a n ¡ 1 y ( k ¡ ¡¢¢¢¡ a 0 y ( k ¡ n ) y ( k ) = b
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3 - Lecture 3 System Representation Transfer Functions...

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