2 - Lecture 2: Discrete-Time Systems and z-Transform...

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1 Lecture 2: Discrete-Time Systems and z -Transform Discrete-time signals v.s. continuous-time signals Discrete-time systems v.s. continuous-time systems z -Transform Inverse z -Transform z -Transform for solving linear difference equations Continuous-Time Signals • A signal that changes continuously in time: • Defined at all times t (no gap) • Signal can take arbitrary values • Example: temperature, position, velocity,… e ( t ) ; ¡1 < t < 1 ; or t ¸ 0 e ( t ) t
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2 • A signal (sequence, or series) whose values are defined only at discrete times: • Signal defined only at integer times k • Signal can still take arbitrary values Discrete-Time Signals e ( k ) ; k = ¢¢¢ ; ¡ 1 ; 0 ; 1 ; ¢¢¢ or k = 0 ; 1 ; 2 ; ¢¢¢ Where do discrete-time signals come from? Some come naturally – Population of a species in different generations – Annual growth percentage of GDP – Results of a numerical algorithm in different rounds of iteration Some arise by sampling continuous-time signals at regular time intervals, say, every T seconds e ( t ) ; ¡1 < t < 1 ) e ( kT ) ; k = ¢¢¢ ; ¡ 1 ; 0 ; 1 ; ¢¢¢ ) e ( k ) ; k = ¢¢¢ ; ¡ 1 ; 0 ; 1 ; ¢¢¢ ; (if T is known from the context)
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3 Signals in Digital Control Systems The process of representing an arbitrary continuous value y ( kT ) with binary bits of finite word length is called quantization (A/D) Quantized signal is discrete in both time and value, called a digital signal , Errors will be incurred as a result of quantization The more binary bits used, the smaller the average quantization error In most of the course, we assume zero quantization error discrete-time signal continuous-time signal digital signal Plant Computer A/D D/A Sensor Data Hold y ( t ) T r ( t ) continuous-time signal discrete-time signal continuous-time system discrete-time system + ¡ A continuous-time system has input and output that are both continuous-time signals Continuous-time linear time-invariant (LTI) system given by differential equation: Model completely determined by the coefficients ® and ¯ Continuous-Time LTI Systems e ( t ) y ( t ) = ¯ m d m e ( t ) dt m + ¢¢¢ + ¯ 1 de ( t ) dt + ¯ 0 e ( t ) ® n d n y ( t ) dt n + ¢¢¢ + ® 1 dy ( t ) dt + ® 0 y ( t )
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4 Taking Laplace transform (assuming zero initial conditions): H ( s ) is called the transfer function For LTI system, H ( s ) is a rational function (fraction of two polynomial functions of s ) Impulse response is the inverse Laplace transform of H ( s ) Transfer Functions of Continuous-Time LTI Systems H ( s ) = Y ( s ) E ( s ) = ¯ m s m + ¢¢¢ + ¯ 1 s + ¯ 0 ® n s n + ¢¢¢ + ® 1 s + ® 0 E ( s ) Y ( s ) H ( s ) A discrete-time system has input and output that are
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This note was uploaded on 04/15/2011 for the course ECE 483 taught by Professor Evens during the Spring '08 term at Purdue University-West Lafayette.

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2 - Lecture 2: Discrete-Time Systems and z-Transform...

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