SP_I_P02_4p - Introduction to Signal Processing 79...

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Introduction to Signal Processing 7 9 Copyright © 2005-2009 – Hayder Radha D. Signal Models There are important signal models that are commonly used to represent a variety of physical phenomenon. For example, applying a constant voltage source ( ) s vt V = to a circuit starting from some time 0 t = is commonly modeled by a step function () () s u t = . Below, we review basic signal models and their functions. Introduction to Signal Processing 8 0 Copyright © 2005-2009 – Hayder Radha 1. Unit Step Function The unit step function is commonly used to represent the occurrence of a signal event that begins or ends at a specific instance of time. The basic unit step function is defined as follows: () 10 00 t ut t = < Introduction to Signal Processing 8 1 Copyright © 2005-2009 – Hayder Radha -2 -1 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 Introduction to Signal Processing 8 2 Copyright © 2005-2009 – Hayder Radha It is quite common to have a signal event commencing at a particular time tT = . In this case, one can use a time- shifted version of the basic unit step function to represent such event: 1 0 ut T −= <
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Introduction to Signal Processing 8 3 Copyright © 2005-2009 – Hayder Radha -5 -4 -3 -2 -1 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 0 T > 0 T < ( ) 2 ut ( ) 2 + Introduction to Signal Processing 8 4 Copyright © 2005-2009 – Hayder Radha Naturally, if 0 T > , then the signal event is taking place after time zero , 0 t = , and vice versa. Furthermore, an event that is taking place for a long time (e.g., we may have been applying a voltage source to a circuit for a long time), and then this event stops at time 0 t = (e.g., we turn the source off); such an event can be represented by a time-inverted version of the unit step function: Introduction to Signal Processing 8 5 Copyright © 2005-2009 – Hayder Radha () 10 00 t t −= > Introduction to Signal Processing 8 6 Copyright © 2005-2009 – Hayder Radha -5 -4 -3 -2 -1 0 1 2 0 0.2 0.4 0.6 0.8 1 ( )
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Introduction to Signal Processing 8 7 Copyright © 2005-2009 – Hayder Radha If a signal event has been applied for a long time, but the event stops abruptly at some time tT = , this phenomenon can be modeled by a time shifter version of the time- inverted signal: () 1 0 ut T u T t −− = −= > Introduction to Signal Processing 8 8 Copyright © 2005-2009 – Hayder Radha -5 -4 -3 -2 -1 0 1 2 0 0.2 0.4 0.6 0.8 1 0 T > ( ) 2 −− ( ) Introduction to Signal Processing 8 9 Copyright © 2005-2009 – Hayder Radha What does the general signal ua t b look like? Introduction to Signal Processing 9 0 Copyright © 2005-2009 – Hayder Radha 2. Rectangular Pulse Many real-time signals have finite duration of time. This can be modeled by an ideal rectangular pulse that begins at some time and ends at some other (later) time. The basic rectangular pulse ( unit pulse ) signal is defined as follows: 11 22 1 rect 0o t h e r w i s e t t ≤< =
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Introduction to Signal Processing 9 1 Copyright © 2005-2009 – Hayder Radha -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 0 0.2 0.4 0.6 0.8 1 1.2 Figure 13: The basic rectangular function. The area is one under this function.
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This note was uploaded on 06/08/2009 for the course ECE 366 taught by Professor Staff during the Spring '08 term at Michigan State University.

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SP_I_P02_4p - Introduction to Signal Processing 79...

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