Mathematical Description of Continuous-Time Signals M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

Mathematical Description of Continuous-Time Signals

M. J. Roberts - All Rights Reserved. Edited

by Dr. Robert Akl

Typical Continuous-Time Signals

M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl

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Continuous vs Continuous-Time Signals

All continuous signals that are functions of time are continuous-time but not all continuous-time signals are continuous


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Continuous-Time Sinusoids


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Continuous-Time Exponentials


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Complex Sinusoids


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The Signum Function

Precise Graph Commonly-Used Graph

The signum function, in a sense, returns an indication ofthe sign of its argument.


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The Unit Step Function


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The Unit Step Function

The unit step function can mathematically describe asignal that is zero up to some point in time and non-zero after that.


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The Unit Ramp Function


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The Unit Ramp Function

Product of a sine wave and a ramp function.


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Introduction to the Impulse


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Introduction to the Impulse


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The Unit Step and Unit Impulse


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Graphical Representation of the Impulse

The impulse is not a function in the ordinary sense because itsvalue at the time of its occurrence is not defined. It is representedgraphically by a vertical arrow. Its strength is either written beside it or is represented by its length.


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Properties of the Impulse


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The Unit Periodic Impulse


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The Periodic Impulse


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The Unit Rectangle Function


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Combinations of Functions


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Shifting and Scaling FunctionsLet a function be defined graphically by


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Shifting and Scaling FunctionsAmplitude Scaling,


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Shifting and Scaling Functions


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Differentiation


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Integration


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Even and Odd Signals


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Even and Odd Parts of Functions


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Even and Odd Parts of Functions


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Products of Even and Odd Functions

Two Even Functions


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An Even Function and an Odd Function


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An Even Function and an Odd Function


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Products of Even and Odd FunctionsTwo Odd Functions


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Integrals of Even and Odd Functions

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Integrals of Even and Odd Functions


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Periodic Signals

A function that is not periodic is aperiodic.


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Sums of Periodic FunctionsThe period of the sum of periodic functions is the least commonmultiple of the periods of the individual functions summed. If the least common multiple is infinite, the sum function is aperiodic.


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ADC Waveforms

Examples of waveforms whichmay appear in analog-to-digital converters. They can be described by a periodic repetition of a ramp returned to zero by a negative step or by a periodic repetition of a triangle-shaped function.


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Signal Energy and Power

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A signal with finite signal energy is called an energy signal.

A signal with infinite signal energy and finite average signal power is called a power signal.



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Mathematical Description of Continuous-Time Signals M. J. Roberts - All Rights Reserved. Edited by Dr. Robert Akl