mathematical description of continuous-time...

Mathematical Description of Continuous-Time Signals

8/2/13 M. J. Roberts - All Rights Reserved 1


Typical Continuous-Time Signals


Continuous vs Continuous-Time Signals


Continuous-Time Sinusoids


Continuous-Time Exponentials


Complex Sinusoids


The Signum Function

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


The Unit Step Function


The Unit Step Function

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


The Unit Ramp Function


The Unit Ramp Function

Product of a sine wave and a ramp function.


Introduction to the Impulse


The Unit Step and Unit Impulse


Graphical Representation of the Impulse

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


Properties of the Impulse


The Unit Periodic Impulse


The Periodic Impulse

8/2/13 M. J. Roberts - All Rights Reserved 21 8/2/13 M. J. Roberts - All Rights Reserved 21

Properties of the Periodic Impulse


The Unit Rectangle Function


Combinations of Functions


Shifting and Scaling Functions


Amplitude Scaling,



Composite Video Signal

The horizontal sync pulse can be described by a rectangle function. The color burst can be described by a sinusoid multiplied by a rectangle function.


BPSK Binary Data Signal

BPSK data transmission is a sequence of sinusoidal cycles whose polarities are determined by the data. The data can be described by a sequence of rectangles. The BPSK signal is the data sequence multiplied by a carrier sinusoid.


EKG Waveform

An EKG waveform can be reasonably approximated by four linear functions in the QRS interval and the P and T pulses can be approximated by Gaussian waveforms.


Differentiation


Integration


Even and Odd Signals


Even and Odd Parts of Functions


Two Even Functions

Products of Even and Odd Functions Two Even Functions


Products of Even and Odd Functions An Even Function and an Odd Function


An Even Function and an Odd Function

Products of Even and Odd Functions


Two Odd Functions

Products of Even and Odd Functions Two Odd Functions


Integrals of Even and Odd Functions


Periodic Signals

A function that is not periodic is aperiodic.


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


Sums of Periodic Functions


ADC Waveforms

Examples of waveforms which may 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.


Transformer Voltage and Current

The voltage and current in a distribution transformer experiencing core saturation are shown above, voltage in blue and current in red. The current can be described by a sinusoid plus alternating periodic pulses which could be narrow triangles or even impulses as an approximate model.


Signal Energy and Power



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.

Scaling and Shifting


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