Lecture2 Mech SU

8/12/2019 Lecture2 Mech SU

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Lecture 2Signal Characteristics

Goals:

Introduce analog and digital signals

Use basic descriptors of time dependent data Understand the relationship between time

and frequency domains

Determine the Fourier series of periodic time

dependent measurands


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Input-Output Signal Concepts

Tasks that face the engineer in the measurement

of physical variable

Selecting a measurement system

Interpreting the output from the measurement

system


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Digital World

What is a digital signal and what differentiates it from other

signals?

All electronic signals (and thus circuits) are either analog or

digital. Both are voltage (or current) signals.

An analog signal is continuous; i.e., infinitely divisible into

smaller and smaller parts.A digital signal is quantized; i.e., the information is divided into

discrete quantities of a finite size.

Up to c. 1950, prior to the advent of the transistor (and

thereafter the computer) all signals were essentially analog.


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Classification of Waveforms

Any signal is either a digital or analog signal the difference is obvious in their

appearance.

Analog Signals

Analog signals are always continuous (there

are no time gaps). The signal is of infinite

resolution.

Discrete Time Signals

Digital signals Information about the signal

magnitude is available only at discrete points

in time

Are particularly useful when data

acqusition and processing performed by

using a digital computer

The magnitude of the signal is continuous

and thus can have any value within the

operating range


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Analog versus Digital Examples

Thermocouple

The thermocouple provides a continuous (analog) signal into themeter, which is digitized and displayed on a LCD panel.


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Sampling: The process of obtaining discretized

information from a continuous variable at finite

time intervals.


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Classification of Waveforms (cont.)

Static: steady in time

Dynamic: changing in time

Simple periodic

Complex periodic

Step

Ramp

Pulse

Step

Ramp

Pulse

Simple periodic

Complex periodic


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Classification of Waveforms (cont.)

Non-deterministic: random or chaotic

Signal Characteristics: DefinitionsMagnitude- generally refers to the maximum value of a signal.

Range - difference between maximum and minimum values of a signal.

Amplitude- indicative of signal fluctuations relative to the mean.

Frequency- describes the time variation of a signal.

Dynamic - signal is time varying.

Static- signal does not change over the time period of interest.

Deterministic- signal can be described by an equation (other than a Fourier series or integralapproximation).

Non-deterministic - describes a signal which has no discernible pattern of repetition and

cannot be described by a simple equation.

Mean - average or static portion of a signal over the time of interest. Sometimes call the dc

component or the dc offset of the signal [Excel tip: Mean = AVERAGE(numbers...)].

RMS - root-mean-square - average value of the square of the signal over the time of interest.[Excel tip: RMS = SQRT(SUMSQ(numbers1 to n)/n)]


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Signal Analysis

There are essentially two ways to analyze a time depending signal (considering that

a time independent signal, i.e., constant is trivial)Fourier Analysis

The signal is reconstructedusing trigonometric terms; this allows one to reproduce

the signal and say something about its spectral (frequency) content.

Statistics

The signal is characterized by its statistics (max, min, mean, median, deviation, etc.)

which allows one to say something about its typicalbehavior

Average or Mean Value

Provides a measure of the static portion of a signal over a period of time.

This is often referred to as the dc component or dc offset.


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Signal Analysis (cont.)

The average value or mean

value of Analog (continuous) isfound by.

The rms value of any continuous

analog signal y(t) over period oftime.

The fluctuating portion alone is often characterized by the a term called the

variance, , or the square root of the variance the standard deviation, .

Where, y', is the true mean value of the signal.

The mean value of discrete time

signal is found by.

The rms value of any discrete time

signal y(t) over period of time.


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Simple Harmonic (SH): data which vary as a sine or cosine function of time

Works well for a simple function. But what about something more complex, such as a

waveform composed of two or more different sine waves?


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Complex Signals can be represented by the addition of a number of simpler periodic

functions, as an example a white light is combined of colors in the spectrum and of simple

Periodic functions

Any complex signal can be thought of as being made up of sines and cosines of differing

periods and amplitudes, which are added together in an infinite trignometric series.


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Complex Signals:

Composed of SH signals of different frequency, amplitude, orboth


This dynamic signal can be represented in thefrequency domain by a frequency spectrum


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Fourier Analysis

The representation of a signal as a series of sines and cosines iscalled Fourier Series

It is used to determine the frequency spectrum of dynamic signals

spectrum analyzer hardware

T = 2/f is the period of the signal, f is the fundamental frequency

(first harmonic), 2f is the second harmonic, etc.

Fourier Series


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Even and Odd Functions

Even functions are symmetric about the origin, oddfunctions are anti-symmetric

Even: g(-t) = g(t) i.e. cos(nt)

Odd: h(-t) = -h(t) i.e. sin(nt)

Therefore if y(t) is even the Fourier series will contain only

cosine functions. If it is odd it will contain only sine

functions.Functions that are neither even nor odd result in Fourier

series that contain both sine and cosine terms.


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The Fourier TransformThe most familiar is that which transforms the time function x(t) into the frequency

function X(f) through the use of the following relationship:

Sometimes this is written as:

The original independent variable is t is for time, usually in seconds, and has the

range of (- to ). The new independent variable f is usually in units of hertz,

abbreviated Hz. The range of f is also (- to ). Some times is used rather

thanf. Remember the variable is in radians per unit time.

The Fourier Transform is reversible

or


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Discrete Fourier AnalysisExample: 1000 point data set sampled at 100 samples/sec

T = 1000/100 sec = 10 seconds

t = 1/1000 secfNyquist = fs/2 = 500 Hz.

f = 1/T = 0.1 Hz .

Example: To resolve a spectra with 0.5 Hz resolution up to 1 KHz what sampling rate, period

and number of samples are required?

f = 1/T = 0.5 Hz therefore the sampling period must be 2 seconds.fNyquist = 1 KHz = fs/2 therefore the sampling rate must be 2 KHz.

f = 1/T = 0.5 Hz = fs/N = 2000 samples/sec/N

therefore

N = 2000/0.5 = 4000 samples

Frequency amplitude ambiguity or spectral leakage is caused by this discrete frequencystep size. If an input signal frequency is not a multiple of the frequency resolution it will be

represented by several data set frequencies of varying amplitudes.

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