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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Signal Analysis (cont.)
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
Signal Analysis (cont.)
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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