02 – Análisis de Fourier

02 – Análisis de Fourier
Diego Andrés Alvarez Marín
Profesor Asociado
Universidad Nacional de Colombia
Sede Manizales
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Contenido
Series de Fourier
Transformada de Fourier
Interpretación física
Implementación en MATLAB
Ventaneo
Teorema del muestreo
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Créditos
Estas diapositivas están basadas en el excelente
material presentado en www.blinkdagger.com y
elaborado por Quan Quach.
Gracias!
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Sistema LTI
(linear time invariant system)
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Escalabilidad
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Aditividad
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Invarianza en el tiempo
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Fidelidad sinusoidal
Si la entrada a un sistema LTI es un sinusoide, la
salida del mismo es otro sinusoide de la misma
frecuencia y forma pero diferente amplitud y fase
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Divide y conquista
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http://www.dspguide.com/CH5.PDF
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●
La lectura de un sensor es un voltaje que
cambia en el tiempo y la cual se asocia a
otro fenómeno físico.
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Análisis de Fourier
Aceleración del
último piso de un
edificio ante un
terremoto.
El espectro de
Fourier permite ver
cuales son las
frecuencias de
vibración del edificio.
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http://research.opt.indiana.edu/Library/FourierBoo
k/toc.html
http://www.complextoreal.com/chapters/fft1.pdf
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La transformada rápida de Fourier
Fast Fourier Transform (FFT)
El comando fft solo opera sobre los "y" no sobre los "t"
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Usando el comando fft
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The function outputs the correct frequency range and the transformed
signal. It takes in as input the signal to be transformed, and the sampling
rate.
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As you can see, this
plot is basically
identical to what we
would expect! We
get peaks at both -4
Hz and +4 Hz, and
the amplitude of the
peaks are 1.
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Como la señal y es
real, el espectro es
simétrico y solo
necesito la mitad
NOTA: usen esta función, no la de www.blinkdagger.com
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The execution time of an FFT algorithm depends on the transform
length. It is fastest when the transform length is a power of two,
and almost as fast when the transform length has only small
prime factors. It is typically slower for transform lengths that are
prime or have large prime factors. Time differences, however,
are reduced to insignificance by modern FFT algorithms
such as those used in MATLAB. Adjusting the transform
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length for efficiency is usually unnecessary in practice.
Potencia de 2
http://www.complextoreal.com/chapters/fft1.pdf
RECOMENDADO POR LOS LECTORES!!!
http://www.dspguide.com/ch8.ht
m
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Uno obtendrá una
única frecuencia en fo
y la amplitud es
correcta.
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Zero padding (rellenado con ceros)
Zero padding is básicamente concatenar al final
del vector actual un vector de ceros antes de
aplicar la FFT.
Se hace por dos razones:
1. Los primeros algoritmos para calcular la FFT
solo funcionaban con un 2^N de datos (este no es
el caso con los algoritmos modernos).
2. Artificialmente mejora la resolución en el
espectro de frecuencia.
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A esta señal le haremos zero padding:
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Se completa la señal con ceros hasta alcanzar 128,
256 y 512 puntos:
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El comando fft de MATLAB hace zero-padding
utilizando un segundo parámetro; el segundo
argumento permite especificar cuantos puntos
debe retornar el comando fft.
Mire las líneas resaltadas en las siguientes
funciones:
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Ahora las funciones en acción:
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Why Does My Output Look Like a
Sinc?
When we pad with zeros, we are effectively
multiplying a rectangular box with the sinusoid in
the time domain. In the frequency domain, this
translates into convolving a sinc function with an
impulse resulting in a sinc-like output!
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As you can see, there are more sampled points as N gets larger, and
it is easier to see the general shape of the spectrum. The larger your
N is, the finer the sampling will be. If I keep on increasing the number
of zeros, it will make my frequency spectrum more refined. So does
this mean I can just zero-pad my signal and get a better FFT
spectrum every time? Well, technically speaking, you get more
frequency bins when you zero pad, so yes, you do get a higher
resolution in the frequency domain. But the caveat is that there is no
new information added when you zero-pad.
When you increase the size of N, all you are really doing is
interpolating the data to obtain more sample points. There is NO new
information added when zero-padding is applied. The resolution has
increased, but I repeat, THERE IS NO NEW INFORMATION ADDED!
This might sound confusing and paradoxical, but it is a very important
point.
If you have the option to take more data, it is ALWAYS better to get
more data than to zero pad. Zero-padding is NOT a substitute for
taking more data!
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In this example, we are going to
use zero padding to help us
distinguish between two peaks
that would otherwise be difficult
to distinguish.
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Aquí el zero padding ayudó a distinguir los dos picos. Al
hacerlo no se agregó más información. The information was
always there, but it was “hidden” in a way. When we took the
FFT of the signal without zero padding, the frequency bins
were not fine enough to differentiate the two peaks. By the
sampling theorem, as long as you sample a bandlimited signal
under the Nyquist rate, you know everything you need to know
to perfectly reconstruct the signal. Therefore, at that point,
you’ve done the best you can do; you can zero-pad all you
want to get more points at the output of your FFT, but you
don’t get any new information, because you already have all
the information there is on the continuous-time signal.
Inherently, there is nothing wrong with zero-padding in itself.
You just have to be careful in its application. Whenever I use
the fft command, I tend to use an N that is 4 times larger than
the amount of data points within my signal. Zero padding
cannot hurt your FFT result.
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Conclusion
It is a common misconception that zero-padding adds more
information. Zero padding adds NO NEW information. The
perceived benefit of zero-padding is increased spectral resolution.
You are getting better resolution, but the key is to realize that there
is NO NEW information added from the zero-padding. Zero-padding
is useful, but it should not be a substitute for taking larger data
samples. If you had to choose between taking twice as much data,
or to zero pad your data, the answer is to ALWAYS take more data.
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Artículos recomendados:
http://zone.ni.com/devzone/cda/tut/p/id/4880
http://www.ele.uri.edu/~hansenj/projects/ele436/fft
.pdf
http://cnx.org/content/m12032/latest
/
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Aliasing
●
Ejemplos sonoros:
–
http://www.sic.rma.ac.be/~xne/el401/aliasing/
–
http://allsignalprocessing.com/aliasing-of-signals-identity-theft-in-the-frequency-domain/
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Aliasing
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Aliasing
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Aliasing
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Otros problemas de la DFT
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●
●
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identificación de frecuencias cercanas con
amplitudes similares
identificación de frecuencia lejanas, una de
gran amplitud y otra frecuencia de amplitud
pequeña
la frecuencia que se quiere identificar puede
caer entre las FFT bins
el spectral leakage causa "ruido" en el resto
del espectro, el cual puede ocultar
frecuencias de interés
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Spectral leakage
Leakage = indication of frequency content where is
none
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Spectral leakage
Every time a window is applied to a signal
(Window = none effectively applies a rectangular
window to the signal), leakage occurs, that is,
power from one spectral component leaks into the
adjacent ones. Leakage from strong spectral
components can result in hiding/masking of
nearby weaker spectral components. Even strong
spectral components can be affected by leakage.
For example, two strong spectral components
close to each other can appear as one due to
leakage.
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El spectral leakage se produce por:
(o razones para usa una ventana)
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The beginning does not match the end of the data
segment we are analyzing (esto no pasa con
synchronous sampling)
We virtually never have an integer number of
periods of any cyclic information in the signal
segment
For aperiodic signals such as modulated signals the
use of a window is highly recommended. The
window will attenuate the signal at both ends of the
signal segment processed to zero. This makes the
signal apear periodic and reduces leakage.
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Spectral leakage + windowing
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Spectral leakage + windowing
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Características espectrales
deseables de una ventana
Existe un trade-off entre ambas, por lo que debe
escoger una u otra:
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Detection (amplitude accuracy): means detecting a
desired signal in the presence of broadband noise (es
decir que identifique con precisión la amplitud, y reduzca
el leakage), especialmente cuando existen componentes
frecuenciales de diferente amplitud.
Frequency (or spectral) resolution: ability to distinguish
spectral components of comparable strength that are
close to each other... además que identifique dichas
frecuencias con precisión. Se mejora usando segmentos
de señal más largas (en caso que sea posible).
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Windows
Vale 0 en f=k*fs/N
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Bin centered components
A sine wave component of our signal is bin
centered if it has an integer number of periods in
the data segment being analyzed, which almost
never happens without planning. We can develop
a formula for testing if a sine wave signal
component is bin centered as follows:
Np = N*k/fs siendo Np un número entero
fs/N = bin frequency spacing
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For a bin centered
signal, all
windows yield the
same peak
amplitude reading
and have
excellent
amplitude
accuracy
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For a bin noncentered signal,
the Hann and Flat
Top windows yield
introduce less
spectral leakage
and have better
amplitude
accuracy than the
uniform window.
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In addition to causing amplitude accuracy errors, spectral leakage can
obscure adjacent frequency peaks
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Main lobe width
Se mide a -3 dB y a -6 dB en FFT bins.
Su ancho se relaciona con la frequency resolution: su
habilidad para reconocer componentes de frecuencia
muy cercanos (y frecuencias puntuales) aumenta a
medida que el ancho del lóbulo disminuye. Sin embargo,
a medida que el ancho del lóbulo disminuye, la energía
de la ventana migra a los side lobes y por lo tanto
aumenta el leakage y la imprecisión en el cálculo de la
amplitud.
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Peak side lobe level
●
●
Se mide como la distancia al main lobe en
dBs.
A medida que aumenta dicha "distancia", el
leakage disminuye (noise supression) y por
lo tanto, se estima mejor la amplitud
(increase of detection ability) de frecuencias
débiles.
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Side lobe roll-off rate
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Se mide en dB/octave o dB/decade
Se desea que este número sea tan grande
como sea posible.
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Una buena ventana debe tener:
●
small main lobe width
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large side lobe levels
●
side lobes fall-off rapidly
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Usar ventanas periódicas para DTF
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Usar ventanas periódicas para DTF
Window functions generated for digital filter design are symmetrical
sequences, usually an odd length with a single maximum at the
center. Windows for DFT/FFT usage, such as in spectral analysis,
are often created by deleting the right-most coefficient of an oddlength, symmetrical window. Such truncated sequences are known
as periodic. The deleted coefficient is effectively restored (by a
virtual copy of the symmetrical left-most coefficient) when the
truncated sequence is periodically extended.
The advantage of this trick is that a 512 length window (for
example) enjoys the slightly better performance metrics of a 513
length design. Such a window is generated by the MATLAB
function hann(512,'periodic'), for instance. To generate it,
the window length (N) is 513, and the 513th coefficient of the
generated sequence is discarded.
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Rectangular window
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It has the largest amount of spectral leakage.
It is useful for analyzing transients that have
a duration shorter than that of the window.
Transients are signals that exist only for a
short time duration.
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Hanning window
●
●
Es la ventana “por defecto”. Se usa en el
95% de los casos.
It is useful for analyzing transients longer
than the time duration of the window and for
general-purpose applications.
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Kaiser-Bessel window
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It is a flexible smoothing window whose
shape you can modify by adjusting the beta
input. Thus, depending on the application,
you can change the shape of the window to
control the amount of spectral leakage.
It is useful for detecting two signals of almost
the same frequency but with significantly
different amplitudes.
Choosing this window often reveals signals
close to the noise floor.
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Barlett (triangle) window
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Flat top window
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●
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●
It has the best amplitude accuracy of all the smoothing
windows at ±0.02 dB for signals exactly between
integral cycles = good detection.
Because it has a wide main lobe, it has poor frequency
resolution = emplea muchos frequency bins para
identificarla.
It is most useful in accurately measuring the amplitude
of single frequency components with little nearby
spectral energy in the signal.
The flat top window is sinusoidal as well, but it actually
crosses the zero line. This causes a much broader peak
in the frequency domain, which is closer to the true
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amplitude of the signal than with other windows.
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Exponential window
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It is useful for analyzing transient response
signals whose duration is longer than the
length of the window. The exponential
window damps the end of the signal,
ensuring that the signal fully decays by the
end of the sample block. You can apply the
exponential window to signals that decay
exponentially, such as the response of
structures with light damping that are excited
by an impact, such as the impact of a
hammer.
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Exact Blackman window
●
It is useful for single tone measurement. The
Exact Blackman window has a lower main
lobe width and a lower maximum side lobe
level than the Blackman window. However,
the Blackman window has a higher side lobe
roll-off rate than the Exact Blackman window.
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Blackman window
●
The Blackman window is useful for single
tone measurement because it has a low
maximum side lobe level and a high side
lobe roll-off rate.
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Blackman-Harris window
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The Blackman-Harris window is useful for single
tone measurement. The Blackman-Harris window
has a wider main lobe and a lower maximum side
lobe level than the Exact Blackman window.
It is similar to Hamming and Hann windows. The
resulting spectrum has a wide peak, but good side
lobe compression. There are two main types of this
window. The 4-term Blackman-Harris is a good
general-purpose window, having side lobe rejection
in the high 90s dB and a moderately wide main
lobe. The 7-term Blackman-Harris window function
has all the dynamic range you should ever need,
but it comes with a wide main lobe.
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Blackman-Nuttall window
●
The Blackman-Nuttall window is useful for
single tone measurement. Among the
Blackman, Exact Blackman, BlackmanHarris, and Blackman-Nuttall windows, the
Blackman-Nuttall window has the widest
main lobe and the lowest maximum side lobe
level.
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Gaussian window
●
The Gaussian window is useful for timefrequency analysis because the Fourier
transform and the derivative of a Gaussian
window both are Gaussian functions. For
example, a Short-Time Fourier Transform
with a Gaussian window is the Gabor
transform.
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How to select a window
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Each window function has its own characteristics and suitability for different
applications. To choose a window function, you must estimate the frequency
content of the signal.
If the signal contains strong interfering frequency components distant from the
frequency of interest, choose a smoothing window with a high side lobe roll-off
rate.
If the signal contains strong interfering signals near the frequency of interest,
choose a window function with a low maximum side lobe level.
If the frequency of interest contains two or more signals very near to each other,
spectral resolution is important. In this case, it is best to choose a smoothing
window with a very narrow main lobe.
If the amplitude accuracy of a single frequency component is more important than
the exact location of the component in a given frequency bin, choose a window
with a wide main lobe.
If the signal spectrum is rather flat or broadband in frequency content, use the
uniform window, or no window.
In general, the Hanning (Hann) window is satisfactory in 95 percent of cases. It
has good frequency resolution and reduced spectral leakage. If you do not know
the nature of the signal but you want to apply a smoothing window, start with86the
Hann window.
How to select a window
These are just a few of the possible window functions. There is no
universal approach for selecting a window function. However, the table
below can help you in your initial choice. Always compare the
performance of different window functions to find the best one for the
application.
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When not to use a window = esto es
ventana rectangular
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In impact modal testing, when analyzing transient
signals such as an excitation signal from hammer
blow (impulse excitation), where most of the energy
is located at the beginning of the recording. Using a
non-rectangular window would attenuate most of
the energy and spread the frequency response
unnecessarily.
In modal analysis (study of the dynamic properties
of structures under vibration excitation), when using
an impulse, a shock response, a sine burst, a chirp
burst, a noise burst, etc. Applying a window function
in this case would just deteriorate the signal-tonoise ratio.
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When not to use a window = esto es
ventana rectangular
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When measuring a repetitive signal locked-in to the
sampling frequency, for example measuring the
vibration spectrum analysis during shaft alignment, fault
diagnosis of bearings, engines, gearboxes etc. Since
the signal is repetitive, all spectral energy is confined to
multiples of the base repetition frequency. Por ejemplo
esto pasa al hacer synchronous sampling.
For periodic signals whose spectral components have
comparable strengths and when the signal segment
processed includes an exact integer multiple of periods.
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When not to use a window = esto es
ventana rectangular
●
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When measuring a signal whose frequencies
are bin aligned, since in this case, no
leakage will exist.
When measuring a repetitive signal locked-in
to the some fundamental base frequency. For
example, in fault diagnosis of bearings, since
in this case the signal is repetitive and all
spectra will be confined to multiples of the
base repetition frequency.
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