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Introduction to Signal
Processing
Professor Mike Brennan
Fundamentals of Signal
Processing
• Classification of signals
• Fourier analysis
• Spectra / Frequency Response Function
Vibration signals
t
( )v t
• Measured quantities (e.g. acceleration, pressure)
which vary with time can be treated as SIGNALS and
can be displayed as a graph of the value of the quantity
against time
Vibration signals
• Signals may be classified into different types
Stationary Non-Stationary
Deterministic Random Transient Continuous
Periodic Quasi-periodic
Signal
Vibration signals
• Stationary – statistical properties do not change with time
•Deterministic – exact value is predictable at all time
•Random – exact value not predictable at any time
•Transient – finite duration
• Periodic – repeats exactly after a period of time
•Quasi-periodic – mixture of periodic signals – non-periodic
Harmonic motion
t
( )x t
angular
displacement
One cycle of motion
2π radians
A
t
Relationship between circular motion in the
complex plane with harmonic motion
Imaginary part – sine wave
Real part – cosine wave
Sinusoidal signals – other descriptions
t
( )x t
T
0
1sin dt
T
avx A tT
For a sine wave
0avx
For a rectified sine wave
0.637avx A
• Average value
Sinusoidal signals – other descriptions
t
( )x t
• Average value
DC
Average value of a signal = DC component of signal
Sinusoidal signals – other descriptions
t
( )x t
T
For a sine wave
2 20.5meanx A
• Mean square value
2
2
0
1sin dt
T
meanx A tT
• Root Mean Square (rms)
2 2rms meanx x A
Many measuring devices, for example a digital voltmeter,
record the rms value
Vibration signals
t
( )x t
• Periodic or deterministic (not sinusoidal)
T T
T is the fundamental period
• Heartbeat
• IC Engine
Vibration signals
t
( )x t
• Transient
• Gunshot
• Earthquake
• Impact
Vibration signals
t
( )x t
• Random
• Uneven Road
• Wind
• Turbulence
Frequency Analysis
• A signal can be represented by its frequency content
• known as its SPECTRUM
• Examples
Frequency Analysis - Filters
• A FILTER “passes” a narrow range of frequency and
“stops” others
• An “ideal” filter:
Gain
Frequency
Frequency Analysis - Filters
• Frequency content can be measured using a set of filters…
Frequency Analysis - Filters
Constant bandwidth filters Constant % bandwidth filters
Frequency Analysis - Filters
• Common constant percentage bandwidth filters are
Octave and Third Octave Filters
• The centre frequencies of an octave filter set are
2nn of f
where fo is the bottom filter in the set
and the bandwidth of filter n is
1, , 2
2L n H L n H nf f f f f f f
Frequency Analysis - Examples
Narrow band analysis Octave band analysis
Fourier Analysis
(Jean Baptiste Fourier 1830)
+
+
+
:
t
( )x t
• Representation of a signal by sines and cosine waves
Fourier Series
• We express this as
1
( ) DC sin 2n n n
n
x t A f t
where
is the of the component
is the
amplitude
frequ
ency
phasis t ehe
th
n
n
n
A n
f
Note that if x(t) has units [V] then An also has units [V]
Fourier Series
• We can represent the values of An and versus fn as
Spectra n
frequency
n
nA• for periodic signals (with period T)
the spacing between the frequency
Components is 1/T Hz
Fourier Composition of a Square wave
frequency
Fourier Composition of a Saw Tooth
Wave
frequency
Fourier Composition of a Pulse Train
frequency
Frequency Analysis
• For non-periodic signals, we allow all frequencies to
be present.
• Effectively, T becomes very large so Δf (=1/T) becomes very small
• i.e., the spacing between frequency components “tends to zero”
but when this happens the amplitudes of the components
gets very small
• So as An→0 as Δf→0
amplitude
bandwidthnA
f
• Now if An has units [V] then An/Δf has units of V/Hz which is
amplitude per unit bandwidth or amplitude density
• So we create and plot this against f
The Fourier Transform
• Exact relationships exists between a time domain signal
and its frequency spectrum
• The FOURIER TRANSFORM:
2( ) ( ) j ftV f v t e dt
calculates the amount of frequency f in signal v(t) by multiplying the
signal by a sine wave at frequency f and integrating over all time
(Fourier Analysis)
• The INVERSE FOURIER TRANSFORM:
2( ) ( ) j ftv t V f e df
expresses the time signal as the sum of an infinite number of sine waves
(Fourier Synthesis)
Relationship between data in the time
and the frequency domain
t
( )x t
f
2/
PSD
X Hz
rms level
AREA
AREA = mean square value
*
T
1PSD = lim , ,k kE X f T X f T
T
rms level AREA
Relationship between data in the time
and the frequency domain - example
t
( )x t rms level
f
2/
PSD
X Hz
f
( ) sin( )x t X t
2
2
X
f
Relationship between data in the time
and the frequency domain
221
1
T X f dfT
x t dtT
Parseval’s Theorem
Mean square value Power Spectral Density (PSD)
Integrated over frequency
Truncation
• Integrating over all time is impossible for measurements
• For transient signals, integrate over duration of signal only..
t
( )x t
T
Truncation
• For continuous signals assume signal is periodic with time
period T and integrate over T
t
( )x t
T
t
( )x t
Fourier Transform
• The Fourier transform gives discrete frequencies at multiples
of 1/T
nA
1 T
• Hence the FREQUENCY RESOLUTION is dependent on T
1
fT
Windowing
• Truncation of a signal leads to a “smearing” of the spectrum
time
• Example – truncated sine wave
frequency
leakage
actual frequency
time
frequency
window
• Leakage can be reduced using a “shaped” WINDOW
Windowing
0 50 100 150 200 250-6
-4
-2
0
2
4
6
8
Time (s)
Am
plit
ude
Total length
Segment Hanning window
Equal Loudness Contours
Frequency Weighting
A, B, and C weighting networks were derived as the inverse of the 40, 70 and 100 dB
Equal Loudness contours
Frequency Weighting
+ =
unweighted spectrum A-weighted spectrum
A-weighting curve
f, Hz f, Hz f, Hz
dB
Frequency Response Function
• The time response of a system to an impulse input is known as
the IMPULSE RESPONSE of the system
system input excitation output response
• The Fourier transform of the impulse response is known as the
FREQUENCY RESPONSE FUNCTION (FRF)
• The FRF can be measured directly from the spectra of the input
and output signals Output( )
FRF = Input( )
f
f
• Either the impulse response or FRF can be used to characterise
any linear system
Summary
• Types of vibration / acoustic signals
• Fourier Series
• Fourier Transform
• Spectra (narrow band / octave band)
• Frequency Response Function