1
Signals – Frequency Domain Analysis
V. Rouillard 2003
Frequency Domain Analysis
2
Signals – Frequency Domain Analysis
V. Rouillard 2003
•
Concerned with analysing the frequency (wavelength) content of a process
•
Application example: Electromagnetic Radiation:
•
Represented by a Frequency Spectrum: plot of intensity vs frequency
3
Signals – Frequency Domain Analysis
V. Rouillard 2003
Solar Radiation Spectrum
0.25
Solar radiation spectrum
(≈1000 W @ equator @noon)
0.1
0.05
Visible
700 nm red
0.15
440 nm violet
2
Irradiance [W/cm /µm]
0.2
Near IR
Infrared
0
0
500
1000
1500
Wavelength [nm]
4
Signals – Frequency Domain Analysis
V. Rouillard 2003
•
Why frequency domain analysis?
•
Consider the following processes:
Vibration on a transport vehicle
2000
5
Signals – Frequency Domain Analysis
V. Rouillard 2003
•
Why frequency domain analysis?
•
Consider the following processes:
Whale sound
6
Signals – Frequency Domain Analysis
V. Rouillard 2003
•
Why frequency domain analysis?
•
Consider the following processes:
Apache helicopter flyover
7
Signals – Frequency Domain Analysis
V. Rouillard 2003
•
Why frequency domain analysis?
•
Consider the following processes:
Ocean surface level fluctuations (waves)
8
Signals – Frequency Domain Analysis
V. Rouillard 2003
•
Why frequency domain analysis?
•
Consider the following processes:
Axle load of passenger vehicle on test track
9
Signals – Frequency Domain Analysis
V. Rouillard 2003
•
Why frequency domain analysis?
•
Consider the following processes:
Vibration of structure due to explosion load (pyrotechnic shock)
10
Signals – Frequency Domain Analysis
V. Rouillard 2003
•
Why frequency domain analysis?
•
Consider the following processes:
11
Signals – Frequency Domain Analysis
V. Rouillard 2003
•
Why frequency domain analysis?
•
Consider the following processes:
12
Signals – Frequency Domain Analysis
V. Rouillard 2003
•
Why frequency domain analysis?
•
Consider the following processes:
13
Signals – Frequency Domain Analysis
V. Rouillard 2003
•
All continuous signals can be shown to be comprised of a summation of individual harmonic
(Fourier) components of various frequencies, amplitudes and phases.
•
Common methods to compute the frequency spectrum of measured data:
•
Filter analysis consists of a number of band-pass frequency filters (analogue or digital)
•
These filters allow only a (narrow) band of frequencies to pass
•
A number of filters with adjacent frequency bands are used to generate a frequency spectrum
Filter (octave band) analysis
0
-10
-20
dB -30
-40
-50
-60
15
31
62
125
250
500
1k
2k
4k
8k
16k
(fractional) Octave bands
14
Signals – Frequency Domain Analysis
V. Rouillard 2003
Digital Fourier Transform (DFT) or Fast Fourier Transform (FFT)
•
Transforming a signal from the time to the frequency domain can be achieved via the Fourier
Transform (also called Fourier Analysis).
•
Information is neither gained or lost when transforming signals from the time domain to the
frequency domain via the FFT.
•
The Fourier transform is reversible – Inverse Fourier Transform (IFT)
•
Although Fourier theory applies strictly to periodic signals, the periodicity of sampled or measured
signals is assumed resulting in a estimate of he frequency spectrum
Magnitude Spectrum
Signal (real)
Complex
FFT
Phase Spectrum
IFT
15
Signals – Frequency Domain Analysis
V. Rouillard 2003
Random signals (Dadisp: Freqa)
•
Broad-band signals contain more sinusoid components than narrow-band signals
•
A sinusoid can be considered as a very-narrow-band random signal.
•
The phase of the sinusoids which make up random signals are values uniformly-distributed
between 0 and 2π.
•
The phase of the sinusoids which make up pulse signals are equal.
Effect of phase on frequency spectrum
•
16
An infinite number of sinusoidal components with equal phase produces the Delta function (very
sharp pulse).
Signals – Frequency Domain Analysis
V. Rouillard 2003
Signal bandwidth examples: Uniformly-distributed random phase.
17
Signals – Frequency Domain Analysis
V. Rouillard 2003
Signal bandwidth examples: Zero bandwidth signals
18
Signals – Frequency Domain Analysis
V. Rouillard 2003
Signal bandwidth examples: Constant phase.
19
Signals – Frequency Domain Analysis
V. Rouillard 2003
Example of constant phase signal: Gaussian wave packet
Elevation [mm]
40
20
0
-20
-40
0
2
4
6
8
10
12
14
16
Time [sec.]
3.5
Energy Spectrum [mm
2
.s]
80
Phase [rad]
60
40
20
0
0.0
-3.5
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
0.0
0.2
0.4
Frequency [Hz]
20
0.6
0.8
1.0
1.2
1.4
1.6
Frequency [Hz]
Signals – Frequency Domain Analysis
V. Rouillard 2003
Digital Fourier Transform (DFT) or Fast Fourier Transform (FFT)
Random signals (Dadisp: Apache_PSD
& Spectral_averag)
•
Each observation is unique – sample of process – one physical realisation of the process
•
The frequency spectrum of each sample is an estimate of the frequency spectrum of the entire
process
•
The estimate of the true frequency spectrum is improved by computing spectral averages.
Important issues when computing the Fourier Transform:
•
Bandwidth: Frequency range to be analysed
•
Frequency Resolution [Hz] = 1/sub-record duration [secs]
•
Spectral estimate accuracy (random error): Std. Deviation of error = 1/√# averages. Spectral
error is reduced by:
•
•
Identifying sub-records within the measured record
•
Computing the spectrum of each sub-record
•
Computing the average spectrum
Given a fixed record length, a compromise has to be reached with respect to frequency
resolution and spectral error.
21
Signals – Frequency Domain Analysis
V. Rouillard 2003
Effects of frequency resolution & spectral averaging. Example: Helicopter fly-by (sound)
22
Signals – Frequency Domain Analysis
V. Rouillard 2003
Effects of frequency resolution & spectral averaging. Example: Heavy vehicle vibrations.
23
Signals – Frequency Domain Analysis
V. Rouillard 2003
•
Spectral averaging must be used carefully
•
When signals are strongly non-stationary (ie. Evident variations in vital characteristics such as RMS
levels or frequencies) spectral averaging will conceal these non-stationary properties.
24
Signals – Frequency Domain Analysis
V. Rouillard 2003
Effects of frequency content variation. Example: Whale cry.
25
Signals – Frequency Domain Analysis
V. Rouillard 2003
Effects of frequency content variation. Example: Whale cry.
26
Signals – Frequency Domain Analysis
V. Rouillard 2003
Effects of frequency content variation. Example: Whale cry.
27
Signals – Frequency Domain Analysis
V. Rouillard 2003
Nyquist Frequency and Sampling Rate (Dadisp: Shannonsine)
Leakage and the effects of widowing functions (Dadisp: leakage_sin
& leakage_rnd)
Overlapping
Zero-padding
28
Signals – Frequency Domain Analysis
V. Rouillard 2003
Effects of spectral leakage and windowing. Example: sinusoid.
29
Signals – Frequency Domain Analysis
V. Rouillard 2003
Effects of spectral leakage and windowing. Example: Heavy vehicle vibrations.
30
Signals – Frequency Domain Analysis
V. Rouillard 2003
Influence of signal clipping on frequency spectrum
No clipping
Clipped
31
Signals – Frequency Domain Analysis
V. Rouillard 2003
Influence of broad-band and narrow-band (power line) noise on frequency spectrum
Clean spectrum
+ Broad-band noise
+ Narrow-band noise
32
Signals – Frequency Domain Analysis
V. Rouillard 2003
Influence of intermittent noise (switchgear interference) on frequency spectrum
Clean spectrum
+ Intermittent noise (sharp pulses)
33
Signals – Frequency Domain Analysis
V. Rouillard 2003
System Analysis
(Excitation – Response Relationships)
34
Signals – Frequency Domain Analysis
V. Rouillard 2003
•
Frequency analysis is useful in determining the frequency characteristics of systems: relationship
between output and input as a function of frequency.
•
Real systems are often assumed to approximate an ideal system.
•
Ideal systems:
•
•
Have constant parameters (no variation in system characteristics wrt time)
Are linear (ie. additive and homogeneous):
•
Additive: Response (output) to sum of excitations (inputs) = sum of responses due to
each individual input:
f ( x1 + x2 ) = f ( x1 ) + f ( x2 )
•
Homogeneous: Response from excitation x constant = response x constant from
excitation:
f ( kx ) = kf ( x )
35
Signals – Frequency Domain Analysis
V. Rouillard 2003
Cross Spectrum
A( f ) = Y
•
Excitation signal
∞
{a( t )} = ∫−∞ a( t )e− j2π ft dt
Response signal
H(f)
B( f ) = Y
System FRF
∞
{b( t )} = ∫−∞ b( t )e − j2π ft dt
A( f ) = Re( f ) + i Im( f )
The cross spectrum of A wrt B is defined as:
A = A e j( 2π ft +Φ A ) ,
A* ( f ) = Re( f ) − i Im( f )
B = B e j( 2π ft +Φ B )
S AB ( f ) = A* ( f ) ⋅ B( f )
= A e − j( 2π ft +Φ A ) ⋅ B e j( 2π ft +Φ B ) = A ⋅ B e j( Φ B −Φ A )
•
Where A*(f) is the complex conjugate of the instantaneous spectrum of a(t) and B(f) is the
instantaneous spectrum of b(t)
•
The amplitude of the cross spectrum is the product of the two amplitudes
•
The phase of the cross spectrum is the difference between the phase of B relative to A.
•
The cross spectrum SBA has the same amplitude but opposite phase.
•
Auto spectra and cross spectra are generally expressed in one-sided form:
36
Signals – Frequency Domain Analysis
V. Rouillard 2003
Cross Spectrum
A( f ) = Y
•
Excitation signal
∞
{a( t )} = ∫−∞ a( t )e − j2π ft dt
Response signal
H(f)
B( f ) = Y
System FRF
∞
{b( t )} = ∫−∞ b( t )e − j2π ft dt
The auto spectrum is obtained in the same way:
S AA( f ) = A* ( f ) ⋅ A( f )
= A e − j( 2π ft +Φ A ) ⋅ A e j( 2π ft +Φ A ) = A
2
•
The autospectrum is the power spectrum which has additive properties useful for averaging.
•
Auto spectra and cross spectra are generally expressed in one-sided form:
G AA( f ) = 0
G AB ( f ) = 0
G AA( f ) = S AA( f )
G AB ( f ) = S AB ( f )
G AA( f ) = 2S AA( f ) G AB ( f ) = 2S AB ( f )
•
f <0
f =0
f >0
As for the auto spectrum the cross spectrum of stationary random signals is best estimated by
averaging over a number of records.
37
Signals – Frequency Domain Analysis
V. Rouillard 2003
Coherence
•
The coherence is used to determine the level of
linear dependence b/w two signals as a function
of frequency
•
The coherence is defined as:
γ 2( f ) =
•
2
G AB ( f )
G AA( f ) ⋅ GBB ( f )
And can be viewed as a squared correlation
coefficient which quantifies the linear relationship
between two variables:
 σ xy 
 σ x ⋅ σ y 


2
2
ρ xy
<1
2
ρ xy
<1
2
ρ xy
=0
Covariance
ρ 2xy = 
•
2
ρ xy
=1
Variance
Note that for a single set of records (no
averaging) the coherence is 1.
38
Signals – Frequency Domain Analysis
V. Rouillard 2003
Coherence
•
In practical cases, reasons for obtaining coherences of less that unity are:
1.
Contamination of either excitation or response signal with noise
2.
The presence on nonlinearities in the relationship between the excitation and response
3.
Spectral leakage due to insufficient resolution or unsuitable windowing function
4.
Time delay between the excitation and response signals
Signal:noise ratio
•
The coherence can be used to determine the signal:noise ratio of the measurement which is
defined as:
S:N =
•
γ2
1−γ 2
If noise contamination of the response signal is assumed to be the only factor influencing the
coherence, then γ2 is proportional to the coherent power while (1- γ2) represents the non-coherent
power which is due to the noise in the response signal.
39
Signals – Frequency Domain Analysis
V. Rouillard 2003
Frequency Response Function
•
The system is excited with a signal of suitable bandwidth (Swept sine [not suited to FFT], band–
limited random signal or impulse) and the system response is measured
•
The excitation and response signals are each transformed to the frequency domain (FFT) to obtain
a complex, instantaneous spectrum which are averaged to produce a mean power spectrum
(autospectrum).
•
Further frequency-domain functions such as the cross-spectrum are then computed and the (linear)
relationship between the excitation and response signals as a function of frequency is established
giving the Frequency Response Function (FRF).
a( t )
8Y
A( f )
H(f)
System FRF
b( t ) = a( t ) ∗ h( t )
8Y
B( f ) = A( f ) ⋅ H( f )
Relationship between excitation and response for an ideal system without noise contamination
•
In the time domain the response is obtained by convolving the system impulse response function
h(t) with the excitation.
•
In the frequency domain, the response spectrum is obtained by multiplying the system Frequency
Response Function H(f) with the excitation spectrum.
40
Signals – Frequency Domain Analysis
V. Rouillard 2003
Frequency Response Function
a( t )
8Y
A( f )
H(f)
System FRF
b( t ) = a( t ) ∗ h( t )
8Y
B( f ) = A( f ) ⋅ H( f )
Relationship between excitation and response for an ideal system without noise contamination
•
For an ideal system without noise, the Frequency Response Function can be determined by:
H( f ) =
B( f )
A( f )
A = A e j( 2π ft +Φ A )
B = B e j( 2π ft +Φ B )
•
H( f ) =
B j( 2π ft +Φ B ) − j( 2π ft +Φ A )
e
⋅e
A
H( f ) =
B j( Φ B −Φ A )
e
A
Which is a complex function in terms of the magnitude ratio and the phase difference.
41
Signals – Frequency Domain Analysis
V. Rouillard 2003
Frequency Response Function
•
In practice it has been shown that when the response signal is contaminated by broad-band
random noise (transducer noise), a better estimate of the FRF is obtained by normalising the cross
spectrum by the auto spectrum of the input:
H( f ) =
•
If the complex conjugate of the output spectrum is used the resulting FRF is improved when noise
contamination is present in the excitation signal:
H( f ) =
•
B( f ) A* ( f ) S AB ( f ) G AB ( f )
⋅
=
=
= H 1( f )
A( f ) A* ( f ) S AA( f ) G AA( f )
B( f ) B* ( f ) S BB ( f ) GBB ( f )
⋅
=
=
= H 2( f )
A( f ) B* ( f ) S BA( f ) GBA( f )
It is interesting to note that the ratio H1:H2 always gives the coherence:
2
G AB ( f )
H 1( f ) G AB ( f ) GBA( f ) G AB ( f ) G AB * ( f )
=
⋅
=
⋅
=
H 2 ( f ) G AA( f ) GBB ( f ) G AA( f ) GBB ( f ) G AA( f ) ⋅ GBB ( f )
H 1( f )
= γ 2( f )
H 2( f )
•
42
Although the magnitude of H1 and H2 will be different for noise contaminated measurements, their
phase is always the same.
Signals – Frequency Domain Analysis
V. Rouillard 2003
Frequency Response Function – effects of noise
•
Consider an ideal system where the measured response signal b(t) is contaminated by extraneous
uncorrelated noise n(t).
•
The noise signal may include some component generated by the system but not caused by the
excitation signal a(t).
n(t)
Given sufficient averaging, the measured cross
spectrum GAB will approximate the true cross
v(t)
h(t)
spectrum GAV and the measured response auto
a(t)
b(t)
Σ
spectrum is the sum of the clean output and
H(f)
the uncorrelated noise:
•
G AB ( f ) = G AV ( f )
2
GBB ( f ) = GVV ( f ) + GNN ( f ) = H( f ) G AA( f ) + GNN ( f )
G AB ( f ) = G AV ( f ) + G AN ( f ) = G AV ( f ) = H( f ) ⋅ G AA( f )
H 1( f ) =
G AB ( f ) G AV ( f )
=
= H( f )
G AA( f ) G AA( f )
← Optimum FRF (effects of noise minimised)
43
Signals – Frequency Domain Analysis
V. Rouillard 2003
n(t)
Frequency Response Function – effects of noise
v(t)
h(t)
a(t)
H(f)
Σ
b(t)
while
2
H 2( f ) =
GBB ( f ) GVV ( f ) + GNN ( f ) H( f ) G AA( f ) + GNN ( f )
=
=
GBA( f )
GVA( f )
H * ( f ) ⋅ G AA( f )
2
= H( f ) +
H( f ) GNN ( f )
GNN ( f )
= H( f ) +
H * ( f ) ⋅ G AA( f )
H * ( f ) ⋅ GVV ( f )
 G ( f )
H 2 ( f ) = H( f ) 1 + NN
GVV ( f ) 

← FRF magnitude overestimated (phase OK)
where Gvv is the coherent output power spectrum and
44
GNN ( f )
is the noise : signal ratio.
GVV ( f )
Signals – Frequency Domain Analysis
V. Rouillard 2003
Frequency Response Function – effects of noise
•
When the measured excitation signal is contaminated by extraneous noise m(t) (measurement
noise), it can be shown that:
u(t)
h(t)
H(f)
m(t)
H 1( f ) =
H( f )
 GMM ( f ) 
1 + G ( f ) 


UU
Σ
b(t)
a(t)
← FRF magnitude underestimated (phase OK)
while
H 2 ( f ) = H( f )
•
← Optimum FRF (effects of noise minimised)
When there is extraneous noise in both the measured excitation and response signals H1 and H2
give the lower and upper bound to the true FRF.
45
Signals – Frequency Domain Analysis
V. Rouillard 2003
Frequency Response Function
Examples:
46
Excitation:
Wave height
System: Ship
Excitation:
Gusts
System: Tower
Excitation:
Pavement
topography
System: Road
vehicle
Response:
Vertical
vibration
Excitation:
Engine
vibrations
System: Military
submarine
Response:
Sound
Excitation:
Aerodynamic
loads
System:
Aeroplane wing
Response:
Stresses or
deflection
Response:
Pitch or roll
Response:
Sway or
deflection
Signals – Frequency Domain Analysis
V. Rouillard 2003
Frequency Response Function
Excitation:
vertical
vibrations
System: Packaged
product
Response:
Product
vibration
Dead Weight
Guided platen
Signal
Analyser
Response accel. signal
Test sample
Vibration
Controller
Input accel. signal (control)
Vibration table
Servo-hydraulic
Actuator
47
Signals – Frequency Domain Analysis
V. Rouillard 2003
Frequency Response Function
Response
accelerometer
Linear bearings
housing
Guided dead
weight
Fibreboard
cushion sample
Vibration table
Guide rod
Input (table)
accelerometer
Signals – Frequency Domain Analysis
V. Rouillard 2003
Frequency Response Function
steady-state Random Nd = 25
10
Transmissiblity
48
5
0
0
5
10
15
Frequency [Hz]
20
25
30
49
Signals – Frequency Domain Analysis
V. Rouillard 2003
Frequency Response Function
Accelerometer
Charge
Amplifier
Dead
Weights
Accelerometer
Packaged Unit
Charge
Amplifier
Table
Servo
Amplifier
PC with
ADC and DAC
Modules.
50
Function Generator with
externally-controlled
frequency & amplitude.
Signals – Frequency Domain Analysis
V. Rouillard 2003
Frequency Response Function
Hydraulic
Servoactuator
51
Signals – Frequency Domain Analysis
V. Rouillard 2003
Frequency Response Function
Response acceleration
Sensor
Excitation acceleration
Sensor
Signals – Frequency Domain Analysis
V. Rouillard 2003
Frequency Response Function
5.0
Before test
After test
Gain
4.0
3.0
2.0
1.0
0.0
00
Phase Difference [Deg]
52
5
10
15
Frequency [Hz]
-30
20
25
Before test
After test
-60
-90
-120
-150
-180
0
5
10
15
Frequency [Hz]
20
25