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