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Acoustics Acoustics Dr. Tamer Elnady – Dr. Wael Akl – Dr. Adel Elsabbagh [email protected] Signal Analysis & Measurement Techniques

Lecture4 - SignalAnalysis

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AcousticsAcoustics

Dr. Tamer Elnady – Dr. Wael Akl – Dr. Adel [email protected]

Signal Analysis & Measurement Techniques

#2: Signal Analysis and Filters

Signal Analysis

Amplitude and Frequency

T

t

peak-peak

(t)

p

~

= 2

^ p p

p

^ p

Signal Analysis in Time Domain

T

t

peak-peak

(t)

p

~

= 2

^ p

p

p

^ p

Hz1T

f

rad/s22T

f

tptp sinˆ

Signal Analysis in Time Domain (cont’d)

ttpfp

2sin1Hz 1Pa 1ˆ

0 1 2 3 4 5-2

-1

0

1

2

t(s)

p(Pa

)

Signal Analysis in Time Domain (cont’d)

tttpffpp

5sin3.02sin1Hz 5.2 , Hz 1Pa 3.0ˆ , Pa 1ˆ

21

21

0 1 2 3 4 5-2

-1

0

1

2

t(s)

p(Pa

)

tfptfptp 2211 2sinˆ2sinˆ

Time and Frequency Domain

T=1/f 0

Signal

Time

Amplitude

Frequency f 0

a) b)

Fourier Analysis

Any Periodic Signal can be considered a sum of a number of harmonic componentsPeriodic Signals are very common in Real life

)()( nTtptp

N

nnn tnfptp

10 )2cos(ˆ)(

1/T

0

Frequency [Hz] Time [s] T

2T/3

T/3

0

2/T 3/T

4/T

Amplitude

Fourier Analysis

Any Periodic Signal can be considered a sum of a number of harmonic componentsPeriodic Signals are very common in Real life

)()( nTtptp

N

nnn tnfptp

10 )2cos(ˆ)(

1/T

0

Frequency [Hz] Time [s] T

2T/3

T/3

0

2/T 3/T

4/T

Amplitude

T=1/f 0

Signal

Time

Amplitude

Frequency f 0

a) b)

Two Sine waves

0 0.01 0.02 0.03 0.04 0.05

Time [s]

0 50 100 150 200

Frequency [Hz] a) b)

Amplitude Signal

Fourier Analysis

0 0.05 0.1 0.15 0.2 0.25 0.3

Time [s]

0 20 40 60 80 100

Frequency [Hz] a) b)

Signal Ampl

Fourier Analysis

Tid [s]

Frekvens [Hz]b)a)

Signal Amp

Frekvens [Hz]Tid [s]a) b)

Signal Amp

t(s)Time [s]

Time [s]

Frequency [Hz]

Frequency [Hz]

Fourier Analysis

11

0 )sin()cos()(n

onn

on tntnta

2

2

2

2

22

20 )(111)(

T

T

T

T

T

T

dttaT

dtdtta

2

20

2

20

22

20 )cos()(2)(cos)cos()(

T

T

T

T

T

Tn dttnta

Tdttndttnta

2

20

2

20

22

20 )sin()(2)(sin)sin()(

T

T

T

T

T

Tn dttnta

Tdttndttnta

Fast Fourier Transform (FFT)

t(s)

Assignment: Read the MATLAB help about the fft function.[put website!!]

Filters

Types of Filters

1

frekvens

Amplitud

Förstärkning

f

(i)

(ii)

(iii)

(iv)

f

f

f

A

f

A

f

A

f

A

f

Förstärkning

Förstärkning

Förstärkning

1

1

1

Amplitude

Frequency

Amplification

Amplification

Amplification

AmplificationLow pass

High pass

Pass band

Stop band

Band Pass Filters

Amplification [dB] Amplification [dB]

0 0

-3

f f Frequency

B=f -f

a) b) l u

u l

f f l u

Band width,

Frequency

CAB Filter

CRB Filter

Octave and Third Octave Filters

Addition of Uncorrelated Sound Fields

Proof of this in sheet 1, problem 2

N

n

Lp

pn

totL

1

1010log10

Narrow band vs. Octave bands

Octave band

Third octave band

Narrow band

frequency (log) [Hz]

L [dB] p

A-weighted SPL

Loudness Curves

A-weighted Sound Pressure Level

Freq (Hz) Level (dB)

31.5 -39.4

63 -26.2

125 -16.1

250 -8.6

500 -3.2

1000 0

2000 1.2

4000 1

8000 -1.1

16000 -6.6