Basic Frequency Analysis Of Sound

Basic Frequency Analysis of Sound

Contents:  Frequency and Wavelength  Frequency Analysis  Perception of Sound BA 7660-06, 1

Abstract The three sections in this Lecture Note give the most fundamental explanations of the concepts behind the use of simple frequency and level measurements of sounds. The basic instrument for sound analysis is a Sound Level Meter.

Copyright© 1998 Brüel & Kjær Sound and Vibration Measurement A/S All Rights Reserved

LECTURE NOTE English BA 7669-11

Frequency Range of Different Sound Sources

1

10

100

1000

10 000

Frequency [Hz]

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The frequency span of the sounds that typically surround human beings vary considerably. Normally, young human beings can detect sounds ranging from 20 to 20000 Hz. However, infrasounds in the range from 1 to 20 Hz and ultrasounds between 20000 to 40000 Hz can affect other human senses and cause discomfort. Note that none of the illustrated sound examples cover the entire frequency range. That is why knowledge of frequency range and the need for frequency analysis is important.

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Audible Range

1

10

100

1000

10 000

Frequency [Hz]

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As can be seen, the range of perception of sound for humans - at a maximum for young people - goes from 20 to 20000 Hz. With age, the human perception of the highest frequencies decreases gradually. When exposed to excessive noise levels, hearing can be damaged, causing reduced sensitivity for hearing low sound levels. The damage can also be restricted to distinct frequency ranges.

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Wavelength Wavelength, λ [m]

Speed of sound, c = 344 m/s

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A sound signal from a loudspeaker mounted at one end of a tube will produce a sound wave that propagates forward at a speed of 344 m/s. If the signal is a single sine signal the sound wave will consist of a number of pressure maxima and minima all separated by one wavelength.

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Wavelength and Frequency

λ=

c f

λ

λ

Wavelength, λ [m] 20

10

10

20

5

50

2

100

1

200

0.2

500

1k

0.1

2k

0.05

5k

10 k

Frequency, f [Hz]

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The wavelength, the speed of sound and the frequency are related according to the formula shown. It is useful to have a rough feeling for which wavelength corresponds to a given frequency. At 1 kHz the wavelength is close to 34 cm or one foot. At 20 Hz it is close to 17 m, and only 1.7 cm at 20 kHz.

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Diffraction of Sound b >> λ

b << λ

b

b

Example: b=1m λ = 0.344 m (≈ f = 1 kHz)

Example : b = 0.1 m λ = 0.344 m (≈ f = 1 kHz) BA 7660-06, 6

Objects placed in a sound field may cause diffraction. But the size of the obstruction should be compared to the wavelength of the sound field to estimate the amount of diffraction. If the obstruction is smaller than the wavelength, the obstruction is negible. If the obstruction is larger than the wavelength, the effect is noticeable as a shadowing effect.

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Diffusion of Sound b >> λ

b << λ

b

b

Example: b = 0.5 m λ = 0.344 m (≈ f = 1 kHz)

Example : b = 0.1 m λ = 0.344 m (≈ f = 1 kHz) BA 7660-06, 7

Diffusion occurs when sound passes through holes in e.g. a wall. If the holes are small compared to the wavelength of the sound, the sound passing will re-radiate in an omnidirectional pattern similar to the original sound source. When the hole has larger dimensions than the wavelength of the sound, the sound will pass through with negligible disturbance.

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Reflection of Sound

Source

Source

Source

Imaginary Source BA 7660-06, 8

When sound hits obstructions large in size compared to it’s wavelength, reflections take place. If the obstruction has very little absorption, all the reflected sound will have equal energy compared to the incoming sound. This is one of the important design principles used when constructing reverberant rooms. If almost all reflected energy is lost due to high absorption in the reflecting surfaces, the situation is close to what is found in an anechoic room.

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Basic Frequency Analysis of Sound

Contents:  Frequency and Wavelength  Frequency Analysis  Perception of Sound BA 7660-06, 9

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Waveforms and Frequencies p

Lp time

p

Lp

Frequency

Lp

Frequency

time

p time

Frequency

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Three examples of the relationship between the waveform of a signal in the time domain compared to its spectrum in the frequency domain. In the top figure a sine wave of large amplitude and wavelength is showing up as a single frequency with a high level at a low frequency. In the middle figure a low amplitude signal with small wavelength is seen to show up in the frequency domain as a high frequency with a low level. At the bottom figure it is shown how a sum of the two signals above also in the frequency domain shows up as a sum.

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Typical Sound and Noise Signals p

Lp time

p

Lp

Frequency

Lp

Frequency

time

p time

Frequency

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Most natural sound signals are complex in shape. The primary result of a frequency analysis is to show that the signal is composed of a number of discrete frequencies at individual levels present simultaneously. The number of discrete frequencies displayed is a function of the accuracy of the frequency analysis which normally can be defined by the user.

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Filters p

Lp Time

Frequency

p

Lp Time

RMS Peak Fast Slow Impulse

Frequency

87.2 BA 7660-06, 12

To analyze a sound signal, frequency filters or a bank of filters are used. If the bandwidths of these filters are small a highly accuracy analysis is achieved. The signal flow chart shown illustrates the elements in a simple sound level meter. On top is a microphone for signal pick up. Then a gain amplification stage followed by a single frequency filter - here shown as an ideal filter. In the following we will look at real filters. After filtering follows a rectifier with the standardised time constants Fast, Slow and Impulse and the signal level is finally converted to dB and shown on the display.

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Bandpass Filters and Bandwidth B 0

Bandwidth = f2 – f1

Ideal filter

Centre Frequency = f0

f1

f0

f2

Frequency Area

Ripple

0 - 3 dB

= Area

Practical filter and definition of Noise Bandwidth

Practical filter and definition of 3 dB Bandwidth

f1

f0

f2

Frequency

f1

f0

f2

Frequency

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Ideal filters are only a mathematical abstraction. In real life, filters do not have a flat top and and vertical sides. The departure from the idealised flat top is described as an amount of ripple. The bandwidth of the filter is described as the difference between the frequencies where the level has dropped 3 dB in level corresponding to 0.707 in absolute measures. It is useful to define a Noise Bandwidth for a filter. This corresponds to an ideal filter of the same level as the real filter, but with its bandwidth (Noise Bandwidth) set to leave the two filters with the same “area”.

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Filter Types and Frequency Scales L B = 400 Hz

0

1k

B = 400 Hz

2k

3k

4k 5k 6k 7k 8k Linear Frequency Axis (primarily used in vibration analysis)

B = 400 Hz

9k

10k

Frequency [Hz]

L B = 1/1 Octave

1

2

4

8

B = 1/1 Octave

B = 1/1 Octave

16 31.5 63 125 250 500 1k 2k 4k 8k 16k Logarithmic Frequency Axis (primarily used in acoustic analysis)

Frequency [Hz]

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The two most used filter banks are: Filters that have the same bandwidth e.g. 400 Hz and displayed using a linear frequency scale. This is what normally is a result of a (FFT) Fast Fourier Transform analysis. Constant bandwidth filters are mainly used in connection with analysis of vibration signals. Filters which all have the same constant percentage bandwidth (CPB filters) e.g.1/1 octave, are normally displayed on a logarithmic frequency scale. Sometimes these filters are also called relative bandwidth filters. Analysis with CPB filters (and logarithmic scales) is almost always used in connection with acoustic measurements, because it gives a fairly close approximation to how the human ear responds.

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1/1 and 1/3 Octave Filters L B = 1/1 Octave

1/1 Octave f2 = 2 × f1 Frequency f2 = 1410 [Hz]

f1 = 708

B = 0 .7 × f 0 ≈ 70%

f0 = 1000

L B = 1/3 Octave

1/3 Octave f2 =

f1 = 891

f2 = 1120 f0 = 1000

Frequency [Hz]

3

2 × f1 = 1.25 × f1

B = 0 .2 3 × f 0 ≈ 2 3 %

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The widest octave filter used has a bandwidth of 1 octave. However, many subdivisions into smaller bandwidths are often used. The filters are often labeled as “Constant Percentage Bandwidth” filters. A 1/1 octave filter has a bandwidth of close to 70% of its centre frequency. The most popular filters are perhaps those with 1/3 octave bandwidths. One advantage is that this bandwidth at frequencies above 500 Hz corresponds well to the frequency selectivity of the human auditory system. Filter bandwidths down to 1/96 octave have been realised.

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3 × 1/3 Oct. = 1/1 Oct. L

B = 1/1 Octave

500

L

1000

2000

Frequency [Hz]

B = 1/3 Octave

800

1000

1250

Frequency [Hz]

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One advantage of constant percentage bandwidth filters are that e.g. two neighbouring filters combine to one filter with flat top, but with double the width. Three 1/3 octave filters combine to equal one 1/1 octave filter.

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Third-octave and Octave Passband Band No.

Nominal Centre Frequency Hz

Third-octave Passband Hz

1 2 3 4 5 6

1.25 1.6 2 2.5 3.15 4

1.12 – 1.41 1.41 – 1.78 1.78 – 2.24 2.24 – 2.82 2.82 – 3.55 3.55 – 4.47

27 28 29 30 31 32

500 630 800 1000 1250 1600

447 – 562 562 – 708 708 – 891 891 – 1120 1120 – 1410 1410 – 1780

40 41 42 43

10 K 1.25 K 16 K 20 K

8910 – 11200 11.2 – 14.1 14.1 – 17.8 K 17.8 – 22.4 K

Octave Passband Hz

1.41 – 2.82 2.82 – 5.62 355 – 708 780 – 1410

11.2 – 22.4 K

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List of Band No., Nominal Centre Frequency, Third-octave Passband and Octave Passband.

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The Spectrogram

1/1 Octave

L

1/3 Octave

Frequency [Hz]

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A detailed signal with many frequency components show up with a filter shape as the dotted curve when subjected to an octave analysis. The solid curve shows the increased resolution with more details when a 1/3 octave analysis is used.

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Basic Frequency Analysis of Sound

Contents:  Frequency and Wavelength  Frequency Analysis  Perception of Sound BA 7660-06, 19

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Sound Frequencies Infra

0.02

0.2

Audio

2

20

200

2000

Ultra

20.000

200K

Hz Frequency

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Only the sounds in the frequency range from 20 to 20000 Hz can be perceived by the human ear and auditory system.

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Auditory Field 140 Threshold of Pain

dB 120

Sound Pressure Level

100

Limit of Damage Risk

80

Music

60

Speech

40 20

Threshold in Quiet

0 20

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50

100

200

500 1k 2k Frequency [Hz]

5k

10k

20 k

960423

This display of the auditory field illustrates the limits of the human auditory system. The solid line denotes, as a lower limit, the threshold in quiet for a pure tone to be just audible. The upper dashed line represents the threshold of pain. However if the Limit of Damage Risk is exceeded for a longer time, permanent hearing loss may occur. This could lead to an increase in the threshold of hearing as illustrated by the dashed curve in the lower right-hand corner. Normal speech and music have levels in the shaded areas, while higher levels require electronic amplification.

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Equal Loudness Contours for Pure Tones

100

130 120 110 100

80

90 80

60

70 60

120

Sound pressure level, Lp (dB re 20 µPa)

50 40

40 30

20

20 10

0

Phon 20 Hz

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100 Hz

1 kHz Frequency

10 kHz

970379

Here are shown the normal equal loudness contours for pure tones. The dashed curve indicates the normal binaural minimum audible field. Note the very non-linear characteristics of the human perception. Almost 80 dB more SPL is needed at 20 Hz to give the same perceived loudness as at 3-4 kHz. This observation together with frequency masking - limitations in the ears capability to discriminate closely spaced frequencies at low sound levels in the presence of higher sounds - is the foundation for the calculation of the loudness of stationary signals. Loudness of non-stationary signals also needs to take the temporal masking of the human perception into account. A correct calculation of these loudness values is crucial for all the following metrics calculations such as Sharpness, Fluctuation Strength and Roughness.

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Perception of Noise

Change in Sound Level (dB)

Change in Perceived Loudness

3

Just perceptible

5

Noticeable difference

10

Twice (or 1/2) as loud

15

Large change

20

Four times (or 1/4) as loud

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This table illustrates how much of a level change in dB there is needed to give different changes in perceived Loudness. This data applies to frequencies around 1 kHz. At higher and lower frequencies, much larger changes in sound level are needed for similar changes in perceived loudness.

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40 dB Equal Loudness Contours and A-Weight Lp 

40 dB Equal Loudness Contour normalized to 0 dB at 1kHz

(dB) 40

40

20 0

Lp 

(dB) 0 40 dB Equal Loudness Contour inverted -20 and compared with A-weighting -40

20 Hz

100

1 kHz

10 kHz

1 kHz

10 kHz

40 A-weighting

20 Hz

100

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Top figure is the equal loudness contour for 40 dB at 1 kHz. The popular A-weighting at the bottom is shown in comparison to the inverted 40 dB equal loudness contour.

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Frequency Weighting Curves Lp

D

[dB]

Lin.

0 D

C B+C

A

-20 A B -40

-60 10

20

50

100

200

500

1k

2k

5k

10 k 20 k

Frequency [Hz]

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The A-weighting, B-weighting and C-weighting curves follow approximately the 40, 70 and 100 dB equal loudness curves respectively. D-weighting follows a special curve which gives extra emphasis to the frequencies in the range 1 kHz to 10 kHz. This is normally used for aircraft noise measurements.

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Calibration and Weighting!

Lp

ΔL = 8.6dB

[dB] 0

-20

-40 A-weighting -60 10

20

50

100

200

500

1k

2k

5k

10 k 20 k

Frequency [Hz]

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Please be careful when calibrating a system with a weighting filter switched in. Using a pistonphone which has a calibration frequency of 250 Hz you get 8.6 dB less signal reading than using a sound level calibrator with a test frequency of 1000 Hz if an A weighting filter is switched in.

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Use of Frequency Weighting

Weighting RMS Peak Fast Slow Impulse

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All sound level meters have built in A-weighting, and some have also others like B and C-weighting.

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Serial Analysis

1

2

3

n

RMS Peak Fast Slow Impulse

87.2 BA 7660-06, 28

For more detailed frequency analysis sound level meters can have a serial filter bank of 1/1 octave and maybe also 1/3 octave bandwidths. As only one filter is active at a time the analysis time is long and requires the sound field to be stationary. The advantage is a lower price.

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Parallel Analysis

1

2

3

n

RMS Peak Fast Slow Impulse

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A sound level meter with a parallel filter bank is the most expensive, but the fastest in operation and does not require the signal to be stationary.

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The Sound Level Analyzer

dB 100 1/1, 1/3 oct

1/3 Octave Analysis

Weighting 80 RMS Peak Fast Slow Impulse

60 40 20

125 250 500 1k

2k

4k

8k

LA

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The most advanced sound level meters act more like an ordinary sound analyzer having both a parallel filter bank and a selection of weighting filters.

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The Spectrogram and Overall levels L

[dB]

1/1 Octave

1/3 Octave

LA [dB(A)] LB [dB(B)] LC [dB(C)] LD [dB(D)] LLin. [dB]

Frequency [Hz]

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The advanced sound level meters often feature comprehensive displays showing all the analysis results from both the parallel analysis filters and the weighting curves simultaneously.

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Conclusion

Headlines of Basic Frequency Analysis of Sound      

Basic introduction to audible frequency range and wavelength of sound Diffraction, reflection and diffusion of sound Frequency analysis using FFT and Digital filters Fundamental concepts for 1/1 and 1/3 octave filters Human perception of sound and background for A,B,C and D-weighting Signal flow and analysis in Sound Level Meters

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Literature for Further Reading 

Acoustic Noise Measurements Brüel & Kjær (BT 0010-12)



Noise Control - Principles and Practice Brüel & Kjær (188-81)

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