February 16, 2020

Higher resolution Octave bands


HIGHER RESOLUTION OCTAVE BANDS


Introduction

The word ‘octave’ is derived from the Latin word meaning ‘eight’. In the musical world where there are 7 notes, the 8th note sounds twice as high as the 1st note. The 8th note is an octave higher than the 1st note. Similarly in the octave band, the upper limit frequency is twice the lower limit frequency. Octave bands are very useful in engineering applications because they reveal the spectral content, meaning they represent the change in noise levels with respect to the frequency of sound. This helps in identifying which frequency is responsible for the noise which helps in nailing down the component in a machine responsible for the frequency.

Octave 1/6th, Octave 1/12th and Octave 1/24th are examples of higher resolution octave bands. They are similar to the octave 1/1 band except that the spectral resolution is exorbitantly high. Octave 1/6th band is similar to octave 1/1 band but each frequency band is divided equally into 6 parts, 12 parts for Octave 1/12th band and 24 parts for Octave 1/24th band. This increases the frequency resolution to an extent that pointing out tonal frequencies becomes easier but at the expense of large processing times and data consumption.


Octave 1/6th Band

Fig .1 1/6th Octave Band

Equations
The relation between the next and the previous center frequency is given by,

n = 6 for 1/6th octave band


n = 12 for 1/12th octave band


n = 24 for 1/24th oc1tave band


CF – center frequency
CFnext – next center frequency
CFprev – previous center frequency

In case of 1/6th Octave band, the next center frequency is 1.12 times the previous. This ratio decreases with increasing spectral resolution. The ratio is 1.06 for 1/12th Octave band and 1.03 for 1/24th Octave band.

The relation between the upper band and lower band frequency limit for a given frequency band is given by,

CL – lower band limit for a given center frequency
CU – upper band limit for a given center frequency


n = 6 for Octave 1/6th band


n = 12 for Octave 1/12th band


n = 24 for Octave 1/24th band

As the spectral resolution keeps increasing or in other words when ‘n’ increases, the number of upper and lower bands also increases.

Visual Representation

The picture below explains how one band of Octave1/1 is divided into 6 equal parts for Octave 1/6 bands. The center frequencies of Octave 1/6 bands are calculated and mentioned in the table below. The corresponding Octave 1/1 center frequencies are also highlighted for understanding the relationship between the two.
Fig .2 Formation of Octave 1/6th band from Octave 1/1 band

Tabular column

Octave 1/6 band
Center frequency
Corresponding Octave 1/1 band

Hz




Band1
16
Band1
Band2
17.95939277
Band3
20.1587368
Band4
22.627417
Band5
25.39841683
Band6
28.50875898
Band7
32
Band2
Band8
35.91878555
Band9
40.3174736
Band10
45.254834
Band11
50.79683366
Band12
57.01751796
Band13
64
Band3
Band14
71.83757109
Band15
80.63494719
Band16
90.50966799
Band17
101.5936673
Band18
114.0350359
Band19
128
Band4
Band20
143.6751422
Band21
161.2698944
Band22
181.019336
Band23
203.1873347
Band24
228.0700718
Band25
256
Band5
Band26
287.3502844
Band27
322.5397888
Band28
362.038672
Band29
406.3746693
Band30
456.1401437
Band31
512
Band6
Band32
574.7005687
Band33
645.0795775
Band34
724.0773439
Band35
812.7493386
Band36
912.2802874
Band37
1024
Band7
Band38
1149.401137
Band39
1290.159155
Band40
1448.154688
Band41
1625.498677
Band42
1824.560575
Band43
2048
Band8
Band44
2298.802275
Band45
2580.31831
Band46
2896.309376
Band47
3250.997354
Band48
3649.12115
Band49
4096
Band9
Band50
4597.60455
Band51
5160.63662
Band52
5792.618751
Band53
6501.994709
Band54
7298.242299
Band55
8192
Band10
Band56
9195.2091
Band57
10321.27324
Band58
11585.2375
Band59
13003.98942
Band60
14596.4846


Octave 1/6 Band Real Time Analysis

Fig .3 Real time Octave 1/6th band


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