A technique for zoom transform and long-time signal analysis

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Canadian Acoustics/Acoustique Canadienne, 11, 3, pp. 45-50, 1983-07

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A technique for zoom transform and long-time signal analysis

Chu, W. T.

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A TECHNIQUE FOR ZOOM TRANSFORM AND LONG-TIME SIGNAL ANALYSIS

by W.T. Chu Appeared in Canadian Acoustics, Vol. 1 1 (3) July 1983 p. 4 5 - 5 0 N * c

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A

TECHNIQUE FOR ZOOM TRANSFORM

AND

LONG-TIME SIGNAL ANALYSIS

W.T. Chu

National Research Council of Canada

Division of Building Research

Ot tawa, Canada KIA OR6

ABSTRACT

This paper reviews a useful technique for performing zoom transform.

The method involves recording a long-time signal and transforming it in

parts, using a smaller transform. Its ability to handle long-time signals

renders the method attractive to both acousticians and vibration engineers.

Examples and the listing of a computer program are given to demonstrate the

technique.

Une technique pratique pour r6aliser la transformation avec zoom

n6cessite l'enregistrement d'un signal de longue dur6e, pour ensuite le

transposer en parties en utilisant une plus petite transformation. La

possibilit6 de traiter des signaux de longue dur6e accroPt l1int6rSt de

cette technique pour les spgcialistes de l'acoustique et des vibrations.

Des exemples ainsi qu'une liste impri.de d'un programme infonnatique sont

donnes pour illustrer cette technique.

The discrete Fourier transform (DFT) has become

a

very important tool for

analysis because of the availability of efficient computer algorithms and low priced

mini- and micro-computers with fast array processors. The efficient method for

computing DFT is called the fast Fourier transform (FFT).

Most machines, however,

are usually limited to a 1 k or 2 k point transform

(k =

1024).

This limitation

often presents problems to acousticians who have to analyse very long time signals

and to vibration engineers who need fine resolution spectra for modal analysis.

Although FFT instruments with zoom features are readily available, software can be

difficult to find. For example, no such programs are listed in "Programs for Digital

Signal Processing" by the IEEE press.

l

Techniques for zoom transform and for long-time signal analysis are available in

the literature,

2-4

but they involve fairly complicated procedures. There is,

however, a straightforward method used by the Bruel and Kjaer Type 2033 "High

Resolution Signal Analyser" to solve both problems. Unfortunately, neither the Bruel

and Kjaer manual nor ~ h r a n e ~

give enough information for other researchers to

implement the technique with their own computers. The procedure involves taking

smaller transforms on selected data points from different segmerits of a pre-recorded

long-time signal. Although yip4 did not promote this procedure in his paper, the

mathematical foundation can be gained from his analysis. This paper reviews this

technique and presents some examples.

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ANALYSIS

Although both the discrete and fast Fourier transform algorithms are, in

general, considered for complex variables, the following discussion is restricted to

real input data since all experimental time functions are real. Consider a time

function x(t) sampled at interval ~t and starting at t

=

0. If N is the total number

of contiguous data points, the finite discrete Fourier transform of x(t) is given by

the following equation:6

where n

=

0, 1,

...

N-1, and Af

=

1/NAt is the spectral resolution.

In practice, the sampling frequency fs(= l/At) is governed by the highest

frequency of interest, fm. The sampling theorem requires that fs

=

2fm. Thus,

As a result, N has to be large for fine resolution analysis at high frequencies. If

the hardware limits N to 1 or 2 k, the usual zoom technique by frequency shift has to

be used. This paper offers a different solution.

Suppose the computer is limited to a P

(=

1 k for example) point transform and

the requirement for N is M

(=

10 for example) times P. For some machines these N

data points have to be stored either in a disk or a tape file first. The proposed

technique involves performing an ordinary 1 k transform ten times using data selected

from different parts of the 10 k samples. After the results from the ten smaller

transforms have been properly combined, the solution is equal to a 10 k transform.

It is also possible to compute, with the same resolution, only a selected portion of

the whole spectrum. Thus, this technique can be considered as a zoom transform also.

The mathematical background is given in the following paragraphs.

Let the N data points be divided into P blocks of M points each, such that

N

= MP.

Using the following index tran~fonnation,~

where r

=

0, 1,

...,

(P-1)

s

=

0, 1,

...,

(M-1).

Equation (1) can be rewritten as

Another index transformation,

with

a =

0,

1,

.

.,

(M-1)

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w i l l r e c a s t Eq. ( 4 ) i n t o

The f i r s t summation on t h e index

r

of Eq. ( 6 ) r e p r e s e n t s a DFT on t h e P sampled p o i n t s , x ( s ) , x(M+s), x(2M+s), e t c . , t o g i v e P complex s p e c t r a l components. The DFT can be c a r r i e d o u t by t h e u s u a l FFT a l g o r i t h m and t h i s o p e r a t i o n h a s t o be r e p e a t e d M t i m e s a s s changes from 0 t o M-1. These i n t e r m e d i a t e r e s u l t s a r e termed p a r t i a l s p e c t r a . That i s

where

B

= 0, 1,

...,

(P-1).

A s p o i n t e d o u t by ~ h r a n e , ~ t h e term exp(-j2rBs/N) r e p r e s e n t s a phase s h i f t c o r r e c t i o n t o t h e frequency components of e a c h of t h e M p a r t i a l s p e c t r a . T h i s i s

governed by t h e i n d e x s and is r e q u i r e d t o compensate f o r t h e time s h i f t between t h e M s e t s of d a t a used i n t h e transform. Thus, Eq. (6) can be r e w r i t t e n a s

where X ; ( B A ~ ) = xS(Bbf) e-j2rBs1N a r e t h e M compensated p a r t i a l s p e c t r a .

There i s no mention of t h e o t h e r term, exp(-j2ras/M), i n Thrane's p a p e r , b u t i t

may be thought of a s a w e i g h t i n g f u n c t i o n of t h e M compensated p a r t i a l s p e c t r a . For a given v a l u e of a , Eq. (8) g e n e r a t e s up t o P s p e c t r a l l i n e s . The number s e l e c t e d w i t h i n t h e range aPAf t o [ ( a + l ) ~ - l ] A f i s determined by t h e c h o i c e of t h e range f o r

B.

T h i s procedure p r o v i d e s a form of zoom t r a n s f o r m . By a l l o w i n g

a

and 8 t o t a k e on a l l v a l u e s from 0 t o M-1 and t o P-1, r e s p e c t i v e l y , t h e f u l l spectrum of N l i n e s c a n be g e n e r a t e d i f n e c e s s a r y .

To minimize s t o r a g e space and computing t i m e , y i p 4 chose t o re-order t h e summation procedure s o t h a t d a t a c o u l d be used i n c h r o n o l o g i c a l o r d e r . H e had t o t r e a t t h e phase s h i f t f a c t o r exp(-j2nBslN) a s u n i t y , however, and t o c o r r e c t t h e r e s u l t s a f t e r t h e zoom spectrum had been computed. A s t h e c o r r e c t i o n f a c t o r depends on t h e type of s i g n a l s being a n a l y s e d , h i s zoom scheme i s l e s s a t t r a c t i v e t h a n t h e method p r e s e n t e d here.

PROGRAMMING HINTS

The M p a r t i a l s p e c t r a a s d e f i n e d by Eq. ( 7 ) can be performed w i t h any a v a i l a b l e FFT program. It i s i m p o r t a n t t o n o t e , however, t h a t t h e r e a r e FFT programs f o r complex i n p u t d a t a and o t h e r programs f o r r e a l i n p u t d a t a o n l y . The two t y p e s w i l l

have d i f f e r e n t i n p u t and o u t p u t format.

For r e a l i n p u t d a t a t h e frequency spectrum o b t a i n e d i s a c o n j u g a t e even f u n c t i o n ; t h a t i s ,

where

*

d e n o t e s complex c o n j u g a t e . Some FFT programs i n t e n d e d f o r u s e

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l i n e s of t h e p a r t i a l s p e c t r a , t h e y must be r e c r e a t e d u s i n g Eq. ( 9 ) . I n g e n e r a l ,

P

i s set by t h e a v a i l a b l e s o f t w a r e o r hardware and i s much l a r g e r t h a n M. The second summation o v e r t h e i n d e x s can be performed i n t h e s t r a i g h t f o r w a r d manner.

It i s i m p o r t a n t t o r e a l i z e t h a t t h e

DFT is

j u s t a n approximation of t h e

continuous F o u r i e r transform, and t h a t t h e r e a r e problems a s s o c i a t e d w i t h i t s usage, f o r example, a l i a s i n g , leakage and p i c k e t - f e n c e e f f e c t . These problems have been d e a l t w i t h i n t h e l i t e r a t ~ r e . ~ I f i t i s n e c e s s a r y t o apply w i n d w i n g such a s Hanning t o t h e d a t a , i t should be a p p l i e d t o t h e o r i g i n a l N d a t a p o i n t s . For t h e

f u l l spectrum, only t h e f i r e t N/2 frequency l i n e 8 a r e independent.

EXAMPLES

To v e r i f y t h e proposed t e c h n i q u e , 512 d a t a p o i n t s are g e n e r a t e d u s i n g a n a n a l y t i c a l f u n c t i o n . F i r s t , a n o r d i n a r y t r a n s f o r m on t h e 512 d a t a p o i n t s was

performed t o g i v e 256 complex frequency r e s u l t s . The same 512 d a t a p o i n t s were t h e n d i v i d e d i n t o 128 b l o c k s of f o u r d a t a p o i n t s e a c h t o be u s e d i n t h e proposed t r a n s f o r m procedure. No s i g n i f i c a n t d i f f e r e n c e s were found between t h e two r e s u l t s .

To i l l u s t r a t e t h e zoom c a p a b i l i t y , a t e s t s i g n a l c o n s i s t i n g of two s i n e waves (198 Hz and 200 Hz) and a band-limited random n o i s e was used. The t e s t s i g n a l was sampled a t a r a t e of 1 kHz. I n i t i a l l y , an o r d i n a r y 512 p o i n t t r a n s f o r m was used. A s t h e s p e c t r a l r e s o l u t i o n f o r t h i s t r a n s f o r m was only 1.95 Hz, i t would n o t be c a p a b l e of r e s o l v i n g t h e two s i n e waves ( s e e Fig. 1 ) . I n c r e a s i n g t h e t o t a l number of d a t a p o i n t s t o 5120 and u s i n g t h e proposed t r a n s f o r m t e c h n i q u e w i t h P = 512 and M = 10, t h e s p e c t r a l r e s o l u t i o n became 0.195 Hz and i t was p o s s i b l e t o r e s o l v e t h e two s i n e waves, a s i n d i c a t e d i n Fig. 2. The l i s t i n g of a sample program i s g i v e n i n t h e Appendix.

F R E Q U E N C Y , H z F R E Q U E N C Y , Hz

F i g u r e 1. Overlapped spectrum o b t a i n e d F i g u r e 2. F i n e r e s o l u t i o n spectrum by t h e o r d i n a r y FFT method u s i n g 512 o b t a i n e d by t h e proposed t r a n s f o r m p o i n t s . S p e c t r a l r e s o l u t i o n = 1.95 Hz method u s i n g 5120 p o i n t s . S p e c t r a l

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CONCLUSION

A s i m p l e t e c h n i q u e f o r zoom t r a n s f o r m and long-time s i g n a l a n a l y s i s h a s b e e n reviewed a n d examples have been g i v e n t o i l l u s t r a t e i t s a p p l i c a t i o n s . It i s hoped t h a t o t h e r a c o u s t i c i a n s and v i b r a t i o n e n g i n e e r s w i l l f i n d i t u s e f u l .

REFERENCES

1. Programs f o r D i g i t a l S i g n a l P r o c e s s i n g . E d i t e d by t h e D i g i t a l S i g n a l P r o c e s s i n g Committee, IEEE P r e s s , New York, 1979.

2. R.K. Otnes and L. Enochson, D i g i t a l Time S e r i e s A n a l y s i s , John Wiley and Sons, New York, 1972.

3. N. Aoshima, New Method of Measuring R e v e r b e r a t i o n Time by F o u r i e r Transforms, J e

Acoust. Soc. Am. 67, 1816-1817, 1980.

4. P.C.Y. Yip, Some A s p e c t s of t h e Zoom Transform, IEEE Trans. on Computers, C-25, 287-296, 1976.

5. N. Thrane, ZoonrFFT, B r u e l and K j a e r Tech. Review No. 2, 1980.

6. E.O. Brigham, The F a s t F o u r i e r Transform, P r e n t i c e H a l l , I n c . , New J e r s e y , 1974. 7. G.D. Bergland, A Guided Tour on t h e F a s t F o u r i e r Transform, IEEE

Spectrum, 6 , 41-52, 1969.

APPENDIX

C SAMPLE PROGRAM FOR A 10 k POINT TRANSFORM USING A 1 k FFT

C SUBROUTINE. THE FFT SUBROUTINE CALLED FAST BY BERGLAND AND DOLm

C I S FOR REAL INPUT DATA ONLY AND I S LISTED I N THE IEEE BOOK ON

C PROGRAMS FOR DIGITAL SIGNAL PROCESSING. COMPLEX C , F, CEXAF, CEXBA

DIMENSION P(10240), PI(1026), C(1024), F(1024) COMMON/CONS/PII, P7, P7TW0, C22, S22, P I 2

C SPLIT THE 10240 POINTS INTO 1024 BLOCKS OF 1 0 DATA POINTS EACH. NP-1024; OF BLOCKS GOVERNED BY THE FFT SIZE.

NM=10; /I _{OF POINTS PER BLOCK.}

NMH=NM/ 2 NPl=NP+l NPHl=NP/2+1 NPH2=NP/2+2

C FOR 'ZOOM' TRANSFORM, ENTER PARTICULAR NAF VALUE FOR ALPHA. C EXAMPLE GIVEN I S FOR THE COMPLETE SPECTRUM.

C NOTE, A 10 k TRANSFORM GIVES 5 k INDEPENDENT FREQUENCI COMPONENTS C ONLY. DO 20 NAF=l,NMH NAFO=NAF-1 DO 25 NBE=l, NP F(NBE)=O.O 25 CONTINUE