Workshop W64 Fourier and Wavelet Analysis · 2012-08-22 · 1 Advanced Lab Fourier Analysis and...

Advanced Lab Fourier Advanced Lab Fourier Analysis and Wavelet Analysis and Wavelet Analysis Using MatlabAnalysis Using Matlab

Workshop W64Workshop W64"Fourier and Wavelet Analysis""Fourier and Wavelet Analysis"

Joseph J. Trout, Ph.D.Richard College of NJAssistant Professor of PhysicsNAMS – Physics ProgramJoseph.trout@stockton.eduAAPT 2012 Summer MeetingPhiladelphia, PennsylvaniaDC06Tue 07/31, 10:20AM - 10:30AM

ABSTRACT:In this laboratory experiment, techniques in data analysis using Fourier and wavelet analysis are introduced. First the students review Fourier series. Sound recordings of tuning forks are then made using Matlab and the students use a simple code to find the amplitudes of the frequencies present in the recording. The students then repeat the analysis using Matlab's Fast Fourier Transform utility and compare the results. Wavelets are then introduced and the recording are then analyzed using wavelet techniques to understand the similarities and differences between Fourier analysis and wavelet analysis. Finally, topics of "noise" and "filtering" are introduced and voice recordings or recordings or music are made and analyzed.

The Fourier TransformThe Fourier TransformIf f is a continuous differentialable function with ∫

−∞

∣f t ∣dt∞ , then:

f x = 12

∫−∞

∞f ei x d

where f (the Fourier Transform of f ) is given by:

f x = 12

∫−∞

f t e−i td t

The Discrete Fourier TransformThe Discrete Fourier TransformLet y={y j }j=−∞

∞ ∈S n ( where S n is the set of n -periodic sequences ofcomplex numbers ). The discrete Fourier transform of y is the sequence

F n {y }k=yk , where:

yk=∑j=0

y jwij with w=e

In detail: the DFT is the sequence:

yk=∑j=0

y j e−2 i k j

Advanced Lab Fourier Advanced Lab Fourier Analysis and Wavelet Analysis and Wavelet Analysis Using MatlabAnalysis Using Matlab Other Wavelets - DaubechiesOther Wavelets - Daubechies

Advanced Lab Fourier Advanced Lab Fourier Analysis and Wavelet Analysis and Wavelet Analysis Using MatlabAnalysis Using Matlab Comparison of DWT to FFTComparison of DWT to FFTf t =2 sin 2 f 1 t 3sin 2 f 2 t sin 2 f 3 t

f 1=5Hzf 2=20Hzf 3=80Hz

Advanced Lab Fourier Advanced Lab Fourier Analysis and Wavelet Analysis and Wavelet Analysis Using MatlabAnalysis Using Matlab Comparison of DWT to FFTComparison of DWT to FFT

f t =2sin 2 f 1 t 3 sin 2 f 2 t sin 2 f 3 t f 1=5Hzf 2=20Hzf 3=80Hz

f t =2sin 2 f 1 t 3 sin 2 f 2 t f 1=5Hzf 2=20Hzf 3=80Hzf t =2sin 2 f 1 t sin 2 f 3 t

f t =2sin 2 f 1 t 3 sin 2 f 2 t f 1=5Hzf 2=20Hzf 3=80Hz

f t =2sin 2 f 1 t sin 2 f 3 t

f t =2sin 2 f 1 t 3 sin 2 f 2 t f 1=5Hzf 2=20Hzf 3=80Hzf t =2sin 2 f 1 t sin 2 f 3 t

Advanced Lab Fourier Advanced Lab Fourier Analysis and Wavelet Analysis and Wavelet Analysis Using MatlabAnalysis Using Matlab Fast Fourier Transform ( FFT)Fast Fourier Transform ( FFT)

Original Signal discretized with 28=256 points. 0≤t≤2

Advanced Lab Fourier Advanced Lab Fourier Analysis and Wavelet Analysis and Wavelet Analysis Using MatlabAnalysis Using Matlab Fast Fourier Transform ( FFT)Fast Fourier Transform ( FFT)

Original Signal discretized with 28=256 points. 0≤t≤2Sun Performance Library FFT used to generate the Discrete Fourier Coefficients yk , k=0,. .. ,255 .Noise has a frequency that is larger than 5 cycles per 2 interval.

Advanced Lab Fourier Advanced Lab Fourier Analysis and Wavelet Analysis and Wavelet Analysis Using MatlabAnalysis Using Matlab Fast Fourier Transform ( FFT) Fast Fourier Transform ( FFT)

Use to Filter out Noise.Use to Filter out Noise.

Function: y t =e−cos t 2

sin 2t2 cos 4t0.4 sin t sin 50 t

Original Signal discretized with 28=256 points. 0≤t≤2Sun Performance Library FFT used to generate the Discrete Fourier Coefficients yk , k=0,. .. ,255 .Noise has a frequency that is larger than 5 cycles per 2 interval.Keep only y k for 0≤k≤5 and set yk=0.0 for 6≤k≤128 .By Theorem, yk=0 for 128≤k≤250 . Applying FFT to filter yk

Filtered by keeping first five coefficients and completinginverse FFT.

Error=∣∣y− yc∣∣l2∣∣y∣∣l2

Advanced Lab Fourier Advanced Lab Fourier Analysis and Wavelet Analysis and Wavelet Analysis Using MatlabAnalysis Using Matlab Using Wavelets to Filter DataUsing Wavelets to Filter Data

Original Signal discretized with 28=256 points. 0≤t≤2Numerical Recipes Daub 8 used to generate wavelet coefficients yk , k=0,. .. ,255 .Keep only wavelet coefficients with a magnitude > 1 .

Pre-Lab: Fourier Series

Computation of Fourier Series Computation of Fourier Series on Interval -on Interval -aa <= x <= + <= x <= +aa

f x =ao∑k=1

∞ [ak coska tbk sin ka t]a0=

12a∫−a

f t dt

ak=1a∫−a

f t cosk ta dtbk=

1a∫−a

f t sink ta dt

Fourier Coefficients of the Function f(x):

Advanced Lab Fourier Advanced Lab Fourier Analysis and Wavelet Analysis and Wavelet Analysis Using MatlabAnalysis Using Matlab Computation of Fourier Series Computation of Fourier Series

on Interval -on Interval -aa <= x <= + <= x <= +aa

f x =ao∑k=1

∞ [ak coska tbk sin ka t]a0=

1N Δ t∑i=0

f iΔ t

N Δ t∑i=0

f i cos(2π F k tN Δ t )dt

bk=1a∫−a

f ( t )sin( k π ta )dt

Approximations Fourier Coefficients of the Function f(x):

f (t )= ∑n=odd

sin (2π n t )

Square Wave

>> t=(0:199)*0.01;

>> y1=(1/1)*sin(2*pi*1*t);>> y3=(1/3)*sin(2*pi*3*t);>> y5=(1/5)*sin(2*pi*5*t);>> y7=(1/7)*sin(2*pi*7*t);>> y9=(1/9)*sin(2*pi*9*t);>> y11=(1/11)*sin(2*pi*11*t);>> y13=(1/13)*sin(2*pi*13*t);>> yt=y1+y3+y5+y7+y9+y11+y13;

>> plot(t,y1,t,y3,t,y5,t,y7,t,y9,t,y11,t,y13);

>> hold all;

>> plot(t,yt,'LineWidth',4);

>> ylabel('y(t)');>> xlabel('t');

0 0 . 2 0 . 4 0 . 6 0 . 8 1 1 . 2 1 . 4 1 . 6 1 . 8 2- 1

- 0 . 8

- 0 . 6

- 0 . 4

- 0 . 2

The first part of the lab uses tuning forks to record the sound. This provides the students a chance to learn about recording sound and gives a physical meaning to the analysis.

These recordings can be made in Matlab using:

audiorecorder

>> recObj1=audiorecorder

recObj1 =

audiorecorder handle

Properties: SampleRate: 8000 BitsPerSample: 8 NumberOfChannels: 1 DeviceID: -1 CurrentSample: 1 TotalSamples: 0 Running: 'off' StartFcn: [] StopFcn: [] TimerFcn: [] TimerPeriod: 0.0500 Tag: '' UserData: [] Type: 'audiorecorder'

You can record using: recordblocking

>> disp('Start recording for 5 seconds')Start recording for 5 seconds>> recordblocking(recObj1,5);>> disp('Recording has finished');Recording has finished>> disp('Play Recording');Play Recording>> play(recObj1);

You can play recording using: play

>> disp('Start recording for 5 seconds')Start recording for 5 seconds>> recordblocking(recObj1,5);>> disp('Recording has finished');Recording has finished>> disp('Play Recording');Play Recording>> play(recObj1);

Make an data array of amplitudes from sound recording: getaudiodata

>> tempRecord = getaudiodata(recObj1);

Caution: Not using semicolon (;) will print all the values.

You can “play” a numerical data array using : sound

>> sound(tempRecord);

You can plot the data array using : plot

>>plot(tempRecord);

0 0 . 5 1 1 . 5 2 2 . 5 3 3 . 5 4

x 1 04

- 0 . 5

- 0 . 4

- 0 . 3

- 0 . 2

- 0 . 1

You can print the data using: fprintf

fprintf('%f \n', tempRecord(1:300));0.000000 0.000000 0.000000 ….

This statement will print first 300 data elements. Note array index starts at 1 (one). Caution: A recording can have zero amplitude for the first several elements.

Should use powers of two to do the analysis. For this part of lab, using tuning forks, it is easier to make long recordings and crop to a power of two data points. Stay away from beginning and end or recording.

>> Record1=tempRecord(10001:12048);

Also, we know the sampling frequency so we can also set up the time variables at the same time.

>> recObj1

recObj1 =

Properties: SampleRate: 8000 BitsPerSample: 8 NumberOfChannels: 1 DeviceID: -1 CurrentSample: 1 TotalSamples: 40000 ….

>> Record1=tempRecord(1:2048);>> Record1=tempRecord(5001:7048);>> Fs=8000;>> Ts=1/Fs;>> N=numel(Record1)

>> t=(0:N-1)*Ts;>>

And you can plot the “cropped” data.

>> plot(t(1:100),Record1(1:100));>> title('512 Hz Tuning Fork');>> xlabel('Time(s)');>> ylabel('Relative Amplitude');

0 0 . 0 0 2 0 . 0 0 4 0 . 0 0 6 0 . 0 0 8 0 . 0 1 0 . 0 1 2 0 . 0 1 4- 0 . 0 8

- 0 . 0 6

- 0 . 0 4

- 0 . 0 2

0 . 0 2

0 . 0 4

0 . 0 6

0 . 0 85 1 2 H z T u n i n g F o r k

T i m e ( s )

0 0 . 0 0 2 0 . 0 0 4 0 . 0 0 6 0 . 0 0 8 0 . 0 1 0 . 0 1 2 0 . 0 1 4- 0 . 0 8

- 0 . 0 6

- 0 . 0 4

- 0 . 0 2

0 . 0 2

0 . 0 4

0 . 0 6

0 . 0 85 1 2 H z T u n i n g F o r k

T i m e ( s )

itude 6 cycles

f =6cycles0.012 s

≈500Hz

f=(Fs)*linspace(0,1,nfft/2+1);for i = 1:(nfft/2+1)a(i)=0;b(i)=0;for j=1:Na(i)=a(i)+Record1(j)*cos(2*pi*f(i)*t(j));b(i)=b(i)+Record1(j)*sin(2*pi*f(i)*t(i));enda(i)=a(i)/N;b(i)=b(i)/N;end

>> for i= 1:(nfft/2+1)A(i)=sqrt(a(i)*a(i)+b(i)*b(i));pow(i)=(2*A(i))*(2*A(i));end

plot(f,pow);

0 1 0 0 0 2 0 0 0 3 0 0 0 4 0 0 0 5 0 0 0 6 0 0 0 7 0 0 0 8 0 0 00

4x 1 0

bar(f(125:150),pow(125:150));title('Power Spectrum');xlabel('Frequency(Hz)');ylabel('Amplitude Squared');

9 5 0 1 0 0 0 1 0 5 0 1 1 0 0 1 1 5 0 1 2 0 00

4x 1 0

- 3 P o w e r S p e c t r u m

F r e q u e n c y ( H z )

l>> for i = 125:140fprintf('Freq= %f Hz Power= %f \n', f(i),pow(i));endFreq= 968.750000 Hz Power= 0.000004 Freq= 976.562500 Hz Power= 0.000007 Freq= 984.375000 Hz Power= 0.000009 Freq= 992.187500 Hz Power= 0.000012 Freq= 1000.000000 Hz Power= 0.000022 Freq= 1007.812500 Hz Power= 0.000058 Freq= 1015.625000 Hz Power= 0.000300 Freq= 1023.437500 Hz Power= 0.003640 Freq= 1031.250000 Hz Power= 0.000121 Freq= 1039.062500 Hz Power= 0.000034 Freq= 1046.875000 Hz Power= 0.000017 Freq= 1054.687500 Hz Power= 0.000011 Freq= 1062.500000 Hz Power= 0.000006 Freq= 1070.312500 Hz Power= 0.000005 Freq= 1078.125000 Hz Power= 0.000004 Freq= 1085.937500 Hz Power= 0.000003

>>nfft=2^(nextpow2(length(Record1)));fftAtemp=fft(Record1,nfft);NumUniqPts=ceil((nfft+1)/2)

NumUniqPts =

fftA=fftAtemp(1:NumUniqPts);fftA=abs(fftA)/length(Record1);fftA=fftA.^2;fftA(2:end-1)=fftA(2:end-1)*2;f=Fs/2*linspace(0,1,nfft/2+1);

bar(f(125:150),fftA(125:150));title('Power Spectrum using Matlab FFT');xlabel('Frequency(Hz)');ylabel('Amplitude Squared');

9 5 0 1 0 0 0 1 0 5 0 1 1 0 0 1 1 5 0 1 2 0 00

2 . 5x 1 0

- 3 P o w e r S p e c t r u m u s i n g M a t l a b F F T

>> for i = 125:140fprintf('Freq= %f Hz Power= %f \n', f(i),fftA(i));endFreq= 968.750000 Hz Power= 0.000001 Freq= 976.562500 Hz Power= 0.000002 Freq= 984.375000 Hz Power= 0.000003 Freq= 992.187500 Hz Power= 0.000004 Freq= 1000.000000 Hz Power= 0.000006 Freq= 1007.812500 Hz Power= 0.000017 Freq= 1015.625000 Hz Power= 0.000077 Freq= 1023.437500 Hz Power= 0.002174 Freq= 1031.250000 Hz Power= 0.000040 Freq= 1039.062500 Hz Power= 0.000010 Freq= 1046.875000 Hz Power= 0.000005 Freq= 1054.687500 Hz Power= 0.000003 Freq= 1062.500000 Hz Power= 0.000002 Freq= 1070.312500 Hz Power= 0.000001 Freq= 1078.125000 Hz Power= 0.000001 Freq= 1085.937500 Hz Power= 0.000001

f(25:30)

187.5000 195.3125 203.1250 210.9375 218.7500 226.5625

Record2=0.2*sin(2*pi*195.3125*t);plot(Record2(1:100));fftAtemp=fft(Record2,nfft);fftA=fftAtemp(1:NumUniqPts);fftA=abs(fftA)/length(Record2);fftA(2:end-1)=fftA(2:end-1)*2;fftA=fftA.^2;bar(f,fftA);bar(f(20:50),fftA(20:50));title('F=195.3125');xlabel('Frequency(Hz)');ylabel('Relataive Amplitude');

1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 00

0 . 0 0 5

0 . 0 1

0 . 0 1 5

0 . 0 2

0 . 0 2 5

0 . 0 3

0 . 0 3 5

0 . 0 4

0 . 0 4 5F = 1 9 5 . 3 1 2 5

f(25:30)

187.5000 195.3125 203.1250 210.9375 218.7500 226.5625

Record2=0.2*sin(2*pi*197*t);plot(Record2(1:100));fftAtemp=fft(Record2,nfft);fftA=fftAtemp(1:NumUniqPts);fftA=abs(fftA)/length(Record2);fftA(2:end-1)=fftA(2:end-1)*2;fftA=fftA.^2;bar(f,fftA);bar(f(20:50),fftA(20:50));title('F=197');xlabel('Frequency(Hz)');ylabel('Relataive Amplitude');

1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 3 5 0 4 0 00

0 . 0 0 5

0 . 0 1

0 . 0 1 5

0 . 0 2

0 . 0 2 5

0 . 0 3

0 . 0 3 5F = 1 9 7

Wavelet Analysis

Trial>> N=20148;Trial>> Fs=8000;Trial>> Ts=1/Fs;Trial>> t=(0:N-1)*Ts;Trial>> y=3*sin(2*pi*247*t);Trial>> signal=struct('val',y,'period',Ts);Trial>> cwtS1(signal,'wavelet','morl');Error using subsindexFunction 'subsindex' is not defined for values of class'struct'. Trial>> cwtS1=cwtft(signal,'wavelet','morl');

Wavelet Prelab:

>> cwtS1

cwtS1 =

cfs: [30x20148 double] scales: [1x30 double] omega: [1x20148 double] meanSIG: 4.0313e-05 dt: 1.2500e-04 wav: 'morl'

Wavelet Prelab:

for i=1:numel(cwtS1.scales)figure;plot(t,real(cwtS1.cfs(i,:)));end

Wavelet Prelab:

Trial>> scales=cwtS1.scales;Trial>> wo=6;Trial>> MorFFac=4*pi/(wo+sqrt(2+wo*wo));Trial>> freq=1./(scales.*MorFFac);Trial>> contour(t*1000,freq,real(cwtS1.cfs));Trial>> Trial>> contour(t*1000,freq,real(cwtS1.cfs));Trial>> axis([10 40 150 400]);Trial>> title('Contour Plot');Trial>> ylabel('Pseudo-Frequency (Hz)');Trial>> xlabel('Time(ms)');

N=2048;Fs=8000;Ts=1/Fs;t=(0:N-1)*Ts;y=3*sin(2*pi*247*t);signal=struct('val',y,'period',Ts);cwtS1=cwtft(signal,'wavelet','morl');cwtS1scales=cwtS1.scales;wo=6;MorFFac=4*pi/(wo+sqrt(2+wo*wo));freq=1./(scales*MorFFac);contour(t*1000,freq,real(cwtS1.cfs));axis([10 40 150 400]);title('Contour Plot');ylabel('Pseudo-Frequency (Hz)');xlabel('Time(ms)');

Trial>> freq

freq =

1.0e+03 *

Columns 1 through 8

3.8721 2.7618 1.9699 1.4050 1.0022 0.7148 0.5098 0.3636

Columns 9 through 16

0.2594 0.1850 0.1320 0.0941 0.0671 0.0479 0.0342 0.0244

Columns 17 through 23

0.0174 0.0124 0.0088 0.0063 0.0045 0.0032 0.0023

Trial>> plot(t,real(cwtS1.cfs(9,:)));Trial>> axis([0 0.05 -18 18]);Trial>>

Trial>> recObj3=audiorecorder

recObj3 =

Properties: SampleRate: 8000 BitsPerSample: 8 NumberOfChannels: 1 DeviceID: -1 CurrentSample: 1 TotalSamples: 0 Running: 'off' StartFcn: [] StopFcn: [] TimerFcn: [] TimerPeriod: 0.0500 Tag: '' UserData: [] Type: 'audiorecorder'

Methods, Events, Superclasses

Data Collection

Trial>> recObj3=audiorecorder

Trial>> Fs=8000;Trial>> Ts=1/Fs;Trial>> recordblocking(recObj3,5)Trial>> tempRecord = getaudiodata(recObj3);Trial>> Record3=tempRecord(1001:3048);

Trial>> N=numel(Record3)

Trial>> t=(0:N-1)*Ts;Trial>> signal3=struct('val',transpose(Record3),'period',Ts);Trial>> cwtS3=cwtft(signal3,'wavelet','morl');Trial>> cwtS3

cwtS3 =

cfs: [23x2048 double] scales: [1x23 double] omega: [1x2048 double] meanSIG: 0.0130 dt: 1.2500e-04 wav: 'morl'

Trial>> scales=cwtS3.scales;Trial>> wo=6;Trial>> MorFFac=4*pi/(wo+sqrt(2+wo*wo));Trial>> freq=1./(scales.*MorFFac);

Trial>> contour(t*1000,freq,real(cwtS3.cfs));Trial>> axis([150 200 0 800])

1 5 0 1 5 5 1 6 0 1 6 5 1 7 0 1 7 5 1 8 0 1 8 5 1 9 0 1 9 5 2 0 00

Trial>> nfft=2^(nextpow2(length(Record3)));Trial>> X=fft(Record3,nfft);Trial>> NumUniqPts=ceil((nfft+1)/2)

NumUniqPts =

Trial>> f=Fs*linspace(0,1,NumUniqPts);Trial>> power=4*(X.*conj(X))/(nfft*nfft);Trial>> power(1)=4*(X(1).*conj(X(1)))/(nfft*nfft);

Trial>> nfft=2^(nextpow2(length(Record3)));Trial>> X=fft(Record3,nfft);Trial>> NumUniqPts=ceil((nfft+1)/2)

NumUniqPts =

Trial>> f=Fs*linspace(0,1,NumUniqPts);Trial>> power=4*(X.*conj(X))/(nfft*nfft);Trial>> power(1)=4*(X(1).*conj(X(1)))/(nfft*nfft);Trial>> plot(f(1:NumUniqPts),power(1:NumUniqPts));Trial>> plot(f(1:300),power(1:300));Trial>> title('Power Spectrum');Trial>> xlabel('Frequency (Hz)');Trial>> ylabel('Relative Amplitudes');

0 5 0 0 1 0 0 0 1 5 0 0 2 0 0 0 2 5 0 00

3x 1 0

- 3 P o w e r S p e c t r u m

Questions?

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