Appendices

Formulas of Discrete Fourier Series

In general application, p(t) is given as pi ¼ p(ti) ¼ p(i t) for i ¼ 0, 1, . . ., N 1, and N ¼ T/ t. Then, the integration for Pk can be changed to summation as follows:

is called the discrete Fourier transform (DFT) and can be computed by using the fft function in MATLAB as follows:

½dk& ¼ fftðpi, NÞ

and therefore:

Pk Ak þ jBk ¼ dk=N ¼ fftðpi, NÞ=N

Example 1: The Cosine Function as Sound Pressure

A pure tone acoustic pressure p(t) of frequency f ¼ 500 [Hz] and amplitude P ¼ 3 [Pa] is transformed into the frequency domain for calculating the A-weighted sound pressure level [dBA]:

pðtÞ ¼ P cos ð2πftÞ

Given:

(a) The number of discretized time domain data is N ¼ 16.

(b) The time increment of time domain data is t ¼ 0.000125 [s].

(c) The time domain pressure (n data) which is the input data for MATLAB fft:

362 Appendices

Frequency [Hz]

Step 4 – Calculate the Nyquist frequency fN:

Step 5 – Replace frequencies after the Nyquist frequency with:

f k ¼ f k 2 f N ¼ f k 8000

Real part coefficients [Pa]: Ak ¼ Re (Pk)

Remarks

Make sure that these are complex conjugate pairs.

Make sure that Ak ¼ 1 is the coefficient of the cosine function:

1

pðtÞ ¼ Ao þ 2 X ½Ak cos ðωktÞ Bk sin ðωktÞ& ¼ 3 cos ð2π ∙ 500 ∙ tÞ½Pa&

k¼1

Step 6 – Calculate the RMS pressure square:

k¼1

Step 7 – Calculate the A-weighted sound pressure level:

q

0 1

2 1:52

¼20 log 10@ 20E 6 A½dB& 3:2 ½dB&

¼97:3 ½dB&

FFT Using MATLAB fft Function

clear all f=500; % Hz P=3; % Pa T=1/f; % s

N=16; % number of data

%

dt=T/N; % time increment df=1/T; % frequency increment DataTime(:,1)=(0:1:N-1)*dt;

DataTime(:,2)=cos(2*pi*f*DataTime(:,1))*P;

%Discrete FFT based on the Fourier series formula DataFreq(:,1)=(0:1:N-1)*df; DataFreq(:,2)=fft(DataTime(:,2),N)/N;

%get the real and imaginary parts DataFreq(:,3)=real(DataFreq(:,2)); % real part DataFreq(:,4)=imag(DataFreq(:,2)); % imaginary part

%plot results

subplot(3,1,1); plot(DataTime(:,1),DataTime(:,2),'-ok','LineWidth',2,'MarkerSize',4); title('Data Time'); xlabel('time [s]'); YLIM([-4 4]);

subplot(3,1,2); plot(DataFreq(:,1),DataFreq(:,3),'-ok','LineWidth',2,'MarkerSize',4); title('Data Freq (Real)'); xlabel('frequency [Hz]'); YLIM([-4 4]) subplot(3,1,3); plot(DataFreq(:,1),DataFreq(:,4),'-ok','LineWidth',2,'MarkerSize',4); title('Data Freq (Imag)'); xlabel('frequency [Hz]'); YLIM([-4 4]) % save the figure

saveas(gcf,'fig','emf')

Example 2: The Sine Function as Sound Pressure

A pure tone acoustic pressure p(t) of frequency f ¼ 1000 [Hz] and amplitude P ¼ 7 [Pa] is transformed into the frequency domain for calculating the A-weighted sound pressure level [dBA]:

pðtÞ ¼ P sin ð2πftÞ

366 Appendices

Frequency [Hz]

Step 4 – Calculate the Nyquist frequency, fN:

Step 5 – Replace frequencies after the Nyquist frequency with:

f k ¼ f k 2 f N ¼ f k 8000

Real part coefficients [Pa]: Ak ¼ Re (Pk)

Data Time

Data Freq (Real)

4

2 0 -2

-4 0 1000 2000 3000 4000 5000 6000 7000 8000

frequency [Hz]

Data Freq (Imag)

4

2 0 -2

-4