Discrete Fourier Transforms

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Presentation transcript:

Discrete Fourier Transforms Domains: Time, t Frequency, f Contineous signal in t and f Sifting via time step Sampled data in t Finite record length Sampled data in t (finite) Sampled data in f Sampled data, periodic in t and f Domains: Time, t Frequency, f

Frequency Convolution Theorem Fourier Transform Pairs contineous signal sampling delta-functions sampled data sampled FT adapted from Brigham (1974)

Aliasing contineous signal sampling delta-functions sampled data Fourier Transform Pairs Fourier Transform Pairs sampled FT adapted from Brigham (1974)

Optimal Sampling at Nyquist Frequency contineous signal sampling delta-functions sampled data Fourier Transform Pairs Fourier Transform Pairs sampled FT adapted from Brigham (1974)

(band-limited signal) Sampling Theorem resolved frequencies (band-limited signal) Fourier Transform Pairs Fourier Transform Pairs Sampled signal (time step) adapted from Brigham (1974)

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TD ~ 4-5 days TD ~ 1 days Degrees of freedom: T/TD TD decorrelation time T record length KS02 red (Canada) KS10 blue KS12 green KS14 black (Greenland)

TD ~ 4-5 days TD ~ 1 days Degrees of freedom: T/TD TD decorrelation time T record length KS02 red (Canada) KS10 blue KS12 green KS14 black (Greenland)

Monthly North-Atlantic Oscillation Index (black) Low-pass Filtered NAO Convolution D(Z)=SIN(Z)/Z C PI = 4.*ATAN(1.) PI2 = 2.*PI IWW = RIWW/DT if (int(iww/2)*2.eq.iww) iww = iww+1 IWW2=(IWW-1)/2 T=FLOAT(IWW-1) OMEGA=PI2/TCO*DT H0=OMEGA/PI CON=PI2/FLOAT(IWW) C COMPUTE WINDOW WEIGHTS SUM=H0 DO 30 I=1,IWW2 H(I)=H0*D(FLOAT(I)*OMEGA)*D(FLOAT(I)*CON) SUM=SUM+2.0*H(I) 30 CONTINUE C C NORMALIZE EACH WEIGHT BY THE SUM OF ALL WEIGHTS H0=H0/SUM DO 35 I=1,IWW2 H(I)=H(I)/SUM 35 CONTINUE DO 55 I=IWW2+1,N-IWW2 K=I SUM=0.0 TEMP=H0*VAL(I) DO 50 J=1, IWW2 TEMP=TEMP+H(J)*(VAL(I+J)+VAL(I-J)) 50 CONTINUE LLPVAL(I)=TEMP TEMP=0 55 CONTINUE DO 56 I=1,N VAL(I) = LLPVAL(I) 56 CONTINUE RETURN END