FFT algorithms There are two classes of FFT algorithms: (1)Algorithms that Nis equal to an integer power of 2,including radix-2 algorithm, radix-4 algorithm,real factor algorithm and Split Algorithm; (2)Algorithms that N is not equal to an integer power of 2,including Prime factor algorithm and Winagrad algortihm
FFT algorithms There are two classes of FFT algorithms : （ 1 ）Algorithms that N is equal to an integer power of 2, including radix-2 algorithm, radix-4 algorithm, real factor algorithm and Split Algorithm; （ 2 ） Algorithms that N is not equal to an integer power of 2, including Prime factor algorithm and Winagrad algortihm
Periodogram A classical power spectral estimation method. PSD (Power Spectral Density): Assume that y(n)is a bounded digital sequence with length N
Periodogram • A classical power spectral estimation method. • PSD (Power Spectral Density) : Assume that y ( n ) is a bounded digital sequence with length N
Computing Periodogram(PSD) ·Direct method $()=2小ep(-2x例 s(oj=2o小em(-am S(Z)=★Y(Z)Y(Z) Where y(Z)=∑y(n)Z n=0
Computing Periodogram (PSD) • Direct method ( ) ( ) ( ) 2 1 0 1 exp 2 N per n S f y n j fn N π Λ − = = − ∑ ( ) ( ) ( ) 2 1 0 1 exp N per n S y n j n N ω ω Λ − = = − ∑ ( ) ( ) ( ) ( ) ( ) 1 1 1 0 per N N n n S Z Y Z Y Z Where Y Z y n Z Λ − − − = = = ∑
Computing Periodogram(PSD) ·Indirect method S()=∑t(k)exp(-2π) k=-(N-1) 3m(o)-(k)exp(-jok) k=-（N-1) s(2=2z k=-（N-1)
Computing Periodogram(PSD) • Indirect method ( ) ( ) ( ) ( ) 1 1 exp 2 N per k N S f r k j fk π Λ Λ − =− − = − ∑ ( ) ( ) ( ) ( ) 1 1 exp N per k N S r k j k ω ω Λ Λ − =− − = − ∑ ( ) ( ) ( ) ( ) 1 1 N k per yy k N S Z r k Z − Λ Λ =− − = ∑
Periodogram Properties Properties: (1)An biased estimator; (2)Not a consistent estimator; (3)Frequency resolution &N
Periodogram Properties Properties: (1) An biased estimator; (2) Not a consistent estimator; (3) Frequency resolution ∝ N