Numerical Analysis – Numerical Analysis – Digital Signal ProcessingDigital Signal Processing

Hanyang University

Jong-Il Park

            

Division of Electrical and Computer Engineering, Hanyang University

Digital Signal Processing Discrete Fourier Transform

Fast Fourier Transform(FFT)

Multi-dimensional FFT

Convolution

            

Division of Electrical and Computer Engineering, Hanyang University

Sampling and aliasing

            

Division of Electrical and Computer Engineering, Hanyang University

Discrete Fourier Transform Fourier Transform

Discrete Fourier Transform

DFT:

IDFT:

            

Division of Electrical and Computer Engineering, Hanyang University

Fast Fourier Transform(FFT)

[Danielson&Lanczos][Cooley&Tukey]

            

Division of Electrical and Computer Engineering, Hanyang University

Decimation-in-time FFTDecimation-in-time FFT

Cooley-Tukey Algorithm

            

Division of Electrical and Computer Engineering, Hanyang University

Sande-Tukey AlgorithmSande-Tukey Algorithm

            

Division of Electrical and Computer Engineering, Hanyang University

Decimation-in-frequency FFT(I)Sande-Tukey Algorithm

            

Division of Electrical and Computer Engineering, Hanyang University

Decimation-in-frequency FFT (II)

            

Division of Electrical and Computer Engineering, Hanyang University

Why FFT?Why FFT?

Further reading: http://en.wikipedia.org/wiki/Cooley-Tukey_FFT_algorithm

            

Division of Electrical and Computer Engineering, Hanyang University

Computation of FFT(I)

input and output of four1() in NR in C

            

Division of Electrical and Computer Engineering, Hanyang University

Computation of FFT(II) Eg. FFT

            

Division of Electrical and Computer Engineering, Hanyang University

2D FFT(I)

            

Division of Electrical and Computer Engineering, Hanyang University

2D FFT(II)

* Generalization to L-dimension

            

Division of Electrical and Computer Engineering, Hanyang University

2D FFT(III) Eg. 2D FFT

            

Division of Electrical and Computer Engineering, Hanyang University

Convolution(I) Def.

            

Division of Electrical and Computer Engineering, Hanyang University

Convolution(II) Convolution theorem

o direct convolution complex computation

o FFT and multiplication less computation

            

Division of Electrical and Computer Engineering, Hanyang University

Convolution(III) Convolution of discrete sampled function

            

Division of Electrical and Computer Engineering, Hanyang University

Convolution(IV) Trouble in using DFT of finite duration

End effects Treated by zero padding

End effect

            

Division of Electrical and Computer Engineering, Hanyang University

Convolution(V) Zero padding

            

Division of Electrical and Computer Engineering, Hanyang University

Convolution(VI) Convolving very large data sets

<Overlap-add method>