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# 11_FFT - Session 13 1 Digital Signal Processing Session...

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1 Digital Signal Processing Session – 13

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2 Digital Signal Processing Session outline This session deals with the various transformation techniques of DSP. The Fast Fourier Transform in the digital domain Continuous and batch processing Real time FFT Windowing and FFT programming
3 Digital Signal Processing FAST FOURIER TRANSFORM

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4 Digital Signal Processing Application of Transformation in Digital Domain Filter Design – Calculating Impulse Response from Frequency Response, Calculating Frequency Response from Impulse Response FFT – The Fast Fourier Transform (FFT) is Simply an Algorithm for Efficiently Calculating the DFT Digital Spectral Analysis – Spectrum Analyzers, Speech Processing, Imaging, Pattern Recognition
5 Digital Signal Processing FFT Vs DFT The FFT is Simply an Algorithm for Efficiently Calculating the DFT Computational Efficiency of an N-Point FFT: DFT: N2 Complex Multiplications FFT: (N/2) log2(N) Complex Multiplications

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6 Digital Signal Processing 8-Point DFT In order to understand the development of the FFT,  consider first the 8-point DFT expansion,
7 Digital Signal Processing Twiddle Factor In order to simplify the diagram, note that the  quantity WN is defined as: This leads to the definition of the  twiddle  factors  as:. The twiddle factors are simply the sine and  cosine basis functions written in polar form.

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8 Digital Signal Processing Applying the Property of WN
9 Digital Signal Processing FFT From DFT In order to understand the basic concepts of the FFT and  its derivation, note that the DFT expansion can be greatly  simplified by taking advantage of the  symmetry and  periodicity of the twiddle factors  as shown in the  previous slide. If the equations are rearranged and factored, the result is  the Fast Fourier Transform (FFT) which requires only  (N/2) log2(N) complex multiplications. The computational  efficiency of the FFT versus the DFT becomes highly  significant when the FFT point size increases to several  thousand.  According to the decimation done FFT can be classified as  •   DIT-FFT (Decimation in Time)   DIF-FFT (Decimation in Frequency)

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10 Digital Signal Processing Radix – 2 FFT The radix-2 FFT algorithm breaks the entire DFT  calculation down into a number of 2-point Dots. Each 2- point DFT consists of a multiply-and-accumulate  operation called a  butterfly.
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11_FFT - Session 13 1 Digital Signal Processing Session...

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