Understanding Intelligent Signal Processing Tests

? Intelligent Signal Processing Test Angelo Ciaramella

Question 8 Question By using zero-padding we obtain a high resolution spectrum a spectrum with all zeros a dense spectrum ISP Verification tests

Discrete Fourier Transform Continuous time transforms Fourier Theorem Continuous Fourier Transform Discrete Time Fourier Transform (DTFT) z-transform Transformation for finite duration sequences Discrete Fourier Transform (DFT) Discrete Cosine Transform (DCT) ISP Verification tests

Roots of the unit circle 2 N=8 N W82 j W83 W81 W80 = W88 0 W84 -1 1 2 W87 -j W85 W86 ISP Verification tests ??= ? ??? The N-th root of the unity ? ?= ? ??? ?? N roots of the unity ?? 0 ? ? 1

DFT and z-Transform It corresponds to sample the z-transform, X (z), in N points equally spaced on the unit circle DFT z plane 2 N ISP Verification tests DTFT

DFT 1 N = 0 kn N ( ) 0 1 x n W k N Analysis = ( ) X k 0 n altrove 1 1 N N ( ) k = k kn 0 1 X W n N N = ( ) x n 0 ISP Verification tests 0 altrove Synthesis

Example of DFT ISP Verification tests 1 0 3 n = ( ) x n DTFT of a 4 points sequence 0 altrove

Example of DFT ISP Verification tests DFT of a 4 points sequence. The DFT is a sampling of the DTFT

Padding We add some zeros to the previous sequence x(n)=[1 1 1 1 0 0 0 0] This operation is named zero-padding ISP Verification tests It is needed to obtain a dense spectrum

DFT ISP Verification tests DFT after zero-padding

High density spectrum We consider the following signal = + ( ) cos( . 0 48 ) cos( . 0 52 ) x n n n ISP Verification tests DFT on 10 points

High density spectrum ISP Verification tests Increasing the points ...

High resolution spectrum ISP Verification tests To obtain an High Resolution Spectrum we increse the sampling points of the source sequence. The estimated frequencies corrispond to the frequencies of the analyzed signal.

Discrete Cosine Transform The Discrete Cosine Transform (DCT) is very similar to DFT but in real domain DCT is used for feature extraction data decorrelation low loss compression the basis functions are orthogonal it is symmetric ISP Verification tests

1D DCT monodimensional signal 0 ? N-1 + N 1 2 ( ) 1 N n k = n = ( ) ( ) ( ) cos X k k x n Analysis 2 0 + N 1 2 ( ) 1 N n k = k = ( ) ( ) ( ) cos x n k X k Synthesis 2 0 ISP Verification tests 1 N = se 0 k = ( ) k 2 N se 0 k

1D DCT bidimensional signal 0 ? N-1 0 ? N-1 Analysis + ) 1 N + N 1 1 2 ( ) 1 2 ( N N i n j m = 0 i = ( , ) ( ) ( ) , ( i x ) cos cos X n m n m j 2 2 = 0 j Synthesis + ) 1 N ISP Verification tests + N 1 1 2 ( ) 1 2 ( N N i n j m = 0 n = , ( i x ) ( ) ( ) ( , ) cos cos j n m X n m 2 2 = 0 m

DFT vs DCT DCT provides the spatial compression, able to detect changes of information between contiguos area avoiding the repetitions ISP Verification tests

References Material Slides Video Lessons Books Signal Processing Book (Ciaramella) free download on the e-learning platform Discrete-time signal processing, A. V. Oppenheim, R. W. Schafer, J.R. Buck, Upper Saddle River, N.J., Prentice Hall, 1999, ISBN 0-13-754920-2 Digital Signal Processing, J. Proakis, D. Manolakis, Prentice Hall, 4 edition, 2006 ISP Verification tests