Lesson_16 - Discrete-Time Signals and Systems Dr. Fred J....

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Discrete-Time Signals and Systems Dr. Fred J. Taylor, Professor Lesson Title: FIR Frequency Response Lesson Number: 16 (Section 6-7 to 6-9) Background: In Chapter 6, the concept of frequency response for FIRs was introduced. Sections 6-7 and 6-8 basically beat it to death beginning with the now too familiar MA filter (a.k.a., running average filter). MA Filter At the end of Chapter 6, the steady-state frequency response of a discrete-time MA FIR system is analyzed. The input-output relationship of an L th order linear phase MA FIR is given by. [ ] [ ] - = - = 1 0 1 L k k n x L n y 1. The frequency response is given by (see previous lessons or textbook for derivation): ( 29 ( 29 ( 29 ( 29 ( 29 ( 29 2 / 1 2 / 1 1 0 2 / sin 2 / sin 1 - - - - - = = = = L j j L L j L k j j e e D e L L e L e H ω 2. where ω is a normalized frequency and D L is called the Dirichlet function (MATLAB diric ). D L (e j ω ) is a real number and MA’s phase response is the linear - ω (L-1)/2 with a group delay of τ = (L-1)/2. Example The Dirchlet function for L=11 is displayed in Figure 1. >> t=0:1/100:2*pi; >> tt=-pi+t; >> d=diric(tt,11); >> plot(d) The corresponding phase slope would be τ =-5 and the phase response is -5 ω . 1
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Discrete-Time Signals and Systems Dr. Fred J. Taylor, Professor Figure 1: Dirichlet function for L=11. The authors spend a few moments discussing the obvious properties of the Dirchlet function (read them) and then embark on study of image processing. Survey of 2-D FIR filtering Image processing is an important element of signal processing. The engineer is often responsible for processing 2-D and 3-D images in order to alter or enhance an image in a manner that will accentuate or suppress image attributes or human interpretability. Digital images are represented by a set of pixels ( viz ; samples) that are separated in space. Spatial domain image processing refers to the direct manipulation of a set of those pixels that compose an image. Spatial images are sampled in terms of pixels/meter metrics instead of samples/second (Sa/s). While spatial domain processing involves procedures that operate directly on pixels, spatial frequency domain image processing involves the use of a Fourier transform or its equivalent. In the spatial domain, harmonics are defined in terms of a fundamental period measured in spatial coordinates (e.g., meters) rather than seconds as is the case for temporal signals. For example, a 1000-point DFT of an image being sampled at 1 pixels/mm, has a fundamental period of 1000 mm or 1 meter. The fundamental spatial frequency is then 1 cycle/m. Consider a 256x256 8-bit B&W image of Albert (see Figure 2). The following Matlab code can be used to import, display, and transform an image. EDU>> image=imread('image_name.bmp','bmp');
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Lesson_16 - Discrete-Time Signals and Systems Dr. Fred J....

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