Linear filtering based on the DFT

Slides:



Advertisements
Similar presentations
DFT & FFT Computation.
Advertisements

Michael Phipps Vallary S. Bhopatkar
DFT properties Note: it is important to ensure that the DFTs are the same length If x1(n) and x2(n) have different lengths, the shorter sequence must be.
Digital Kommunikationselektronik TNE027 Lecture 5 1 Fourier Transforms Discrete Fourier Transform (DFT) Algorithms Fast Fourier Transform (FFT) Algorithms.
The Discrete Fourier Transform. The spectrum of a sampled function is given by where –  or 0 .
Block Convolution: overlap-save method  Input Signal x[n]: arbitrary length  Impulse response of the filter h[n]: lenght P  Block Size: N  we take.
Chapter 8: The Discrete Fourier Transform
MM3FC Mathematical Modeling 3 LECTURE 3
Lecture #17 INTRODUCTION TO THE FAST FOURIER TRANSFORM ALGORITHM Department of Electrical and Computer Engineering Carnegie Mellon University Pittsburgh,
FFT-based filtering and the Short-Time Fourier Transform (STFT) R.C. Maher ECEN4002/5002 DSP Laboratory Spring 2003.
Sampling, Reconstruction, and Elementary Digital Filters R.C. Maher ECEN4002/5002 DSP Laboratory Spring 2002.
20 October 2003WASPAA New Paltz, NY1 Implementation of real time partitioned convolution on a DSP board Enrico Armelloni, Christian Giottoli, Angelo.
Finite Impuse Response Filters. Filters A filter is a system that processes a signal in some desired fashion. –A continuous-time signal or continuous.
Discrete-Time and System (A Review)
1 Chapter 8 The Discrete Fourier Transform 2 Introduction  In Chapters 2 and 3 we discussed the representation of sequences and LTI systems in terms.
CHAPTER 3 Discrete-Time Signals in the Transform-Domain
8.1 representation of periodic sequences:the discrete fourier series 8.2 the fourier transform of periodic signals 8.3 properties of the discrete fourier.
Finite-Length Discrete Transform
1 BIEN425 – Lecture 8 By the end of the lecture, you should be able to: –Compute cross- /auto-correlation using matrix multiplication –Compute cross- /auto-correlation.
The Discrete Fourier Transform 主講人:虞台文. Content Introduction Representation of Periodic Sequences – DFS (Discrete Fourier Series) Properties of DFS The.
Zhongguo Liu_Biomedical Engineering_Shandong Univ. Chapter 8 The Discrete Fourier Transform Zhongguo Liu Biomedical Engineering School of Control.
Hossein Sameti Department of Computer Engineering Sharif University of Technology.
Real time DSP Professors: Eng. Julian S. Bruno Eng. Jerónimo F. Atencio Sr. Lucio Martinez Garbino.
Digital Signal Processing Chapter 3 Discrete transforms.
Fourier Analysis of Discrete Time Signals
Digital Signal Processing
Z TRANSFORM AND DFT Z Transform
Copyright ©2010, ©1999, ©1989 by Pearson Education, Inc. All rights reserved. Discrete-Time Signal Processing, Third Edition Alan V. Oppenheim Ronald W.
Lecture 6: DFT XILIANG LUO 2014/10. Periodic Sequence  Discrete Fourier Series For a sequence with period N, we only need N DFS coefs.
LIST OF EXPERIMENTS USING TMS320C5X Study of various addressing modes of DSP using simple programming examples Sampling of input signal and display Implementation.
Chapter 5 Finite-Length Discrete Transform
Dan Ellis 1 ELEN E4810: Digital Signal Processing Topic 3: Fourier domain 1.The Fourier domain 2.Discrete-Time Fourier Transform (DTFT) 3.Discrete.
EEE 503 Digital Signal Processing Lecture #2 : EEE 503 Digital Signal Processing Lecture #2 : Discrete-Time Signals & Systems Dr. Panuthat Boonpramuk Department.
ES97H Biomedical Signal Processing
Digital Signal Processing
1 Lecture 4: March 13, 2007 Topic: 1. Uniform Frequency-Sampling Methods (cont.)
Technological Educational Institute Of Crete Department Of Applied Informatics and Multimedia Neural Networks Laboratory Slide 1 DISCRETE SIGNALS AND SYSTEMS.
DTFT continue (c.f. Shenoi, 2006)  We have introduced DTFT and showed some of its properties. We will investigate them in more detail by showing the associated.
ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete Objectives: FIR Filters Design of Ideal Lowpass Filters Filter Design Example.
The Discrete Fourier Transform
Analysis of Linear Time Invariant (LTI) Systems
Finite Impuse Response Filters. Filters A filter is a system that processes a signal in some desired fashion. –A continuous-time signal or continuous.
EE345S Real-Time Digital Signal Processing Lab Fall 2006 Lecture 17 Fast Fourier Transform Prof. Brian L. Evans Dept. of Electrical and Computer Engineering.
Dr. Michael Nasief Digital Signal Processing Lec 7 1.
بسم الله الرحمن الرحيم Digital Signal Processing Lecture 14 FFT-Radix-2 Decimation in Frequency And Radix -4 Algorithm University of Khartoum Department.
بسم الله الرحمن الرحيم Lecture (12) Dr. Iman Abuel Maaly The Discrete Fourier Transform Dr. Iman Abuel Maaly University of Khartoum Department of Electrical.
1 Chapter 8 The Discrete Fourier Transform (cont.)
Lecture 16 Outline: Discrete Fourier Series and Transforms Announcements: Reading: “5: The Discrete Fourier Transform” pp HW 5 posted, short HW (2.
 Carrier signal is strong and stable sinusoidal signal x(t) = A cos(  c t +  )  Carrier transports information (audio, video, text, ) across.
Homework 3 1. Suppose we have two four-point sequences x[n] and h[n] as follows: (a) Calculate the four-point DFT X[k]. (b) Calculate the four-point DFT.
DIGITAL SIGNAL PROCESSING ELECTRONICS
Digital Signal Processing
Lecture 12 Linearity & Time-Invariance Convolution
FFT-based filtering and the
The Discrete Fourier Transform
Fast Fourier Transform
Lecture 14 Outline: Discrete Fourier Series and Transforms
4.1 DFT In practice the Fourier components of data are obtained by digital computation rather than by analog processing. The analog values have to be.
Chapter 8 The Discrete Fourier Transform
Sampling the Fourier Transform
APPLICATION of the DFT: Convolution of Finite Sequences.
Z TRANSFORM AND DFT Z Transform
Lecture 16 Outline: Linear Convolution, Block-by Block Convolution, FFT/IFFT Announcements: HW 4 posted, due tomorrow at 4:30pm. No late HWs as solutions.
Chapter 8 The Discrete Fourier Transform
Chapter 8 The Discrete Fourier Transform
Signals and Systems Lecture 18: FIR Filters.
Fast Fourier Transform
CE Digital Signal Processing Fall Discrete Fourier Transform (DFT)
Image Enhancement in Spatial Domain: Neighbourhood Processing
Presentation transcript:

Linear filtering based on the DFT Lecture 3 Linear filtering based on the DFT Advanced Digital Signal Processing Dr Dennis Deng Department of Electronic Engineering, La Trobe University 1998,1999

Contents Linear filter and circular convolution Overlap and save method Overlap and add method Self test exercises

Linear Filtering A sequence x(n) of length L filtered by an FIR filter h(n) of length M The length of y(n) is L+M-1 Example x(n) =[1 1 1] , h(n) =[-1 1]

Linear Filtering In the frequency domain Sampling both side by an interval The kth sample is given by Using the DFT notation X(k) and H(k) are the DFT (with zero padding) of x(n) and h(n), respectively.

Linear Filtering Performing the inverse DFT Is the same as y(n) ? YES. Because is the Fourier transform of y(n), to recover y(n), an N-point sampling of and an N-point IDFT is necessary. An equivalent question is under what condition circular convolution becomes linear convolution ? A 4-point circular convolution of the previous example in the following slide

Linear Filtering This example shows that by zero padding such that the length of the circular convolution is greater than or equal to L+M-1, circular convolution is the same as linear convolution.

Linear Filtering The multiplication in DFT domain corresponds to circular convolution in the time domain. By zero padding the two sequences, circular convolution becomes linear convolution To perform linear convolution using the DFT (1) zero padding x(n) and h(n) such that the length of them is N=L+M-1 (2) Calculate an N-point DFT for both sequences and multiply them point-by-point (3) perform an N-point inverse DFT

Linear Filtering The following example shows a 3-point circular convolution

Linear Filtering The following example shows a 5-point circular convolution These examples show that when circular convolution is the same as linear convolution

Overlap-save method When the DFT is used to implement linear filtering, a signal is processed in blocks. Due to the real-time requirement (low delay) and the limitation of physical memory, the size of the block can not be arbitrarily large. The length of the FIR filter is M and the length of on block of data is L (L>M) Each time a block of data of length L+M-1 is filtered by using the DFT method.

Overlap-save method The method is shown in the following diagram

Overlap-save method Each step consists: (1) calculating N-point DFT of x(n) and multiply it with the N-point DFT of h(n). (2) calculating N-point IDFT and discarding the first M-1 output sample. The last L samples are the desired filter output (3) append the last M-1 samples to the beginning of the new block of signal (4) goto (1) until reach the end of the signal.

Overlap-save method The reason for discarding the first M-1 output samples is that they are the result of circular convolution. Linear convolution starts at the Mth sample. For the same reason, the last M-1 samples of the processed block should be appended to the beginning of the new block

Overlap-save method This is can be easily seen from a simple example: x(n)=[1 2 2 1]; h(n)=[1 -1]. (L=4, M=2). A 6-point circular convolution (x(n) represents a block of data)

Overlap-add method This method is shown in the following diagram

Overlap-add method In each step: (1) perform the L+M-1 point DFT, multiplication and IDFT (2) the last M-1 samples of the previous output is added to the first M-1 samples of the current output

Summary The DFT can be used to implement linear filtering A key point is to choose the N-point DFT such that circular convolution is the same as linear convolution The overlap-save and overlap-add methods can be used in real time signal processing.

Self test exercises For a signal x(n)=[1 2 3 2 1] and a filter h(n)=[1 -2 1], (1) Calculate its circular convolution. (2) Verify your result by using Matlab (hint: real(ifft(fft(x).*fft(h,6)))) (3) Calculate its N-point (N=8,9,10,128) circular convolution. Which one is the same as linear convolution For a signal x(n)=[1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4] and a filter h(n)=[1 2 1], calculate the filter output using (1) Overlap-save (2) Overlap-add