Digital Signal Processing Module 6 Discrete Fourier Transform (DFT)

Transcription

1 Objective: Digital Signal Processing Module 6 Discrete Fourier Transform (DFT) 1. To analyze finite duration sequences using Discrete Fourier Transform. 2. To understand the characteristics and importance of properties of DFT 3. To understand the relation between transforms Introduction: We have seen how to represent a sequence in terms of a linear combination of complex exponentials using the discrete-time Fourier transform (DTFT) and how the sequence values may be used as the coefficients in a power series expansion of a complexvalued function of z. For finite-length sequences there is another representation, called the Discrete Fourier Transform (DFT). Unlike the DTFT, which is a continuous function of a continuous variable, ω, the DFT is a sequence that corresponds to samples of the DTFT. Such a representation is very useful for digital computations and for digital hardware implementations. Description: Discrete Fourier Transform The DFT is an important decomposition for sequences that are finite in length. Whereas the DTFT is a mapping from a sequence to a function of a continuous variable, ω, the DFT is a mapping from a sequence, x(n), to another sequence, X(k), The DFT may be easily developed from the discrete Fourier series representation for periodic sequences. Let x(n) be a finite-length sequence of length N that is equal to zero outside the interval [0, N - 1]. A periodic sequence x n may be formed from x(n) as follows: This periodic extension may be expressed as follows: where (n mod N ) and ((n))n are taken to mean "n modulo N." That is to say, if n is written in the form n = kn +l, where 0 l < N, For example, ((13))8 = 5 A periodic sequence may be expanded using the DFS as in

2 Because x n = x n for n = 0, 1,..., N - 1, x(n) may similarly be expanded as follows: Because the DFS coefficients are periodic, if we let X(k) be one period of X k and replacex k in the sum with X (k), then we have The sequence X(k) is called the N-point DFT of x(n). These coefficients are related to x(n) as follows: The above two equations form a DFT pair, and we write This expansion is valid for complex-valued as well as real-valued sequences. Properties of Discrete Fourier Transform Since discrete Fourier transform is similar to the discrete Fourier series representation, the properties are similar to DFS representation. We use the notation to say that {X[k]} are DFT coefficient of finite length sequence x[n] The Properties of DFT are summarized below in Table 6.1

3 Relation between DFT, DTFT and Z-transform Comparing the definition of the DFT of x(n) to the DTFT, it follows that the DFT coefficients are samples of the DTFT: Alternatively, the DFT coefficients correspond to N samples of X ( z ) that are taken at N equally spaced points around the unit circle: Let us consider a sequence x(n) of finite duration N with z- transform

4 We have Substituting, we get Illustrative Examples: Problem 1: Compute the N-point DFT of each of the following sequences Solution: For x 3 (n), the DFT may be found directly as follows: Problem 2: Find the N -point DFT of the sequence. Solution: The DFT of this sequence may be evaluated by expanding the cosine as a sum of complex exponentials: Using the periodicity of the complex exponentials, we may write x(n) as follows: Therefore, the DFT coefficients are

5 Problem 3: Find the 10-point inverse DFT of Solution: To find the inverse DFT, note that X(k) may be expressed as follows: Written in this way, the inverse DFT may be easily determined. Specifically, note that the inverse DFT of a constant is a unit sample: Similarly, the DFT of a constant is a unit sample: Therefore, it follows that Summary: The discrete Fourier transform (DFT) is a fundamental transform in digital signal processing, with applications in frequency analysis, fast convolution, image processing, etc. Assignment: Problem 1: Consider the finite-length sequence Find the DFT of x(n) Problem 2: Consider the sequence Find the four-point DFT of x(n). Problem 3: Determine IDFT of X(k)={12,-1.5+j2.598,,-1.5+j0.866,0, -1.5-j0.866,-1.5-j2.598}

6 Simulation: % Clearing and Closing Commands clc; % To clear Command Window clear all; % To clear the workspace close all; % To close the previous Waveforms/Graphs if any % To Compute DFT/IDFT x=input('enter the sequence:'); N=length(x); X=[]; %DFT for k=0:n-1 y1=0; for n=0:n-1 y1=y1+x(n+1)*exp(-i*2*pi*n*k/n); X=[X,y1]; disp('the result of DFT is:');x % Inverse DFT y=[]; for n=0:n-1 y2=0; for k=0:n-1 y2=y2+x(k+1)*exp(i*2*pi*n*k/n); y=[y,y2]; disp('the result of IDFT is:');y/n Input: Enter the sequence:[ ] Output: The result of DFT is: X = [ i i i i i i i i] The result of IDFT is: ans = [ i i i i i i i i] References: 1. Digital Signal Processing, Principles, Algorithms and Applications John G Proakis, Dimitris G Manolakis, Pearson Education / PHI, Discrete Time Signal Processing A V Oppenheim and R W Schaffer, PHI, Digital Signal Processing Monson H.Hayes Schaum s Outlines, McGraw-Hill, Fundamentals of Digital Signal Processing using Matlab Robert J Schilling, Sandra L Harris, Thomson Digital Signal processing A Practical Approach, Emmanuel C Ifeachor and Barrie W Jervis, 2 nd Edition, PE Digital Signal Processing A Computer Based Approach, Sanjit K.Mitra, McGraw Hill,2 nd Edition, 2001