Create Presentation
Download Presentation

Download Presentation
## Chapter 8 The Discrete Fourier Transform

- - - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - - -

**Chapter 8 The Discrete Fourier Transform**Biomedical Signal processing Zhongguo Liu Biomedical Engineering School of Control Science and Engineering, Shandong University 1 Zhongguo Liu_Biomedical Engineering_Shandong Univ.**Chapter 8 The Discrete Fourier Transform**• 8.0 Introduction • 8.1 Representation of Periodic Sequence: the Discrete Fourier Series • 8.2 Properties of the Discrete Fourier Series • 8.3 The Fourier Transform of Periodic Signal • 8.4 Sampling the Fourier Transform • 8.5 Fourier Representation of Finite-Duration Sequence: the Discrete Fourier Transform • 8.6 Properties of the Discrete Fourier Transform • 8.7 Linear Convolution using the Discrete Fourier Transform**Filter Design Techniques**8.0 Introduction**8.0 Introduction**• Discrete Fourier Transform (DFT)for finite duration sequence • DFT is a sequence rather than a function of a continuous variable • DFT corresponds to sample, equally spaced in frequency, of the Fourier transform of the signal.**8.0 Introduction**• The relationship between periodic sequence and finite-length sequences： • The Fourier series representation of the periodic sequence corresponds to the DFT of the finite-length sequence.**Given a periodic sequence with period N so that**8.1 Representation of Periodic Sequence: the Discrete Fourier Series • The Fourier series representation can be written as • Fourier series representation of continuous-time periodic signals require infinite many complex exponentials • Not that for discrete-time periodic signals we have**8.1 Representation of Periodic Sequence: the Discrete**Fourier Series • Due to the periodicity of the complex exponential we only need N exponentials for discrete time Fourier series • No need**To obtain the Fourier series coefficients we multiply both**sides by for 0nN-1 and then sum both the sides , we obtain Discrete Fourier Series Pair • A periodic sequence in terms of Fourier series coefficients**Discrete Fourier Series Pair**Problem 8.51, HW**The Fourier series coefficients of is**8.1 Representation of Periodic Sequence: the Discrete Fourier Series • a periodic sequence with period N,**The sequence is periodic with period N**8.1 Representation of Periodic Sequence: the Discrete Fourier Series**Synthesis equation:**Discrete Fourier Series (DFS) • Let • Analysis equation:**N points**…… …… …… 2 N-1 N N+1 N+2 -N -N+1 -2 -1 0 1 Ex. 8.1 DFS of a impulse train • Consider the periodic impulse train n**N points**k …… …… …… 2 N-1 N N+1 N+2 -N -N+1 -2 -1 0 1 Ex. 8.1 DFS of a impulse train**N points**1 k …… …… …… 2 N-1 N N+1 N+2 -N -N+1 -2 -1 0 1 N points 1 n …… …… …… 2 N-1 N N+1 N+2 -N -N+1 -2 -1 0 1**-**N 1 ~ N points [ ] [ ] ~ å = kn X k x n W N N = n 0 …… …… …… -N N 1 2 -2 -1 0 Example 8.2 Duality in the Discrete Fourier Series • The discrete Fourier series coefficients is the periodic impulse train**N points**N …… …… …… 2 N-1 N N+1 N+2 -N -N+1 -2 -1 0 1 N points 1 …… …… …… 2 N-1 N N+1 N+2 -N -N+1 -2 -1 0 1 k n**N points**1 k …… …… …… 2 N-1 N N+1 N+2 -N -N+1 -2 -1 0 1 N points N N points …… …… 1 …… 2 N-1 N N+1 N+2 -N -N+1 -2 -1 0 1 N points 1 n …… …… …… 2 N-1 N N+1 N+2 -N -N+1 -2 -1 0 1 …… …… …… 2 N-1 N N+1 N+2 -N -N+1 -2 -1 0 1**Example 8.3 The Discrete Fourier Series of a Periodic**Rectangular Pulse Train • Periodic sequence with period N=10 1**magnitude**phase**magnitude**phase**8.2 Properties of the Discrete Fourier Series**• Linearity: two periodic sequence, both with period N**8.2 Properties of the Discrete Fourier Series**• Shift of a sequence Problem 8.52, HW**N**k …… 2 N-1 0 1 1 1 n 1 …… 2 N-1 0 1 k …… N-1 2 0 1 n …… 2 N-1 0 1 8.2 Properties of the Discrete Fourier Series • Duality**8.2.4 Symmetry**Problem 8.53, HW**and are two periodic sequences, each with period N**and with discrete Fourier series and 8.2.5 Periodic Convolution**The sum is over the finite interval**• The value of in the interval repeat periodically for m outside of that interval 8.2.5 Periodic Convolution**The Fourier series coefficients of is**8.1 Representation of Periodic Sequence: the Discrete Fourier Series • a periodic sequence with period N, Review**Synthesis equation:**Discrete Fourier Series (DFS) • Let • Analysis equation:**8.2 Properties of the Discrete Fourier Series**• Shift of a sequence**N**k …… 2 N-1 0 1 1 1 n 1 …… 2 N-1 0 1 k …… N-1 2 0 1 n …… 2 N-1 0 1 8.2 Properties of the Discrete Fourier Series • Duality**8.3 The Fourier Transform of Periodic Signal**• Periodic sequences are neither absolutely summable nor square summable, hence they don’t have a strict Fourier Transform**8.3 The Fourier Transform of Periodic Signal**• We can represent Periodic sequences as sums of complex exponentials: DFS • We can combine DFS and Fourier transform • Fourier transform of periodic sequences • Periodic impulse train with values proportional to DFS coefficients**8.3 The Fourier Transform of Periodic Signal**This is periodic with 2 since DFS is periodic • The inverse transform can be written as**N points**1 • The DFS was calculated previously to be …… …… …… N points -N N 1 2 -2 -1 0 1 …… …… …… n 2 N-1 N -N -2 -1 0 1 • Therefore the Fourier transform is Ex. 8.5 Fourier Transform of a periodic impulse train • Consider the periodic impulse train**1**…… …… -N N 1 2 -2 -1 0 Relation between Finite-length and Periodic Signals • Consider finite length signal x[n] spanning from 0 to N-1 • Convolve with periodic impulse train • The Fourier transform of the periodic sequence is**Relation between Finite-length and Periodic Signals**• This implies that • DFS coefficients of a periodic signal can be thought as equally spaced samples of the Fourier transform of one period**If is periodic with period N, the DFS are**• If for and otherwise Relation between Finite-length and Periodic Signals then**Ex. 8.5 Relation between FS coefficients and FT**• Consider the sequence • The Fourier transform**Ex. 8.5 Relation between FS coefficients and FT**• The Fourier transform • The DFS coefficients • Consider the sequence**The Fourier transform**• The DFS coefficients Ex. 8.5 Relation between FS coefficients and FT • Consider the sequence**Consider an aperiodic sequence with Fourier transform**,and assume that a sequence is obtained by sampling at frequency • is Fourier series coefficients of periodic sequence 8.4 Sampling the Fourier Transform**N points**1 …… …… …… -N N 1 2 -2 -1 0 Sampling the Fourier Transform**Sampling the Fourier Transform**• Samples of the DTFT of an aperiodic sequence • can be thought of as DFS coefficients • of a periodic sequence • obtained through summing periodic replicas of original sequence • If the original sequence is of finite length, • and we take sufficient number of samples of its DTFT, • then the original sequence can be recovered by