Zhongguo Liu_Biomedical Engineering_Shandong Univ. Chapter 8 The Discrete Fourier Transform Zhongguo Liu Biomedical Engineering School of Control Science and Engineering, Shandong University Biomedical Signal processing

2 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

3 Filter Design Techniques 8.0 Introduction

4  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.

5 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.

6  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  Given a periodic sequence with period N so that  The Fourier series representation can be written as

7 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

8 Discrete Fourier Series Pair  A periodic sequence in terms of Fourier series coefficients  To obtain the Fourier series coefficients we multiply both sides by for 0  n  N-1 and then sum both the sides, we obtain

9 Discrete Fourier Series Pair Problem 8.51, HW

Representation of Periodic Sequence: the Discrete Fourier Series  a periodic sequence with period N,  The Fourier series coefficients of is

Representation of Periodic Sequence: the Discrete Fourier Series  The sequence is periodic with period N

12 Discrete Fourier Series (DFS)  Let  Analysis equation:  Synthesis equation:

13 Ex. 8.1 DFS of a impulse train  Consider the periodic impulse train n 01 2 …… N-1NN+1N+2 …… -2 …… -N+1-N N points

14 Ex. 8.1 DFS of a impulse train 01 2 …… N-1NN+1N+2 …… -2 …… -N+1-N N points k

15 k 01 2 …… N-1NN+1N+2 …… -2 …… -N+1-N N points 1 n 01 2 …… N-1NN+1N+2 …… -2 …… -N+1-N N points 1

16 Example 8.2 Duality in the Discrete Fourier Series  The discrete Fourier series coefficients is the periodic impulse train      1 0 ~ ~ N n kn N WnxkX 0 12 … N … -2 … -N N points N

17 k n 01 2 …… N-1NN+1N+2 …… -2 …… -N+1-N N points N 01 2 …… N-1NN+1N+2 …… -2 …… -N+1-N N points 1

18 k 01 2 …… N-1NN+1N+2 …… -2 …… -N+1-N N points 1 n 01 2 …… N-1NN+1N+2 …… -2 …… -N+1-N N points …… N-1NN+1N+2 …… -2 …… -N+1-N N points N 01 2 …… N-1NN+1N+2 …… -2 …… -N+1-N N points 1

19 Example 8.3 The Discrete Fourier Series of a Periodic Rectangular Pulse Train  Periodic sequence with period N=10 1

20 magnitude phase

21 magnitude phase

Properties of the Discrete Fourier Series  Linearity: two periodic sequence, both with period N

Properties of the Discrete Fourier Series  Shift of a sequence Problem 8.52, HW

Properties of the Discrete Fourier Series  Duality 01 2 …… N-1 n …… N-1 k …… N-1 n 01 2 …… N-1 k N

Symmetry Problem 8.53, HW

Periodic Convolution  and are two periodic sequences, each with period N and with discrete Fourier series and

Periodic Convolution  The sum is over the finite interval  The value of in the interval repeat periodically for m outside of that interval

28 Example 8.4 Periodic Convolution

Periodic Convolution

Representation of Periodic Sequence: the Discrete Fourier Series  a periodic sequence with period N,  The Fourier series coefficients of is Review

31 Discrete Fourier Series (DFS)  Let  Analysis equation:  Synthesis equation:

Properties of the Discrete Fourier Series  Shift of a sequence

Properties of the Discrete Fourier Series  Duality 01 2 …… N-1 n …… N-1 k …… N-1 n 01 2 …… N-1 k N

Periodic Convolution

35 Example 8.4 Periodic Convolution

The Fourier Transform of Periodic Signal  Periodic sequences are neither absolutely summable nor square summable, hence they don ’ t have a strict Fourier Transform

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

The Fourier Transform of Periodic Signal This is periodic with 2  since DFS is periodic  The inverse transform can be written as

39  Consider the periodic impulse train Ex. 8.5 Fourier Transform of a periodic impulse train 0 12 … N … -2 … -N N points 1  Therefore the Fourier transform is  The DFS was calculated previously to be N points n 01 2 … N-1N … -2 … -N 1

40 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 0 12 N … -2-2 … -N 1

41 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

42 Relation between Finite-length and Periodic Signals  If is periodic with period N, the DFS are  If for and otherwise then

43 Ex. 8.5 Relation between FS coefficients and FT  Consider the sequence  The Fourier transform

44  Consider the sequence  The DFS coefficients  The Fourier transform Ex. 8.5 Relation between FS coefficients and FT

45  Consider the sequence  The DFS coefficients  The Fourier transform Ex. 8.5 Relation between FS coefficients and FT

Sampling the Fourier Transform  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

47 Sampling the Fourier Transform

48 Sampling the Fourier Transform 0 12 … N … -2 … -N N points 1

49 Sampling the Fourier Transform

50  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 Sampling the Fourier Transform

51 Sampling the Fourier Transform  It is not necessary to know the DTFT at all frequencies  To recover the discrete-time sequence in time domain  Discrete Fourier Transform is used in  Representing a finite length sequence by samples of DTFT

Fourier Representation of Finite-Duration Sequence: Discrete Fourier Transform  Consider a finite-length sequence of length N samples such that outside the range  To each finite-length sequence of length N, we can associate a period sequence

53 Discrete Fourier Transform  For, the DFS is with period N  The Discrete Fourier Transform of is

54 Discrete Fourier Transform

55 Discrete Fourier Transform pairs  Analysis equation  Synthesis equation

56 Discrete Fourier Transform TimeFrequency Fourier transform (FT) continuous Fourier series (FS)continuous periodic Continuous impulse train Discrete-time Fourier transform (DTFT) discretecontinuous periodic Discrete Fourier series (DFS) discrete periodic continuous impulse train, periodic Discrete Fourier transform (DFT) discrete

57 四种傅立叶变换

58 Ex. 8.7 The DFT of a Rectangular Pulse  x[n] is of length 5  We can consider x[n] of any length greater than 5  Let ’ s pick N=5  Calculate the DFS of the periodic form of x[n]

59 Ex. 8.7 The DFT of a Rectangular Pulse  If we consider x[n] of length 10  We get a different set of DFT coefficients  Still samples of the DTFT but in different places

60 Review Relation of DTFT,DFS, DFT DTFT N sampling DFS DFT DFS  Let

61 Discrete Fourier Transform

62 Review Relation of DTFT,DFS, DFT DTFT N sampling DFS DFT DFS

63 Sampling of DTFT of Linear Convolution  Consider of length L and of length P Linear Convolution  The inverse DFT of is ：

Properties of the Discrete Fourier Transform  If has length and has length, Linearity

Circular Shift of a Sequence

66 Figure 8.12 circular shift Ex. 8.8 Circular Shift of a Sequence

Circular Shift of a Sequence

Circular Shift of a Sequence

Circular Shift of a Sequence

Duality

71 Ex.8.9 The Duality Relationship for the DFT

Symmetry Properties

Symmetry Properties

Symmetry Properties

Symmetry Properties

Symmetry Properties

习题答案修正  Problem 8.56 的证明上式中应该没有 一 项，并且该式后面加上限制 0 ≦ n ≦ N-1 ，也正因为 0 ≦ n ≦ N-1 ，所以在下式中， 也应该没有此项 。其他涉及此项的也该去除，因为 0 ≦ n ≦ N-1

Circular Convolution  For two finite-duration sequences and, both of length N, with DFTs and

Circular Convolution

Circular Convolution

81 Ex Circular Convolution with a Delayed Impulse Sequence

82 Ex Circular Convolution with a Delayed Impulse Sequence

83 Example 8.11 Circular Convolution of Two Rectangular Pulses

84 Ex Circular Convolution of Two Rectangular Pulses

Summary of Properties of the Discrete Fourier Transform

Summary of Properties of the Discrete Fourier Transform

Linear Convolution using the Discrete Fourier Transform  1. Compute the N-point DFT and of the two sequence and  2. Compute for  3. Compute as the inverse DFT of  Implement a convolution of two sequences by the following procedure:

Linear Convolution using the Discrete Fourier Transform  In most applications, we are interested in implementing a linear convolution of two sequence.  To obtain a linear convolution, we will discuss the relationship between linear convolution and circular convolution.

Linear Convolution of Two Finite-Length Sequences  for  is maximum length of length LP

Circular Convolution as Linear Convolution with Aliasing and linear convolution  circular convolution corresponding to DFTs:, Whether they are same?  depends on the length of the DFT in relation to the length of and

Linear Convolution using the Discrete Fourier Transform  1. Compute the N-point DFT and of the two sequence and  2. Compute for  3. Compute as the inverse DFT of  Implement a convolution of two sequences by the following procedure: Review

Circular Convolution as Linear Convolution with Aliasing  For finite sequence  The inverse DFT of is one period of :  If N ≧ length of x[n], then x p [n]= x[n]

Circular Convolution as Linear Convolution with Aliasing  The Fourier transform of is  Linear convolution:  Define a DFT  The inverse DFT of is ：

Circular Convolution as Linear Convolution with Aliasing  And  From  The circular convolution of two-finite sequences is equivalent to linear convolution of the two sequences, followed by time aliasing as above.  Linear convolution:

Circular Convolution as Linear Convolution with Aliasing  if N, the length of the DFTs, satisfies  If has length L and has length P, then has maximum length  The circular convolution corresponding to is identical to the linear convolution corresponding to DFT DTFT

96 linear convolution 6 points shift right of the linear convolution 6 points shift left of the linear convolution 6 points circular convolution= linear convolution with aliasing 12 points circular convolution = linear convolution Ex Circular Convolution as Linear Convolution with Aliasing. N=6 N=12

97 Which points of Circular Convolution equal that of Linear Convolution when Aliasing?  Consider of length L and of length P, where P < L Linear Convolution Circular Convolution Fig.8.19 Fig.8.20

98

Implementing Linear Time- Invariant Systems Using the DFT  Linear time-invariant systems can be implemented by linear convolution.  Linear convolution can be obtained from the circular convolution.  So, circular convolution can be used to implement linear time-invariant systems.

100 Zero-Pading  Consider an L-point input sequence and a P-point impulse response  The linear convolution of these two sequence has finite duration with length (L+P-1)  For the circular convolution and linear convolution to be identical, the circular convolution must have a length of at least (L+P-1) points.

101 Zero-Pading  The circular convolution can be achieved by multiplying the DFTs of and.  Since the length of the linear convolution is (L+P-1) points, the DFTs that we compute must also be of at least that length, i.e., both and must augmented with sequence values of zero.  The process is called Zero-Pading

102  Each section can be convolved with the finite-length impulse response and the output sections fitted together in an appropriate way. Block Convolution  If the input signal is of indefinite duration, the input signal to be processed is segmented into sections of length L.  The processing of each section can then be implemented using the DFT.

103 Block Convolution overlap-add method (1) segment into sections of length L; (2) fill 0 into and some section of, then do L+P-1 points FFT ; (3) calculate

104 P-1 points (4)add the points n=0…P-2 in y[n] to the last P-1 points in the former section y[n] ， the output for this section is the points n=0…L-1 (3) calculate (1) segment into sections of length L; (2) fill 0 into and some section of, then do L+P-1 points FFT overlap-add method L=16

Circular Convolution as Linear Convolution with Aliasing

106 overlap-save method P-1 points (4) the output for this section is L-P+1 points of y[n] n=P-1,…L-1 (1) segment into sections of length L, overlap P-1 points; (2) fill 0 into and some section of, then do L points FFT (3) calculate L=25

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