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 determine the DFS of a impulse train  Consider the periodic impulse train n 01 2 …… N-1NN+1N+2 …… -2 …… -N+1-N N points Solution:

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

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  If the discrete Fourier series coefficients is the periodic impulse train, determine the signal.      1 0 ~ ~ N n kn N WnxkX 0 12 … N … -2 … -N N points N Solution:

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 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 k 01 2 …… N-1NN+1N+2 …… -2 …… -N+1-N N points 1 Duality in Discrete Fourier Series

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  Shift of a sequence

Properties of the Discrete Fourier Series  Shift of a sequence

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

Symmetry Problem 8.53, HW

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

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

The Fourier Transform of discrete-time 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 with period N, and the impulses are spaced at integer multiples of 2  /N.  The inverse transform can be written as

32  the periodic impulse train Ex. 8.5 determine the 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 Solution:

33 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 12N … -2 … -N 1

34 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.

35 Relation between Finite-length and Periodic Signals —— verification:  If is periodic with period N, the DFS are  If is one period of, i.e.  then (1) (2)  compare (1) and (2),we get:

36 Ex. 8.6 Relation between FS coefficients and Fourier transform of one period of Periodic Signal  Consider the sequence and one period:  The Fourier transform Solution:

37  The DFS coefficients  The Fourier transform Ex. 8.6 Relation between FS coefficients and Fourier transform of one period and

38  The DFS coefficients  The Fourier transform Ex. 8.6 Relation between FS coefficients and Fourier transform of one period and

Sampling the Fourier Transform  is Fourier series coefficients of periodic sequence  Consider an aperiodic sequence with Fourier transform,and assume that a sequence is obtained by sampling at frequency

40 Sampling the Fourier Transform to recover x[n] x[n]x[n]

… N … -2 … -N N points 1 Sampling the Fourier Transform to recover x[n] x[n] = x[n]

42 Sampling the Fourier Transform to recover x[n] ≠ x[n] x[n]

43  Samples of DTFT X(e -j w ) of an aperiodic sequence x[n]  are thought of as DFS coefficients  of a periodic sequence  obtained through summing periodic replicas of original sequence x[n].  If the original sequence x[n] is of finite length,  and we take sufficient number N of samples of its DTFT X(e -j w )  then the original sequence x[n] can be recovered by Sampling the Fourier Transform to recover x[n]

44 Discrete Fourier Transform or DFT  Given a finite-length sequence x[n], we can form a periodic sequence, which in tum can be represented by a DFS  Given a sequence of Fourier coefficients, we can find, and then obtain x[n].  When the Fourier series is used in this way to represent finite-length sequences, it is called the Discrete Fourier Transform or DFT.  It is not necessary to know the DTFT X(e -jw ) at all frequencies, to recover the discrete-time sequence x[n] in time domain.

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

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

47 Discrete Fourier Transform DFS: DFT:  Analysis equation :  Synthesis equation: Usually DFT pairs is written as:

48 Discrete Fourier Transform pairs  Analysis equation  Synthesis equation

49 Recall different transforms between Time-Frequency domains TimeFrequency Fourier transform (FT) continuous Fourier series (FS)continuous periodic discrete Discrete-time Fourier transform (DTFT) discretecontinuous periodic Discrete Fourier series (DFS) discrete periodic discrete, periodic Discrete Fourier transform (DFT) discrete

50 四种傅立叶变换

51 Ex. 8.7 Calculate the DFT of a Rectangular Pulse  x[n] is of length 5 Solution:  To form from x[n] we can consider x[n] of any length greater than 5.  Let ’ s pick N=5  Calculate the DFS of x[n]

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

53 Review Relation of DTFT,DFS, DFT DTFT N sampling DFS DFT DFS  Let = x[n] x[n]

54 Discrete Fourier Transform

55 Review Relation of DTFT,DFS, DFT DTFT N sampling DFS DFT DFS ≠ x[n] x[n]

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

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

Circular Shift of a Sequence

Circular Shift of a Sequence--proof Circular Shift linear shift Proof: Recall: Problem: Given,

Circular Shift of a Sequence

Circular Shift of a Sequence

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

Duality Time domain Frequency domain

64 Ex.8.9 The Duality Relationship for the DFT

Symmetry Properties periodic conjugate symmetric components periodic conjugate-antisymmetric components

Symmetry Properties

Symmetry Properties conjugate symmetric components conjugate-antisymmetric components It can be shown in Problem 56

Symmetry Properties defination

Symmetry Properties

70 习题 8.56 答案说明  Problem 8.56 的证明中注意上式中相加的信号，只在 时间 0 ≦ n ≦ N-1 之间相加，对于周期信号 的形式 ，即时间 -n 为负值的情况，必须转换成时间为正的形 式 ，即 0 ≦ N-n ≦ N-1.  对于非周期信号 ，即周期信号 的一个周期的 时间反转信号，其在时间大于 0 而小于 N-1 的期间， 除了在时刻 0 有值外，其他时刻无值。  对于非周期信号 x[n-N] ，其在时间大于 0 而小于 N-1 的期间无值，对于 ，当 n=0 时，包含了 x * [N] 的 值，所以需要减去 。

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

Circular Convolution

Circular Convolution

74 Ex Circular Convolution with a Delayed Impulse Sequence n 0 = 1 N=5 Solution1 ：

75 Ex Circular Convolution with a Delayed Impulse Sequence Solution2 ：

76 Example 8.11 Circular Convolution of Two Rectangular Pulses Solution1 ： DFT length

77 Ex Circular Convolution of Two Rectangular Pulses Solution2 ： DFT length

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:  FFT algorithms are available for computing the DFT of a finite-duration sequence.

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 P

83  circular convolution corresponding to DFTs: ， Circular Convolution as Linear Convolution with Aliasing as linear convolution, 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] N sampling

Circular Convolution as Linear Convolution with Aliasing  The Fourier transform of is  Linear convolution:  Define a DFT The inverse DFT of is ：  If N ≧ length of x 3 [n], then N sampling

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:

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

89 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

90 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

91 View the process of forming the circular convolution x 3p [n] through linear convolution plus aliasing, as taking the (P - 1) values of x 3 [n] from n=L to n=L+P -2 and adding them to the first (P - 1) values of x 3 [n]. circular convolution “ wraps around ".

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.

93 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.

94 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

95  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.

96 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

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

Circular Convolution as Linear Convolution with Aliasing

99 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 圆周卷积中后 L-P+1 个点结果与线性卷积相等

the discrete cosine transform(DCT)

101

102 DCT-1 DCT-2 symmetric and periodic extension of signal, then do DFS and get DCT by taking the dominant period 。

103 relationship between 2N- poinsts DFT of extended sequence and N-points DCT of original sequence

104 Compare with DFT:energy compaction property DCT DFT

105 summary 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 series 8.4 fourier representation of finite-duration sequences: Definition of the discrete fourier transform 8.5 sampling the fourier transform (point of sampling) 8.6 properties of the fourier transform 8.7 linear convolution using the discrete fourier transform 8.8 the discrete cosine transform (DCT)

106 requirements ： definition, calculation and properties of DFS; derivation of definition of DFT ： DFS or spectral sampling; concepts of spectral sampling, ， time-domain periodic extension; properties of DFT ： linearity 、 circular shift, symmetry, circular convolution 、 paswal’s theory; relationship between linear and circular convolution; derivation of definition DCT and comparison with DFT. key and difficulty ： spectral sampling and properties of DFT

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