Chapter 8 The Discrete Fourier Transform Biomedical Signal processing Chapter 8 The Discrete Fourier Transform Zhongguo Liu Biomedical Engineering School of Control Science and Engineering, Shandong University 山东省精品课程《生物医学信号处理(双语)》 http://course.sdu.edu.cn/bdsp.html 2019/5/4 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 8.8 the discrete cosine transform (DCT)

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 samples, equally spaced in frequency, of the Discrete-time Fourier transform (DTFT) of the signal.

8.0 Introduction Derivation and interpretation of DFT is based on 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.

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 Fourier series representation of continuous-time periodic signals require infinitely many complex exponentials Not that for discrete-time periodic signals, since

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

Discrete Fourier Series Pair The Fourier series representation of a periodic sequence: To obtain the Fourier series coefficients we multiply both sides by for 0nN-1 and then sum both the sides , we obtain

Discrete Fourier Series Pair Problem 8.51, HW

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

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

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

Ex. 8.1 determine the DFS of a impulse train Consider the periodic impulse train n 1 2 …… N-1 N N+1 N+2 -1 -2 -N+1 -N N points Solution:

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

N points n …… N points k …… 1 2 N-1 N N+1 N+2 -1 -2 -N+1 -N 1 2 N-1 N 1 2 …… N-1 N N+1 N+2 -1 -2 -N+1 -N N points k 1 2 …… N-1 N N+1 N+2 -1 -2 -N+1 -N N points

Example 8.2 Duality in the Discrete Fourier Series If the discrete Fourier series coefficients is the periodic impulse train, determine the signal. 1 2 …… N -1 -2 -N N points Solution: DFS [ ] å - = 1 ~ N n kn W x k X

…… …… 1 2 N-1 N N+1 N+2 -1 -2 -N+1 -N N points k 1 2 N-1 N N+1 N+2 -1 1 2 …… N-1 N N+1 N+2 -1 -2 -N+1 -N N points k 1 2 …… N-1 N N+1 N+2 -1 -2 -N+1 -N N points n

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

Periodic sequence with period N=10 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 2019年5月4日9时51分 8.2 Properties of the Discrete Fourier Series 8.2.1 Linearity two periodic sequence, both with period N:

8.2 Properties of the Discrete Fourier Series 8.2.2 Shift of a sequence

8.2 Properties of the Discrete Fourier Series 8.2.3 Duality 1 2 …… N-1 n 1 2 …… N-1 n 1 2 …… N-1 k 1 2 …… N-1 k N 8.2.4 Symmetry Properties

8.2.4 Symmetry Problem 8.53, HW

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

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

Example 8.4 Periodic Convolution

8.3 The Fourier Transform (DTFT) of discrete-time Periodic Signal Periodic sequences are neither absolutely summable nor square summable, hence they don’t have a strict Fourier Transform (DTFT) :

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

Ex. 8.5 determine the Fourier Transform of a periodic impulse train the periodic impulse train 1 2 …… N -1 -2 -N N points n Solution: The DFS was calculated previously to be N points K 1 2 …… N-1 N -1 -2 -N Therefore the Fourier transform is N points ω 1 2 …… N-1 N -1 -2 -N

Relation between Finite-length and Periodic Signals 1 2 N …… -1 -2 -N 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 1 2 N …… -1 -2 -N The Fourier transform of the periodic sequence is N points ω 1 2 …… N-1 N -1 -2 -N

Discrete Fourier Series (DFS) Let Analysis equation: Synthesis equation: Review

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.

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

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:

Ex. 8.6 Relation between FS coefficients and Fourier transform of one period x[n] of The DFS coefficients The Fourier transform

8.4 Sampling the Fourier Transform x[n] Length M, may be ∞ Consider an aperiodic sequence with Fourier transform ,and assume that a sequence is obtained by sampling at frequency Length M, N may not be the same could be Fourier series coefficients of a periodic sequence with period N .

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

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

Sampling the Fourier Transform to recover x[n]

Sampling the Fourier Transform to recover x[n] Samples of DTFT X(e jw) 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 M, and we take sufficient number N (N≥M) of samples of its DTFT X(e j w) then X(e jw) can be recovered from these samples, and the original sequence x[n] can be recovered by infinite shifted replicas

Discrete Fourier Transform or DFT Given a finite-length sequence x[n], we can form a periodic sequence , which in turn can be represented by a DFS Given a sequence of Fourier coefficients , we can find , and then obtain x[n] .

Discrete Fourier Transform or DFT to represent finite length x[n] by FS : 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.

2019年5月4日9时51分 8.5 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 -∞<n<∞; notation,用有限长度序列x[m](0<m<N)表示 ；((n))N 表示n以N为模.

Discrete Fourier Transform For , the DFS is with period N The Discrete Fourier Transform of is DFT: -∞<k<∞

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

Discrete Fourier Transform pairs Analysis equation Synthesis equation

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

四种傅立叶变换

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

Ex. 8.7 The DFT of a Rectangular Pulse 2019年5月4日9时51分 Ex. 8.7 The DFT of a Rectangular Pulse x[n] If we consider x[n] of length 10, form We get a different set of DFS coefficients Still samples of the DTFT but in different places Different N Different

8.6 Properties of the Discrete Fourier Transform 8.6.1 Linearity If has length and has length ,

8.6.2 Circular Shift of a Sequence

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

8.6.2 Circular Shift of a Sequence ？

8.6.2 Circular Shift of a Sequence

Ex. 8.8 Circular Shift of a Sequence Figure 8.12

8.6.3 Duality Time domain Frequency domain

Ex.8.9 The Duality Relationship for the DFT 2019年5月4日9时51分 Ex.8.9 The Duality Relationship for the DFT

8.6.4 Symmetry Properties of x[n] periodic conjugate symmetric components periodic conjugate-antisymmetric components of x[n]

8.6.4 Symmetry Properties

xep[n] and xop[n] are not equivalent to xe[n] and xo[n] as in: 8.6.4 Symmetry Properties xep[n] and xop[n] are not equivalent to xe[n] and xo[n] as in: conjugate symmetric components conjugate-antisymmetric components It can be shown in Problem 56

8.6.4 Symmetry Properties Defination as before: so compare

8.6.4 Symmetry Properties

8.6.4 Symmetry Properties

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

8.6.5 Circular Convolution

8.6.5 Circular Convolution

Ex. 8.10 find Circular Convolution of x2[n] and x1[n](Delayed Impulse Sequence). Solution1： N=5

Ex. 8.10 Circular Convolution with a Delayed Impulse Sequence Solution2：

Example 8.11 Circular Convolution of Two Rectangular Pulses Solution1： DFT length Circular convolution length ǂ linear convolution

Ex. 8.11 Circular Convolution of Two Rectangular Pulses Solution2： DFT length Circular convolution length = linear convolution

8.6.6 Summary of Properties of the Discrete Fourier Transform

8.6.6 Summary of Properties of the Discrete Fourier Transform

8.7 Linear Convolution using the Discrete Fourier Transform FFT algorithms are available for computing the DFT of a finite-duration sequence. Implement a convolution of two sequences by the following procedure: 1. Compute the N-point DFT and of the two sequence and 2. Compute for 3. Compute as the inverse DFT of

8.7 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.

8.7.1 Linear Convolution of Two Finite-Length Sequences P length 8.7.1 Linear Convolution of Two Finite-Length Sequences P length L for is maximum length of

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

8.7.2 Circular Convolution as Linear Convolution with Aliasing For finite sequence N sampling The inverse DFT of is one period of : If N≥length of x[n], then xp[n]= x[n]

8.7.2 Circular Convolution as Linear Convolution with Aliasing DTFT of is N sampling Define a DFT: The inverse DFT of is ： If N≥length of x3[n], then

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

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

Ex. 8.12 Circular Convolution as Linear Convolution with Aliasing. DFT 6 points shift right of the linear convolution N=6 6 points shift left of the linear convolution 6 points circular convolution= linear convolution with aliasing N=6 N=12 12 points circular convolution = linear convolution

Which sequence values in L-point Circular Convolution (when Aliasing occurs )equal that of Linear Convolution? Fig.8.19 Linear Convolution Consider L-point circular convolution of xl[n] (length L ) and x2[n] (length P ), where P < L. Fig.8.20 Circular Convolution N点圆周卷积没有时间混叠的点

circular convolution “ wraps around ". View the process of forming the circular convolution x3p[n] through linear convolution plus aliasing, as taking the (P - 1) values of x3[n] from n=L to n=L+P -2 and adding them to the first (P - 1) values of x3[n].

8.7.3 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.

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.

Zero-Pading The circular convolution can be achieved by multiplying the DFTs of and . L P N 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

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

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

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

Circular Convolution as Linear Convolution with Aliasing

overlap-save method input (1) segment into sections of length L, overlap P-1 points; (2) fill 0 into and some section of , then do L points FFT L=25 P-1 points (3) calculate P-1 points P-1 (4) the output for this section is L-P+1 points of y[n] n=P-1,…L-1 points 圆周卷积中后L-P+1个点结果与线性卷积相等

8.8 the discrete cosine transform (DCT) The DFT is a class of finite-length transform representations of the form , basis sequences is complex Are there real basis sequences that would yield a real-valued transform ?

8.8 the discrete cosine transform(DCT) An orthogonal transform for real sequences is the discrete cosine transform (DCT). The DCT is closely related to DFT and has become especially useful in signal- processing applications, particularly speech and image compression.

8.8.1 Definitions of the DCT DFT DCT-1 DCT-2 just as the DFT involves an implicit assumption of periodicity, the DCT involves implicit assumptions of both periodicity and even symmetry. symmetric and periodic extension of signal, then do DFS and get DCT by taking the dominant period。 DFT DCT-1 DCT-2

8.8.1 Definitions of the DCT DCT-1 DCT-2 DCT-3 DCT-4

8.8.2 Definitions of the DCT-1, DCT-2

8.8 the discrete cosine transform(DCT) other Definitions of DCT-1, DCT-2 , DCT-3 , DCT-4

other Definitions of DCT-1, DCT-2 , DCT-3 , DCT-4

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

8.8.3 Relationship between the DFT and the DCT-1 one period of the periodic sequence 𝑥 1[n] defines the finite-length sequence (2N-2)-point DFT of the (2N-2)-point sequence x1[n]:

8.8.4 Relationship between the DFT and the DCT-2 one period of the periodic sequence 𝑥 2[n] defines the finite-length sequence 2N-point DFT of the 2N-point sequence x2[n]:

8.8.4 Relationship between the DFT and the DCT-2

8.8 the discrete cosine transform(DCT) 8.8.5 Energy Compaction Property of the DCT-2 Parseval's theorem for DCT-1 DCT-2

8.8.5 Energy Compaction Property of the DCT-2 Ex. 8.1.3 Energy Compaction in the DCT-2

Ex. 8.1.3 Energy Compaction in the DCT-2 (a) Real part of 32-point DFT (symetric) (b) Imaginary part of 32-point DFT (c) 32-point DCT-2 of the test signal

Ex. 8.1.3 Energy Compaction in the DCT-2 (a) Real part of 32-point DFT (symetric) (b) Imaginary part of 32-point DFT

Ex. 8.1.3 Energy Compaction in the DCT-2 (c) 32-point DCT-2 of the test signal

Ex. 8.1.3 Energy Compaction in the DCT-2 Comparison of truncation errors for DFT and DCT-2

8.8.6 Applications of the DCT DCT is useful in a number of signal-processing applications, particularly speech and image compression.

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

Zhongguo Liu_Biomedical Engineering_Shandong Univ. Chapter 8 HW 8.4, 8.7, 8.10, 8.14, 8.17 8.51,8.52, 8.53, Zhongguo Liu_Biomedical Engineering_Shandong Univ. 124 2019/5/4 返 回 上一页 下一页