数字信号处理a(双语)chapter 5-finite-length discrete transformb-1140318.ppt
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1、Chapter 5,Finite-Length Discrete Transforms,Part B: Operations on Finite-Length Sequences and DFT Properties,Circular Time-reversal of a Sequence (Section 2.3.1 ) Circular Shift of a Sequence(Section 2.3.2 and 5.7 ) Circular Convolution(Section 5.4 and 5.7 ) Classification of Finite-Length Sequences
2、 ( Section 5.5) DFT Symmetry Relations and Theorems (Section 5.6 and 5.7) Fourier-Domain Filtering (Section 5.8) Computation of the DFT of Real Sequences (Section 5.9) Linear Convolution Using the DFT( Section 5.10),Operations on Finite-Length Sequences,Consider the length-N sequence xn defined for
3、0nN-1 A time-reversal operation on xn will result in a length-N sequence defined for -(N-1)n0 A linear time-shift of xn by integer-valued M will result in a length-N sequence xn + M no longer defined for 0nN-1 A convolution sum of two length-N sequences defined for 0nN-1 will result in a sequence of
4、 length 2N-1 defined for 0n2N-2,Operations on Finite-Length Sequences,We need to define new type of time reversal, time-shifting and convolution operation, so that the resultant length-N sequences are also are in the range 0nN-1,1、Circular Time-Reversal Operation,See in textbook Section 2.3.1,Modulo
5、 Operation(取模运算),The time-reversal operation on a finite-length sequence is obtained by the modulo operation N= m modulo N where m and N is any integer, and 0,1,N-1 be a set of N positive integers r= N is called the residue(余数), which is an integer with a value between 0 and N-1 r=m+lN, where l is a
6、 positive or negative integer chosen to make m+lN an integer between 0 and N-1,Modulo Operation(取模运算),Example For N = 7 and m = 25, we have r=25+7l=25-7*3=4,Thus 7 =4 For N = 7 and m = -15, we get r=-15+7l=-15+7*3=6,Thus 7 =6,1、Circular Time-Reversal Operation,The circular time-reversal version yn o
7、f a length-N sequence xn defined for 0nN-1 is given by yn=xN Example Consider xn=x0, x1, x2, x3, x4 Its circular time-reversed version is given by yn=x5=x0, x4, x3, x2, x1,2、Circular Shift of a Sequence,See in Section 2.3.2 and 5.7,2、Circular Shift of a Sequence,This property is analogous to the tim
8、e-shifting property of the DTFT, but with a subtle difference Consider length-N sequences defined for 0nN1,2、Circular Shift of a Sequence,2、Circular Shift of a Sequence,A right circular shift by n0 is equivalent to a left circular shift by Nn0 sample periods.,2、Circular Shift of a Sequence,A circula
9、r shift by an integer number n0 greater than N is equivalent to a circular shift by N,Example: 6=1, x6=x6,2、Circular Shift of a Sequence,The length-N sequence is displayed on a circle at N equally spaced points The circular shift operation can be viewed as a clockwise or anti-clockwise rotation of t
10、he sequence by n0 sample spacings,Compare the shift and circular shift,shift,circularshift,周期延拓,shift,取主值区间0N-1,2、Circular Shift of a Sequence,2、Circular Shift of a Sequence,3、Circular Convolution,See in Section 5.4 and 5.7,3. Circular Convolution,Circular convolution is analogous to linear convolut
11、ion, but with a subtle difference Consider two length-N sequences, g(n) and h(n) respectively,3、Circular Convolution,3、Circular Convolution,3、Circular Convolution,Example Determine the 4-point circular convolution of the two length-4 sequences: gn=1 2 0 1, hn=2 2 1 1 as skecthed below,3、Circular Con
12、volution,The result is a length-4 sequence yCn From the above we observe,N,3、Circular Convolution,Likewise,3、Circular Convolution,yCn,(a) yc0,循环卷积过程图解,3、Circular Convolution,The circular convolution can also be computed using a DFT-based approach The N-point circular convolution can be written in ma
13、trix form as,3、Circular Convolution,Note: 1、The element in each row of the matrix are obtained by circularly rotating the elements of the previous row to the right by one position. Such a matrix is called a circulant matrix(轮换矩阵、 循环行列式矩阵) 2、使用矩阵形式计算循环卷积前,需要通过补零把参与循环卷积的两个输入序列扩充成相同长度,且此长度等于DFT的点数,3、Ci
14、rcular Convolution,Example Now let us extend the two length-4 sequences to length 7 by appending each with three zero-valued samples, i.e.,3、Circular Convolution,We next determine the 7-point circular convolution of gen and hen:,Matrix method:,3、Circular Convolution,As can be seen from the above tha
15、t yn is precisely the sequence yLn obtained by a linear convolution of gn and hn Try to think: What is the relation between the circular convolution and the linear convolution?,yCn,3. Circular Convolution,3、Circular Convolution,3、Circular Convolution,3、Circular Convolution,3、Circular Convolution,3、C
16、ircular Convolution,3、Circular Convolution,In this case, hence, the procedure of circular convolution is equivalent to that of linear convolution over the region of principle value Obviously, this conclusion always holds when the length of Circular Convolution is not less than 7 Summary Provided tha
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