第4章快速傅里葉變換

1、第第4章章 快速傅里葉變換快速傅里葉變換 (fast fourier transformfft)4.1 引言引言4.2 基基2fft算法算法4.1 引言引言) 1() 1 ()0()()0()1(01000100nxwxwxwwnxxnnnnnnnn) 1() 1 ()0()() 1 ()1(11101101nxwxwxwwnxxnnnnnnnn) 1() 1 ()0()()()1(1010nxwxwxwwnxkxnknknknnnnknnkwnxenxnxkxnnknnnnknnjn0 )()()(dft)(10102) 1() 1 ()0() 1() 1 ()0()1()1(2)1(1)1

2、(0)1()1(1211101)1(0201000nxxxwwwwwwwwwwwwnxxxnnnnnnnnnnnnnnnnnnnnnwkxekxnkxnxnknknnknknj0 )()(1)(idft)(10102)1() 1 ()0(1)(1)0()1(01000100nxwxwxwnwkxnxnnnnnkkn)1() 1 ()0(1)(1) 1 ()1(11101101nxwxwxwnwkxnxnnnnnkkn)1() 1 ()0(1)(1)()1(1010nxwxwxwnwkxnnxnnnnnnnnkknn) 1() 1 ()0(1) 1() 1 ()0()1()1(2)1(1)1(0

3、)1()1(1211101)1(0201000nxxxwwwwwwwwwwwwnnxxxnnnnnnnnnnnnnnnnnnn) 1() 1 ()0() 1() 1 ()0()1()1(2)1(1)1(0)1()1(1211101)1(0201000nxxxwwwwwwwwwwwwnxxxnnnnnnnnnnnnnnnnnnn) 1() 1 ()0(1) 1() 1 ()0()1()1(2)1(1)1(0)1()1(1211101)1(0201000nxxxwwwwwwwwwwwwnnxxxnnnnnnnnnnnnnnnnnnn計(jì)算計(jì)算x(k)的一個(gè)值需要的一個(gè)值需要n次復(fù)數(shù)乘法和次復(fù)數(shù)乘法和

4、(n-1)次復(fù)數(shù)加法，計(jì)算次復(fù)數(shù)加法，計(jì)算x(k)的所有的所有n個(gè)值需要個(gè)值需要nn次復(fù)數(shù)乘法和次復(fù)數(shù)乘法和n(n-1)次復(fù)數(shù)加法。次復(fù)數(shù)加法。nnwnxekxnkxnxnkwnxenxnxkxnknknnkknnjnnnknnnnknnjn0)()(1)()(0)()()()(1010210102 idft dft一、一、時(shí)域抽取法基時(shí)域抽取法基2fft原理原理 4.2 基基2fft算法算法將長度為將長度為n的序列的序列x(n) 按奇偶分解為兩個(gè)按奇偶分解為兩個(gè)n/2點(diǎn)的子序列點(diǎn)的子序列 則則x(n)的的dft為為12 1 0 ) 12()(12 1 0 )2()(21nrrxrxnrrxr

5、x，1 1 0 , )()()()()()() 12()2()()()()(dft)(12/0212/212/02/112/02212/02112/0)12(12/0210nkkxwkxwrxwwrxwrxwwrxwrxwrxwnxwnxwnxnxkxnrknkrnknnrkrnnrkrnknnrkrnnrrknnrkrnnknnnknnnnknnn，奇數(shù)偶數(shù)2/212/02/222/112/02/112110)(dft)()()(dft)()(1, 1 , 0 )()()()(nnrkrnnnrkrnknnnknnrxwrxkxrxwrxkxnkkxwkxwnxkx，1 1 0 , )()(

6、)()()()() 12()2()()()()(dft)(12/0212/212/02/112/02212/02112/0)12(12/0210nkkxwkxwrxwwrxwrxwwrxwrxwrxwnxwnxwnxnxkxnrknkrnknnrkrnnrkrnknnrkrnnrrknnrkrnnknnnknnnnknnn，奇數(shù)偶數(shù)x(k)按前按前n/2點(diǎn)和后點(diǎn)和后n/2點(diǎn)分開表示點(diǎn)分開表示12 1 0)()()2()()()(2121nkkxwkxnkxkxwkxkxknkn，abca cba cb圖圖4.2.2 n點(diǎn)點(diǎn)dft一次時(shí)域抽取分解運(yùn)算流圖（一次時(shí)域抽取分解運(yùn)算流圖（n=8）n/2

7、點(diǎn)dftwn0n/2點(diǎn)dftwn1wn2wn3x(0)x1(0)x(2)x(4)x(6)x(1)x(3)x(5)x(7)x1(1)x1(2)x1(3)x2(0)x2(1)x2(2)x2(3)x(0)x(1)x(2)x(3)x(4)x(5)x(6)x(7)12 1 0)()()2()()()(2121nkkxwkxnkxkxwkxkxknkn，abca cba cb)2()(1rxrx) 12()(2rxrx12 1 0nr，圖圖4.2.2 n點(diǎn)點(diǎn)dft一次時(shí)域抽取分解運(yùn)算流圖（一次時(shí)域抽取分解運(yùn)算流圖（n=8）n/2點(diǎn)dftwn0n/2點(diǎn)dftwn1wn2wn3x(0)x1(0)x(2)x(4

8、)x(6)x(1)x(3)x(5)x(7)x1(1)x1(2)x1(3)x2(0)x2(1)x2(2)x2(3)x(0)x(1)x(2)x(3)x(4)x(5)x(6)x(7)圖圖4.2.2包括兩個(gè)包括兩個(gè)n/2點(diǎn)點(diǎn)dft和和n/2個(gè)蝶形，每個(gè)個(gè)蝶形，每個(gè)n/2點(diǎn)點(diǎn)dft需要需要(n/2)(n/2)次復(fù)數(shù)乘法和次復(fù)數(shù)乘法和(n/2-1)(n/2)次復(fù)數(shù)加法運(yùn)算，每個(gè)次復(fù)數(shù)加法運(yùn)算，每個(gè)蝶形只有一次復(fù)數(shù)乘法運(yùn)算和兩次復(fù)數(shù)加法運(yùn)算。所以，總的復(fù)蝶形只有一次復(fù)數(shù)乘法運(yùn)算和兩次復(fù)數(shù)加法運(yùn)算。所以，總的復(fù)數(shù)乘法次數(shù)為：數(shù)乘法次數(shù)為：2/2/ ) 1(2/2)2/(22nnnnn總的復(fù)數(shù)加法次數(shù)為：總的復(fù)

9、數(shù)加法次數(shù)為：2/2)2/(2)2/() 12/(2nnnn圖圖4.2.3 n點(diǎn)點(diǎn)dft二次時(shí)域抽取分解運(yùn)算流圖（二次時(shí)域抽取分解運(yùn)算流圖（n=8） )(3rx )(4rx )(5rx )(6rx圖圖4.2.4 n點(diǎn)點(diǎn)fft運(yùn)算流圖（運(yùn)算流圖（n=8）分組分組方法方法原始數(shù)據(jù)原始數(shù)據(jù)序列序列n/2n/2分組分組n/4n/4分組分組( (最終分組結(jié)果最終分組結(jié)果) )計(jì)算計(jì)算結(jié)果結(jié)果按按奇奇偶偶分分組組x(0)x(0)x x1 1(0)(0)x(0)x(0)x x3 3(0)(0)x(0)x(0)x(0)x(0)x(1)x(1)x x1 1(1)(1)x(2)x(2)x x3 3(1)(1)x(

10、4)x(4)x(1)x(1)x(2)x(2)x x1 1(2)(2)x(4)x(4)x x4 4(0)(0)x(2)x(2)x(2)x(2)x(3)x(3)x x1 1(3)(3)x(6)x(6)x x4 4(1)(1)x(6)x(6)x(3)x(3)x(4)x(4)x x2 2(0)(0)x(1)x(1)x x5 5(0)(0)x(1)x(1)x(4)x(4)x(5)x(5)x x2 2(1)(1)x(3)x(3)x x5 5(1)(1)x(5)x(5)x(5)x(5)x(6)x(6)x x2 2(2)(2)x(5)x(5)x x6 6(0)(0)x(3)x(3)x(6)x(6)x(7)x(7)x x2 2(3)(3)x(7)x(7)x x6 6(1)(1)x(7)x(7)x(7)x(7)離散數(shù)據(jù)點(diǎn)按奇偶分組過程離散數(shù)據(jù)點(diǎn)按奇偶分組過程 【例例】假設(shè)時(shí)域連續(xù)信號(hào)假設(shè)時(shí)域連續(xù)信號(hào)x(t)=x1(t)+x2(t)+x3(t)，其中，其中x1(t)=3sin(30t)， x2(t)=2sin(40t)， x3(t)=sin(60t)。(1)如果用如果用fft對(duì)對(duì)x(t)進(jìn)行頻譜分析，問采樣頻率進(jìn)行頻譜分析，問采樣頻率fs和采樣點(diǎn)數(shù)和采樣點(diǎn)數(shù)n應(yīng)如何選應(yīng)如何選擇，才能精確求出擇，才能精確求出x1(t)、x2(t)、x3