數(shù)字信號(hào)處理與DSP器件：第11章 DFT和FFT處理

1、第11章 DFT和FFT處理DFT基礎(chǔ)與傅里葉變換關(guān)系與傅里葉級(jí)數(shù)關(guān)系DFT窗效應(yīng)頻譜圖FFT基礎(chǔ)discrete Fourier transform (DFT) 離散傅里葉變換inverse DFT 逆離散傅里葉變換phase spectrum 相位頻譜frequency spacing頻率間隔resolution分辨率smear模糊spectral leakage 頻譜泄漏spectrogram頻譜圖fast Fourier transform (FFT) 快速傅里葉變換butterfly 蝶形11.1 DFT 基礎(chǔ)DTFT定義所需采樣點(diǎn)是無限的，在計(jì)算機(jī)上無法實(shí)現(xiàn)DFT定義輸入和輸出個(gè)數(shù)

2、都等于N幅度頻譜是偶函數(shù)相位頻譜是奇函數(shù)二者都是周期函數(shù)，周期為NIDFT逆變換也具有N為周期的周期性。FIGURE 11-1 Relationship between DTFT and DFT, signal and IDFT.Joyce Van de VegteFundamentals of Digital Signal ProcessingCopyright 2002 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458All rights reserved.FIGURE 11-1 Relationship betw

3、een DTFT and DFT, signal and IDFT.Joyce Van de VegteFundamentals of Digital Signal ProcessingCopyright 2002 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458All rights reserved.FIGURE 11-1 Relationship between DTFT and DFT, signal and IDFT.Joyce Van de VegteFundamentals of Digital Signa

4、l ProcessingCopyright 2002 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458All rights reserved.FIGURE 11-1 Relationship between DTFT and DFT, signal and IDFT.Joyce Van de VegteFundamentals of Digital Signal ProcessingCopyright 2002 by Pearson Education, Inc.Upper Saddle River, New Jers

5、ey 07458All rights reserved.當(dāng)它們對(duì)同一組采樣值進(jìn)行運(yùn)算時(shí)，必有：當(dāng)K從0-(N-1)時(shí)，相應(yīng)數(shù)字頻率從0- 那么，DFT的前N個(gè)點(diǎn)覆蓋了0到采樣頻率fs之間的模擬頻率。Review Correlations of sequencesIt is a measure of the degree to which two sequences are similar. Given two real-valued sequences x(n) and y(n) of finite energy,CrosscorrelationAutocorrelationThe index

6、 l is called the shift or lag parameter.The special case: y(n)=x(n)由歐拉恒等式：信號(hào)與正弦越相關(guān)，頻譜分量就越大。DFT的每個(gè)分量可看成是窄帶濾波器的輸出，窄帶濾波器中心位于 弧度。DFT整體可看成是由窄帶毗鄰濾波器構(gòu)成的濾波器組。Crosscorrelation頻率采樣點(diǎn)以fs/N為間隔，該頻率間隔被稱為DFT的分辨率，因?yàn)樗枋隽薉FT分辨相鄰信號(hào)頻率的程度。頻率間隔越小，分辨率越好。DFT頻率間隔=DFT分辨率=fs/N假定采樣頻率保持不變，當(dāng)采樣點(diǎn)越多時(shí)，DFT分辨率越好。DFT分量Xk位于以下頻率處：K=0到k=N/

7、2的DFT點(diǎn)攜帶了DFT全部必要的幅度和相位信息。其余點(diǎn)只是基帶重要信號(hào)頻率的鏡像副本，是采樣的人為結(jié)果。例：為了強(qiáng)調(diào)DFT的濾波器組解釋。下圖給出了兩個(gè)40秒的余弦波，這兩個(gè)信號(hào)疊加并與隨機(jī)噪聲一起構(gòu)成了信號(hào)x(t)該信號(hào)實(shí)質(zhì)上包含了兩個(gè)主要的頻率成分1/16Hz和3/8Hz,現(xiàn)以fs=6.4Hz進(jìn)行采樣。采樣信號(hào)中兩個(gè)主要頻率分量的數(shù)字頻率為： FIGURE 11-9 Two cosine signals for Example 11.4.Joyce Van de VegteFundamentals of Digital Signal ProcessingCopyright 2002 by

8、 Pearson Education, Inc.Upper Saddle River, New Jersey 07458All rights reserved.FIGURE 11-11 First 16 DFT frequency components of Example 11.4.Joyce Van de VegteFundamentals of Digital Signal ProcessingCopyright 2002 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458All rights reserved.F

9、IGURE 11-13 DFT magnitude spectrum from k = 0 to k = N/2 for Example 11.4.Joyce Van de VegteFundamentals of Digital Signal ProcessingCopyright 2002 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458All rights reserved.由于dft可看成是一組毗鄰窄帶濾波器，每個(gè)濾波器以數(shù)字頻率 弧度為中心，因而頻譜峰值應(yīng)位于 k=2.5和k=15處。由于k 必須是整數(shù)， k

10、=2.5處的峰又分為k=2和k=3處的兩個(gè)小峰。頻譜后半部分的峰是基帶頻率的鏡像。DFT不能超過分辨率所允許的范圍而去準(zhǔn)確定位頻率。當(dāng)DFT中沒有頻率與所分析信號(hào)的重要頻率相符時(shí)，DFT就導(dǎo)致了真實(shí)頻譜的模糊。DFT并不區(qū)分周期和非周期信號(hào)。但有助于揭示信號(hào)是否具有周期性。非周期信號(hào)的DFT頻譜包絡(luò)將呈現(xiàn)大小變化以及間隔變化的許多凸起，但沒有清晰的尖峰。周期信號(hào)的幅度頻譜中固定間隔上有確定的窄的尖峰。與傅里葉變換的關(guān)系傅里葉變換給出了模擬信號(hào)的頻譜特性，不受采樣和量化的影響，計(jì)算中需要準(zhǔn)確的信號(hào)數(shù)學(xué)函數(shù)，多數(shù)情況下難以實(shí)現(xiàn)數(shù)字域傅里葉分析的目的是盡可能的逼近傅里葉變換所給出的信息DTFT提供了

11、模擬信號(hào)采樣形式的頻譜信息，但受頻譜混疊和量化等誤差的影響DFT只是獲得DTFT采樣形式的適合處理器的方法,DFT也接近原始傅里葉變換FIGURE 11-28 Deriving the DFT from the Fourier transform.Joyce Van de VegteFundamentals of Digital Signal ProcessingCopyright 2002 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458All rights reserved.FIGURE 11-28 Derivin

12、g the DFT from the Fourier transform.Joyce Van de VegteFundamentals of Digital Signal ProcessingCopyright 2002 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458All rights reserved.FIGURE 11-28 Deriving the DFT from the Fourier transform.Joyce Van de VegteFundamentals of Digital Signal P

13、rocessingCopyright 2002 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458All rights reserved.FIGURE 11-28 Deriving the DFT from the Fourier transform.Joyce Van de VegteFundamentals of Digital Signal ProcessingCopyright 2002 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458A

14、ll rights reserved.DFT計(jì)算的三個(gè)主要步驟: 1時(shí)域采樣 2時(shí)域加窗 3頻域采樣例:一段音樂以44.1kHz進(jìn)行采樣,DFT窗的長度為23.22毫秒.求:窗內(nèi)將有多少個(gè)時(shí)域采樣點(diǎn)?這些時(shí)域采樣點(diǎn)將產(chǎn)生多少個(gè)DFT采樣值？DFT的分辨率是多少？與傅里葉級(jí)數(shù)的關(guān)系定義式完全相同實(shí)質(zhì)區(qū)別僅是解釋不同DFS分析的是自然的周期信號(hào)DFT分析的是周期或非周期信號(hào)的加窗部分，不過它的IDFT總是周期的周期信號(hào)的幅度頻譜用這兩種方法計(jì)算過程和結(jié)果都相同DFT窗效應(yīng)當(dāng)所研究的信號(hào)是時(shí)不變信號(hào)，而且采樣頻率一定時(shí)，用長的窗可以給出非常準(zhǔn)確的信號(hào)頻譜，DFT分辨率盡可能小時(shí)，模糊會(huì)最小如果信號(hào)的

15、特性隨時(shí)間變化，長的窗可能會(huì)使計(jì)算結(jié)果發(fā)生混淆DFT處理數(shù)字?jǐn)?shù)據(jù)，不可避免地將出現(xiàn)由采樣引起的頻譜混疊和量化誤差加窗是DFT誤差的另一主要來源FIGURE 11-31 Ideal magnitude spectrum for pure sinusoid.Joyce Van de VegteFundamentals of Digital Signal ProcessingCopyright 2002 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458All rights reserved.FIGURE 11-32 Spec

16、trum of sinusoid using rectangular windows of different lengths.Joyce Van de VegteFundamentals of Digital Signal ProcessingCopyright 2002 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458All rights reserved.對(duì)于特性不隨時(shí)間變化的信號(hào)，窗口越長對(duì)頻譜的近似越好對(duì)一定的采樣頻率，窗加長，DFT的分辨率改善，標(biāo)志正弦頻率的峰變窄；而窗口越短，峰越寬所有對(duì)數(shù)頻譜表明旁瓣值

17、大約比主峰值小13dB.這些旁瓣就是頻譜泄漏的實(shí)例窗函數(shù)的邊沿越陡峭，泄漏越大頻譜泄漏有時(shí)又稱為振鈴它是由采樣信號(hào)的頻譜和窗的頻譜卷積引起的可用邊界平滑的窗來減小振鈴FIGURE 11-33 Spectrum of sinusoid using nonrectangular windows.Joyce Van de VegteFundamentals of Digital Signal ProcessingCopyright 2002 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458All rights reserve

18、d.加窗沒有使得頻譜峰變窄，事實(shí)上反而將其加寬了，但它們極大的減小了峰兩邊的頻譜泄漏要使峰變窄，可用能集合大量采樣值的更長的窗口FIGURE 11-34 Approximation to ideal cosine spectrum.Joyce Van de VegteFundamentals of Digital Signal ProcessingCopyright 2002 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458All rights reserved.當(dāng)旁瓣比主瓣值至少小40dB時(shí)，才可獲得清晰正弦峰的較好

19、近似11.5 頻譜圖單個(gè)DFT只能提供信號(hào)的有限信息，更有幫助的是DFT的集合，每個(gè)DFT給出不同時(shí)間間隔的信息頻譜圖就是這樣的集合它是頻譜對(duì)時(shí)間的圖形FIGURE 11-36 Spectogram of “Mary Had a Little Lamb.”Joyce Van de VegteFundamentals of Digital Signal ProcessingCopyright 2002 by Pearson Education, Inc.Upper Saddle River, New Jersey 07458All rights reserved. 圖中每一個(gè)豎段包含了時(shí)間上的一個(gè)窗的DFT幅度幅度越大顯示越黑，幅度越小顯示越淺每一間隔中的最低頻率處的條紋為基頻各間隔中基頻上方等間距的條紋為其他諧波11.6 FFT基礎(chǔ)FFT與DFT的輸出相同，但運(yùn)算量要小得多最常用的FFT是基2時(shí)域抽取法FFT.這種FFT的基本原理是將一個(gè)N點(diǎn)的計(jì)算分解為兩個(gè)N/2點(diǎn)的計(jì)算，每個(gè)N/2點(diǎn)的計(jì)算在進(jìn)一步分解為N/4點(diǎn)的計(jì)算，