【正文】 由此引起的誤差稱為量化誤差 Basic of Digital Signal Processing數(shù)字信號處理基礎(chǔ) Quantization is realized by A/D transducer. Quantization error is decided by its resolution. If number of bit (word length) of a A/D transduc。 整周期采樣的結(jié)果是使得因頻域抽樣后所拓展的周期時域信號完全等同于實際的周期信號。inti r l] periods is that periodical time domain signal prolonged by being sampled in frequency domain is pletely equal to the real periodic signal. 上式表明：只有信號的截斷長度 T為待分析信號周期的整數(shù)倍時，才可能使譜線落在 f0，獲得準(zhǔn)確的頻譜。 Basic of Digital Signal Processing數(shù)字信號處理基礎(chǔ) TNfNTf ss 11 ????Equation above indicates: So long as truncation length T of signal is integral multiple of periods of signal to be analysed, it can make spectral line existing at f0 and obtain exact is truncation of integral periods. Result of truncation[tr 39。對于簡諧信號，為了得到特定頻率 f0的譜線，必須滿足 整數(shù)?? ff 0 整數(shù)?0TTT： time length of signal 。 為了解決上述矛盾，可以采用 ZOOMFFT或 ChipZ變換，或采用基于模型的現(xiàn)代譜分析技術(shù)。tin mi]矛盾 above, ZOOMFFT, ChipZ transform or modelbased modern spectrum analysis technology may be adopted. 顯然，可以通過降低 fs或提高 N以提高 ?f。 ?f 越小，分辨力越高，被擋住的頻率成分越少。intid frequency truncation 頻率分辨力、整周期截斷 Interval[39。 Basic of Digital Signal Processing數(shù)字信號處理基礎(chǔ) In the time domain , as long as satisfying sampling theorem, fence effect doesn’t lose signal ，只要滿足采樣定理，柵欄效應(yīng)不會丟失信號信息 Basic of Digital Signal Processing數(shù)字信號處理基礎(chǔ) In frequency domain, fence effect may lose important or characteristic frequency ponents (due to leakage, existence of other frequency near the lost frequency ponent .) It will produce results of spectrum analysis without signification意義 . 在頻域，則有可能丟失的重要的或具有特征的頻率成分（由于泄漏，丟失頻率成分附近的頻率有可能存在），導(dǎo)致譜分析結(jié)果失去意義。w :did] 擋住 and not shown (regarded as 0.). This phenomenon is called fence effect. Obviously ,sampling must bring fence effect. 經(jīng)頻域采樣后的頻譜僅在各采樣點上存在，而非采樣點的頻譜則被“擋住”無法顯示（視為 0），這種現(xiàn)象稱為 柵欄效應(yīng) 。indeks] of several typical window functions 幾種典型窗函數(shù)的技術(shù)指標(biāo) T y p e of w i n d o w fu n ct i o n 窗函數(shù)類型 W i d t h o f p ri n c i p al p e t al主瓣寬度 T h e b i g g e s t a m p l i t u d e o f s i d e p et al 最大旁瓣幅度 A t t e n u at i o n s p e ee d of s i d e p et al 旁瓣衰減速度 R ecta n g l e w i n d o w矩形窗 2/T 13 dB 6 d B/ o c t T ri an g l e w i n d o w 三角形窗 4/T 26 dB 12 d B / o ct H an n i ng w i n d o w 漢寧窗 4/T 32 dB 1 8 d B / o ct Basic of Digital Signal Processing數(shù)字信號處理基礎(chǔ) sampling in frequency domain and fence effect 頻域采樣與柵欄效應(yīng) Sampling in frequency domain is similar as that in time in frequency would result in signal truncated in time domain prolongating 延拓periodically. Signal truncated in time domain is reformed as a periodic 域采樣與時域采樣類似，頻域采樣導(dǎo)致對時域截斷信號進行周期延拓，將時域截斷信號“改造”為周期信號。trai l] windsow 三角窗 ?????????20221)(TtTttTtw T )2(si nc2)(2 fTTfW T ??Basic of Digital Signal Processing數(shù)字信號處理基礎(chǔ) W(f) 2 T w(t) 1 0 T/2 T/2 t T/2 0 2 T f Hanning window 漢寧窗（余弦窗） ???????????????2022c os2121)(TtTtTttw?? ?)1()1(41)(21)( TfWTfWfWfW RRR ????? )(sinc)( fTTfWR ??其中 Basic of Digital Signal Processing數(shù)字信號處理基礎(chǔ) 2 T w(t) 1 0 T/2 T/2 t W(f) T/2 2 T 0 f exponent[eks39。一個好的窗函數(shù)應(yīng)當(dāng)： 主瓣盡可能窄 （提高頻率分辨力）、 旁瓣相對于主瓣盡可能小 ，且 衰減快 （減小泄漏）。petl]of sinc function will bee narrow. Side petals花瓣 move closer to pricinpal petal. As side petals attenuate rapidly, it could reduce leakage. But obviously number of sampling points increases, it adds calculation burden. ?提高截斷信號長度，即提高矩形窗寬度，此時 sinc函數(shù)主瓣變窄，旁瓣向主瓣密集，由于旁瓣衰減較快，故可減小泄漏，但顯然采樣點數(shù)隨之提高，增加計算負擔(dān)。tr keitid] .The phenomena that signal energy distributes and extends at frequency axis is called leakage. So no matter how high sampling frequency is, signal will produce aliasing inevitably不可避免 . 矩形窗函數(shù)的頻譜為 無限帶寬 的 sinc函數(shù)－即使 x(t)為帶限信號，經(jīng)截斷后必然成為 無限帶寬 信號，這種信號的能量在頻率軸分布擴展的現(xiàn)象稱為 泄漏 。截斷相當(dāng)于對信號進行 加窗處理 ，截斷即是將信號乘以時域的有限寬矩形窗函數(shù)： ????????2021)(TtTttwThat is:sampled x(t)s(t) bees x(t)s(t)w(t). 即：采樣后信號 x(t)s(t)經(jīng)截斷成為 x(t)s(t)w(t)。從理論上講，由于抗混濾波器的非理想特性，信號中高頻分量不可能完全衰減，因此不可能徹底消除混疊。但提高采樣頻率將導(dǎo)致在同樣信號長度下，采樣點數(shù)隨之提高，增加計算負擔(dān)。此即為 采樣定理 。 ? ???????? ???? ?? ???????? r ssns TrfTfSnTtts ??1)()()()()()()( fSfXtstx ?? ? ???????? ???? ????r ss TrfXTfSfXnxF1)()()]([So:所以 Basic of Digital Signal Processing數(shù)字信號處理基礎(chǔ) A,B,C are mistaken as a same line. A、 B、 C 被誤認為是一條曲線 High frequency sine signal is mistaken as a low frequency sine signal. 高頻正弦信號被誤認為是低頻正弦信號 Basic of Digital Signal Processing數(shù)字信號處理基礎(chǔ) x(t) 0 1 2 3 t A B C Ts Aliasing[39。 2|)(|)()( fHfSfS xy ??Signal Analysis in Frequency Domain信號的頻域分析 )()(fHth)()(fYty)()(fXtx}|)(|{log)]([log)}({log 2fHFfSFfSF xy ???? ?? ??qCqCqChxy ??Signal Analysis in Frequency Domain信號的頻域分析 2） Diagnose gear fault by cepstrum 用倒頻譜診斷齒輪故障 )sin()cos1()( 0 ??? ??? ttmAty m?? ? ? ? ?? ? ? ?? ????????? ????????? tmAtmAtAty mm 000 sin2sin2sinGear’s oscillation 齒輪的振動 Frequency of axis of rotation轉(zhuǎn)軸頻率 mesh frequency 嚙合頻率 Power spectrum 功率譜 Cepstrum 倒譜 Signal Analysis in Frequency Domain信號的頻域分析 1/q=1/= 1/q=1/= Basic of Digital Signal Processing數(shù)字信號處理基礎(chǔ) ? basic steps of digital signal processing 數(shù)字信號處理的基本步驟 pretreatment預(yù)處理 A/Dtransform A/D轉(zhuǎn)換 x(t) Digital signal prcessor Or puter 數(shù)字信號 處理器 或 計算機 pretreatment預(yù)處理 A/Dtransform A/D轉(zhuǎn)換 x(t) Results display 結(jié)果顯示 Diagram of digital signal processing system 數(shù)字信號處理系統(tǒng)簡圖 Basic of Digital Signal Processing數(shù)字信號處理基礎(chǔ) ? Signal pretreatment:amplitude modulation, filtering, isolating direct current ponent,demodulation. 信號預(yù)處理 ：幅值調(diào)理、濾波、隔離直流分量、解調(diào)等 ? A/D transfrom: sampling ,quantifying into digital quantity. A/D轉(zhuǎn)換 ：采樣、量化為數(shù)字量 ?Digital signal processor or puter: signal