【正文】 frequency resolution分辨力 ), side petals are as relatively small as possible pared with the principal petal and attenuate rapidly( to decrease aliasing).采用其它窗函數。一個好的窗函數應當： 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）。 Common window functions 常用窗函數 rectangle window 矩形窗 ????????2021)(TtTttw )(sinc)( fTTfW ??Basic of Digital Signal Processing數字信號處理基礎 triangle[39。trai l] windsow 三角窗 ?????????20221)(TtTttTtw T )2(si nc2)(2 fTTfW T ??Basic of Digital Signal Processing數字信號處理基礎 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數字信號處理基礎 2 T w(t) 1 0 T/2 T/2 t W(f) T/2 2 T 0 f exponent[eks39。p un nt] window 指數窗 ???????? ?000)(ttetw at22 )2(1)(fafW???（ a0） Basic of Digital Signal Processing數字信號處理基礎 w(t) 1 0 t W(f) 1/a 0 f Technical indexes[39。indeks] of several typical window functions 幾種典型窗函數的技術指標 T y p e of w i n d o w fu n ct i o n 窗函數類型 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數字信號處理基礎 sampling in frequency domain and fence effect 頻域采樣與柵欄效應 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。 Basic of Digital Signal Processing數字信號處理基礎 x(t) w(t) 0 t f0 f0 0 T T T s2(t) 0 S2(f) 0 f f [X(f)*W(f)] S2(f) [x(t) w(t)]* s2(t) T Sampling in frequency domain 頻域采樣 f0 f0 0 f 0 Spectrum after sampling in frequency domain exists only at each sampled point. Spectrum of unsampled points is warded[39。w :did] 擋住 and not shown (regarded as 0.). This phenomenon is called fence effect. Obviously ,sampling must bring fence effect. 經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），這種現象稱為 柵欄效應 。顯然，采樣必然帶來柵欄效應。 Basic of Digital Signal Processing數字信號處理基礎 In the time domain , as long as satisfying sampling theorem, fence effect doesn’t lose signal ，只要滿足采樣定理，柵欄效應不會丟失信號信息 Basic of Digital Signal Processing數字信號處理基礎 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），導致譜分析結果失去意義。 X(f)?S0(f) x(t)s0(t) s0(t) x(t) S0(f) X(f) w(t) W(f) x(t)s0(t)w(t) X(f)?S0(f)?W(f) s1(t) S1 (f) [X(f)?S0(f)?W(f) ]S1 (f ) [x(t)s0(t)w(t) ]* s1(t) Discrete Fourier transform illustration 離散傅里葉變換圖解說明 ? Frequency resolution , integer[39。intid frequency truncation 頻率分辨力、整周期截斷 Interval[39。int v l] of frequency sampling Δf decides frequency resolution. When Δf is smaller ,resolution will bee higher. Warded frequency ponents will bee fewer. Since DFT(discrete Fourier transformation) exports N effective spectral values in a period (1/Ts), the frequency interval is: 頻率采樣間隔 ?f決定了頻率分辨力。 ?f 越小，分辨力越高，被擋住的頻率成分越少。 由于 DFT在頻域的一個周期內（周期為： 1/Ts）輸出 N個有效譜值，故頻率間隔為： TNfNTf ss 11 ????Basic of Digital Signal Processing數字信號處理基礎 Obviously ,?f can be improved by decreasing fs or increasing N. But fs is restricted by sampling theorem and impossible to be decreased arbitrarily任意地 . Increasing N certainly adds calculating work. To solve antinomy[ 39。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。但前者受采樣定理的限制，不可能隨意降低，后者必然增加計算量。 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代譜分析技術。 Basic of Digital Signal Processing數字信號處理基礎 With spectral lines discrete , frequency value corresponding to spectral line of the spectrum is an integral multiple of ?f. For simple harmonic signals, in order to obtain particular frequency f0, the following equation must be satisfied:由于譜線是離散的，因此頻譜譜線對應的頻率值都是 ?f整數倍。對于簡諧信號，為了得到特定頻率 f0的譜線，必須滿足 整數?? ff 0 整數?0TTT： time length of signal 。 T0： periods of signal with frequency f0信號的周期。 Basic of Digital Signal Processing數字信號處理基礎 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。kei n]截斷 of integral 完整的 [ 39。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倍時，才可能使譜線落在 f0，獲得準確的頻譜。此即為 整周期截斷 。 整周期采樣的結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。 Basic of Digital Signal Processing數字信號處理基礎 ? Quantization and quantization errors 量化與量化誤差 ?Process that discrete signals sampled from analog signals are translated into digital signals (amplitude discretization) is called quantization. Error produced by quantization is called quantization error. ?模擬信號經采樣后得到的離散信號轉變?yōu)閿底中盘枺ǚ惦x散化）的過程稱為量化。由此引起的誤差稱為量化誤差 Basic of Digital Signal Processing數字信號處理基礎 Quantization is realized by A/D transducer. Quantization error is decided by its resolution. If number of bit (word length) of a A/D tran