|
數(shù)字信號(hào)處理中通常是取其有限的時(shí)間片段進(jìn)行分析，而不是對(duì)無(wú)限長(zhǎng)的信號(hào)進(jìn)行測(cè)量和運(yùn)算。具體做法是從信號(hào)中截取一個(gè)時(shí)間片段，然后對(duì)信號(hào)進(jìn)行傅里葉變換、相關(guān)分析等數(shù)學(xué)處理。信號(hào)的截?cái)喈a(chǎn)生了能量泄漏，而用FFT算法計(jì)算頻譜又產(chǎn)生了柵欄效應(yīng)，從原理上講這兩種誤差都是不能消除的。在FFT分析中為了減少或消除頻譜能量泄漏及柵欄效應(yīng)，可采用不同的截取函數(shù)對(duì)信號(hào)進(jìn)行截短，截短函數(shù)稱為窗函數(shù)，簡(jiǎn)稱為窗。
泄漏與窗函數(shù)頻譜的兩側(cè)旁瓣有關(guān)，對(duì)于窗函數(shù)的選用總的原則是，要從保持最大信息和消除旁瓣的綜合效果出發(fā)來(lái)考慮問題，盡可能使窗函數(shù)頻譜中的主瓣寬度應(yīng)盡量窄，以獲得較陡的過渡帶；旁瓣衰減應(yīng)盡量大，以提高阻帶的衰減，但通常都不能同時(shí)滿足這兩個(gè)要求。頻譜中的如果兩側(cè)瓣的高度趨于零，而使能量相對(duì)集中在主瓣，就可以較為接近于真實(shí)的頻譜。不同的窗函數(shù)對(duì)信號(hào)頻譜的影響是不一樣的，這主要是因?yàn)椴煌拇昂瘮?shù)，產(chǎn)生泄漏的大小不一樣，頻率分辨能力也不一樣。信號(hào)的加窗處理，重要的問題是在于根據(jù)信號(hào)的性質(zhì)和研究目的來(lái)選用窗函數(shù)。圖1是幾種常用的窗函數(shù)的時(shí)域和頻域波形，其中矩形窗主瓣窄，旁瓣大，頻率識(shí)別精度最高，幅值識(shí)別精度最低，如果僅要求精確讀出主瓣頻率，而不考慮幅值精度，則可選用矩形窗，例如測(cè)量物體的自振頻率等；布萊克曼窗主瓣寬，旁瓣小，頻率識(shí)別精度最低，但幅值識(shí)別精度最高；如果分析窄帶信號(hào)，且有較強(qiáng)的干擾噪聲，則應(yīng)選用旁瓣幅度小的窗函數(shù)，如漢寧窗、三角窗等；對(duì)于隨時(shí)間按指數(shù)衰減的函數(shù)，可采用指數(shù)窗來(lái)提高信噪比。表1 是幾種常用的窗函數(shù)的比較。 如果被測(cè)信號(hào)是隨機(jī)或者未知的，或者是一般使用者對(duì)窗函數(shù)不大了解，要求也不是特別高時(shí)，可以選擇漢寧窗，因?yàn)樗男孤⒉▌?dòng)都較小，并且選擇性也較高。但在用于校準(zhǔn)時(shí)選用平頂窗較好，因?yàn)樗耐◣Р▌?dòng)非常小，幅度誤差也較小。
表1 幾種常用的窗函數(shù)的比較
下面是幾種窗函數(shù)歸一化DTFT幅度的MATLAB程序: 附上DTFT函數(shù)（dtft.m）： function [ X ] = dtft( x,n,w ) % Computes Discrete-time Fourier Transform % [X] = dtft(x,n,w) % X = DTFT values computed at w.frequencies % x = finite duration sequence over n % n = sample position vector % w = frequency location vector X = x*exp(-j*n'*w); % end
矩形窗: %DTFT of a Rectangular Window, M=10,25,50,101 clc; close all; Hf_1=figure; set(Hf_1,'NumberTitle','off','Name','P0304a'); w=linspace(-pi,pi,501); wtick=[-1:0.5:1]; magtick=[0:0.5:1.1]; % M=10 M=10; n=0:M; x=ones(1,length(n)); X=dtft(x,n,w); magX=abs(X); magX=magX/max(magX); subplot(2,2,1); plot(w/pi,magX,'LineWidth',1.5); axis([-1 1 0 1.1]); ylabel('|X|'); title(['M=10']); set(gca,'XTick',wtick,'YTick',magtick); % M=25 M=25; n=0:M; x=ones(1,length(n)); X=dtft(x,n,w); magX=abs(X); magX=magX/max(magX); subplot(2,2,2); plot(w/pi,magX,'LineWidth',1.5); axis([-1 1 0 1.1]); title(['M=25']); set(gca,'XTick',wtick,'YTick',magtick); % M=50 M=50; n=0:M; x=ones(1,length(n)); X=dtft(x,n,w); magX=abs(X); magX=magX/max(magX); subplot(2,2,3); plot(w/pi,magX,'LineWidth',1.5); axis([-1 1 0 1.1]); xlabel('\omega/\pi'); ylabel('|X|'); title('M=50'); set(gca,'XTick',wtick,'YTick',magtick); % M=101 M=101; n=0:M; x=ones(1,length(n)); X=dtft(x,n,w); magX=abs(X); magX=magX/max(magX); subplot(2,2,4); plot(w/pi,magX,'LineWidth',1.5); axis([-1 1 0 1.1]); xlabel('\omega/\pi'); ylabel('|X|'); title(['M=101']); 
三角窗: % Triangular Window: % DTFT of a Triangular Window,M = 10,25,50,101 clc; close all; Hf_1=figure; set(Hf_1,'NumberTitle','off','Name','P0304b'); w=linspace(-pi,pi,501); wtick=[-1:0.5:1]; magtick=[0:0.5:1.1]; % M = 10 M=10; n=0:M; x=(1-(abs(M-1-(2*n))/(M+1))); x=dtft(x,n,w); magX=abs(X); magX=magX/max(magX); subplot(2,2,1); plot(w/pi,magX,'LineWidth',1.5); axis([-1 1 0 1.1]); ylabel('|X|'); title(['M = 10']); set(gca,'XTick',wtick,'YTick',magtick); % M = 25 M=25; n=0:M; x=(1-(abs(M-1-(2*n))/(M+1))); X=dtft(x,n,w); magX=abs(X); magX=magX/max(magX); subplot(2,2,2); plot(w/pi,magX,'LineWidth',1.5); axis([-1 1 0 1.1]); title(['M = 25']); set(gca,'XTick',wtick,'YTick',magtick); % M = 50 M=50; n=0:M; x=(1-(abs(M-1-(2*n))/(M+1))); X=dtft(x,n,w); magX=abs(X); magX=magX/max(magX); subplot(2,2,3); plot(w/pi,magX,'LineWidth',1.5); axis([-1 1 0 1.1]); xlabel('\omega/\pi'); ylabel('|X|'); title(['M = 50']); set(gca,'XTick',wtick,'YTick',magtick); % M = 100 M=101;n=0:M; x=(1-(abs(M-1-(2*n))/(M+1))); X=dtft(x,n,w); magX=abs(X); magX=magX/max(magX); subplot(2,2,4); plot(w/pi,magX,'LineWidth',1.5); axis([-1 1 0 1.1]); xlabel('\omega/\pi'); title(['M = 101']); set(gca,'XTick',wtick,'YTick',magtick); 
海寧窗: % Hann Window % DTFT of a Hann Window, M = 10,25,50,101 clc;close all; Hf_1 = figure; set(Hf_1,'NumberTitle','off','Name','P0304c'); w=linspace(-pi,pi,501); wtick=[-1:0.5:1]; magtick=[0:0.5:1.1]; % M = 10 M=10;n=0:M; x=0.5*(1-cos((2*pi*n)/(M-1))); X=dtft(x,n,w); magX=abs(X); magX=magX/max(magX); subplot(2,2,1); plot(w/pi,magX,'LineWidth',1.5); axis([-1 1 0 1.1]); ylabel('|X|'); title(['M = 10']); set(gca,'XTick',wtick,'YTick',magtick); % M = 25 M=25;n=0:M; x=0.5*(1-cos((2*pi*n)/(M-1))); X=dtft(x,n,w); magX=abs(X); magX=magX/max(magX); subplot(2,2,2); plot(w/pi,magX,'LineWidth',1.5); axis([-1 1 0 1.1]); title(['M = 25']); set(gca,'XTick',wtick,'YTick',magtick); % M = 50 M=50;n=0:M; x=0.5*(1-cos((2*pi*n)/(M-1))); X=dtft(x,n,w); magX=abs(X); magX=magX/max(magX); subplot(2,2,3); plot(w/pi,magX,'LineWidth',1.5); axis([-1 1 0 1.1]); xlabel('\omega/\pi'); ylabel('|X|'); title(['M = 50']); set(gca,'XTick',wtick,'YTick',magtick); % M = 101 M=101;n=0:M; x=0.5*(1-cos((2*pi*n)/(M-1))); X=dtft(x,n,w); magX=abs(X); magX=magX/max(magX); subplot(2,2,4); plot(w/pi,magX,'LineWidth',1.5); axis([-1 1 0 1.1]); xlabel('\omega/\pi'); title(['M = 101']); set(gca,'XTick',wtick,'YTick',magtick); 
哈明窗: % Hamming Window: clc; close all; Hf_1=figure; set(Hf_1,'NumberTitle','off','Name','P0304d'); w=linspace(-pi,pi,501); wtick=[-1:0.5:1]; magtick=[0:0.5:1.1]; % M = 10 M=10; n=0:M; x=(0.54-0.46*cos((2*pi*n)/(M-1))); X=dtft(x,n,w); magX=abs(X); magX=magX/max(magX); subplot(2,2,1); plot(w/pi,magX,'LineWidth',1.5); axis([-1 1 0 1.1]); ylabel('|X|'); title(['M = 10']); set(gca,'XTick',wtick,'YTick',magtick); % M = 25 M=25; n=0:M; x=(0.54-0.46*cos((2*pi*n)/(M-1))); X=dtft(x,n,w); magX=abs(X); magX=magX/max(magX); subplot(2,2,2); plot(w/pi,magX,'LineWidth',1.5); axis([-1 1 0 1.1]); title(['M = 25']); set(gca,'XTick',wtick,'YTick',magtick); % M = 50 M=50; n=0:M; x=(0.54-0.46*cos((2*pi*n)/(M-1))); X=dtft(x,n,w); magX=abs(X); magX=magX/max(magX); subplot(2,2,3); plot(w/pi,magX,'LineWidth',1.5); axis([-1 1 0 1.1]); xlabel('\omega/\pi'); ylabel('|X|'); title(['M=50']); set(gca,'XTick',wtick,'YTick',magtick); % M = 101 M=101; n=0:M; x=(0.54-0.46*cos((2*pi*n)/(M-1))); X=dtft(x,n,w); magX=abs(X); magX=magX/max(magX); subplot(2,2,4); plot(w/pi,magX,'LineWidth',1.5); axis([-1 1 0 1.1]); xlabel('\omega/\pi');title(['M=101']); set(gca,'XTick',wtick,'YTick',magtick);  |
|
|
