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# ç¬¬07ç«  æ•°å­—æ»¤æ&su

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7.2 Design of FIR filters by windowing 7.2.1 design ideas 7.2.2 properties of commonly used windows 7.2.3 effect of these windows to frequency response 7.2.4 design step FIR filters are almost entirely restricted to discrete-time implementations . Consequently, the design techniques for FIR filters are based on directly approximating the desired frequency response of the discrete-time systems. Furthermore, most techniques for approximating the magnitude response of an FIR system assume a linear phase constraint . The simplest method of FIR filter design is called the window method （窗函数法） .

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7.2.1 design ideas 1 The window method generally begins with an ideal desired frequency response, it’s inverse transform is 1 [] ( ) 2 jj n dd hn H e e d π ωω ω = Many idealized systems are defined by piecewise- constant or piecewise-functional frequency responses with discontinuities at the boundaries between bands. As a results, these systems have impulse responses that are non-causal and infinitely long .
2 The most straightforward approach to obtaining a causal FIR approximation to such systems is to truncate the ideal response: [] [][] d hn h nwn = () 1 ( ) ( ) 2 jj j d He H e We d π ωθ ω θ = where, the window w[n] is the rectangular window or others. The corresponding effect in the frequency domain is 加上不同 形状 长度 的窗，形成对理想频响的不同程度的逼近

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] [ n w ] [ ] [ ] [ n w n h n h M d = /2 sin[ ( 1) / 2] () sin( / 2) jj M M We e ωω ω + = 11 | | 22 j j j j M dd A He H e W e e ππ =∗ = 1 2 + M π Gibbs （吉布斯）效应 phenomenon 3 1 2( 1) 3 j M M + + 广义幅度 0 0 0 0 n n n πω 0 0 M/2 M M ,| | 0, | | jM j c d c e ωπ < = << ( / 2)) [ ] , ,..., (/ 2 ) c d nM hn n == c 1 1 1 M+1
(1) The width of the resulting transition band （过渡带） is determined by the width of the main lobe （主瓣） of the Fourier transform of the window. (2)The pass band and stop band ripples （波动） are determined by the side lobes （旁瓣） of the Fourier transform of the window. The ripples in the pass band and the stop band are approximately the same, and are not dependent on M and can be changed only by changing the shape of the window . 滤波器的过渡带宽 Å 窗频谱的主瓣宽 Å 窗长和窗形状。 滤波器的阻带衰减 Å 窗频谱的旁瓣相对幅度 Å 窗形状。 采用时域缓变的窗形状（其高频成分少，低频多，即旁瓣相对 幅度小，主瓣宽）能增加阻带率减。 阻带率减

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10 (1)rectangular : [ ] 0 nM wn other =
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