1.Truncating the Impulse 2.Impulse Response of Ideal 2.Impulse Response of Ideal Response Lowpass Filters Lowpass Filters The group delay of h(n)is Msamples The ideal lowpass filter has a zero-phase ·Truncating to range-M≤n≤Mand delaying r@=-d（-oM=M frequency response (1. with Msamples yields the causal FIR lowpass Hzp(em）= filter do sin(@(n-M)) where the linear phase response is-@M 0,包.<回sπ The corresponding impulse response ir(n)= .0≤n≤2M (n-M) coefficients 0 otherwise h(n)= sin on -,-00≤n≤∞ The truncation of the impulse response is doubly infinite,not absolutely summable, coefficients of the ideal filters exhibit an and therefore unrealizable oscillatory behavior in the respective magnitude responses 3.Gibbs Phenomenon 3.Gibbs Phenomenon 3.Gibbs Phenomenon Gibbs phenomenon-Oscillatory behavior in the magnitude responses of causal FIR filters As can be seen,as the length of the lowpass .Truncation of hn)can be expressed by obtained by truncating the impulse response filter is increased,the number of ripples in windowing operation,i.e.,by multiplying the coefficients of ideal filters both passband and stopband increases,with a h(n)sequence with a finite-length sequence corresponding decrease in the ripple widths Impact of the langth of the window function w(n) (1)Narrower transition band Height of the largest ripples remain the same h(n)=h(n)w(n) 2)Mare ripp电es independent of length where w(n)is a window function (3)Smaller rpplo width Similar oscillatory behavior observed in the 周Same largest poak中pb magnitude responses of the truncated versions The perommance is beftor. of other types of ideal filters Haw to reduce the highest ripple?7 1. Truncating the Impulse Response The group delay of h(n) is M samples where the linear phase response is ˉ¹M () ( ) d M M d     8 2. Impulse Response of Ideal Lowpass Filters The ideal lowpass filter has a zero-phase frequency response The corresponding impulse response coefficients is doubly infinite doubly infinite, not absolutely not absolutely summable summable, and therefore unrealizable unrealizable 1, ( ) 0, j c LP c c H e         sin () , c LP n hn n n     9 2. Impulse Response of Ideal Lowpass Filters Truncating to range ˉMİnİM and delaying with M samples yields the causal FIR lowpass filter The truncation of the impulse response coefficients of the ideal filters exhibit an oscillatory behavior oscillatory behavior in the respective magnitude responses sin ( )   ,0 2 ˆ ( ) ( ) 0, otherwise c LP n M n M h n n M          10 0 0.2 0.4 0.6 0.8 1 0 0.5 1 Normalized Frequency Magnitude N=20 N=60 3. Gibbs Phenomenon Gibbs phenomenon - Oscillatory behavior in the magnitude responses of causal FIR filters obtained by truncating the impulse response coefficients of ideal filters Impact of the length of the window function (1) Narrower transition band (2) More ripples (3) Smaller ripple width (4) Same largest peak ripple The performance is better. How to reduce the highest ripple? 11 3. Gibbs Phenomenon As can be seen, as the length of the lowpass filter is increased, the number of ripples in both passband and stopband increases, with a corresponding decrease in the ripple widths Height of the largest ripples remain the same independent of length Similar oscillatory behavior observed in the magnitude responses of the truncated versions of other types of ideal filters 12 3. Gibbs Phenomenon Truncation of hd(n) can be expressed by windowing operation, i.e., by multiplying the hd(n) sequence with a finite-length sequence w(n) where w(n) is a window function () () () t d h n h n wn