11.5 FIR Filter Design 683
whereα=(N− 1 )/ 2. Using a window w[n]of lengthNand centered at(N− 1 )/ 2 , the windowed
impulse response ish[n]=hd[n]w[n], and the designed FIR filter is
H(z)=
N∑− 1
n= 0
h[n]z−n
n The design using windows is a trial-and-error procedure. Different trade-offs can be obtained by using
various windows and various lengths of the windows.
n The symmetry of the impulse responseh[n]with respect to(N− 1 )/ 2 , independent of whether this is an
integer or not, guarantees the linear phase of the filter.
11.5.2 Window Functions
In the previous section, the windowed impulse responsehw[n] was written as
hw[n]=hd[n]w[n]
where
w[n]=
{
1 −(N− 1 )/ 2 ≤n≤(N− 1 )/ 2
0 otherwise
(11.56)
is arectangular windowof lengthN. If we wishHw(ejω)=Hd(ejω), we would need a rectangular win-
dow of infinite length so that the impulse responseshw[n]=hd[n] (i.e., no windowing). This ideal
rectangular window has a discrete-time Fourier transform
W(ejω)= 2 πδ(ω) −π≤ω < π (11.57)
Sincehw[n]=w[n]hd[n], thenHw(ejω)is the convolution ofHd(ejω)andW(ejω)in the frequency
domain—that is,
Hw(ejω)=
1
2 π
∫π
−π
Hd(ejθ)W(ej(ω−θ))dθ
=
∫π
−π
Hd(ejθ)δ(ω−θ)dθ=Hd(ejω)
Thus, forN→∞, the result of this convolution isHd(ejω), but ifNis finite the convolution in the
frequency domain would give a distorted version ofHd(ejω). Thus, to obtain a good approximation
ofHd(ejω)using a finite window w[n] the window must have a spectrum approximating that of the
ideal rectangular window. That is, an impulse in frequency in−π≤ω < πas in Equation (11.57)
with most of its energy concentrated in the low frequencies. The smoothness of the window makes
this possible.
