Signals and Systems - Electrical Engineering

682 C H A P T E R 11: Introduction to the Design of Discrete Filters

that causes the truncation ofhd[n]. The windowed impulse responsehw[n] has a discrete-time Fourier

transform of

Hw(ejω)=

(N∑− 1 )/ 2

n=−(N− 1 )/ 2

hw[n]e−jωn

For a large value ofNwe have thatHw(ejω)must be a good approximation ofHd(ejω)—that is,

|Hw(ejω)|≈|Hd(ejω)|

∠Hw(ejω)=∠Hd(ejω)= 0

It is not clear how the value ofNshould be chosen—this is what we meant by this design is a

trial-and-error method.

To makeHw(z)a causal filter, we shift to the right the impulse responsehw[n] by(N− 1 )/2 (assume

N is chosen to be an odd number so that this division is an integer) samples to obtain

Hˆ(z)=Hw(z)z−(N−^1 )/^2 =

(N∑− 1 )/ 2

m=−(N− 1 )/ 2

hw[m]z−(m+(N−^1 )/^2 )

=

N∑− 1

n= 0

hd[n−(N− 1 )/ 2 )]w[n−(N− 1 )/ 2 )]z−n

after lettingn=m+(N− 1 )/2. For a large value ofN, we have

|Hˆ(ejω)|=|Hw(ejω)e−jω(N−^1 )/^2 |=|Hw(ejω)|≈|Hd(ejω)|

∠Hˆ(ejω)=∠Hw(ejω)−

N− 1

2

ω=−

N− 1

2

ω (11.53)

since∠Hw(ejω)=∠Hd(ejω)=0. That is, the magnitude response of the FIR filterHˆ(z)is approxi-

mately (depending on the value ofN) the desired response and its phase response is linear. These

results can be generalized as follows.

n If the desired low-pass frequency response has a magnitude

|Hd(ejω)|=

{

1 −ωc≤ω≤ωc

0 otherwise

(11.54)

and a linear phase

θ(ω)=−ω(N− 1 )/ 2

the corresponding impulse response is given by

hd[n]=

{

sin(ωc(n−α))/(π(n−α)) n6=α

ωc/π n=α

(11.55)