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where w N(n) is a specially selected time window (e.g., Hanning, Hamming, Blackman, or Kaiser window),<br />

used in order to reduce the so-called Gibbs phenomenon [37]. Depending on filter type, the ideal filter<br />

coefficients can be calculated using equations listed in Table 27.4 [9].<br />

Another, more advanced, method for the design of FIR filters, is an optimization procedure developed<br />

by Parks and McClellan (also known as the Remez method) [9]. This method is implemented in the<br />

MATLAB environment with two functions: remezord to estimate the filter order and remez to compute<br />

the filter coefficients [30]. This optimization method should be used, if a relatively high stopband<br />

attenuation is required, e.g., with a 20-bit resolution for representation of signal samples, we usually<br />

need a stopband attenuation of approximately 120 dB. As a design example, Fig. 27.20 presents the<br />

frequency response of a lowpass FIR filter designed with the Parks–McClellan method, with the normalized<br />

cutoff frequency of π/64. This filter can be used as a prototype filter for the design of analysis and<br />

synthesis filter banks for audio coders, according, e.g., to the MUSICAM and MPEG-1 standards.<br />

© 2002 by CRC Press LLC<br />

Gain [dB]<br />

20<br />

0<br />

-20<br />

-40<br />

-60<br />

-80<br />

-100<br />

-120<br />

-140<br />

TABLE 27.4 Impulse Response of Ideal Filters<br />

Filter Type Impulse Response<br />

Lowpass<br />

Highpass<br />

Passband<br />

Stopband<br />

sin(<br />

nω c)<br />

--------------------- n ≠ 0<br />

nπ<br />

hd( n)<br />

ω ⎧<br />

⎪<br />

= ⎨<br />

⎪ c<br />

⎩<br />

----- n = 0<br />

π<br />

hd( n)<br />

hd( n)<br />

hd( n)<br />

sin(<br />

nω c)<br />

– --------------------- n ≠ 0<br />

nπ<br />

1 ω ⎧<br />

⎪<br />

= ⎨<br />

⎪<br />

c<br />

⎩<br />

– ----- n = 0<br />

π<br />

sin( nω 2)<br />

– sin nω 1<br />

( )<br />

-------------------------------------------------- n ≠ 0<br />

nπ<br />

ω ⎧<br />

⎪<br />

= ⎨<br />

⎪<br />

2 – ω1 ⎩<br />

----------------- n = 0<br />

π<br />

sin( nω 1)<br />

– sin nω 2<br />

( )<br />

-------------------------------------------------- n ≠ 0<br />

nπ<br />

1 ω ⎧<br />

⎪<br />

= ⎨<br />

⎪<br />

2 – ω1 ⎩<br />

+ ----------------- n = 0<br />

π<br />

-160<br />

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4<br />

Normalized frequency<br />

FIGURE 27.20 Frequency response of an FIR lowpass filter designed with the Parks–McClellan method.

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