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63 delays, 63 additions, and 32 multiplications.
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+ a,+ az, the structure is and
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No, the filter is unstable because r3= 1.2.
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G ( z ) = H (z2).
The system function will have generalized linear phase, H ( z ) = fz-PH(z-I).
6. (b) $.
CHAP. 81
IMPLEMENTATION OF DISCRETE-TIME SYSTEMS
The variance of the output noise will be larger if the pole closest to the unit circle is the second filter in the cascade. Thus, the output noise variance will be larger if the order of the cascade is reversed. (a) a = 2.70;. (b) o = 2.20:. : : The variance of the output noise is simply the noise variance, a = .h2-2B = 12-'O : 12
= 0.20770; where 0 = h2-30. :
8.61 8.62 8.63 8.64
1 O.~Z-~ 1 - 2.4 cos(0.75n)z-I Hl(z) = and H2(z) = 1 - 1.4cos(0.25n)z-' 0.49zr2' 1 - 1.8cos(0.9n)z-I
+ 1.44~-~ +0.81~-~
9
Filter Design
9.1 INTRODUCTION
This chapter considers the problem of designing a digital filter. The design process begins with the filter specifications, which may include constraints on the magnitude and/or phase of the frequency response, constraints on the unit sample response or step response of the filter, specification of the type of filter (e.g., FIR or IIR), and the filter order. Once the specifications have been defined, the next step is to find a set of filter coefficients that produce an acceptable filter. After the filter has been designed, the last step is to implement the system in hardware or software, quantizing the filter coefficients if necessary, and choosing an appropriate filter structure (Chap. 8).
9.2 FILTER SPECIFICATIONS
Before a filter can be designed, a set of filter specifications must be defined. For example, suppose that we would like to design a low-pass filter with a cutoff frequency w,.. The frequency response of an ideal low-pass filter with linear phase and a cutoff frequency w,. is
which has a unit sample response
hd(n) =
sin(n - a ) w , .
n(t7 - a )
Because this filter is unrealizable (noncausal and unstable), it is necessary to relax the ideal constraints on the frequency response and allow some deviation from the ideal response. The specifications for a low-pass filter will typically have the form
as illustrated in Fig. 9-1. Thus, the specifications include the passband cutoff frequency, w,, the stopband cutoff frequency, w,, the passband deviation, 6,. and the stopband deviation, 6,. The passband and stopband deviations
Passband
Stopband
Fig. 9-1.
Filter specifications for a low-pass filter,
CHAP. 91
FILTER DESIGN
are often given in decibels (dB) as follows:
a,. = -2010g(6,~)
The interval [w,, w,] is called the trunsitiotr hand. Once the filter specifications have been defined, the next step is to design a filter that meets these specificatlons.
9.3 FIR FILTER DESIGN
The frequency response of an N th-order causal FIR filter is
and the design of an FIR filter involves finding the coefficients h ( n ) that result in a frequency response that satisfies a given ser of filter specifications. FIR filters have two important advantages over 1IR filters. First, they are guaranteed to be stable, even after the filter coefficients have been quantized. Second, they may be easily constrained to have (generalized) linear phase. Because FIR filters are generally designed to have linear phase, in the following we consider the design of linear phase FIR filters.
Linear Phase FIR Design Using Windows
Let hd(n) be the unit sample response of an ideal frequency selective filter with linear phase,
Because hd(n) will generally be infinite in length, it is necessary to find an FIR approximation to Hd(ejw).With the window design method, the filter is designed by windowing the unit sample response,
where w(n) is a finite-length window that is equal to zero outside the interval 0 its midpoint: w(n) = w(N - n )
n 5 N and is symmetric about
The effect of the window on the frequency response may be seen from the complex convolution theorem,
Thus, the ideal frequency response is snzoothed by the discrete-time Fourier transform of the window, W(ejW). There are many different types of windows that may be used in the window design method, a few of which are listed in Table 9- 1 . How well the frequency response of a filter designed with the window design method approximates a desired response, H~(&"). is determined by two factors (see Fig. 9-2):
The width of the main lobe of W (el").
2. The peak side-lobe amplitude of W(ejU).
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