ssrs 2d barcode Linear Phase Filters in Software

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8.3.3 Linear Phase Filters
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Linear phase filters have a unit sample response that is either symmetric,
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IMPLEMENTATION OF DISCRETE-TIME SYSTEMS
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[CHAP 8
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Fig. 8-3. An FIR tilter implemented as a cascade of second-order systems.
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or antisymmetric (see Sec. 5.3). h ( n ) = -h(N - n) This symmetry may be exploited to simplify the network structure. For example, if N is even and h(n) is symmetric (type 1 filter),
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Therefore, forming the sums [x(n - k) x(n - N k)] prior to multiplying by h(k) reduces the number of multiplications. The resulting structure is shown in Fig. 8-4(a). If N is odd and h(n) is symmetric (type I1 filter), the structure is as shown in Fig. 8-4(h). There are similar structures for the antisymmetric (types 111 and IV) linear phase filters.
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CHAP. 81
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8.3.4 Frequency Sampling
The frequency sampling structure is an implementation that parameterizes a filter in terms of its DFT coefficients. Specifically, let H ( k ) be the N-point DFT of an FIR filter with h ( n ) .= 0 for n < 0 and n 2 N.' Because the unit sample response of the filter is 1 N-I h ( n )= ~ ( k ) J ~ ~ ~ ~ / ~ u
the system function may be written as
H ( z )=
n =O
h(n)z-" =
x [i x
nO =
k =O
~z - f l j
Evaluating the sum over n , this becomes
which corresponds to a cascade of an FIR filter
k(1 - z - ~ with a parallel network of one-pole filters: )
For a narrowband filter that has most of its DFT coefficients equal to zero, the frequency sampling structure will be an efficient implementation. The frequency sampling structure is shown in Fig. 8-5. If h ( n ) is real, H ( k ) = H * ( N - k ) , and the structure may be simplified. For example, if N is even,
where
A ( k ) = H ( k )+ H ( N - k ) B ( k ) = ~ ( k - j) n k / N e 2
+ H(N
- k ) eJZnklN
A similar simplification results when N is odd.
8.4 STRUCTURES FOR IIR SYSTEMS
The input x ( n ) and output y ( n ) of a causal IIR filter with a rational system function
'Note that here we are assuming that h ( n ) is of length N, instead of N I as in the previous sections. This is consistent with the convention that the frequency sampling filter is based on an N-point DFT of h(n).
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[CHAP. 8
Fig. 8-5. Frequency sampling filter structure.
is described by the linear constant coefficient difference equation
In the following sections, several different implementations of this system are presented, including the direct form structures, the cascade and parallel forms, and the transposed filter structures.
Direct Form
There are two direct form filter structures, referred to as direct form I and direct form 11. The direct form I structure is an implementation that results when Eq. (8.3)is written as a pair of difference equations as follows:
The first equation corresponds to an FIR filter with input x ( n ) and output w ( n ) , and the second equation corresponds to an all-pole filter with input w ( n ) and output y ( n ) . Therefore, this pair of equations represents a
CHAP. 81
IMPLEMENTATION OF DISCRETE-TIME SYSTEMS
cascade of two systems,
as illustrated in Fig. 8-6. The computational requirements for a direct form I structure are as follows: Number of multiplications: p Number of additions: p Number of delays: p
+ q + 1 per output sample
+ q per output sample
Fig. 8-6. Direct form I realization of an IIR filter.
The directform IJ structure is obtained by reversing the order of the cascade of B(z) and 1/A(z) as illustrated in Fig. 8-7. With this implementation, x ( n ) is first filtered with the all-pole filter l / A ( z ) and then with B(z):
Fig. 8-7. Reversing the order of the cascade in the direct form I filter structure.
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[CHAP. 8
If we denote the output of the all-pole filter 1 /A(z) by w(n),this structure is described by the following pair of coupled difference equations:
This structure may be simplified by noting that the two sets of delays are delaying the same sequence. Therefore, they may be combined as illustrated in Fig. 8-8 for the case in which p = q. The computational requirements for a direct form I1 structure are as follows: Number of multiplications: p Number of additions: p
+ q + I per output sample
+ q per output sample
Number of delays: max(p. q ) The direct form I1 structure is said to be canonic because it uses the minimum number of delays for a given H(z).
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