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Round-off Effects in FIR Digital Filters 579
10.3.1 FIXED-POINT ARITHMETIC
We will consider the effects on two realizations: direct-form and cascadeform.
There is no parallel-form realization for FIR filters since we do
not have a partial fraction expansion, except for the frequency sampling
realization, which can be analyzed using IIR filter techniques. The analysis
of FIR filters is much simpler than that for IIR because there are no
feedback paths. One consequence of this is the absence of limit cycles.
Direct-form realization Consider an FIR filter of length M (i.e.,
there are M samples in the impulse response), which is realized using the
direct form as shown in Figure 10.26a. The filter coefficients are the samples
of the impulse response h(n). We have to introduce quantizers in the
vertical branches. If we use the implementation in which each multiplier
output is quantized, then we obtain the model shown in Figure 10.26b.
On the other hand if we implement the filter in a typical DSP chip, then
the final sum is quantized as shown in Figure 10.26c. We will separately
consider the effects of round-off noise and scaling (to avoid overflow).
Round-off noise Let the output of the filter in Figure 10.26b due to
round-off errors be ŷ(n) =y(n)+q(n). Then
q(n) =
M−1
∑
k=0
e k (n) (10.80)
where e k (n) are the noise sources introduced in each vertical branch to
account for the rounding operations. Since these noise sources are all
independent and identical, the noise power in q(n) isgiven by
σ 2 q =
M−1
∑
0
σ 2 e k
( ) 2
= Mσe 2 −2B
= M = M 12 3 2−2(B+1) (10.81)
In Figure 10.26c the output due to the rounding operation is ŷ(n) =
y(n)+e(n). Hence the noise power in this case is given by
σ 2 q = σ 2 e = 1 3 2−2(B+1) (10.82)
which is smaller by a factor of M compared to (10.81) as expected.
Scaling to avoid overflow We assume that the fixed-point numbers
have the two’s-complement form representation, which is a reasonable
assumption. Then we will have to check only the overflow of the total sum.
Thus, this analysis is the same for both implementations in Figure 10.26
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