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Round-off Effects in FIR Digital Filters 589
Subtracting y(n) = ∑ M−1
k=0
h(k) x(n − k) from (10.99), we obtain
q(n) =
∑
{A(n, k) − 1} h(k) x(n − k) (10.100)
M−1
k=0
Now from (10.98), the average value of A(n, k) isEA(n, k) =1andthe
average power of A(n, k) is
E[A 2 (n, k)] =
(1+ 1 ) M+1−k
3 2−2B
≈ 1+(M +1− k) 2−2B
3
for small 2 −2B (10.101)
Assuming that the input signal x(n)isalso a white sequence with variance
σ 2 x, then from (10.101) the noise power is given by
σ 2 q =
k
(M + 1)2−2B
3
M−1
∑
σx
2 k=0
Since (1 −
M+1 ) ≤ 1 and using σ2 y = σx
2
is upper bounded by
or the SNR is lower bounded by
σ 2 q ≤ (M +1) 2−2B
3
(
|h(k)| 2 1 − k )
M +1
(10.102)
∑ M−1
k=0 |h(k)|2 the noise power σ 2 q
σ 2 y (10.103)
SNR ≥ 3
M +1 22B (10.104)
Equation (10.104) shows that it is best to compute products in order of
increasing magnitude.
□ EXAMPLE 10.15 Again consider the 4th-order FIR filter given in Example 10.13 in which
M =5,B = 12, and h(n) ={0.1, 0.2, 0.4, 0.2, 0.1}. From (10.104), the SNR is
lower bounded by
( ) 3
SNR dB ≥ 10 log 10
M +1 224 =69.24 dB
and the approximate value from (10.102) is 71 dB, which is comparable to the
fixed-point value of 72 dB. Note that the fixed-point results would degrade with
less than optimum scaling (e.g., if signal amplitude were 10 dB down), whereas
the floating point SNR would remain the same. To counter this, one could put a
variable scaling factor A on the fixed-point system, which is then getting close to
the full floating-point system. In fact, floating-point is nothing but fixed-point
with variable scaling—that is, a scaling by a power of two (or shifting) at each
multiplication and addition.
□
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