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Round-off Effects in FIR Digital Filters 581
signals use
1
X max ≤
max |H(e jω )|
and wideband random signals use
X max ≤
1
4σ x
√∑ |h(n)|
2 .
Using (10.84) and assuming that x(n) isuniformly distributed over
[−X max , +X max ], the input signal power is given by
σ 2 x = X2 max
3
=
1
3( ∑ |h(n)|) 2 (10.85)
Furthermore, assuming that x(n) isalso a white sequence, the output
signal power is given by
∑
σy 2 = σx
2 |h(n)| 2 = 1 ∑ |h(n)|
2
3 ( ∑ |h(n)|) 2 (10.86)
Thus, the output SNR is
SNR = σ2 y
σ 2 q
= 22(B+1)
A
[ ∑ ]
|h(n)|
2
( ∑ |h(n)|) 2
(10.87)
where A = M for the model in Figure 10.26b or A =1for the model in
Figure 10.26c. The corresponding SNR in dB is
( ∑ )
|h(n)|
2
SNR dB =6.02+6.02B +10log 10
( ∑ |h(n)|) 2 − 10 log 10 A (10.88)
10.3.2 ANALYSIS USING MATLAB
This simulation in MATLAB can be done in parallel fashion since there
is no feedback path for the multiplication quantization errors. Using the
function Qfix function with ’round’ mode, we will compute the quantized
multiplier output. In the case of M quantizers, assuming two’scomplement
format, we will use the ’twosc’ mode for each quantizer.
Only the final sum will be quantized and saturated. In the case of one
quantizer, we need the ’satur’ mode. These simulation steps are detailed
in the following example.
□ EXAMPLE 10.13 Let a fourth-order (M =5)FIR filter be given by
H(z) =0.1+0.2z −1 +0.4z −2 +0.2z −3 +0.1z −4 (10.89)
which is implemented as a direct form with B = 12 fractional bit quantizers.
Compute SNRs for models in Figure 10.26b and c and verify them using
MATLAB simulations.
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