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Round-off Effects in IIR Digital Filters 567
Using (10.41) and (10.47), the output SNR is
SNR = △ σ2 y (1 −|α|)2 12 ( 1 −|α| 2)
σq
2 =
3(1−|α| 2 ) 2 −2B
=4 ( (10.48)
2 2B) (1 −|α|) 2 =2 2(B+1) (1 −|α|) 2
or the SNR in dB is
SNR dB
△
=10log10 (SNR) = 6.02 + 6.02B +20log 10 (1 −|α|) (10.49)
Let δ =1−|α|, which is the distance of the pole from the unit circle.
Then
SNR dB =6.02+6.02B +20log 10 (δ) (10.50)
which is a very informative result. First, it shows that the SNR is directly
proportional to B and increases by about 6 dB for each additional bit
added to the word length. Second, the SNR is also directly proportional
to the distance δ. The smaller the δ (or nearer the pole to the unit circle),
the smaller is the SNR, which is a consequence of the filter characteristics.
As an example, if B =6and δ =0.05, then SNR = 16.12 dB and if B =12
and δ =0.1, then SNR = 58.26 dB.
10.2.6 ANALYSIS USING MATLAB
To analyze the properties of the round-off errors in IIR filters we will
simulate them using the MATLAB function QFix with quantization mode
’round’ and overflow mode ’satur’.Ifproper scaling to avoid overflow is
performed, then only the multiplier output needs to be quantized at each
n without worrying about the overflow. However, we will still saturate
the final sum to avoid any unforeseen problems. In previous simulations,
we could perform the quantization operations on vectors (i.e., perform
parallel processing). Since IIR filters are recursive filters and since each
error is fed back into the system, vector operation is generally not possible.
Hence the filter output will be computed sequentially from the first to the
last sample. For a large number of samples, this implementation will slow
the execution speed in MATLAB since MATLAB is optimized for vector
calculations. However, for newer fast processors, the execution time is
within a few seconds. These simulation steps are detailed in the following
example.
□ EXAMPLE 10.11 Consider the model given in Figure 10.19b. We will simulate this model in
MATLAB and investigate its output error characteristics. Let a =0.9, which
will be quantized to B bits. The input signal is uniformly distributed over
the [−1, +1] interval and is also quantized to B bits prior to filtering. The
scaling factor X max is computed from (10.44). Using 100, 000 signal samples
and B =6bits, the following MATLAB script computes the true output y(n),
the quantized output ŷ(n), the output error q(n), and the output SNR.
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