The Impact of Finite Wordlength Implementation - Digital Signal Processing: An Introduction with MATLAB and Applications - page 192

quantizers have been introduced to represent the appropriate round-off/truncatoin
effects. The origins of these different types of quantization sources are:
1. A/D conversion Q I .
2. D/A conversion Q O .
3. Coefficient quantization Q C .
4. Arithmetic operation quantization Q A .
Because of the nonlinearity introduced into DSP systems via quantization,
spontaneous oscillations can sometimes occur, and these oscillations are often
referred to as limit cycles. Conveniently, these limit cycles only tend to occur in
recursive (IIR) digital filters with poles that are operating close to the margin of
stability.
Unfortunately, precise analysis of the model given in Fig. 4.4 b is almost
impossible, since the quantization is a nonlinear process and depends on the input.
The latter, of course, is unknown a priori. In addition, the possibility of overflow
exacerbates the difficulty of analysis. One can perform an approximate analysis
through the adoption of some assumptions based on using a stochastic model that
eventually linearizes the problem. This, however, is only an approximation and
needs to be used with care.
MATLAB: The Fixed-point numeric object fi can be used to convert the
double-precision representation normally used in MATLAB to fixed-point format.
The fi object has the following three general types of properties:
• Data Properties.
• Mathematical ( fimath ) Properties.
• Numeric type ( numerictype ) Properties
Each one of these classes of properties automatically generates several relevant
options that can be accessed to set the required fixed-point operations.
Example 1 Design an FIR digital lowpass filter using fixed-point representation
with 12 bit wordlength and a 32-bit accumulator. The required filter specifications
are as follows:
Sampling frequency fs = 4000 Hz
The passband-edge frequency, fpass = 300 Hz
The stopband-edge frequency, fstop = 1000 Hz
maximum peak-to-peak ripple, Rpass = 0.015
minimum stopband attenuation, Rstop = 0.2
Solution : The first step is to design the filter in the default MATLAB double-
precision representation. The Parks-McClellan optimal equiripple FIR order esti-
mator firpmord can be used to find the approximate order M (see MATLAB
signal processing toolbox documentation):