Lab 3: Prelab (Part 2)
1.1
2006/07/26 11:21:23.721 GMT-5
2006/08/20 22:20:28.067 GMT-5
Thomas
Shen
tbshen@uiuc.edu
Douglas
L.
Jones
dl-jones@uiuc.edu
Thomas
Shen
tbshen@uiuc.edu
coefficient quantization
DSP
fixed-point
IIR filter
integer
stability
You will design a fourth-order notch filter and investigate the effects of filter-coefficient quantization. You will compare the response of the filter having unquantized coefficients with that of a filter having coefficients quantized as a single, fourth-order stage and with that of a filter having coefficients quantized as a cascade of two, second-order stages.
Filter-Coefficient Quantization
One important issue that must be considered when IIR filters
are implemented on a fixed-point processor is that the filter
coefficients that are actually used are quantized from the
"exact" (high-precision floating point) values computed by
MATLAB. Although quantization was not a concern when we
worked with FIR filters, it can cause significant deviations
from the expected response of an IIR filter.
By default, MATLAB uses 64-bit floating point numbers in all
of its computation. These floating point numbers can typically
represent 15-16 digits of precision, far more than the DSP can
represent internally. For this reason, when creating filters
in MATLAB, we can generally regard the precision as
"infinite," because it is high enough for any reasonable task.
Not all IIR filters are necessarily "reasonable"!
The DSP, on the other hand, operates using 16-bit fixed-point
numbers in the range of -1.0 to
1.0
2
-15
. This gives the DSP only 4-5 digits of precision
and only if the input is properly scaled to occupy the full
range from -1 to 1.
For this section exercise, you will examine how this
difference in precision affects a notch filter
generated using the `butter`

command:
`[B,A] = butter(2,[0.07 0.10],'stop')`

.
Quantizing coefficients in MATLAB
It is not difficult to use MATLAB to quantize
the filter coefficients to the 16-bit precision used on the
DSP. To do this, first take each vector of filter
coefficients (that is, the
A
and
B
vectors) and divide by the smallest power of two
such that the resulting absolute value of the largest filter
coefficient is less than or equal to one. This is an easy
but fairly reasonable approximation of how numbers outside
the range of -1 to 1 are actually handled on the DSP.
Next, quantize the resulting vectors to 16 bits of precision
by first multiplying them by
2
15
32768
, rounding to the nearest integer (use
`round`

), and then dividing the resulting vectors
by 32768. Then multiply the resulting numbers, which will be
in the range of -1 to 1, back by the power of two that you
divided out.
Effects of quantization
Explore the effects of quantization by quantizing the filter
coefficients for the notch filter. Use the
`freqz`

command to compare the response of the
unquantized filter with two quantized versions: first,
quantize the entire fourth-order filter at once, and second,
quantize the second-order ("bi-quad") sections separately
and recombine the resulting quantized sections using the
`conv`

function. Compare the
response of the unquantized filter and the two quantized
versions. Which one is "better?" Why do we always
implement IIR filters using second-order sections instead of
implementing fourth (or higher) order filters directly?
Be sure to create graphs showing the difference between the
filter responses of the unquantized notch filter, the notch
filter quantized as a single fourth-order section, and the
notch filter quantized as two second-order sections. Save
the MATLAB code you use to generate these graphs, and be
prepared to reproduce and explain the graphs as part of your
quiz. Make sure that in your comparisons, you rescale the
resulting filters to ensure that the response is unity (one)
at frequencies far outside the notch.