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Unformatted text preview: ECE426 SPRING2007 HOMEWORK SET 3 Special Due Date: Thursday April 12, 2007 5PM H11 STABILITYITRIANGLE The figure below shows a secondorder digital filter and the "stability triangle" in the
"coefficient plane" as discussed in lecture, " (a) Demonstrate that the network has stable poles if A1 and A0 are chosen inside
the stability triangle (Region 8). (b) Find the subregion (C) inside the stability triangle that corresponds to
complex poles (poles with a nonzero imaginary part)“ ' '"'(c)"Sh'owth'at'Si'de""a":'C'orre'sp'0ndS‘to‘d'i'gital‘si‘n‘e'wave'“oscillators—4 'c‘o‘m'pleX‘conj'ugate "“ "' poles on the unit circle. What happens 'at Ao=1 and A1=12? I (d) Since thestabiiity boundary, the entire unit circle in the z—plane, corresponds
to Side "a" itself (a portion of the boundary in the coefficient plane), how do
you explain Side Y‘b” and Side "c"? (e) What is going on in the region SC (S outside of C)? H12 LIMIT CYCLES A secondorder digital filter section is shown below. The feedback coefficients are 1 .8
and «0.9as shown. >¢C5i=o (a) Determine if this filter is stable, assuming no quantization. (b) Determine if this filter has a small scale limitcycle if it starts with an initial
oonditiOn y(n1)=—20 y(n2)= —20, and the output summer is subject to rounding
to the nearest integer. If it has a limit cycle, show that the particular rounding
case for this limit cycle results in a pole appropriate for this "oscillation ." ' (c) Now suppose that the output of the summer is subject to two's complement
(Mickey Mouse) arithmetic. Find the twostep overflow oscillation for this
case.  {Notez' Refer to the FWL notes on the blackboard .} H12 SCALECHANGE INTERMODULATION DISTORTION A signal consists of two components:
x1 (t) = sin.(2n*1000*t)
x2(t) = 1000 {0.5 + 0.5 sgn[sin(2n*100*t)] } That is, we have a small sinewave of frequency 1000 Hz and a much larger square
function that is either 1000 or zero, at a frequency of 100 Hz. Suppose that the signal
is sampled and converted to a floatingpoint format of: .xys X 10W where xyz are three decimal places and w is an exponent that can be any integer.
Find the spectrum of the sampled, quantized signal. You may calculate this
analytically, approximate it with Matlab, or both. Assume that the sampling frequency
is sufficiently high.  ' H14 NOISE GAIN In the figure below, we have a secondorder filter consisting of two firstorder
sections in cascade, For this network, both feedback coefficients are the same value
. “a,” and N1, N2, and N3 all represent a noise power of (IQ? Find a closed form
expression for the total noise power at the output. It would be a good idea to verify this result using an experimental test with Matlab, In the calculation above, you have needed to sum the impulse response of the overall
filter‘in addition to that of the firstorder section. Show that the result for the overall
response agrees with the results given in the AN295 reference, equation (23) on the blackboard. ...
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 Spring '06
 HUTCHINS
 Signal Processing

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