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# hw3 - ECE426 SPRING2007 HOMEWORK SET 3 Special Due Date...

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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 second-order 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 sub-region (C) inside the stability triangle that corresponds to complex poles (poles with a non-zero 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 S-C (S outside of C)? H12 LIMIT CYCLES A second-order digital filter section is shown below. The feedback coefficients are 1 .8 and «0.9-as shown. >¢C5i=o (a) Determine if this filter is stable, assuming no quantization. (b) Determine if this filter has a small- scale limit-cycle if it starts with an initial oonditiOn y(n-1)=—20 y(n-2)= —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 two-step overflow oscillation for this case. - {Notez' Refer to the FWL notes on the blackboard .} H12 SCALE-CHANGE 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 floating-point 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 second-order filter consisting of two first-order 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 first-order section. Show that the result for the overall response agrees with the results given in the AN-295 reference, equation (23) on the blackboard. ...
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hw3 - ECE426 SPRING2007 HOMEWORK SET 3 Special Due Date...

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