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Problems 591
P10.6 Consider the 1st-order recursive system y(n) =0.75 y(n − 1) + 0.125δ(n) with zero initial
conditions. The filter is implemented in 4-bit (including sign) fixed-point
two’s-complement fractional arithmetic. Products are rounded to 3-bits.
1. Determine and plot the first 20 samples of the output using saturation limiter for the
addition. Does the filter go into a limit cycle?
2. Determine and plot the first 20 samples of the output using two’s-complement overflow
for the addition. Does the filter go into a limit cycle?
P10.7 Repeat Problem P10.6 when products are truncated to 3 bits.
P10.8 Consider the 2nd-order recursive system y(n) =0.125δ(n) − 0.875 y(n − 2) with zero
initial conditions. The filter is implemented in 5-bit (including sign) fixed-point
two’s-complement fractional arithmetic. Products are rounded to 4-bits.
1. Determine and plot the first 30 samples of the output using a saturation limiter for the
addition. Does the filter go into a limit cycle?
2. Determine and plot the first 30 samples of the output using two’s-complement overflow
for the addition. Does the filter go into a limit cycle?
P10.9 Repeat Problem P10.8 when products are truncated to 4 bits.
P10.10 Let x(n) = 1 [sin(n/11) + cos(n/13) + sin(n/17) + cos(n/19)] and c =0.7777. For the
4
following parts use 500, 000 samples of x(n) and the StatModelR function.
1. Quantize cx(n) toB =4bits, and plot the resulting distributions for the error signals
e 1(n) and e 2(n). Comment on these plots.
2. Quantize cx(n) toB =8bits, and plot the resulting distributions for the error signals
e 1(n) and e 2(n). Comment on these plots.
3. Quantize cx(n) toB =12bits, and plot the resulting distributions for the error signals
e 1(n) and e 2(n). Comment on these plots.
P10.11 Let x(n) =bearandom sequence uniformly distributed between −1 and 1, and let
c =0.7777. For the following parts, use 500, 000 samples of x(n) and the StatModelR
function.
1. Quantize cx(n) toB =4bits and plot the resulting distributions for the error signals
e 1(n) and e 2(n). Comment on these plots.
2. Quantize cx(n) toB =8bits and plot the resulting distributions for the error signals
e 1(n) and e 2(n). Comment on these plots.
3. Quantize cx(n) toB =12bits and plot the resulting distributions for the error signals
e 1(n) and e 2(n). Comment on these plots.
P10.12 Consider an LTI system with the input x(n) and output y(n)
y(n) =b 0x(n)+b 1x(n − 1) + a 1y(n − 1) (10.106)
1. Draw the direct-form I structure for the above system.
2. Let e b0 (n) denote the multiplication quantization error resulting from the product
b 0x(n), e b1 (n − 1) from the product b 1x(n − 1), and e a1 (n − 1) from the product
a 1y(n − 1) in the direct-form I realization. Draw an equivalent structure that contains
only one noise source.
3. Draw an equivalent system that can be used to study multiplication quantization error
for the system in (10.106). The input to this system should be the noise source in
part 2, and the output should be the overall output error q(n).
4. Using the model in part 3, determine an expression for the variance of the output error
e(n).
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