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298 Chapter 6 IMPLEMENTATION OF DISCRETE-TIME FILTERS
your function on the following numbers. In each case the numbers to be considered are
both positive and negative. Also, in each case select the appropriate number of bits B.
(a) x = ±0.5625 (b) x = ±0.40625 (c) x = ±0.953125
(d) x = ±0.1328125 (f) x = ±0.7314453125
Compare your representations with those in Problem P6.26, part 1.
2. Develop a MATLAB function x = tc2sm(y, B) that converts the B-bit two’s-complement
format decimal equivalent number y into the sign-magnitude format fraction x. Verify
your function on the following fractional binary representations:
(a) y = 110110 (b) y = 0.011001 (c) y = 100110011
(d) y = 111101110
(e) y = 000010001
Compare your representations with those in Problem P6.26, part 2.
P6.28 Determine the 10-bit sign-magnitude, one’s-complement, and two’s-complement
representation of the following decimal numbers:
(a) 0.12345 (b) −0.56789 (c) 0.38452386 (d) −0.762349 (e) −0.90625
P6.29 Consider a 32-bit floating-point number representation with a 6-bit exponent and a 25-bit
mantissa.
1. Determine the value of the smallest number that can be represented.
2. Determine the value of the largest number that can be represented.
3. Determine the dynamic range of this floating-point representation and compare it with
the dynamic range of a 32-bit fixed-point signed integer representation.
P6.30 Show that the magnitudes of floating-point numbers in a 32-bit IEEE standard range from
1.18 × 10 −38 to 3.4 × 10 38 .
P6.31 Compute and plot the truncation error characteristics when B =4for the sign-magnitude,
one’s-complement, and two’s-complement formats.
P6.32 Consider the 3rd-order elliptic lowpass filter:
H(z) = 0.1214 ( 1 − 1.4211z −1 + z −2)( 1+z −1)
(1 − 1.4928z −1 +0.8612z −2 )(1− 0.6183z −1 )
1. If the filter is realized using a direct-form structure, determine its pole sensitivity.
2. If the filter is realized using a cascade-form structure, determine its pole sensitivity.
P6.33 Consider the filter described by the difference equation
y(n) = 1 √
2
y(n − 1) − x(n)+ √ 2x(n − 1) (6.84)
1. Show that this filter is an all-pass filter (i.e., |H(e jω )|) isaconstant over the entire
frequency range −π ≤ ω ≤ π. Verify your answer by plotting the magnitude response
|H(e jω )| over the normalized frequency range 0 ≤ ω/π ≤ 1. Use subplot(3,1,1).
2. Round the coefficients of the difference equation in (6.84) to 3 decimals. Is the filter still
all-pass? Verify your answer by plotting the resulting magnitude response, |Ĥ1(e jω )|,
over the normalized frequency range 0 ≤ ω/π ≤ 1. Use subplot(3,1,2).
3. Round the coefficients of the difference equation in (6.84) to 2 decimals. Is the filter still
all-pass? Verify your answer by plotting the resulting magnitude response, |Ĥ2(e jω )|,
over the normalized frequency range 0 ≤ ω/π ≤ 1. Use subplot(3,1,3).
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