hw9 - ECE Users Pages - Georgia Institute of Technology

GEORGIA INSTITUTE OF TECHNOLOGY
SCHOOL OF ELECTRICAL AND COMPUTER ENGINEERING
ECE 2026
Summer 2014
Problem Set #9
Assigned: July 11, 2014
Due: July 21 or July 22, 2014 (in rec)
Reading: Chapter 7 on The z-Transform, and Chapter 8 on IIR Filters.
Reminder: Only turn in starred problems, a subset of which will be randomly selected for grading.
Your homework is due in recitation at the beginning of recitation. Late homeworks will be given a zero.
Decompose your solution to each starred problem into the following two parts: Approach, and Details.
PROB. 9.1.* This problem concerns the cascade connection of three LTI systems, as shown below:
x[ n ]
3-POINT
RUNNING
AVERAGE
h1 [ n ]
y1[ n ] = x2[ n ]
LTI
SYSTEM
#2
y2[ n ] = x3[ n ]
h2 [ n ]
LTI
SYSTEM
#3
y[ n ]
h3 [ n ]
where:
• the first system is a 3-point running average filter;
• the second system is a first-difference filter defined by the difference equation:
y[ n ] = x2[ n ] – x2[ n – 1 ];
• and the third system has the following frequency response:
H3(e jˆ ) = 1 + e –jˆ .
(a)
(b)
(c)
(d)
(e)
Find an equation for the system function H1( z ) of the first LTI system.
Find an equation for the system function H2( z ) of the second LTI system.
Find an equation for the system function H3( z ) of the third LTI system.
Find an equation for the system function H( z ) of the overall system (from x[ n ] to y[ n ]).
Find an equation for the impulse response h[ n ] of the overall system (from x[ n ] to y[ n ]).
PROB. 9.2.* Consider a linear time-invariant system whose difference equation is:
y[ n ] = x[ n ] + b1x[n – 1] + x[n – 2].
We want to use this system to remove an unwanted pure tone at 3200 Hz from a digital
recording of music, where the sampling rate was 10 kHz. Find a numerical value for the
unspecificed filter coefficient b1 so that the unwanted tone is completely eliminated.
PROB. 9.3.* Consider the following system which performs discrete-time processing of a continuoustime signal:
x( t )
IDEAL
C-to-D
CONVERTER
x[ n ]
LTI
SYSTEM
y[ n ]
IDEAL
D-to-C
CONVERTER
y( t )
fs = 4800 Hz
The sampling rate for both the C-D and D-C converters is fs = 4800 Hz, and the discretetime system is defined by the difference equation:
y[ n ] = x[ n ] + x[ n – 1 ] + x[n – 2 ].
This is an FIR filter with coefficients {b0, b1, b2} = {, 1, }. Find a numerical value for
the parameter  so that the overall response to the following continuous-time input signal:
is:
x( t ) =
3 – 2cos(800t – /7)
y( t ) =
3 – 2.
PROB. 9.4.* Plot all of the poles and zeros of the system function H( z ) in the complex z-plane for a
linear time-invariant system defined by the following difference equation:
y[ n ] = x[ n ] + 0.2x[n – 1] – 0.4y[n – 1] – 0.16y[n – 2].
PROB. 9.5.* Write the difference equation for an LTI system whose system function H( z ) has the
following pole-zero plot:
Im{z}
(unit circle)
Re{z}
(The locations of the poles are not specified precisely, but they can be easily estimated to
within say10%.)
PROB. 9.6.
A second-order IIR system is defined by the following system function:
1 + 0.35z – 1
H( z ) = --------------------------------------------------- .
1 + 0.7z – 1 + 0.49z – 2
(a)
(b)
(c)
(d)
(e)
PROB. 9.7.
Write the difference equation that relates the input x[ n ] to the output y[ n ].
Plot all the poles and zeros of H( z ) in the complex z-plane.
Is it possible for a sinusoidal input of the form x[ n ] = cos(
ˆ1n) to produce the all-zero
output y[ n ] = 0 for all n? (Answer YES or NO.)
ˆ1   such that the
If the anser to part (c) is YES, specify all values of 
ˆ1 in the range 0  
output would be zero.
If the answer to part (c) is NO, explain why it is impossible.
It would take some work to find the step response s[ n ] of this system for all n, so we
won’t try. But eventually, for large enough values of n, the step response settles down.
In the limit of large n, the step response approaches a constant. What is this constant?
Let us start with a first-order IIR system whose difference equation is:
y1[ n ] = 0.5y1[ n – 1 ] + x1[ n ].
Suppose that we use this system as a building block to create a more complicated system.
In particular, as shown below, suppose that we form a serial cascade of five such systems:
x[ n ]
1st-ORDER
IIR
1st-ORDER
IIR
1st-ORDER
IIR
1st-ORDER
IIR
1st-ORDER
IIR
y[ n ]
Assume that each IIR system is initially at rest, so that the overall system is LTI.
(a)
(b)
Find the difference equation that relates the overall output y[ n ] to the overall input x[ n ].
P
Hint: Use the z-transform and the binomial expansion (a+b)n = n ( nk )akbn – k.
k=0
A sinusoidal input of the form x[ n ] = cos(
ˆ 1 n) will result in a sinusoidal output of the
form y[ n ] = Acos(
ˆ 1 n + ), where the value of A ≥ 0 and  will depend on the value
ˆ1 .
of the frequency 
Find the value for 
ˆ 1 in the range 0  
ˆ1  that results in an output amplitude of A = 1;
ˆ 1 so that the response to x[ n ] = cos(
ˆ 1 n) is y[ n ] = cos(
ˆ 1 n + ) for some .
i.e., find 
PROB. 9.8.
(a)
(b)
True or False. Explain the reasoning behind your response. (Hint: Only one is true!)
The cascade connection of two LTI FIR systems is necessarily FIR.
The cascade connection of two LTI IIR systems is necessarily IIR.
PROB. 9.9.
Consider a linear time-invariant system whose system function is:
 1 + z –1   1 – e j/3 z – 1   1 – e –j /3 z – 1 
H( z ) = -------------------------------------------------------------------------------------------------- .
 1 + 0.8e j2/3 z –1   1 + 0.8e – j2/ 3 z – 1 
(a)
(b)
(c)
(d)
(e)
(f)
Write the difference equation that relates the input x[ n ] to the output y[ n ] .
Hint: Multiply out the factors.
Plot all the poles and zeros of H( z ) in the complex z-plane.
Hint: Express H( z ) as a ratio of factored polnomials in positive power of z instead of z–1.
If a sinusoidal input of the form x[ n ] = 2.3cos(
ˆ1n + /5) results in the all-zero output
y[ n ] = 0 for all n, what are the possible values for the sinusoid frequency 
ˆ1 in the range
0
ˆ1  ?
Is it possible for a sinusoidal input of the form x[ n ] = cos(
ˆ1n) to produce the all-zero
output y[ n ] = 0 for all n? (Answer YES or NO.)
If the anser to part (c) is YES, specify all values of 
ˆ1 in the range 0  
ˆ1   such that the
output would be zero.
If the answer to part (c) is NO, explain why it is impossible.
It would take some work to find the step response s[ n ] of this system for all n, so we
won’t try. But eventually, for large enough values of n, the step response settles down.
In the limit of large n, the step response approaches a constant. What is this constant?
PROB. 9.10. So far we have characterized an FIR or IIR LTI system in four different ways:
(i)
(ii)
(iii)
(iv)
The difference equation relating y[ n ] to x[ n ].
The impulse response h[ n ].
The frequency response H(e jˆ ).
The system function H( z ).
Knowing any one of the four is enough to determine the other three.
In each part below we specify one of the four. In each case, specify the other three.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
h[ n ] = [n – 7].
y[ n ] = x[ n ] + 2x[n – 1] + 2x[n – 2] + x[n – 3].
H(e jˆ ) = 5e –j8ˆ .
H( z ) = z–1.
y[ n ] = x[ n ] – 0.1y[n – 1].
ˆ ).
H(e jˆ ) = 7je –j8ˆ sin(3
H( z ) = (1 – z–1)(1 + z–1)2.
ˆ
sin  2.5
H(e jˆ ) = 7e –j8ˆ ------------------------- .
ˆ
sin  0.5
1
H( z ) = ------------------------- .
1 – 0.3z – 1

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