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PROBLEM:
Consider the following system for sampling, filtering, and reconstruction of a continuous-time signal:
x(t)
-
Ideal
C-to-D
Converter
6
f s = 1/Ts
x[n]
-
LTI
System
H (z)
y[n]
-
Ideal
D-to-C
Converter
-
y(t)
6
f s = 1/Ts
where f s = 1000 samples/sec, the LTI system has the system function H (z) = 3z −2 , and the continuoustime input signal is
x(t) = 2 cos(750πt + π/4) + 3 cos(1600πt − 3π/5).
(a) Plot the complete frequency spectrum for x[n] in the region −π < ω̂ ≤ π.
(b) Determine an expression for the output y(t) of this system for the input x(t) indicated.
PROBLEM:
Factor the following polynomial and plot its roots in the complex plane.
P(z) = 1 + 12 z −1 + 21 z −2 + z −3
Note the symmetry of the coefficients. In M ATLAB see the functions called roots and zplane.
PROBLEM:
Suppose that three systems are hooked together in “cascade.” In other words, the output of S 1 is the input
to S2 , and the output of S2 is the input to S3 . The three systems are specified as follows:
S1 :
y1 [n] = x 1 [n] − x 1 [n − 1]
S2 :
y2 [n] = x 2 [n] + x 2 [n − 2]
S3 :
y3 [n] = x 3 [n − 1] + x[n − 2]
NOTE: the output of Si is yi [n] and the input is x i [n].
Determine the equivalent system that is a single operation from the input x[n] (into S 1 ) to the output
y[n] which is the output of S3 . Thus x[n] is x 1 [n] and y[n] is y3 [n].
Write one difference equation that defines the overall system in terms of x[n] and y[n] only..
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PROBLEM:
The input to the C-to-D converter in the figure below is
x(t) = 7 + 9 cos(1600πt − π/4) − 11 cos(12000πt − π/3)
The system function for the LTI system is
H (z) =
1
(1 + z −5 )
5
If f s = 8000 samples/second, determine an expression for y(t), the output of the D-to-C converter.
x(t)
-
Ideal
C-to-D
Converter
6
T = 1/ f s
x[n]
-
LTI
System
H (z)
y[n]
-
Ideal
D-to-C
Converter
6
T = 1/f
-
y(t)
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PROBLEM:
The diagram in Fig. 1 depicts a cascade connection of two linear time-invariant systems; i.e., the output of
the first system is the input to the second system, and the overall output is the output of the second system.
x[n]
-
LTI
System #1
h 1 [n]
w[n]
-
LTI
System #2
h 2 [n]
-
y[n]
Figure 1: Cascade connection of two LTI systems.
(a) Suppose that LTI System #1 is described by the difference equation
w[n] = x[n] − αx[n − 1].
Determine the impulse response h 1 [n] of the first system.
(b) The LTI System #2 is described by the impulse response
n
h 2 [n] = α (u[n] − u[n − L]) =
L−1
X
k=0
k
α δ[n − k] =
αn
0
n = 0, 1, . . . , L − 1
otherwise.
For the special case of L = 6, use convolution to show that the impulse response sequence of the
overall cascade system is
h[n] = h 1 [n] ∗ h 2 [n] = δ[n] − α 6 δ[n − 6].
(c) Generalize your result in part (b) for the general case of L any integer value.
(d) Obtain a single difference equation that relates y[n] to x[n] in Fig. 1.
(e) Assuming that 0 < α < 1, how would you choose L so that y[n] = x[n] in Figure 1; i.e., how would
you choose L so that the second system “undoes” the effect of the first system?
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PROBLEM:
Consider the following cascade system:
x[n]
X (z)
-
LTI
System #1
H1 (z)
w[n]
W (z)
-
LTI
System #2
H2 (z)
-
y[n]
Y (z)
It is known that
H (z) = (1 − z −2 )(1 − 0.8e jπ/4 z −1 )(1 − 0.8e− jπ/4 z −1 )(1 + z −2 )
(a) Determine the poles and zeros of H (z) and plot them in the complex z-plane.
(b) It is possible to determine two possible system functions H1 (z) and H2 (z) so that: (1) the overall
cascade system has the given system function H (z) and (2) w[n] = x[n] − x[n − 4]. Find H1 (z) and
H2 (z).
(c) Determine the difference equation that relates y[n] to w[n] for your answer in part (b).
PROBLEM:
The following is a simple problem that might be posed about an LTI system:
Given the input sequence x[n] find the output sequence y[n] of a 5-point running average filter
for all values of n.
The following is a partial list of possible approaches to solving this problem:
1. Use the difference equation representation of the system to compute (e.g., using M ATLAB) the output
y[n] for all required values of n.
2. Multiply the z-transform of the input by the system function and determine y[n] as the inverse ztransform of Y (z).
3. Break the input into a sum of complex exponential signals, use the frequency response function to
determine the output due to each complex exponential signal separately, and finally, add the individual
outputs together to get y[n].
4. Some combination of the above methods. Remember, when the input is a sum of two or more signals
and the system is linear, we can solve the problem separately for each of the input components and
then superimpose the outputs. We can therefore use the method that is most appropriate for each of
the components of the input.
In each of these solution methods you would use one or more of the basic representations of the 5-point
running average filter. Which method is easiest will have a lot to do with the nature of the input signal. This
may require that you convert a given representation of the system into one of the other forms. For example,
if you are given the difference equation and you want to use approach #2, you will have to determine the
system function H (z) from the difference equation coefficients.
Now in each of the following cases, the input will be given. In each case, determine which representation
of the system and which of the above approaches will lead to the easiest solution of the problem, and detail
the steps in using that approach to solve the problem. For example, if you choose approach #2 to solve the
problem, your answer should be something like the following:
Step 1 Find X (z), the z-transform of x[n].
Step 2 Find H (z), the system function of the 5-point running averager.
Step 3 Multiply X (z)H (z) to get Y (z).
Step 4 Find the inverse z-transform of Y (z) to get y[n].
Now here are some possible inputs. In each case, state which of the above (#1, #2, or #3) approaches
you would use. There may not be a clear cut answer. Give the approach that you think will yield the solution
with least effort. Outline your approach to solving the problem of finding the output of the 5-point moving
averager. You do not have to actually find the output—just tell how you would solve it in a step-by-step
procedure described as illustrated above.
(a) x[n] is a sampled audio signal. It is represented by a vector of 100000 numbers.
(b) x[n] = 4 cos(0.1πn + π/2) + 3 cos(0.4πn − π ) for −∞ < n < ∞.
1
0 ≤ n ≤ 10
(c) x[n] =
0
otherwise.
(d) x[n] = 10δ[n − 50].
(e) x[n] = 10δ[n − 50] + 4 cos(0.1πn + π/2) + 3 cos(0.4πn − π) for −∞ < n < ∞.
PROBLEM:
We now have four ways of describing an LTI system: the difference equation; the impulse response, h[n];
the frequency response, H (e j ω̂ ); and the system function, H (z). In the following, you are given one of these
representations and you must find the other three.
(a) y[n] = (x[n] + 2x[n − 2] + x[n − 4]).
(b) h[n] = δ[n] + δ[n − 1] + δ[n − 2] + δ[n − 3] + δ[n − 4].
(c) H (e j ω̂ ) = [1 + cos(2ω̂)]e− j ω̂3 . Hint: Expand the cosine using Euler’s formula.
(d) H (z) = 1 − 2z −2 + z −4 + z −7 .
PROBLEM:
The input to the C-to-D converter in the figure below is
x(t) = 10 + 6 cos(2000πt + π/8) + 4 cos(6000πt − π/3)
The system function of the LTI system is
H (z) = (1 − z −2 )
x(t)
-
Ideal
C-to-D
Converter
6
T = 1/ f s
x[n]
-
LTI
System
H (z)
y[n]
-
Ideal
D-to-C
Converter
-
y(t)
6
T = 1/ f s
(a) If f s = 10000 samples/second, determine an expression for y(t), the output of the D-to-C converter.
(b) If f s = 5000 samples/second, determine an expression for y(t), the output of the D-to-C converter.
Note that even when aliasing distortion occurs, we can still determine the effect of the system on the
input x[n] and therefore we can determine y(t) from y[n].
PROBLEM:
Consider the following cascade system:
x[n]
X (z)
-
LTI
System #1
H1 (z)
w[n]
W (z)
-
LTI
System #2
H2 (z)
-
y[n]
Y (z)
Suppose that
H1 (z) = (1 − j z −1 )(1 + j z −1 )(1 + z −1 ) and
h 2 [n] = δ[n] + δ[n − 4]
(a) Determine the system function, H (z), for the overall cascade system (i.e., from input X (z) to output
Y (z).)
(b) Determine and plot the impulse response h[n] of the overall cascade system.
(c) Write down the difference equation that relates y[n] to x[n].