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<font size="4"> Communication Networks Signal and Image processing (CS) </font> | <font size="4"> Communication Networks Signal and Image processing (CS) </font> | ||
| − | <font size="4"> [[QE637_sol2012|Question 5, August 2012(Published on May 2017)]] | + | <font size="4"> [[QE637_sol2012|Question 5, August 2012(Published on May 2017)]]</font> |
| + | </center> | ||
<font size="4">[[ QE637_sol2012_Q1 | Problem 1]],[[ QE637_sol2012_Q2 |2]]</font> | <font size="4">[[ QE637_sol2012_Q1 | Problem 1]],[[ QE637_sol2012_Q2 |2]]</font> | ||
| − | |||
---- | ---- | ||
===Solution1:=== | ===Solution1:=== | ||
| − | a) | + | a) |
| + | |||
<math> | <math> | ||
\begin{align} | \begin{align} | ||
& {{h}_{1}}(m,n)=\sum\limits_{j=-N}^{N}{{a}_{j}\delta (m,n-j)=}{a}_{n}\delta (m) \\ | & {{h}_{1}}(m,n)=\sum\limits_{j=-N}^{N}{{a}_{j}\delta (m,n-j)=}{a}_{n}\delta (m) \\ | ||
& \delta (m,n)=\left\{ \begin{matrix} | & \delta (m,n)=\left\{ \begin{matrix} | ||
| − | 1\ m=n=0 \\ | + | 1\quad m=n=0 \\ |
0\qquad O.W \\ | 0\qquad O.W \\ | ||
| − | \end{matrix}, \right. \delta (m,n-j)=\left\{ \begin{matrix} | + | \end{matrix}, \right. \quad \delta (m,n-j)=\left\{ \begin{matrix} |
| − | 1\ | + | 1\quad m=0;\ n=j \\ |
0\qquad O.W \\ | 0\qquad O.W \\ | ||
\end{matrix} \right. \\ | \end{matrix} \right. \\ | ||
| Line 33: | Line 34: | ||
</math> | </math> | ||
| − | b) | + | b) |
| + | |||
<math> | <math> | ||
\begin{align} | \begin{align} | ||
& {{h}_{2}}(m,n)=\sum\limits_{i=-N}^{N}{{b}_{i}\delta (m-i,n)=}{b}_{m}\delta (n) \\ | & {{h}_{2}}(m,n)=\sum\limits_{i=-N}^{N}{{b}_{i}\delta (m-i,n)=}{b}_{m}\delta (n) \\ | ||
& \delta (m,n)=\left\{ \begin{matrix} | & \delta (m,n)=\left\{ \begin{matrix} | ||
| − | 1\ m=n=0 \\ | + | 1\quad m=n=0 \\ |
0\qquad O.W \\ | 0\qquad O.W \\ | ||
| − | \end{matrix}, \right. \delta (m-i,n)=\left\{ \begin{matrix} | + | \end{matrix}, \right. \quad \delta (m-i,n)=\left\{ \begin{matrix} |
| − | 1\ m=i;\ n=0 \\ | + | 1\quad m=i;\ n=0 \\ |
0\qquad O.W \\ | 0\qquad O.W \\ | ||
\end{matrix} \right. \\ | \end{matrix} \right. \\ | ||
| Line 47: | Line 49: | ||
</math> | </math> | ||
| − | c) | + | c) |
| + | |||
<math> | <math> | ||
\begin{align} | \begin{align} | ||
| Line 53: | Line 56: | ||
& z(m,n)=\sum\limits_{i=-N}^{N}{{{b}_{i}}\ y(m-i,n)=}\sum\limits_{i=-N}^{N}{{{b}_{i}}\ \left( \sum\limits_{j=-N}^{N}{{{a}_{j}}\ x(m-i,n-j)} \right)=\sum\limits_{i=-N}^{N}{\sum\limits_{j=-N}^{N}{{{b}_{i}}\ {{a}_{j}}\ x(m-i,n-j)}}} \\ | & z(m,n)=\sum\limits_{i=-N}^{N}{{{b}_{i}}\ y(m-i,n)=}\sum\limits_{i=-N}^{N}{{{b}_{i}}\ \left( \sum\limits_{j=-N}^{N}{{{a}_{j}}\ x(m-i,n-j)} \right)=\sum\limits_{i=-N}^{N}{\sum\limits_{j=-N}^{N}{{{b}_{i}}\ {{a}_{j}}\ x(m-i,n-j)}}} \\ | ||
& \delta (m-i,n-j)=\left\{ \begin{matrix} | & \delta (m-i,n-j)=\left\{ \begin{matrix} | ||
| − | 1\ m=i;\ n=j \\ | + | 1\quad m=i;\ n=j \\ |
0\qquad O.W \\ | 0\qquad O.W \\ | ||
\end{matrix} \right. \\ | \end{matrix} \right. \\ | ||
| Line 59: | Line 62: | ||
</math> | </math> | ||
| − | d) | + | d) |
| − | Number of multiplies per output point to implement each individual system = 2N+1 | + | |
| − | So, | + | Number of multiplies per output point to implement each individual system = 2N+1. |
| + | So, the number of multiplies per output point to implement each of the two individual systems is 2(2N+1) = 4N+2. | ||
Number of multiplies per output point to implement the complete system with a single convolution is <math>\left( 2N+1 \right)\left( 2N+1 \right)\text{ }=4{{N}^{2}}+4N+1</math> | Number of multiplies per output point to implement the complete system with a single convolution is <math>\left( 2N+1 \right)\left( 2N+1 \right)\text{ }=4{{N}^{2}}+4N+1</math> | ||
e) | e) | ||
| + | |||
Implementing the two systems in sequence requires less computation, but it is more complex and more sensitive to noise. | Implementing the two systems in sequence requires less computation, but it is more complex and more sensitive to noise. | ||
Implementing the two systems in a single complete system requires more computation, but it is simpler, less sensitive to noise, and more stable. | Implementing the two systems in a single complete system requires more computation, but it is simpler, less sensitive to noise, and more stable. | ||
| Line 72: | Line 77: | ||
== Solution 2: == | == Solution 2: == | ||
| − | a) | + | a) |
| + | |||
<math> | <math> | ||
\begin{align} | \begin{align} | ||
| − | & {{h}_{1}}(m,n)=\sum\limits_{j=-N}^{N}{{a}_{j}\delta (m,n-j)=}\sum\limits_{j=-N}^{N}{{a}_{j}\delta (m)\ \delta(n-j)=} | + | & {{h}_{1}}(m,n)=\sum\limits_{j=-N}^{N}{{a}_{j}\delta (m,n-j)=}\sum\limits_{j=-N}^{N}{{a}_{j}\delta (m)\ \delta(n-j)=} {a}_{n}\ \delta (m) \\ |
\end{align} | \end{align} | ||
</math> | </math> | ||
| − | + | <span style="color:green"> The answer is correct, but it's not obvious how the solution is derived. (See Solution 1) </span> | |
| + | |||
| + | b) | ||
| + | |||
<math> | <math> | ||
\begin{align} | \begin{align} | ||
| − | & {{h}_{2}}(m,n)=\sum\limits_{i=-N}^{N}{{b}_{i}\delta (m-i, | + | & {{h}_{2}}(m,n)=\sum\limits_{i=-N}^{N}{{b}_{i}\delta (m-i,n)=}\sum\limits_{i=-N}^{N}{{b}_{i}\delta (m-i)\ \delta(n)=}{b}_{m}\ \delta (n) \\ |
\end{align} | \end{align} | ||
</math> | </math> | ||
| − | + | <span style="color:green"> The answer is correct, but it's not obvious how the solution is derived. (See Solution 1) </span> | |
| + | |||
| + | c) | ||
| + | |||
<math> | <math> | ||
\begin{align} | \begin{align} | ||
| Line 92: | Line 104: | ||
\end{align} | \end{align} | ||
</math> | </math> | ||
| + | |||
| + | <span style="color:green"> The answer is correct, but it's not obvious how the solution is derived. (See Solution 1) </span> | ||
d) | d) | ||
| + | |||
Individually: <math> 2(2N+1)=4N+2 </math> <br \> | Individually: <math> 2(2N+1)=4N+2 </math> <br \> | ||
| − | Complete system: <math> | + | Complete system: <math>{( 2N+1)}^{2}=4{{N}^{2}}+4N+1</math> |
| − | + | ||
| − | + | ||
| − | + | ||
e) | e) | ||
| + | |||
Fewer multipliers are required when implementing individually, but the system is more complicated. | Fewer multipliers are required when implementing individually, but the system is more complicated. | ||
| + | |||
More complete for the complete system. | More complete for the complete system. | ||
| + | <span style="color:green"> The Student didn't mention advantage/disadvantage of the complete system, it is just mentioned that "More complete for the complete system". (See Solution 1) </span> | ||
---- | ---- | ||
===Related Problem=== | ===Related Problem=== | ||
| − | Consider | + | Consider the following 2-D LSI systems. The first system (S1) has input x(m, n) and output y(m, n), and the second system (S2) has input y(m, n) and output z(m, n). |
| − | + | ||
| − | The | + | <center> <math>\begin{align} |
| − | + | & y(m,n)=ay(m,n-1)+x(m,n)\qquad (S1) \\ | |
| − | h(m,n) | + | & z(m,n)=bz(m-1,n)+y(m,n)\qquad (S2) \\ |
| − | + | \end{align}</math></center> | |
| − | + | ||
| − | + | The third system (S3) is formed by the composition of (S1) and (S2) with input x(m, n) and output z(m,n) and impulse response <math>{{h}_{3}}(m,n)</math>. | |
| − | + | ||
| − | </math> | + | a) Calculate the 2-D impulse response, <math>{{h}_{1}}(m,n)</math>, of the first system (S1). |
| + | |||
| + | b) Calculate the 2-D impulse response, <math>{{h}_{2}}(m,n)</math>, of the second system (S2). | ||
| − | + | c) Calculate the 2-D impulse response, <math>{{h}_{3}}(m,n)</math>, of the complete system (S3). | |
| − | + | d) Calculate the 2-D transfer function, <math>{{H}_{1}}({z}_{1},{z}_{2})</math>, of the first system (S1). | |
| − | + | ||
| − | + | ||
| − | + | e) Calculate the 2-D transfer function, <math>{{H}_{3}}({z}_{1},{z}_{2})</math>, of the first system (S3). | |
| + | Refer to [https://engineering.purdue.edu/~bouman/ece637/previous/ece637S2014/exams/exam1/exam.pdf <u>ECE637 Spring 2014 Exam1 Problem1</u>].<br> | ||
---- | ---- | ||
[[ECE_PhD_Qualifying_Exams|Back to ECE QE page]]: | [[ECE_PhD_Qualifying_Exams|Back to ECE QE page]]: | ||
Latest revision as of 20:43, 2 May 2017
Communication Networks Signal and Image processing (CS)
Solution1:
a)
$ \begin{align} & {{h}_{1}}(m,n)=\sum\limits_{j=-N}^{N}{{a}_{j}\delta (m,n-j)=}{a}_{n}\delta (m) \\ & \delta (m,n)=\left\{ \begin{matrix} 1\quad m=n=0 \\ 0\qquad O.W \\ \end{matrix}, \right. \quad \delta (m,n-j)=\left\{ \begin{matrix} 1\quad m=0;\ n=j \\ 0\qquad O.W \\ \end{matrix} \right. \\ \end{align} $
b)
$ \begin{align} & {{h}_{2}}(m,n)=\sum\limits_{i=-N}^{N}{{b}_{i}\delta (m-i,n)=}{b}_{m}\delta (n) \\ & \delta (m,n)=\left\{ \begin{matrix} 1\quad m=n=0 \\ 0\qquad O.W \\ \end{matrix}, \right. \quad \delta (m-i,n)=\left\{ \begin{matrix} 1\quad m=i;\ n=0 \\ 0\qquad O.W \\ \end{matrix} \right. \\ \end{align} $
c)
$ \begin{align} & h(m,n)=\sum\limits_{i=-N}^{N}{\sum\limits_{j=-N}^{N}{{{b}_{i}}\ {{a}_{j}}^{{}}\delta (m-i,n-j)}}={{b}_{m}}\ {{a}_{n}} \\ & z(m,n)=\sum\limits_{i=-N}^{N}{{{b}_{i}}\ y(m-i,n)=}\sum\limits_{i=-N}^{N}{{{b}_{i}}\ \left( \sum\limits_{j=-N}^{N}{{{a}_{j}}\ x(m-i,n-j)} \right)=\sum\limits_{i=-N}^{N}{\sum\limits_{j=-N}^{N}{{{b}_{i}}\ {{a}_{j}}\ x(m-i,n-j)}}} \\ & \delta (m-i,n-j)=\left\{ \begin{matrix} 1\quad m=i;\ n=j \\ 0\qquad O.W \\ \end{matrix} \right. \\ \end{align} $
d)
Number of multiplies per output point to implement each individual system = 2N+1. So, the number of multiplies per output point to implement each of the two individual systems is 2(2N+1) = 4N+2.
Number of multiplies per output point to implement the complete system with a single convolution is $ \left( 2N+1 \right)\left( 2N+1 \right)\text{ }=4{{N}^{2}}+4N+1 $
e)
Implementing the two systems in sequence requires less computation, but it is more complex and more sensitive to noise. Implementing the two systems in a single complete system requires more computation, but it is simpler, less sensitive to noise, and more stable.
Solution 2:
a)
$ \begin{align} & {{h}_{1}}(m,n)=\sum\limits_{j=-N}^{N}{{a}_{j}\delta (m,n-j)=}\sum\limits_{j=-N}^{N}{{a}_{j}\delta (m)\ \delta(n-j)=} {a}_{n}\ \delta (m) \\ \end{align} $
The answer is correct, but it's not obvious how the solution is derived. (See Solution 1)
b)
$ \begin{align} & {{h}_{2}}(m,n)=\sum\limits_{i=-N}^{N}{{b}_{i}\delta (m-i,n)=}\sum\limits_{i=-N}^{N}{{b}_{i}\delta (m-i)\ \delta(n)=}{b}_{m}\ \delta (n) \\ \end{align} $
The answer is correct, but it's not obvious how the solution is derived. (See Solution 1)
c)
$ \begin{align} & h(m,n)=\sum\limits_{i=-N}^{N}{\sum\limits_{j=-N}^{N}{{{b}_{i}}\ {{a}_{j}}\ \delta (m-i,n-j)}}=\sum\limits_{i=-N}^{N}{\sum\limits_{j=-N}^{N}{{{b}_{i}}\ {{a}_{j}}\ \delta (m-i)\ \delta (n-j)}}={{b}_{m}}\ {{a}_{n}} \\ \end{align} $
The answer is correct, but it's not obvious how the solution is derived. (See Solution 1)
d)
Individually: $ 2(2N+1)=4N+2 $
Complete system: $ {( 2N+1)}^{2}=4{{N}^{2}}+4N+1 $
e)
Fewer multipliers are required when implementing individually, but the system is more complicated.
More complete for the complete system.
The Student didn't mention advantage/disadvantage of the complete system, it is just mentioned that "More complete for the complete system". (See Solution 1)
Related Problem
Consider the following 2-D LSI systems. The first system (S1) has input x(m, n) and output y(m, n), and the second system (S2) has input y(m, n) and output z(m, n).
The third system (S3) is formed by the composition of (S1) and (S2) with input x(m, n) and output z(m,n) and impulse response $ {{h}_{3}}(m,n) $.
a) Calculate the 2-D impulse response, $ {{h}_{1}}(m,n) $, of the first system (S1).
b) Calculate the 2-D impulse response, $ {{h}_{2}}(m,n) $, of the second system (S2).
c) Calculate the 2-D impulse response, $ {{h}_{3}}(m,n) $, of the complete system (S3).
d) Calculate the 2-D transfer function, $ {{H}_{1}}({z}_{1},{z}_{2}) $, of the first system (S1).
e) Calculate the 2-D transfer function, $ {{H}_{3}}({z}_{1},{z}_{2}) $, of the first system (S3).
Refer to ECE637 Spring 2014 Exam1 Problem1.
