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| + | [[Category:ECE301Spring2013JVK]] [[Category:ECE]] [[Category:ECE301]] [[Category:Probability]] [[Category:Problem_solving]] | ||
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| + | [[Category:LTI_systems]] | ||
<br> 1. Impulse response examples for each of the following systems : linear and non-linear, causal and non-causal, with and without memory, invertible/non-invertible, stable/non-stable, time variant and time invariant. | <br> 1. Impulse response examples for each of the following systems : linear and non-linear, causal and non-causal, with and without memory, invertible/non-invertible, stable/non-stable, time variant and time invariant. | ||
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Causal: <span class="texhtml">''h''(''t'') = (''t'' − 1) * ''u''(''t'' − 1)</span> | Causal: <span class="texhtml">''h''(''t'') = (''t'' − 1) * ''u''(''t'' − 1)</span> | ||
| − | Noncausal: <span class="texhtml">''h''(''t'') = '' | + | Noncausal: <span class="texhtml">''h''(''t'') = ''ln''( − ''t'')</span> |
With memory: <span class="texhtml">''h''(''t'') = 1 − ''u''(''t'' + 1)</span> | With memory: <span class="texhtml">''h''(''t'') = 1 − ''u''(''t'' + 1)</span> | ||
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Invertible: <span class="texhtml">''h''(''t'') = 2''u''(''t'' − 5)</span> | Invertible: <span class="texhtml">''h''(''t'') = 2''u''(''t'' − 5)</span> | ||
| − | Noninvertible: <span class="texhtml">''y''[''n''] = '' | + | Noninvertible: <span class="texhtml">''y''[''n''] = ''cos''(''x''[''n''])</span> |
| − | Stable: <span class="texhtml">''h(t) = [e<sup>-t</sup>]u(t)''</span> | + | Stable: <span class="texhtml">''h(t) = [e<sup>-t</sup>]u(t)''</span> |
| − | Nonstable: <span class="texhtml">''y''(''t'') = ''d''/'' | + | Nonstable: <span class="texhtml">''y''(''t'') = ''d''/''dt''' '''x''(''t'')</span> |
Time variant: <span class="texhtml">''y''[''n''] = ''n'' * ''x''[''n'' − 1]</span> | Time variant: <span class="texhtml">''y''[''n''] = ''n'' * ''x''[''n'' − 1]</span> | ||
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2. Example of graphical convolution. | 2. Example of graphical convolution. | ||
| − | ... | + | [https://engineering.purdue.edu/Intranet/Users/naputt.areethamsirikul.1/convolution_graphic.jpg engineering.purdue.edu/Intranet/Users/naputt.areethamsirikul.1/convolution_graphic.jpg]<br> |
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| + | [[Category:Period]] | ||
3. Example question related to fundamental period. | 3. Example question related to fundamental period. | ||
| − | <span class="texhtml">''x''[''n''] = ( − 1)<sup>''n''</sup> * '' | + | <span class="texhtml">''x''[''n''] = ( − 1)<sup>''n''</sup> * ''cos(''pi'' * ''n'' − ''p''i'' / 2)) + ''cos''[''pi'' * ''n''] * ''s''in''[''p''i'' * ''n'']</span><br> |
The first term is always zero because of the cosine. The second term uses trigonometric properties to convert it to sin(2pi*n)/2 whose period is 1.<br>Fundamental period = 1 | The first term is always zero because of the cosine. The second term uses trigonometric properties to convert it to sin(2pi*n)/2 whose period is 1.<br>Fundamental period = 1 | ||
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| + | [[Bonus_point_1_ECE301_Spring2013|Back to first bonus point opportunity, ECE301 Spring 2013]] | ||
Latest revision as of 08:10, 11 February 2013
1. Impulse response examples for each of the following systems : linear and non-linear, causal and non-causal, with and without memory, invertible/non-invertible, stable/non-stable, time variant and time invariant.
Linear: y[n] = 2x[3n − 4] + ( − 1)n * x[n]
Nonlinear: y(t) = x2[t]
Causal: h(t) = (t − 1) * u(t − 1)
Noncausal: h(t) = ln( − t)
With memory: h(t) = 1 − u(t + 1)
Without memory: h[n] = u[n] − u[n − 1]
Invertible: h(t) = 2u(t − 5)
Noninvertible: y[n] = cos(x[n])
Stable: h(t) = [e-t]u(t)
Nonstable: y(t) = d/dt x(t)
Time variant: y[n] = n * x[n − 1]
Time invariant: y[n] = ( − j)n * x[n]
2. Example of graphical convolution.
engineering.purdue.edu/Intranet/Users/naputt.areethamsirikul.1/convolution_graphic.jpg
3. Example question related to fundamental period.
x[n] = ( − 1)n * cos(pi * n − pi / 2)) + cos[pi * n] * sin[pi * n]
The first term is always zero because of the cosine. The second term uses trigonometric properties to convert it to sin(2pi*n)/2 whose period is 1.
Fundamental period = 1
