EE422 Cheat Sheet I
Table of Single-Sided Laplace Transforms
|
Signal |
Laplace Transform |
|
1. d (n)(t) |
sn |
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2. 1 or u(t) |
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3. |
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4. cos(w 0t) u(t) |
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5. sin(w 0t) u(t) |
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6. exp(-a t) cos(w 0t) u(t) |
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7. exp(-a t) sin(w 0t) u(t) |
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8. Square wave: u(t) - 2u(t-T/2) + 2u(t-T) - … |
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9. [sin(w 0t) - w 0t cos(w 0t)] u(t) |
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10. [w 0t sin(w 0t)] u(t) |
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11. w 0t exp(-a t) sin(w 0t) u(t) |
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12. exp(-a t)[sin(w 0t) - w 0t cos(w 0t)] u(t) |
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Laplace Transform of x(t) periodic for t>0
:
, where x1(t) is defined as x(t) over one period.
Table of Laplace Transform Theorems
|
Name |
Time Domain |
Frequency Domain |
|
1. Linearity |
a1x1(t) + a2x2(t) |
a1X1(s) + a2X2(s) |
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2. Differentiation |
|
snX(s) - sn-1x(0-) - … - x(n-1)(0-) |
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3. Integration |
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4. s-shift |
x(t) exp(-at) |
X(s+a) |
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5. Delay |
x(t-t0) u(t-t0) |
X(s) exp(-st0) |
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6. Convolution |
|
X1(s) X2(s) |
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7. Product |
x1(t) x2(t) |
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8. Initial value (provided limit exists) |
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9. Final value (provided limit exists) |
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10. Time scaling |
x(at), a > 0 |
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Bode Plots
Decibel = 20 log(| H(jw) |)
Common conversion factors:
|
Number |
dB |
Rectangular |
Polar |
|
1 |
= 0 dB |
1 + j |
|
|
|
= 3 dB |
1 + j2 |
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2 |
= 6 dB |
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|
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3 |
= 9.5 dB |
||
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5 |
= 14 dB |
||
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8 |
= 18 dB |
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10 |
= 20 dB |
||
|
100 |
= 40 dB |
Frequency Domain Models
Time Domain Frequency Domain Time Domain Frequency Domain
Block Diagrams
Û
Û
Continuous State Variables:
Model: 
Solution: 
Transfer Function Matrix: 
