1. Signals and Systems
EE235
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2. Futile
Q: What did the monsterous voltage source say to the chunk of wire? A: "YOUR RESISTANCE IS FUTILE!"
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3. Today’s menu
Communications (intro)
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4. Communications Practical problem Dividing the wire – how?
One wire vs. hundreds of channels
One room vs. hundreds of people
Dividing the wire – how?
Time
Frequency
Orthogonal signals (like CDMA)
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5. FDM (Frequency Division Multiplexing)
Focus on Amplitude Modulation (AM)
From Fourier Transform: X
x(t)
m(t)=ejw0t
y(t)
Y(w)=X(w-w0) w0 w
X(w)
Time
FOURIER
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6. FDM (Frequency Division Multiplexing)
Amplitude Modulation (AM)
Frequency change – NOT LTI!
F(w) w w -5 5
Multiply by cosine!
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7. Double Side Band Amplitude Modulation
FDM – DSB modulation in time domain
x(t)
x(t)+B
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8. Double Side Band Amplitude Modulation
FDM – DSB modulation in freq. domain
For simplicity, let B=0 !
X(w) 1 !
–!C !C
1/2
Y(w)
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9. Modulation (Low-Pass First! Why?)
DSB – How it’s done.
Modulation (Low-Pass First! Why?) !
X3(w)
X1(w) 1
X2(w)
x1(t)
cos(w1t)
y(t) !1 ! !2 !3
1/2
Y(w)
x2(t)
cos(w2t)
x3(t)
cos(w3t)
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10. Band-pass, Mix, Low-Pass
DSB – Demodulation
Band-pass, Mix, Low-Pass
m(t)=cos(w0t)
z(t) = y(t)m(t) = x(t)[cos(w0t)]2
= 0.5x(t)[1+cos(2w0t)]
y(t)=x(t)cos(w0t)
Z(w) x
Y(w) w
-2w0
2w0
-w0 w0 w
LPF
What assumptions?
-- Matched phase of mod & demod cosines
-- No noise
-- No delay
-- Ideal LPF
X(w) w
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11. DSB – Demodulation (signal flow)
Band-pass, Mix, Low-Pass
LPF
BPF1
BPF2
BPF3 !1 ! !2 !3
1/2
Y(w)
X3(w)
X1(w) 1
X2(w)
cos(w1t)
cos(w2t)
cos(w3t)
y(t)
x1(t)
x3(t)
x2(t)
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12. DSB in Real Life (Frequency Division)
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KPQ 560 kHz Day DAN WENATCHEE WA US 5.0 kW
KVI 570 kHz Unl ND1 SEATTLE WA US 5.0 kW
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KONA 610 kHz Day DA2 KENNEWICK-RICHLAND-P WA US 5.0 kW
KCIS 630 kHz Day DAN EDMONDS WA US 5.0 kW
KAPS 660 kHz Day DA2 MOUNT VERNON WA US 10.0 kW
KOMW 680 kHz Day NDD OMAK WA US 5.0 kW
KXLX 700 kHz Day DAN AIRWAY HEIGHTS WA US 10.0 kW
KIRO 710 kHz Day DAN SEATTLE WA US 50.0 kW
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13. Laplace Transform Focus on: Doing (Definitions and properties)
Understanding its possibilities (ROC)
Poles and zeroes (overlap with EE233)
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14. Laplace Transform Definition: Where Inverse:
Good news: We don’t need to do this, just use the tables.
Fourier Series coefficient dk differs from its F(w) equivalent by 2pi
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15. Laplace Transform Definition: Where Inverse:
Good news: We don’t need to do this, just use the tables.
Fourier Series coefficient dk differs from its F(w) equivalent by 2pi
Inverse Laplace expresses f(t) as sum of exponentials with fixed s  has specific requirements
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16. Region of Convergence Example: Find the Laplace Transform of:
Fourier Series coefficient dk differs from its F(w) equivalent by 2pi
We have a problem: the first term for t=∞ doesn’t always vanish!
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17. Region of Convergence Example: Continuing… In general: for
In our case if: then
For what value of s does:
Pole at s=-3. Remember this result for now!
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18. Region of Convergence
A very similar example: Find Laplace Transform of:
For what value does:
This time: if then
Same result as before!
Note that both cases have the region dissected at s=-3, which is the ROOT of the Laplace Transform.
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19. Region of Convergence Laplace transform not uniquely
Comparing the two:
Laplace transform not uniquely
invertible without region of convergence
ROC -3
ROC -3
Non-casual,
Left-sided
Casual,
Right-sided
Laplace transform not uniquely invertible without region of convergence
s-plane
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20. Finding ROC Example Example: Find the Laplace Transform of:
From table:
ROC: Re(s)>-6
ROC: Re(s)>-2
Combined:
ROC: Re(s)>-2
Causal signal: Right-sided ROC (at the roots).
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21. Laplace and Fourier
Very similar (Fourier for Signal Analysis, Laplace for Control and System Designs)
ROC includes the jw-axis, then Fourier Transform = Laplace Transform (with s=jw)
If ROC does NOT include jw-axis but with poles on the jw-axis, FT can still exist!
Example:
But Fourier Transform still exists:
No Fourier Transform if ROC is Re(s)<0 (left of jw-axis)
ROC: Re(s) > 0
Not including jw-axis
Laplace transform not uniquely invertible without region of convergence
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22. Laplace and Fourier No Fourier Transform Example:
ROC exists: Laplace ok
ROC does not include jw-axis, no Fourier Transform
ROC: Re(s)>-3
ROC: Re(s)<-1
Combined:
-3<ROC<-1
No Laplace Transform since there is no overlapped ROC!
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23. Laplace and Fourier No Laplace Example: ROC: Re(s)>-1
Combined:
ROC: None!
No Laplace Transform since there is no overlapped ROC!
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24. Summary Laplace intro Region of Convergence Causality
Existence of Fourier Transform
No Laplace Transform since there is no overlapped ROC!
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