Signals and Systems EE235 Leo Lam © 2010-2012

Futile Q: What did the monsterous voltage source say to the chunk of wire? A: "YOUR RESISTANCE IS FUTILE!" Leo Lam © 2010-2012

Today’s menu Communications (intro) Leo Lam © 2010-2012

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) Leo Lam © 2010-2012

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 Leo Lam © 2010-2012

FDM (Frequency Division Multiplexing) Amplitude Modulation (AM) Frequency change – NOT LTI! F(w) w w -5 5 Multiply by cosine! Leo Lam © 2010-2012

Double Side Band Amplitude Modulation FDM – DSB modulation in time domain x(t) x(t)+B Leo Lam © 2010-2012

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) Leo Lam © 2010-2012

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) Leo Lam © 2010-2012

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 Leo Lam © 2010-2012

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) Leo Lam © 2010-2012

DSB in Real Life (Frequency Division) KARI 550 kHz Day DA2 BLAINE WA US 5.0 kW KPQ 560 kHz Day DAN WENATCHEE WA US 5.0 kW KVI 570 kHz Unl ND1 SEATTLE WA US 5.0 kW KQNT 590 kHz Unl ND1 SPOKANE WA US 5.0 kW 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 Leo Lam © 2010-2012

Laplace Transform Focus on: Doing (Definitions and properties) Understanding its possibilities (ROC) Poles and zeroes (overlap with EE233) Leo Lam © 2010-2012

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 Leo Lam © 2010-2012

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 Leo Lam © 2010-2012

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! Leo Lam © 2010-2012

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! Leo Lam © 2010-2012

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. Leo Lam © 2010-2012

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 Leo Lam © 2010-2012

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). Leo Lam © 2010-2012

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 Leo Lam © 2010-2012

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! Leo Lam © 2010-2012

Laplace and Fourier No Laplace Example: ROC: Re(s)>-1 Combined: ROC: None! No Laplace Transform since there is no overlapped ROC! Leo Lam © 2010-2012

Summary Laplace intro Region of Convergence Causality Existence of Fourier Transform No Laplace Transform since there is no overlapped ROC! Leo Lam © 2010-2012