1. Leo Lam © 2010-2012 Signals and Systems EE235 Lecture 29

2. Leo Lam © 2010-2012 Today’s menu The lost Sampling slides Communications (intro)

3. Sampling Leo Lam © 2010-2012 Convert a continuous time signal into a series of regularly spaced samples, a discrete-time signal. Sampling is multiplying with an impulse train 3 t t t multiply = 0 TSTS

4. Sampling Leo Lam © 2010-2012 Sampling signal with sampling period T s is: Note that Sampling is NOT LTI 4 sampler

5. Sampling Leo Lam © 2010-2012 Sampling effect in frequency domain: Need to find: X s () First recall: 5 timeT Fourier spectra 0 1/T

6. Sampling Leo Lam © 2010-2012 Sampling effect in frequency domain: In Fourier domain: 6 distributive property Impulse train in time  impulse train in frequency, dk=1/Ts What does this mean?

7. Sampling Leo Lam © 2010-2012 Graphically: In Fourier domain: No info loss if no overlap (fully reconstructible) Reconstruction = Ideal low pass filter 0 X() bandwidth

8. Sampling Leo Lam © 2010-2012 Graphically: In Fourier domain: Overlap = Aliasing if To avoid Alisasing: Equivalently: 0 Shannon’s Sampling Theorem Nyquist Frequency (min. lossless)

9. Sampling (in time) Leo Lam © 2010-2012 Time domain representation cos(2  100t) 100 Hz Fs=1000 Fs=500 Fs=250 Fs=125 < 2*100 cos(2  25t) Aliasing Frequency wraparound, sounds like Fs=25 (Works in spatial frequency, too!)

10. Summary: Sampling Leo Lam © 2010-2012 Review: –Sampling in time = replication in frequency domain –Safe sampling rate (Nyquist Rate), Shannon theorem –Aliasing –Reconstruction (via low-pass filter) More topics: –Practical issues: –Reconstruction with non-ideal filters –sampling signals that are not band-limited (infinite bandwidth) Reconstruction viewed in time domain: interpolate with sinc function

11. Leo Lam © 2010-2012 Onto… Communications (intro)

12. Communications Leo Lam © 2010-2012 Practical problem –One wire vs. hundreds of channels –One room vs. hundreds of people Dividing the wire – how? –Time –Frequency –Orthogonal signals (like CDMA)

13. FDM (Frequency Division Multiplexing) Leo Lam © 2010-2012 Focus on Amplitude Modulation (AM) From Fourier Transform: X x(t) m(t)=e j 0 t y(t) Y()=X( 0 ) 00  X()  TimeFOURIER

14. FDM (Frequency Division Multiplexing) Leo Lam © 2010-2012 Amplitude Modulation (AM) Frequency change – NOT LTI!  -55  F Multiply by cosine!

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

16. Double Side Band Amplitude Modulation Leo Lam © 2010-2012 FDM – DSB modulation in freq. domain For simplicity, let B=0 ! 0 X(w) 1 ! –!C–!C !C!C 0 1/2 Y(w)

17. DSB – How it’s done. Leo Lam © 2010-2012 Modulation (Low-Pass First! Why?) y(t) !1!1 ! 0 !2!2 !3!3 1/2 Y(  ) ! 0 ! 0 ! 0 X3()X3() X1()X1() 1 1 1 X2()X2() x 2 (t) x 1 (t) x 3 (t) cos(w 3 t) cos(w 1 t) cos(w 2 t)

18. DSB – Demodulation Leo Lam © 2010-2012 Band-pass, Mix, Low-Pass x y(t)=x(t)cos(  0 t) m(t)=cos(  0 t)z(t) = y(t)m(t) = x(t)[cos(  0 t)] 2 = 0.5x(t)[1+cos(2  0 t)] 00  0  0  0 LPF Y(  ) Z(  ) X(  )    What assumptions? -- Matched phase of mod & demod cosines -- No noise -- No delay -- Ideal LPF

19. DSB – Demodulation (signal flow) Leo Lam © 2010-2012 Band-pass, Mix, Low-Pass LPF BPF1 BPF2 BPF3 !1!1 ! 0 !2!2 !3!3 1/2 Y(  ) ! 0 ! 0 ! 0 X3()X3() X1()X1() 1 1 1 X2()X2() cos(  1 t) cos(  2 t) cos(  3 t) y(t) x 1 (t) x 3 (t) x 2 (t)

20. DSB in Real Life (Frequency Division) Leo Lam © 2010-2012 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