Leo Lam © Signals and Systems EE235 Lecture 29

Leo Lam © Today’s menu The lost Sampling slides Communications (intro)

Sampling Leo Lam © 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

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

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

Sampling Leo Lam © 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?

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

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

Sampling (in time) Leo Lam © 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!)

Summary: Sampling Leo Lam © 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

Leo Lam © Onto… Communications (intro)

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

FDM (Frequency Division Multiplexing) Leo Lam © 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

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

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

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

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

DSB – Demodulation Leo Lam © 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

DSB – Demodulation (signal flow) Leo Lam © 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() X2()X2() cos(  1 t) cos(  2 t) cos(  3 t) y(t) x 1 (t) x 3 (t) x 2 (t)

DSB in Real Life (Frequency Division) Leo Lam © 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