BandPass (4A) Young Won Lim 1/11/14

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1 BandPass (4A)

2 Copyright (c) 22 Young W. Lim. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version.2 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover exts, and no Back-Cover exts. A copy of the license is included in the section entitled "GNU Free Documentation License". Please send corrections (or suggestions) to youngwlim@hotmail.com. his document was produced by using OpenOffice and Octave.

3 Binary ASK (Amplitude Shift Keying) s (t) = 2 E (t) cos(ω t + Φ) s (t) = 2 E (t) cos(ω t + Φ) Baseband (3A) 3

4 Binary PSK (Phase Shift Keying) s (t) = 2 E cos(ω t + 2 π 2 ) s (t) = 2 E cos(ω t + 2 π 2 ) Baseband (3A) 4

5 Binary FSK (Frequency Shift Keying) s (t) = 2 E cos(ω t + Φ) s (t) = 2 E cos(ω t + Φ) Baseband (3A) 5

6 Coherent and Non-Coherent Detection Coherent Detection he Sinusoidal Reference Signal is synchronous in phase with the carrier wave used in the modulator Phase Locked Loop Synchronous Phase Symbol Interval Cost Non-Coherent Detection Abandon phase synchronization Baseband (3A) 6

7 Coherent Detection of BPSK signals source NRZ Line Encoder Product Modulator s(t) BPSK Signal Input binary data stream 2 cos(2π f c t) Decision LPF Product Modulator BPSK Signal hreshold 2 cos(2π f c t) Baseband (3A) 7

8 Non-Coherent Detection of BASK signals Decision Envelope Detector BPF BASK Signal hreshold f c Baseband (3A) 8

9 Non-Coherent Detection of BFSK signals Envelope Detector BPF Decision f BFSK Signal hreshold Envelope Detector BPF f 2 Baseband (3A) 9

10 M-ary PAM he amplitude of transmitted pulses is varied in a discrete manner in accordance with an input stream of digital data source Input binary data stream M-ary PAM Bit Rate = b log 2 M Line Encoder ransmit Filter G(f) {b k } {a k } s(t) Clock Pulse Level encoded signal Channel H(f) M possible amplitude level (M>2) M symbols ransmits sequence of symbols : Symbol duration / : Symbol rate Binary PAM b : Bit duration /b : Bit rate Baseband (3A)

11 Inner Product of Vectors b = (b, b 2,,b n ) a = ( a, a 2,, a n ) b = ( b, b 2,, b n ) θ a = (a, a 2,, a n ) a b = a b + a 2 b a n b n b θ a a cos θ θ a b = a b cosθ b cosθ orthogonal cos(±π/2) = a b = Baseband (3A)

12 Inner Product of Functions b = (b, b 2,,b n ) a = ( a, a 2,, a n ) b = ( b, b 2,, b n ) θ a = (a, a 2,, a n ) a b = a b + a 2 b a n b n ϕ (t) = cos(ωt) ϕ (t) = sin(ωt) ϕ (t) ϕ 2 (t) ϕ (t),ϕ 2 (t) = ϕ (t) ϕ 2 (t) d t Baseband (3A) 2

13 Inner Product of rigonometric Functions ϕ (t) = cos(ωt) ϕ 2 (t) = sin(ωt) ϕ (t), ϕ 2 (t) = cos(ωt)sin(ωt) dt = sin(2ωt) 2 dt = ϕ (t), ϕ (t) = cos 2 (ωt) dt = +cos(2ωt) 2 dt = 2 ϕ 2 (t), ϕ 2 (t) = sin 2 (ωt) dt = cos(2ωt) 2 dt = 2 Baseband (3A) 3

14 Orthonormal Basis () ϕ 2 (t) = 2sin(ω c t) e 2 = (,) (+,+) (+,+) e = (,) ϕ (t) = 2cos(ω c t) e e 2 = e e = ϕ (t), ϕ 2 (t) = ϕ (t)ϕ 2 (t) dt = e 2 e 2 = ϕ (t), ϕ (t) = ϕ 2 (t) dt = ϕ 2 (t), ϕ 2 (t) = ϕ 2 2 (t) dt = Baseband (3A) 4

15 Orthonormal Basis (2) cos(ω c t)cos(θ) + sin(ω c t)sin(θ) = cos(ω c t θ) cos(ω c t)cos(θ) sin(ω c t)sin(θ) = cos(ω c t + θ) ϕ 2 (t) = 2sin(ω c t) ϕ 2 (t) = 2sin(ω c t) (cosθ, sin θ) θ θ ϕ (t) = 2cos(ω c t) ϕ (t) = 2cos(ω c t) (cosθ, sin θ) Baseband (3A) 5

16 Amplitude v(t) = Acos(ω c t + θ) = 2 A rms cos(ω c t + θ) Symbol ime in BPSK P (t) = v2 (t) R Unit Resistance R = P = v 2 (t) dt Average Power E = P Average Energy = A2 cos 2 (ωt+θ) dt = A2 + cos(2ωt+2θ) 2 dt = A2 2 2 = A rms A = 2 P A = 2 E Baseband (3A) 6

17 Signal Space 2 E cos(ω ct + θ) = ϕ (t) = + 2 cos(ω c t) 2 cos(ω ct) E cos(θ) 2 sin(ω c t) E sin(θ) ϕ 2 (t) = 2 sin(ω c t) ϕ 2 (t) = 2 sin(ω ct) ( E cosθ, E sin θ) θ ϕ (t) = 2 cos(ω c t) Baseband (3A) 7

18 Signal Space QPSK () θ (t) = 3π/4 θ (t) = π/4 ( 2, + 2 ) ( + 2, + 2) θ 2 (t) = 5π/4 θ 3 (t) = 7π/4 ( 2, + 2 ) ( + 2, + 2 ) s (t) = 2 E s (t) = 2 E s 2 (t) = 2 E s 3 (t) = 2 E cos(ω c t + θ ) = 2 E cos(ω c t + θ ) = 2 E cos(ω c t + θ 2 ) = 2 E cos(ω c t + θ 3 ) = 2 E cos(ω c t + π/4) cos(ω c t + 3π/4) cos(ω c t + 5π/4) cos(ω c t + 7π/4) = 2 E [cos(ω t)cos(π /4) sin(ω c ct)sin(π /4)] = 2 E = 2 E = 2 E [cos(ω t)cos(3π/4) sin(ω c c t)sin(3 π/4)] [cos(ω t)cos(5π/4) sin(ω c c t)sin(5 π/4)] [cos(ω t)cos(7π/4) sin(ω c c t)sin(7 π/4)] Baseband (3A) 8

19 Signal Space Dibit ( E, + E ) 2 2 ( + E, + E ) 2 2 ( E, + E ) 2 2 ( + E, + E ) 2 2 ( E, E ) 2 2 ( + E, E ) 2 2 ( E, E ) 2 2 ( + E, E ) 2 2 s (t) = + E 2 2 cos(ω c t) E 2 2 sin(ω c t) s (t) = E 2 2 cos(ω c t) E 2 2 sin(ω c t) s 2 (t) = E 2 2 cos(ω c t) + E 2 2 sin(ω c t) Gray Code s 3 (t) = + E 2 2 cos(ω c t) + E 2 2 sin(ω c t) Baseband (3A) 9

20 Signal Space QPSK (3) (, +) (+, +) bit (, +) bit bit (+, +) bit s (t) = E [cos(ω c t) (+) sin(ω c t) (+)] s (t) = E [cos(ω c t) ( ) sin(ω c t) (+)] s 2 (t) = E [cos(ω c t) ( ) sin(ω c t) ( )] s 3 (t) = E [cos(ω c t) (+) sin(ω c t) ( )] π/4 3π/4 5π/4 7π/4 Baseband (3A) 2

21 ISI Baseband (3A) 2

22 Autocorrelation of Random and Power Signals Baseband (3A) 22

23 ime Averaging and Ergodicity Baseband (3A) 23

24 Autocorrelation of Random and Power Signals Baseband (3A) 24

25 ime Averaging and Ergodicity Baseband (3A) 25

26 References [] [2] [3] B. Sklar, Digital Communications: Fundamentals and Applications

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