Tes 2 Review Tes 2 Review Professor Deepa Kundur Universiy of Torono Reference: Secions: 4.1, 4.2, 4.3, 4.4, 4.6, 4.7, 4.8 of 5.1, 5.2, 5.3, 5.4, 5.5 6.1, 6.2, 6.3, 6.4, 6.5, 6.6 S. Haykin and M. Moher, Inroducion o Analog & Digial Communicaions, 2nd ed., John Wiley & Sons, Inc., 2007. ISBN-13 978-0-471-43222-7. Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 1 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 2 / 57 Angle Modulaion Consider a sinusoidal carrier: Chaper 4: Angle Modulaion c() = A c cos(2πf c + φ }} c ) = A c cos(θ i ()) angle θ i () = 2πf c + φ c = 2πf c for φ c = 0 f i () = 1 dθ i () = f c 2π d Angle modulaion: he message signal m() is piggy-backed on θ i () in some way. Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 3 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 4 / 57
Angle Modulaion Phase Modulaion (PM): Frequency Modulaion (FM): θ i () = 2πf c + k p m() f i () = 1 dθ i () = f c + k p dm() 2π d 2π d s PM () = A c cos[2πf c + k p m()] PM vs. FM s PM () = A c cos[2πf c + k p m()] ] s FM () = A c cos [2πf c + 2πk f m(τ)dτ s PM () = 0 [ ] dg() A c cos 2πf c + k p d s FM () = A c cos [2πf c + 2πk f g()] θ i () = 2πf c + 2πk f m(τ)dτ f i () = 1 dθ i () = f c + k f m() 2π d ] s FM () = A c cos [2πf c + 2πk f m(τ)dτ 0 0 Modulaing wave Modulaing wave Inegraor Differeniaor Phase Modulaor Frequency Modulaor FM wave PM wave Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 5 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 6 / 57 carrier carrier message message ampliude modulaion ampliude modulaion phase modulaion phase modulaion frequency modulaion frequency modulaion Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 7 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 8 / 57
carrier Angle Modulaion message m() Inegraor Phase Modulaor s () FM ampliude modulaion phase modulaion m() Differeniaor Frequency Modulaor s () PM frequency modulaion Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 9 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 10 / 57 Properies of Angle Modulaion Consancy of Transmied Power: PM 1. Consancy of ransmied power 2. Nonlineariy of angle modulaion 3. Irregulariy of zero-crossings 4. Difficuly in visualizing message 5. Bandwidh versus noise rade-off Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 11 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 12 / 57
Consancy of Transmied Power: FM Consancy of Transmied Power: AM Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 13 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 14 / 57 Nonlineariy of Angle Modulaion Irregulariy of Zero-Crossings Consider PM (proof also holds for FM). Suppose Le m 3 () = m 1 () + m 2 (). s 1 () = A c cos [2πf c + k p m 1 ()] s 2 () = A c cos [2πf c + k p m 2 ()] Zero-crossing: insans of ime a which waveform changes ampliude from posiive o negaive or vice versa. s 3 () = A c cos [2πf c + k p (m 1 () + m 2 ())] s 1 () + s 2 () cos(2πf c + A + B) cos(2πf c + A) + cos(2πf c + B) Therefore, angle modulaion is nonlinear. Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 15 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 16 / 57
Zero-Crossings: PM Zero-Crossings: FM Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 17 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 18 / 57 Zero-Crossings: AM Difficuly of Visualizing Message Visualizaion of a message refers o he abiliy o glean insighs abou he shape of m() from he modulaed signal s(). Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 19 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 20 / 57
Visualizaion: PM Visualizaion: FM Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 21 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 22 / 57 Visualizaion: AM Bandwidh vs. Noise Trade-Off Noise affecs he message signal piggy-backed as ampliude modulaion more han i does when piggy-backed as angle modulaion. The more bandwidh ha he angle modulaed signal akes, ypically he more robus i is o noise. Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 23 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 24 / 57
carrier AM vs. FM message ampliude modulaion phase modulaion AM is an older echnology firs successfully carried ou in he mid 1870s han FM was developed in he 1930s (by Edwin Armsrong). FM has beer performance han AM because i is less suscepible o noise. FM akes up more ransmission bandwidh han AM; Recall, B T,FM = 2 f + 2f m vs. B T,AM = 2W or W AM is lower complexiy han FM. frequency modulaion Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 25 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 26 / 57 Narrow Band Frequency Modulaion Suppose m() = A m cos(2πf m ). f i () = f c + k f A m cos(2πf m ) = f c + f cos(2πf m ) f = k f A m frequency deviaion θ i () = 2π 0 f i (τ)dτ = 2πf c + f f m sin(2πf m ) = 2πf c + βsin(2πf m ) β = f f m s FM () = A c cos [2πf c + βsin(2πf m )] For narrow band FM, β 1. Narrowband FM Modulaion: Modulaing wave s FM () A c cos(2πf c ) β A }} c sin(2πf c ) sin(2πf }} m ) }} carrier 90 o shif of carrier 2πfm Am 0 } m(τ)dτ } DSB-SC signal Inegraor Produc Modulaor -90 degree Phase Shifer - + Narrow-band FM wave carrier Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 27 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 28 / 57
Transmission Bandwidh of FM Waves A significan componen of he FM signal is wihin he following bandwidh: ( B T 2 f + 2f m = 2 f 1 + 1 ) β called Carson s Rule f is he deviaion of he insananeous frequency f m can be considered o be he maximum frequency of he message signal Generaion of FM Waves Narrowband FM modulaor m() Narrow band s() Frequency s () Inegraor Modulaor Muliplier wideband FM wave Crysal Conrolled Oscillaor frequency is very sable For β 1, B T 2 f = 2k f A m For β 1, B T 2 f 1 β = 2 f f /f m = 2f m Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 29 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 30 / 57 Demodulaion of FM Waves d d Ideal Envelope Deecor Chaper 5: Pulse Modulaion Frequency Discriminaor: uses posiive and negaive slope circuis in place of a differeniaor, which is hard o implemen across a wide bandwidh Phase Lock Loop: racks he angle of he in-coming FM wave which allows racking of he embedded message Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 31 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 32 / 57
Pulse Modulaion Pulse Ampliude Modulaion (PAM) he variaion of a regularly spaced consan ampliude pulse sream o superimpose informaion conained in a message signal T T s A Noe: T < T s m() T T s m(-t ) s m(t ) s m(2t ) s Three ypes: 1. pulse ampliude modulaion (PAM) 2. pulse duraion modulaion (PDM) 3. pulse posiion modulaion (PPM) Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 33 / 57 m() T m(-t ) s m(t ) s m(2t ) s Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 34 / 57 T s s() Pulse Duraion Modulaion (PDM) Pulse Posiion Modulaion (PPM) s T T s m() m(-t ) s m(t ) s m(2t ) s m() m(-t ) s m(t ) s m(2t ) s PDM m() m(-t ) s m(t ) s m(2t ) s s() PDM m() m(-t ) s m(t ) s m(2t ) s s() PPM m() s() Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 35 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 36 / 57
Summary of Pulse Modulaion Le g() be he pulse shape. PAM: s PAM () = k a m(nt s )g( nt S ) n= where k a is an ampliude sensiiviy facor; k a > 0. PDM: ( ) nt s s PDM () = g k d m(nt s ) + M d n= where k d is a duraion sensiiviy facor; k d m() max < M d. PPM: s PPM () = g( nt s k p m(nt s )) n= where k p is a posiion sensiiviy facor; k p m() max < (T s /2). Pulse-Code Modulaion Mos basic form of digial pulse modulaion SOURCE Transmier PCM Daa Sequence Tranmission Pah Channel Oupu Receiver DESTINATION Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 37 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 38 / 57 PCM Transmier PCM Transmier: Sampler Coninuous-ime Message Low-pass Filer Ani-aliased Cs-ime Discree-ime Digial signal Source Sampler Quanizer Encoder PCM Daa Sequence Coninuous-ime Message Low-pass Filer Ani-aliased Cs-ime Discree-ime Digial signal Source Sampler Quanizer Encoder PCM Daa Sequence Ani-aliasing Filer Sampling above Nyquis wih Narrow Recangular PAM Pulses Using a Non-uniform Quanizer Maps Numbers o Bi Sequences Ani-aliasing Filer Sampling above Nyquis wih Narrow Recangular PAM Pulses Using a Non-uniform Quanizer Maps Numbers o Bi Sequences m() m(-t s ) m(t s) m(2t s) m(-t s ) m(t s) m(2t s) T s() Ts Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 39 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 40 / 57
PCM Transmier: Non-Uniform Quanizer PCM Transmier: Encoder Coninuous-ime Message Low-pass Filer Ani-aliased Cs-ime Discree-ime Digial signal Source Sampler Quanizer Encoder PCM Daa Sequence Coninuous-ime Message Low-pass Filer Ani-aliased Cs-ime Discree-ime Digial signal Source Sampler Quanizer Encoder PCM Daa Sequence Ani-aliasing Filer Normalized oupu v 1.0 0.75 0.5 0.25 Sampling above Nyquis wih Narrow Recangular PAM Pulses large mu 0 0.25 0.5 0.75 1.0 Normalized inpu m Ampliude Compressor Using a Non-uniform Quanizer -3-2 - 3 2 - -2-3 v Uniform Quanizer v[n] m[n] 1 Maps Numbers o Bi Sequences 2 3 n -3-2 -1 0 1 2 3 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 41 / 57 m Ani-aliasing Filer Sampling above Nyquis wih Narrow Recangular PAM Pulses Using a Non-uniform Quanizer Quanizaion-Level Index Binary Codeword (R = 3) 0 000 1 001 2 010 3 011 4 100 5 101 6 110 7 111 Maps Numbers o Bi Sequences Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 42 / 57 Q: The signal g() has a peak volage of 8 V (i.e., ranges beween ±8 V). Wha is he power of he uniform quanizaion noise e q ()? Noe: The power of quanizaion noise from a uniform quanizer is: Q: A signal g() wih bandwidh 20 Hz is sampled a he Nyquis rae is uniformly quanized ino L = 16, 777, 216 levels and hen binary coded. (a) How many bis are required o encode one sample? (b) Wha is he number of bis/second required o encode he audio signal? E QN = 2 12 Was where separaion of quanizaion levels. x [n] q x[n] 1 n -3-2 -1 0 1 2 3 A: (a) Number of bis per sample = log 2 (16777216) = 24 bis. A: (b) Number of bis per second = Number of bis per sample Number of samples/second = 24 2 20 = 960 bis per second A: To compue : = range of signal L = 8 ( 8) 16777216 = 1 1048576 = 1 2 20 E QN = 2 12 Was = (2 20 ) 2 = 2 40 12 12 Was Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 43 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 44 / 57
PCM: Transmission Pah Q: The original un-quanized signal has average power of 10 W. Wha is he resuling raio of he signal o quanizaion noise (SQNR) of he uniform quanizer oupu in decibels? SOURCE Transmier PCM Daa Sequence Tranmission Pah Channel Oupu Receiver DESTINATION A: I is given ha E S = 10. Therefore, ( ) ( ES SQNR = 10 log 10 = 10 log E 10 QN 10 2 40 12 ) 141.2 db PCM Daa Shaped for Transmission Tranmission Line Regeneraive Repeaer Tranmission Line... Regeneraive Repeaer Tranmission Line Channel Oupu Disored PCM Wave Amplifier- Equalizer Decision-making Device Regeneraed PCM Wave Timing Circui Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 45 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 46 / 57 PCM: Regeneraive Repeaer Disored PCM Wave Amplifier- Equalizer Decision-making Device Regeneraed PCM Wave PCM: Receiver Two Sages: Timing Circui Original PCM Wave 1. Decoding and Expanding: 1.1 regenerae he pulse one las ime 1.2 group ino code words 1.3 inerpre as quanizaion level 1.4 pass hrough expander (opposie of compressor) THRESHOLD 1 0 0 1 1 0 2. Reconsrucion: 2.1 pass expander oupu hrough low-pass reconsrucion filer (cuoff is equal o message bandwidh) o esimae original message m() BIT ERROR Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 47 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 48 / 57
Baseband Transmission of Digial Daa Chaper 6: Baseband Daa Transmission Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 49 / 57 Source 0011100010 Binary inpu sequence Line Encoder Transmi- Filer G(f) Channel H(f) Receivefiler Q(f) +1 if bk is symbol 1 b k = 0, 1} and a k = 1 if b Pulse Specrum k is symbol 0 P(f) s() = a k g( kt b ) k= x() = s() h() Sample a ime Sample a ime y() = x() q() = s() h() q() = a k g( kt b ) h() q() = k= Decision- Making Device Decision- Making Device k= Oupu binary daa Desinaion a k p( kt b ) Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 50 / 57 Baseband Transmission of Digial Daa Baseband Transmission of Digial Daa Source 0011100010 Binary inpu sequence Line Encoder Transmi- Filer G(f) Channel H(f) Receivefiler Q(f) Sample a ime Decision- Making Device Oupu binary daa Desinaion Source 0011100010 Binary inpu sequence Line Encoder Transmi- Filer G(f) Channel H(f) Receivefiler Q(f) Sample a ime Decision- Making Device Oupu binary daa Desinaion Pulse Specrum P(f) y() = k= Sample a ime a k p( kt b ) where p() = g() h() q() P(f ) = G(f ) H(f ) Q(f ). Decision- Making Device Pulse Specrum P(f) P(f) = G(f)H(f)Q(f) Sample a ime Decision- Making Device Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 51 / 57 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 52 / 57
Baseband Transmission of Digial Daa The Nyquis Channel Minimum bandwidh channel Pulse Specrum P(f) P(f) = G(f)H(f)Q(f) Sample a ime y i = y(it b ) and p i = p(it b ) y i = Eai + a }} k p i k k=,k i signal o deec }} inersymbol inerference Decision- Making Device To avoid inersymbol inerference (ISI), we need p i = 0 for i 0. for i Z Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 53 / 57 Opimum pulse shape: p op () = Esinc(2B 0 ) P op (f ) = E 2B 0 B 0 < f < B 0 0 oherwise, B 0 = 1 2T b Noe: No ISI. p i = p(it b ) = Esinc(2B 0 it b ) ( ) 1 Esinc 2 2T b it b = Esinc(i) = 0 for i 0. Disadvanages: (1) physically unrealizable (sharp ransiion in freq domain); (2) slow rae of decay leaving no margin of error for sampling imes. Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 54 / 57 Raised-Cosine Pulse Specrum has a more graceful ransiion in he frequency domain more pracical pulse shape: p() = ( ) cos(2παb0 ) Esinc(2B 0 ) 1 16α 2 B 02 2 E 2B 0 ]} 0 f < f 1 P(f ) = E [ π( f f1 ) (B 0 f 1 ) 4B 0 1 + cos f 1 < f < 2B 0 f 1 0 2B 0 f 1 f α = 1 f 1 B 0 B T = B 0 (1 + α) where B 0 = 1 2T b and f v = αb 0 Raised-Cosine Pulse Specrum Raised-Cosine, α=1 Raised-Cosine, α=0.5-2b 0 A= sqr(e)/2b A/2 0 B 0 Nyquis Pulse B 0 /2 B 0 /2 B 0 /2 B 0 /2 Bandwidh -B 2B0 0 Nyquis Pulse, α=0 f (khz) Noe: No ISI. p i = 0 for i 0. Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 55 / 57 Trade-off: larger bandwidh han Nyquis pulse. Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 56 / 57
The Eye Paern Slope dicaes sensiiviy o iming error Bes sampling ime Disorion a sampling ime NOISE MARGIN ZERO-CROSSING DISTORTION Time inerval over which wave is bes sampled. Noe: an open eye denoes a larger noise margin, lower zero-crossing disorion and greaer robusness o iming error. Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 57 / 57