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 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 2 / 56 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 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 4 / 56
Angle Modulaion Phase Modulaion (PM): 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()] Frequency Modulaion (FM): 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 m() m() Inegraor Differeniaor Phase Modulaor Frequency Modulaor s () FM s () PM Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 5 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 6 / 56 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 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 8 / 56
carrier Properies of Angle Modulaion message ampliude modulaion phase modulaion 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 frequency modulaion Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 9 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 10 / 56 Consancy of Transmied Power: PM Consancy of Transmied Power: FM Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 11 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 12 / 56
Consancy of Transmied Power: AM Nonlineariy of Angle Modulaion Consider PM (proof also holds for FM). Suppose s 1 () = A c cos [2πf c + k p m 1 ()] s 2 () = A c cos [2πf c + k p m 2 ()] Le m 3 () = m 1 () + m 2 (). 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 13 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 14 / 56 Irregulariy of Zero-Crossings Zero-Crossings: PM Zero-crossing: insans of ime a which waveform changes ampliude from posiive o negaive or vice versa. Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 15 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 16 / 56
Zero-Crossings: FM Zero-Crossings: AM Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 17 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 18 / 56 Difficuly of Visualizing Message Visualizaion: PM 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 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 20 / 56
Visualizaion: FM Visualizaion: AM Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 21 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 22 / 56 Bandwidh vs. Noise Trade-Off carrier Noise affecs he message signal piggy-backed as ampliude modulaion more han i does when piggy-backed as angle modulaion. message ampliude modulaion The more bandwidh ha he angle modulaed signal akes, ypically he more robus i is o noise. phase modulaion frequency modulaion Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 23 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 24 / 56
AM vs. FM Narrow Band Frequency Modulaion Suppose m() = A m cos(2πf m ). 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. 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. Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 25 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 26 / 56 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 - + Narrow-band FM wave 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 -90 degree Phase Shifer carrier 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 27 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 28 / 56
Generaion of FM Waves Demodulaion of FM Waves Narrowband FM modulaor m() Narrow band s() Frequency s () Inegraor Modulaor Muliplier wideband FM wave d d Ideal Envelope Deecor Crysal Conrolled Oscillaor frequency is very sable 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 29 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 30 / 56 Pulse Modulaion he variaion of a regularly spaced consan ampliude pulse sream o superimpose informaion conained in a message signal Chaper 5: Pulse Modulaion T T s A Noe: T < T 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 31 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 32 / 56
Pulse Ampliude Modulaion (PAM) Pulse Duraion Modulaion (PDM) T T s T T s m() m(-t ) s m(t ) s m(2t ) s m() m(-t ) s m(t ) s m(2t ) s m() T m(-t ) s m(t ) s m(2t ) s Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 33 / 56 T s s() PDM m() m(-t ) s m(t ) s m(2t ) s Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 34 / 56 s() Pulse Posiion Modulaion (PPM) m() PDM m() PPM m() s m(-t ) s m(-t ) s m(t ) s m(t ) s m(2t ) s m(2t ) s s() s() 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). Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 35 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 36 / 56
Pulse-Code Modulaion Mos basic form of digial pulse modulaion SOURCE Transmier PCM Daa Sequence Tranmission Pah Channel Oupu Receiver DESTINATION PCM Transmier Coninuous-ime Message Low-pass Filer Ani-aliased Cs-ime Discree-ime Digial signal Source Sampler Quanizer Encoder Ani-aliasing Filer Sampling above Nyquis wih Narrow Recangular PAM Pulses Using a Non-uniform Quanizer Maps Numbers o Bi Sequences PCM Daa Sequence Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 37 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 38 / 56 PCM Transmier: Sampler PCM Transmier: Non-Uniform Quanizer 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) T Ts m(-t s ) m(t s) m(2t s) s() Normalized oupu v 1.0 0.75 0.5 0.25 large mu Ampliude Compressor -3-2 - Uniform Quanizer 3 2 - -2-3 v[n] v m[n] 1 2 3 n -3-2 -1 0 1 2 3 m Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 39 / 56 0 0.25 0.5 0.75 1.0 Normalized inpu m Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 40 / 56
PCM Transmier: Encoder Coninuous-ime Message Low-pass Filer Ani-aliased Cs-ime Discree-ime Digial signal Source Sampler Quanizer Encoder 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 41 / 56 PCM Daa Sequence 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? 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 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 42 / 56 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: E QN = 2 12 Was where separaion of quanizaion levels. 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? A: To compue : x [n] q x[n] 1 n -3-2 -1 0 1 2 3 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 = 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 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 44 / 56
PCM: Transmission Pah SOURCE Transmier PCM Daa Sequence Tranmission Pah Channel Oupu Receiver DESTINATION PCM: Regeneraive Repeaer Disored PCM Wave Amplifier- Equalizer Decision-making Device Timing Circui Regeneraed PCM Wave PCM Daa Shaped for Transmission Tranmission Line Regeneraive Repeaer Tranmission Line... Regeneraive Repeaer Tranmission Line Channel Oupu Original PCM Wave Disored PCM Wave Amplifier- Equalizer Decision-making Device Timing Circui Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 45 / 56 Regeneraed PCM Wave THRESHOLD 1 0 0 1 1 0 BIT ERROR Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 46 / 56 PCM: Receiver Two Sages: 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) Chaper 6: Baseband Daa Transmission 2. Reconsrucion: 2.1 pass expander oupu hrough low-pass reconsrucion filer (cuoff is equal o message bandwidh) o esimae original message m() Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 47 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 48 / 56
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) +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 ) Source 0011100010 Binary inpu sequence Line Encoder Transmi- Filer G(f) Channel H(f) Pulse Specrum P(f) y() = k= Receivefiler Q(f) Sample a ime Sample a ime a k p( kt b ) where p() = g() h() q() P(f ) = G(f ) H(f ) Q(f ). Decision- Making Device Decision- Making Device Oupu binary daa Desinaion Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 49 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 50 / 56 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 Pulse Specrum P(f) P(f) = G(f)H(f)Q(f) Sample a ime Decision- Making Device Pulse Specrum P(f) P(f) = G(f)H(f)Q(f) Sample a ime Decision- Making Device 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 for i Z To avoid inersymbol inerference (ISI), we need p i = 0 for i 0. Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 51 / 56 Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 52 / 56
The Nyquis Channel Minimum bandwidh channel 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 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 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 53 / 56 α = 1 f 1 B 0 B T = B 0 (1 + α) where B 0 = 1 2T b and f v = αb 0 Noe: No ISI. p i = 0 for i 0. Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 54 / 56 Raised-Cosine Pulse Specrum The Eye Paern Raised-Cosine, α=0.5 A= sqr(e)/2b A/2 0 Nyquis Pulse, α=0 Slope dicaes sensiiviy o iming error Bes sampling ime NOISE MARGIN Disorion a sampling ime Raised-Cosine, α=1-2b 0 0 B 0 Nyquis Pulse B 0 /2 B 0 /2 B 0 /2 B 0 /2 Bandwidh -B 2B0 Trade-off: larger bandwidh han Nyquis pulse. f (khz) Professor Deepa Kundur (Universiy of Torono) Tes 2 Review 55 / 56 Time inerval over which wave is bes sampled. ZERO-CROSSING DISTORTION 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 56 / 56