Communication Systems PDF - Physics Wallah

Communication System Published By: Physics Wallah ISBN: 978-93-94342-39-2 Mobile App: Physics Wallah (Available on Play Store) Website: www.pw.live Email: [email protected] Rights All rights will be reserved by Publisher. No part of t...

Communication System Published By: Physics Wallah ISBN: 978-93-94342-39-2 Mobile App: Physics Wallah (Available on Play Store) Website: www.pw.live Email: [email protected] Rights All rights will be reserved by Publisher. No part of this book may be used or reproduced in any manner whatsoever without the written permission from author or publisher. In the interest of student's community: Circulation of soft copy of Book(s) in PDF or other equivalent format(s) through any social media channels, emails, etc. or any other channels through mobiles, laptops or desktop is a criminal offence. Anybody circulating, downloading, storing, soft copy of the book on his device(s) is in breach of Copyright Act. Further Photocopying of this book or any of its material is also illegal. Do not download or forward in case you come across any such soft copy material. Disclaimer A team of PW experts and faculties with an understanding of the subject has worked hard for the books. While the author and publisher have used their best efforts in preparing these books. The content has been checked for accuracy. As the book is intended for educational purposes, the author shall not be responsible for any errors contained in the book. The publication is designed to provide accurate and authoritative information with regard to the subject matter covered. This book and the individual contribution contained in it are protected under copyright by the publisher. (This Module shall only be Used for Educational Purpose.)  Design Against Static Load COMMUNICATION SYSTEM INDEX 1. Amplitude Modulation........................................................................................................ 8.1 – 8.10 2. Angle Modulation................................................................................................................ 8.11 – 8.18 3. Random Variable and Random Process............................................................................... 8.19 – 8.45 4. Digital Communication........................................................................................................ 8.46 – 8.55 5. Digital Receiver.................................................................................................................... 8.56 – 8.70 6. Information Theory............................................................................................................. 8.71 – 8.81 7. Miscellaneous..................................................................................................................... 8.82 – 8.86 GATE-O-PEDIA ELECTRONICS & COMMUNICATION ENGINEERING  Design Against Static Load 1 AMPLITUDE MODULATION 1.1. Introduction Band limiting : Bandun limited to bandlimited (LPF) Base band signal : Message signals, low cut off fre = 0 Hz or very close to 0 Hz. Bandpass signal : By shifting baseband signal to very high freq. fH Wideband signal :  1 (Base band signal) fL fH Narrowband signal : 1 (Bandpass signal) fL Modulated Signal: C (t ) = Ac cos(ct + ) = Ac cos ct Carrier signal (carrier before modulation) S (t ) = A(t )cos[ ct + (t )] Modulated signal Instantaneous phase Instantaneous Instantaneous amplitude frequency Amplitude Modulation: DSB-FC (Double side band full carrier) C (t ) = Ac cos ct carrier before modulation S AM (t ) = Ac cos ct + m (t )cos ct  S AM (t ) = [ Ac + m (t )]cos ct carrier after modulation [m (t )] Modulation Index  = max Ac (1)  < 1 (under modulation) Am = 1 Ac GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.1  Communication Systems [ E (t )]max − [ E (t )]min = [ E (t )]max + [ E (t )]min   1, A(t )  0, E(t ) = A(t ) Recovery through E, D possible. S (t )max = E (t ) |max = Ac (1 + ) S (t )min = E (t ) |min = Ac (1 − ) (2) Critical Modulation:-  = 1, A(t )  0 , E (t ) = A (t ) , m(t) can be recovered with envelope detector. (3) Over modulation:   1, A (t )  0, E(t ) =| A(t ) |, not possible by E.D Frequency Related Parameters (1) (2) C (t ) → f max = f m (3) PAM = PC + PSB Pm PUSB = PLSB = 4 Modulation efficiency PSB P /2 = = m PAM P + Pm c 2 Share of sideband power in total power ka [Amplitude sensitivity of amplitude modulator] 1 ka = (per volt), Ac A (t ) = Ac [1 + ka m (t )] A (t) > 0, E. D. Applicable GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.2  Communication Systems For single tone sinusoidal signal Am2 fmax = fm , BW = 0Hz, Pm = → for message signal 2 Am = Ac S AM (t ) = [ Ac + Am cos 2f mt ]cos 2fct = Ac [1 +  cos 2f mt ]cos 2fct Ac A S AM (t ) = Ac cos2fct + cos[2( fc + fm ) t ] + c cos[2( fc − f m )t ]  2  2  carrier USB LSB Pm Ac2  2  Pc2 PAM = Pc + = 1 + =  cP + → PSB 2 2  2   2 Pc Pc2 2 2 =  %  =  100% Pc2 2 + 2 Pc + 2 If ka given →  = ka | m (t ) |max | m (t ) |max If ka not given →  = Ac Important Points: =0 =1 % Change PAM = Pc PAM = 1.5Pc 50 % 1 =0  = = 33.33 % 0 % to 33.33 % 3 GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.3  Communication Systems (2)  →   (3) PAM → Will be constant if Pc  and   If m (t ) is multiple single tone signal-  Am Am  S (t ) = Ac 1 + 1 cos 2f m1t + 2 cos 2f m2 t  cos 2fct  Ac Ac  Am1 Am2 1 = , 2 = − 1  2 Ac Ac m(t) → fm1 , fm2 → fmax = fm2 fm2  fm1 BW = fm2 − fm1 S (t ) = f c BW = 2 × Max. Freq. component of m (t ) Power Related Parameters Am21 Am2 2 Pm = + 2 2 Pm Ac2  12 22  PAM = Pc + = 1 + +  2 2  2 2  Pc12 P 2 PUSB1 = PLSB1 = , PUSB2 = PLSB2 = c 2 2 2 Pc12 P 2 PUSB1 = PLSB1 = , PUSB2 = PLSB2 = c 2 4 4 PSB 2 = = T2 PAM 2 + T T = 12 + 22 + 32 + − − − − GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.4  Communication Systems Important Points: m(t) (volt) PAM Prod  2  Pc  2  (1) Sinusoidal Pc 1 +  1 +    2  R  2  Pc (2) Square wave Pc (1 + 2 ) (1 + 2 ) R  2  Pc  2  (3) Triangular wave Pc 1 +  1 +   3  R  3   2 2 VAM = Vc 1 + , I AM = Ic 1 + for sinusoidal 2 2 DSB- FC [AM] Modulator (1) Square law Modulator: a1 Ac2 a1 Ac2 y(t ) = a0 m (t ) + a0 Ac cos 2f ct + a1m2 (t ) + + cos 4f ct + 2m (t ) Ac cos 2f ct 2 2 (1) (2) (3) (4) (5) (6) Only (2) and (6) are desirable  2a  Z (t ) = a0 Ac cos 2fct 1 + 1 m (t )  DSB –FC  a0  Z (t ) = Ac' [1 + ka m (t )]cos2fct only when fc  3 f m fc  (2 + 1) f m 2a1 Ac' = a0 Ac , ka = ,  = ka | m (t ) |max a0 (2) Switching Modulator: Ac  4  Z (t ) = 1 + m (t )cos 2fct  DSB − FC 2  Ac  Z (t ) = Ac' [1 + ka m (t )]cos2fct Ac 4 Ac' = , ka =  = ka | m (t ) |max 2 Ac GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.5  Communication Systems DSB- FC Demodulator- (1) Square law demodulator- a1 Am2 a1 Am2 Y (t ) = a1 Ac Am cos2fmt + + cos4fmt 4 4 Y (t ) = B0 + B1 cos 0t + B2 cos 20t B2  2nd harmonic distortion D2 = = B1 4 ( D2 )max % = 25% Practically not used S 2 I =  min  Envelope Detector: A2 + B2 cos ct → E.D → A2 + B2 (1) x (t ) = A cos 0t + B sin 0t → E(t ) = A2 + B2 (2) x (t ) = A cos(0t + ) + B sin 0t → E(t ) = A2 + B2 − 2 AB sin  (3) x (t ) = A (t )cos ct → E (t ) =| A(t ) | (4) x (t ) = ( Ac + m (t ))cos ct → E (t ) =| Ac + m (t ) | Important Points: Used only when   1 1 Tc = RS C , (charging time constant) fc 1 Tc = RS C  → Peaks are not detected. fc 1 Diagonal clipping → RLC = fm 1 1 − 2 To avoid diagonal clipping RLC  , RLC  fm m 1 Ta = RLC  fluctuation is output fc 1 To remove fluctuation RLC  fc Proper choice of discharging time constant RLC - GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.6  Communication Systems No fluctuation No diagonal clipping (7) m(t): Multitone f m → f max = Max. freq. component of m(t) 1.2. Synchronous Detector    Recovery  =0 0  (2n + 1) ✓ 2  =0 0 = (2n + 1) Q.N.E 2 0 =0 =0  =0 =0 =0 ✓ DSB-SC : S DSB−SC (t ) = m (t ) Ac (cos 2fct ) E (t ) =| A (t ) |=| Ac m (t ) |. = 2  max. freq. component of m (t ) BW PSB Pm Pc PDSB = Pm Pc = PSB → PUSB = PLSB = + 2 2 Pc2 PDSB = = PSB 2 Single tone modulation. Ac Am AA S (t ) = cos[2( fc + f m ) t ] + c m cos[2( fc − f m )t ] 2 2 Ac2 Am2 PDSB = Pm Pc = 4 Ac2  Am1 Am2  2 2 Multiline PDSB = Pc Pm =  +  2  2 2     A2  Square wave - PDSB = Pc Pm =  c  Am2  2     A2  A2  Triangular wave PDSB = Pc Pm =  c  m  a  2  3      A2  A2  Saw-toothed wave- PDSB = Pc Pm =  c  m   2  3     (1) Balanced Modulator- S DSB (t ) = 2 Ac ka m (t )cos fct (2) Ring Modulator- y (t )  m (t )cos ct GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.7  Communication Systems  = 0,   0, y (t ) = 0 QNE Ac AC'   0,  = 0, y (t ) = m (t )cos(t ) → distorted m (t ) 2 Ac AC'  = 0,  = 0, y (t ) = m (t ) → Attenuated 2 Hilbert Transformation. 1 h (t ) = , t 1 mh (t ) = m (t )  t − j 0  H () = − j sgn () = 0 =0 j  0  M h ( f ) = M ( f )[− j sgn ( f )] H.T H.T [cos (t )] = sin (t ) ⎯⎯⎯ →− cos (t ) Non causal LTI system. H.T. x (t ) ⎯⎯ → xh (t ) H.T. xh (t ) ⎯⎯ →−x(t ) Magnitude spectrum of x(t) and xh (t ) will be same If x(t): Band limited then xh (t ) is also bandlimited. If x(t) is non periodic then xh (t ) is also non periodic x(t) and xh (t ) are orthogonal signal. Drawback of DSB-SC 2 sideband Txed. If receiver is designed in such a way that it may recover the complete message signal from single SB then DSB-SC S/S becomes impractical. SSB- SC (Single sideband suppressed carrier) (1) Point to point communication Phase derserimination (2) Two methods of generation Frequency discrimination GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.8  Communication Systems (a) Phase Discrimination: Ac m(t ) A m (t ) +  LSB S (t )SSB = cos 2fct  c h sin 2fct 2 2 −  USB Problem Solving (1) Identify the phase descrimination setup (2) Phase descrimination setup (3) Phase descrimination setup: +  SDSB (t ) → LSB −  SDSB (t ) → USB Spectral gap in D.S.B BPF Signal 0 Hz Ideal SSB-SC 0 Hz Practical VSB-SC  0Hz Ideal SSB -SC VSB − SC depends on  0Hz Practical practical BPF SSB − SC SSB- SC can be demodulated by Synchronous detection. (1)  = 0,   0, m (t ) recovery not possible → freq. synchronization (2)   0  = 0 , m (t ) recovery not possible → Phase synchronization (3) Perfect syne,  = 0,  = 0 can be recovered  (4)  = 0,  = → NoQNE 2 Note: (1) When video signal is transmitted through SSB- SC modular VSB- SC is generated. (2) Synchronous detector can not recover m (t ) video signal from the above generated VSB- SC. GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.9  Communication Systems Percentage Power Saved (1) % power saved in DSB- SC as compare to DSB-FC. Psaved % Psaved = 100% PTotal Pc 2 % Psaved = = = (1 − )  2  2 + 2 Pc 1 +   2  (2) % power saved in SSB- SC as compare to DSB-FC- 4 + 2 % Psaved = 4 + 22 (3) % power saved in SSB-SC as compared to DSB-SC. % Psaved = 50% Modulation B.W Power Application (1) DSB-FC 2 f max PC + PSB Broadcatting (2) DSB- SC 2 f max PSB  (3) PSB Point to point voice SSB-SC f max 2 communication PSB Point to point video (4) VSB-SC f max  f  2 f max  PVSB  PSB 2 communication. Pre envelope and Complex Envelope (1) Pre Envelope calculated for both baseband and bandpass signal. Let x (t ) is real signal. x+ (t ) = Pre envelope of x (t ) x+ (t ) = x (t ) + j xˆ (t ) xˆ (t ) = HT [ x (t )] x+ ( f ) = x ( f )1 + sgn( f ) Complex Envelope: For bandpass only but result in low pass only x (t ) → Bandpass signal. Step-1. Calculate x+ (t ) = x (t ) + j xˆ (t ) xc (t ) = x+ (t )e− jct Step 2. left shift of pre envelope by fc X c ( f ) = X + ( f + fc )  GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.10  Communication Systems 2 MODULATION ANGLE 2.1. Introduction Signal = x(t) |x (t)|max (1) A cos 0t + B cos 0t A+ B (2) A sin 0t + B sin 0t A+ B (3) A sin 0t + B cos 0t A2 + B2 (4) A cos 1t + B cos 2t A+ B (5) A sin 1t + B sin 2t A+ B (6) A cos 1t + B sin 2t A + B if A = B  A + B if A  B 2.1.1. Instantaneous Angle and Instantaneous frequency- S (t ) = Ac cos[i (t )] i (t ) → Instantaneous angle (rad) d i (t ) = i (t ) → instantaneous angular frequency. dt 1 d i (t )  (t ) fi (t ) = or fi (t ) = i 2 dt 2 t i (t ) =  i (t ) − Angle Modulation : Frequency Modulation Phase Modulation Frequency Modulation : Sangle (t ) = Ac cos[ct + (t )] GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.11  Communication Systems d (t ) If angle Modulation is FM,  m(t ) dt d (t ) = K f m(t ), K f = frequency sensitivity of frequency modulator dt i (t ) = c + K f m(t )  i (t ) = c + (t )  frequency deviation t i (t ) = c +  K f m(t) dt (t ) = K f m(t ) − d (t ) (t ) = dt Few Important Results rad Hz For Important Results For K f : Kf : V - sec Volt 1. Instantaneous frequency i (t ) = ct + K f m(t ) fi (t ) = fc + K f m(t ) 2. Instantaneous frequency deviation (t ) = K f m(t ) f (t ) = K f m(t ) Hz 3. Frequency deviation in +ve direction [(t)]max = K f [m(t)]max [f ]max = K f [m(t )]max 4. Frequency deviation in –ve direction [(t)]min = K f [m(t)]min [f (t)]min = K f [m(t)]min 5. Maximum value of instantaneous frequency [i (t )]max = c + [(t )]max [ fi (t )]max = fc + [f (t )]max 6. Minimum value of instantaneous frequency [i (t )]min = c + [(t )]min [ fi (t )]min = fc + [f (t )]min 7. Peak to peak frequency deviation [] p− p = [i (t)]max − [i (t)]min [f ] p− p = [ f (t )]max − [ f (t )]min 8. Maximum frequency deviation 9. Modulation index or deviation ratio of FM (t ) max = K f [m(t )]max f (t ) max = K f [m(t )]max Maximum frequency deviation K f m(t ) max K f m(t ) max BFM = BFM = BFM = Maximum frequency component of m(t ) max f max  rad  Hz Important Phase Calculation Kf   Kf :  V-sec  Volt 1. Instantaneous phase deviation in FM t t (t ) = K f  m() d  (t ) = 2K f  m() d  − − 2. Maximum phase deviation in FM t t (t ) max = K f  m() d  2K f  m() d  − − max GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.12  Communication Systems General expression for FM rad Kf : V-sec  t  Sangle (t ) = Ac cos ct +  K f m() d   −  Hz For K f : Volt  t  SFM = Ac cos ct + 2K f  m() d   −  For m(t ) = Am cos2fmt – fmax = fm ,[m(t)]max = + Am , [m(t )]min = − Am , m(t ) max = Am SFM (t ) = Ac cos c (t ) + BFM sin(2fmt ) For m(t ) = Am1 cos2fm1t + Am2 cos2fm2 t – fmax = ( fm1 , fm2 )max SFM (t ) = Ac cos c (t ) + B1 sin 2f m1t + B2 sin 2f m2 t  K f Am1 K A B1 = , B2 = f m2 fm1 fm2 Phase Modulation – (t )  m(t ) (t ) = K p m(t )   rad Volt K p : Phase sensitivity of phase modulator rad Kp = Volt Phase Calculation : K p : rad/Volt i (t ) = ct + K pm(t ) 1. Instantaneous phase deviation = (t ) = K p m(t ) 2. Maximum phase deviation = (t ) max = K p m(t ) max GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.13  Communication Systems Frequency Calculation i (t ) = c + t dm(t ) (t ) = K p dt dm(t ) (t ) max = K p dt max dm(t ) (t ) min = K p dt min i (t ) max = c + [(t )]max i (t ) min = c + [(t )]min p− p = [i (t )]max − [i (t )]min dm(t ) (t ) max = K p dt max dm(t ) Kp (t ) max dt max FM = = max max SFM (t ) = Ac cos c (t ) + KPM (t ) When m(t ) = Am cos2fmt m(t ) max = Am , fmax = fm , (t ) = −K p Amm sin mt [(t )]max = K p Amm {(t )}min = −K p Amm [i (t )]max = c + K p Amm , [i (t )]min = c − K p Amm () p− p = 2K p Amm (t ) max = K p Amm  = K p Am = (t ) max SPM (t ) = Ac cos c (t ) +PM cos 2fmt  SPM (t ) = Ac cos c (t ) + 1 cos 2f m1 t + 2 cos 2f m2 t  GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.14  Communication Systems Types of FM – Narrow Band (  1) Wide Band Ac A SFM (t ) = Ac cos 2fc (t )+ cos[2 ( fc + f m ) t ] − c cos[2 ( f c − f m ) t ]  2  2  Carrier USB LSB SFM (t ) = SNBFM (t ) B.W = 2 fm  2  PNBFM = PC 1 +    1, 2  1  2 Ac2 PNBFM  PC = 2 Relation between DSB-FC and NBFM – Ac A S AM (t ) = Ac cos2fc (t ) + cos[2 ( fc + fm ) t ] + c cos[2 ( fc − fm ) t] 2 2 Ac A SNBFM (t ) = Ac cos2fc (t ) + cos[2 ( fc + fm ) t ] − c cos[2 ( fc − fm ) t ] 2 2 1. Frequency Component Strength AM Strength NBFM fc Ac Ac 2 2 fc + f m Ac Ac 4 4 fc − f m Ac −Ac 4 4 2. SNBFM (t ) + SAM (t ) = SSB-SC → USB-FC SAM (t ) − SNBFM (t ) = SSB-SC → LSB-FC 3. LSB in NBFM is 1800 inverted w.r.t to LSB in AM  SFM (t ) = Ac  J n ()cos[2 ( fc + nf m ) t ] n=− For aby value of  Jn () = (−1)n Jn () GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.15  Communication Systems  J n=− 2 n () = 1 J0 () = 0,  = 2.4,5.5,8.6,11.8 as n , → J n ()    1: S (t ) → 1 Carrier + 2 SB NB Angle Modulation If   1: S (t ) : 1 Carrier + Infinite SB Wide Band Angle Modulation Ideal BW of WBFM =  Carson’s Rule – BW = (+1)2 fm PM for PM FM for FM Ac2 Power of Carrier before modulation = = Pc 2 Power of Carrier after modulation P = Pc  J 0  + 2( J1 () + J 2 () +  2 2 2 Power of Carrier component in modulated signal = Pc J02 () PSB = 2Pc  J12 () + J 22 () +  PSB = PTotal → Pc  J () + 2( J12 () + J 22 () + )  2 0 If J0 () = 0 then  = 100% For Infinite sidebands PWB = Pc For Non sinusoidal –  SFM = Ac  Cn cos  2 ( fc + nf m ) t + Cn  n=− m(t ) BW Singletone sinusoidal → (+1)2 fm ⎯⎯ Non sinusoidal → (+1)2 fm , fm = fundamental frequency ⎯⎯ periodic signal Other Cases → (+1)2 fmax ⎯⎯ BW = (1+)2 fm or 2(f + fmax ) GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.16  Communication Systems Frequency Mixture and Multiplier Mixture/Multiplier Input Mixture Output (Multiplied by n) Multiplier Output Ac Ac' Ac' fc fC − f L or fC + f L = fC' nfc   n fm fm fm f f n f BW BW (n+1)2 fm Spactral spacing fm fm Frequency components fc' , fc'  fm , fc'  2 fm nfc , nfc  fm , nfc  2 fm Wideband Angle Modulation generation –  dm(t )  PM [m(t )] = FM  If K p = K f = K  dt  t  FM [m(t )] = PM   m() d  −  Wideband FM Generation Methods 1. Armstrong Method (Indirect Method) 2. Direct Method VCO (Voltage Controlled Oscillator is used). It is modified version of Hartley oscillator  C = c 2C0 GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.17  Communication Systems FM Demodulator 1. Theoretical method 2. Practical method. PLL (Phase Locked Loop) Kf (1) v(t ) = m(t ) KV (2) Lock mode → Frequency lock Capture mode → Phase lock (3) L.R  C.R Super Hetrodyne Receiver f L = Local oscillator frequency f S = Desired frequency f Si = Frequency of image station Case 1 : If relation between fl and fs is not mentioned. Assume : fl  f s 1. fl = fs + IF 2. fSi = fl + IF 3. fSi = fs + 2IF Case 2 : When relation between fi and fs is given If fSi  fl  fs If f S  f s  f sl ' then Case 1 then 1. fs = fl + IF 2. fl = fSi + IF 3. fs = fSi + 2IF Image Rejection Ratio IRR = 1 + P2Q2 Q : Quality factor of Oscillator f Si2 − f s2 P= fSi  fs P2Q2  1 f Si f s IRR = PQ  GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.18  Communication Systems 3 RANDOM VARIABLE AND RANDOM PROCESS 3.1. Introduction Random variable → Real and complex R.V. is a function performing mapping from sample space of R.E. to real line. X () : Random variable Domain of R.V. →  (Sample point) Range of R.V. → Subset of real line One to one or many to one mapping P{X  a} → Probability of set in which all the comes satisfy x ()  a. CDF of R.V. Let random variable X, x →Values taken by R.V. (1) FX ( x) = P{X  x} = 1 − P{X  x} (2) FX (a) = P{ X  a} = 1 − P{ X  a} (3) F| X | ( y) = P{| X | y} = P{− y  X  y} Properties (1) FX () = 1 (2) FX (−) = 0 (3) FX () + FX (−) = 1 (4) FX ( x) = P{ X  x}  0  FX ( x)  1 (a) CDF always non negative. (b) Lower bound: FX ( x) = 0 , upper Bound = 1  dF ( x)  (5) CDF is monotonically non decreasing function of x  X  0  dx  (6) Graph of CDF is always amplitudes continuous from right. Key point : GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.19  Communication Systems (1) P{a  x  b} = FX (b+ ) − FX (a+ ) (2) P{a  X  b} = FX (b+ ) − FX (a− ) (3) P{a  X  b} = FX (b− ) − FX (a+ ) (4) P{a  X  b} = FX (b− ) − FX (a− ) P ( x = a) CDF is continuous at x = a CDF is continuousat x = a + − P{x = a } = FX ( a ) − FX ( a ) P{ x = a} = FX ( a + ) − f X ( a − ) P{ x = a} = 0 0 P{ x = a} = Size of Jump Probability Density Function Random variable X x →Variable taken by R.V. f X ( x) → Symbol dFX ( x) f X ( x) = dx x f X ( x) =  f X ( x) dX − a f X ( x) =  f X () d  − Properties : (1) f X ( x)  0 → Non negative (2) 0  f X ( x)   Upper bound Lower bound  (3) FX () =  f X ( x) dx = 1 − (4) Graph of PDF can be even or NENO but cannot be odd. x (5) P{−  X  x} =  f X () d  − b+ (6) P{a  X  b} = + f X ( x) dx a GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.20  Communication Systems b+ (7) P{a  X  b} = − f X ( x) dx a b− (8) P{a  X  b} = + f X ( x) dx a P (x = a ) Graph having impulse at x = a No impulse + P{ x = 0} = FX ( a ) − FX ( a ) − at x = a + a f a+ P{x = a } = ( x) dx f Y P{x = a } = Y ( x) dx a− P{ x = a} = 0 − a = Total area Discrete Random Variable: (1) PDF should have impulses only. (2) CDF should have staircase only. (1) Probability mass function of DRV : Let X is D.R.V. PX ( x) = P( X = x) probability such that X = x 0  PX ( x) = 1  PX ( x) = 1 x (2) PDF of a D.R.V : Let X is O.R.V. f X ( x) =  PX ( xi )  ( x − xi ) =  P( x = xi )  ( x − xi ) i i (3) CDF of a D.R.V. : Let X is D.R.V. x FX ( x) =  f X ( x) dx − FX ( x) =  P{X = xi }u{x − xi } x (4) P{X = a} may or may not be zero. Continuous Random Variable Maps sample point to continuous range of values on real axis. (1) PDF of C.R.V should not contain impulses at all. (2) CDF of C.R.V Should not contain jump type discontinuity It should be amplitude continuous every where (3) PMF not defined for C.R.V because for CRV P{ X = a} will always be zero. GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.21  Communication Systems P( A  B) P( A / B) = → Conditional probability of A given B. P( B)  B P( A  B) = P(B)P( A / B) = P( A)P   = Joint probability.  A Expectation operator : Performs operations on R.V. only. Linear Operator    xf X ( x) dx X : CRV  E [C] = C, E [C 2 ] = C 2 E ( X ) =  − xi P{X = xi } X : DRV  i E [aX ] = a E [ X ] E [aX + b] = a E [ X ] + E[b] E [ag ( X ) + bH ( y)) = a E [ g ( x)] + b E [ H ( y)]  E[ X 2 ] = x 2 f X ( x) dx −  E  g ( x)  =  g ( x) f X ( x) dx − Gaussian Random Variable CRV X is having Gaussian or random distribution. X is having Gaussian PDF, X is called G.R.V. E [ X ] =  X , E [( X − x )2 ] = Variance = 2X −( x− x )2 1 22X X N{ X , 2X } f X ( x) = e −  x   22X Key Point : −( x − X )2  1 22X (1)  22X e dx = 1 − −( x − X )2  1 22X (2)  22X e dx =  X = E [ X ] − −( x − X )2  X 1 −( x −  X )2 1 22X 1 (3)  22X e 22X dx =  22X e dx = 2 X − GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.22  Communication Systems Zero mean Gaussian distribution- X N ( X ,2X )  X N [0, 2X ]  E [ X ] = 0 − x2 1 2 f X ( x) = e 2 X 22X Zero Mean, unit variance : − x2  − x2 1 1 1 X N (0.1) f y ( x) = 2 e 2 ,  2 e 2 dx = 2 0 Q- function :  1 2 Q ( x) =  e− z /2dz as x , Q ( x)  2 0 Q () = 0, Q(−) = 1, Q (0) = 0.5, Q( x) + Q(− x) = 1  z − X  P [ X  z ] = Q ( P) = Q    X   z − X  f X ( z) = P( X  z ) = 1 − P( X  z ) = 1 − Q    X  Statistical averages of a R.V. nth order moment about origin-  n   x f X ( x) dx X : CRV  E[( X − 0)n ] = E [ X n ] =  −  xn P{X = x } X : DRV  i i i 1st order moment about origin  E[ X ] =  xf X ( x)dx =i xi P{X = xi } − E [ X ] = X =  X = mi → dc value, avg. value Mean value [E [ X ]]2 → d.c. power GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.23  Communication Systems 2nd order moment about origin-   2   x f X ( x) X : CRV E [( X − 0)2 ] = E [ X 2 ] = X 2  −  x P{ X = x } X : DRV i i i E [ X 2 ] = Mean square value of R.V. X = Total power of R.V. x 1st order moment about mean - E [( X −  X )] = 0 2nd order moment about mean −E[( X −  X )2 ] = E[ X 2 ] − 2X  2X = E [ X 2 ] − u2X A.C. Total dc Power Power Power Important point: (1) 2X  0, E [ X 2 ]  2X (2) If X is zero mean R.V. E[ X 2 ] = 2X , MSV( X ) = Var( X ) (3) Standard deviation Variance = 2X =  X (4) Y = aX + b E[Y 2 ] = a2 E [ X 2 ] + b2 + 2ab E [ X ] Y2 = a22X Standard Distribution of R.V. (1) Uniform distribution X U [a, b]  1  a X b f X ( x) =  (b − a)  0  otherwise GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.24  Communication Systems a+b a2 + b2 + ab (b − a)2 E[ X ] = , E[ X 2 ] = , 2X = 2 3 12 (2) Triangular distribution X tri (a, m, b) a+m+b E[ X ] = 3 (3) Rayleigh Distribution X → CRV  − x2   x 22X f X ( x) =  2 e x0   X   0 else − x2  x 2  2 e 2 X dx = 1 0 X If X and Y are two G.R.V. Then Z = X 2 + Y 2 will have reyleigh distribution. (4) Exponential Distribution : If CRV has exponential distribution then it will have PDF e−x  x0 −x f X ( x) =   0 x0  e dx = 1 0 Laplacian Distribution X → CRV f X ( x) = ae−b|x| −  x   a If = 1 , a  0, b  0 b GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.25  Communication Systems   ae bx x0 f X ( x) =  −bx  ae x0 Discrete Random variable-Binomial, Position distribution Binomial distribution necessary condition:- (1) The no of trials n showed be finite. (2) Trials are independent (3) Each trials should result in 2 outcomes success or failure. (4) Prob of success in each trial should be constant. PMF: P{X = r success} = ncr pr qn−r E [ X ] =  xi p{X = xi } = n p 2X = npq i E[ X 2 ] = npq + (np)2 Std deviation  X =  npq Position Distribution Specific type of binomial distribution where n → n →very large, p → very small, np → finite  = np r e −  p{X = r} = probability of X = r (success) r! E [ X ] = , 2X =  If Y = g ( X ) is having monotonic TX. PDF Given X ⎯⎯⎯ → f X ( x) ,  dx  fY [ y] =  f X [ x]  function of y  dy  (1) case a > 0  y −b  1  y −b  FY [ y] = FX   , fY ( y) = f X   a  a  a  (2) Y = −aX + b a  0 GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.26  Communication Systems  y −b  1  y −b  FY ( y) = 1 − FX   , fY ( y) = f X   −a  a  −a  Monotonic linear TX: y = aX + b X U [m1 , m2 ] → Y U [am1 + b, am2 + b] X  [m1, m2 , m3 ] → Y  [am1 + b, am2 + b, am3 + b] X N [ X , 2X ] → Y N [Y , 2y ] Y N[a X + b, a22X ] Monotonic Non-Linear TX: X → f X ( x) Y → X 3 , fY ( y) = ?  dx  fY ( y) =  f X ( x)   dy  1 fY ( y) = f ( y1/3 ) 2/3 X 3y Non - Monotonic TX: Y = y, g ( X ) = y, X = g −1( y) → x1  → x2 → x  3 dx1 dx fY ( y) = f X ( x1 ) + f X ( x2 ) 2 + − − − − dy dy 2D Random variable : → FX ,Y ( x, y) = Joint CDF  ( X , Y ) → 2DR.V. → f X ,Y ( x, y) = Joint PDF  → PXY ( xi , yi ) = Joint PMF If A and B are independent  A  B P   = P( A), P   = P(B), P( A  B) = P( A)P(B)  B  A GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.27  Communication Systems If X and Y are independent R.V. FXY ( x, y) = f X ( x) fY ( y)  y  x f XY ( x, y) = f X ( x) f Y   = fY ( y ) f X   X x Y   y If X and Y are independent R.V. f XY ( x, y) = f X ( x) fY ( y)  yj   xi  PXY ( xi , y j ) = PX ( xi ) PY   = PY ( y j ) PX    yj  X  i  x   Y If X and Y are independent R.V. PXY ( xi , y j ) = PX ( xi )PY ( y j ) Joint CDF = Let ( X , Y ) are BIVARIATE R.V. FXY ( x, y) = P{X  x)  (Y  y) = P{X  x; Y  y} Properties: (1) 0  FXY ( x, y)  1 (2) FXY (−, y) = P{( X  −)  (Y  y)} = 0 (3) FXY ( x, −) = 0 (4) FXY (−, −) = 0 (5) FXY (, ) = 1 (6) FXY ( x1, y1 ) = P{( X  x1 )  (Y  y1 )} (7) P{( x1  X  x2 )  ( y1  Y  y2 )} = FXY ( x1+. y1+ ) + FXY ( x2+ , y2+ ) − FXY ( x1+ , y2+ ) − FXY ( x2+ , y1+ )  y x (8) FXY ( x, y) = FX ( x) FY   = FY ( y) FX   X x Y   y (9) X and Y are independent R.V. FXY ( x, y) = FX ( x) FY ( y) (10) FX ( x, y) = FXY ( x, ), FY ( y) = FXY (, y) GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.28  Communication Systems Conditional CDF  x  F ( x, y) FX   = XY of FY ( y)  0 Y  y FY ( y)  x  P[( X  x)  (Y  y)] FX   = P [( X  )  (Y  y)] Y   y Joint PDF 2 FXY ( x, y) f XY ( x, y) = X Y x y FXY ( x, y) =   f XY (u, v)du dv − −   FXY ( x, y) =   f XY ( x, y)dx dy = 1 − − Marginal PDF   (1) f X ( x) =  f XY ( x, y)dy, fY ( y) =  f XY ( x, y)dx − − If X and Y are independent f XY ( x, y) = f X ( x) fY ( y)  y  x f XY ( x, y) = f X ( x) f Y   = fY ( y ) f X   X x Y   y Conditional PDF x y f ( x, y) − −   f XY ( x, y)dxdy f XY ( x, y) = XY =  fY ( y)  f XY ( x, y)dx − Probability Calculation in 2-D region Given Joint PDF P{(a  X  b)  (c  y  d )} = ? R1 : Region in which probability has to be calculated. Method: GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.29  Communication Systems P ( X , Y  R1 ) =   f XY ( x, y)dxdy ( R = R1  R2 ) R (1) X and Y not independent R.V. P ( X ,Y  R1) =   f X ,Y ( x) fY (Y ) dxdy R = ( R1  R2 ) Rr (Central Limit Theorem) If X and Y are D.R.V  PXY ( xi , y j ) = 1 i j PXY ( xi , y j ) = P{( X = xi )  (Y = y j )} Joint PMF Marginal PMF : PX ( xi ) =  PXY ( xi , y j ) j PY ( y j ) =  PXY ( xi , y j ) i Minimum of 2 independent R.V. X, Y are two I.R.V min ( X ,Y )  Z = ( X  Z )  (Y  Z ) P [min ( X , Y )  Z ] = P [ X  Z ]P [Y  Z ] =   f XY ( x, y)dxdy R P[min ( X ,Y )  Z ] = 1 − P[min ( X , Y )  Z ] P[min( X ,Y )  Z ] =  f X ( x) fY ( y)dxdy R R Let Z = Max ( X , Y ) → R.V. CDF of Z FZ (Z ) = FX (Z )  FY (Z ) PDF of Z f Z (Z ) = FX (Z ) fY (Z ) + FY (Z ) f X (Z ) Let Z = min[ X ,Y ] → R.V. CDF of Z FZ (Z ) = f X (Z ) + gY (Z ) + fY (Z ) FX (Z ) PDF of Z f Z (Z ) = f X (Z ) + fY (Z ) − FX (Z ) fY (Z ) − FY (Z ) f X (Z ) Statistical parameters of 2D R.V. (1) (k , r )th order joint moment about origin E [ X kY r ] (1,1)st order joint moment about origin. E[ X 1,Y1] = E [ XY ] = RXY → Cross correlation between R.V. X and V. GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.30  Communication Systems E [ XY ] = RXY = 0 → R.V. X and Y are orthogonal. (2) (k , r )th order joint moment about mean- E [( X − X )k (Y − Y )r ] (1,1)st order joint Moment about mean- E [( X − X )(Y − Y )] = E [ XY ] − XY = cov( X , Y ) cov( X ,Y ) = XY = E[ XY ] − E[ X ] E[Y ] = RXY −x  y When 2 R.V. X and Y are uncorrelated- cov( X , Y ) = 0 , E [ X ,Y ] = E [ X ] E [Y ] E [ X kY r ] = E [ X k ]E [Y r ] = X ,Y are independent. If 2 R.V. are independent then they has to be uncorrelated but converse is not necessarily true. One function of two R.V. W = aX + bY (1) E [W ] = a E [ X ] + bE [Y ] (2) E[W 2 ] = a2 E [ X 2 ] + b2 E [Y 2 ] + 2ab RXY (3) W 2 = a22X + b2Y2 + 2ab cov( X ,Y ) One function of Three R.V. W = aX1 + bX 2 + cX 3 (1) E [W ] = a X1 + b X 2 + a X3 (2) E[W 2 ] = a2 X12 + bX 22 + c2 X32 + 2abX1X 2 + 2bcX 2 X3 + 2caX1X3 (3) W 2 = a22X1 + b22X 2 + c22X 3 + 2ab cov( X1, X 2 ) + 2bc cov( X 2 , X 3 ) + 2ca cov( X1, X 3 ) Var (X + Y) = Var (X – Y) Only when X, Y are → uncorrelated and independent Correlation coefficient  XY cov( X ,Y ) ( X ,Y ) = =  X Y (Std.dev.of X )  (std.dev.of Y ) −1    1 ( X , X ) = 1, ( X , − X ) = −1 X, Y are independent  ( X , Y ) = 0 Let X, Y are two R.V. GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.31  Communication Systems   g (Y )   y E =  g ( y) f Y  dy  X = x  − X  x   g( X )   x E  =  g ( x) f X   dx  Y = y  − Y   y Calculation of probability in n-D region Theorem -1 If X1, X 2 , X 3 − − − X n are statistically independent random variables. Let Z = X1 + X 2 − − − − X n    f X1 ( z) f X 2 ( z) f X n ( z) f Z ( z) = f X1 ( z)* f X 2 ( z) − − − − * f X n ( z) When and only when all the R.V. are statistically independent. R.V. are linearly combined. Theorem-2 X1, X 2 , X 3 − − − X n are statically independent non Gaussian R.V. Z = X1 + X 2 + X 3 − − − − X n f Z ( z) = f X1 ( z)* f X 2 ( z) − − − − * f X n ( z) If n → f Z ( z ) = Gaussian irrespective of nature of ( X i )in=1 Theorem-3 X1, X 2 , X 3 − − − X n are statistically independent G.R.V. Z = X1 + X 2 − − − + X n fZ ( z) = f X1 ( z) + f X 2 ( z) + − − − − f X n ( z) n → Finite| infinite, → Z: GRV Problem Solving Technique : Case 1 : X1, X 2 − − − − X n are statistically independent G.R.V.  a P[ X1 + X 2 + X 3  a] = P(Z  a) =  f Z ( z )dz = 1 −  f Z ( z )dz  a − Non GRV Where Z = X1 + X 2 + X 3 GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.32  Communication Systems f Z ( z) = f X1 ( z)  f X 2 ( z)  f X3 (z) Case 2 : If X1, X 2 , X3 are statistically independent G.R.V.  a − z  P( X1 + X 2 + X 3  a) = P [Z  a] = Q    z  Z = X1 + X 2 + X 3 2 =  X1 +  X 2 +  X 3 , 2z = 2X1 + 2X 2 + 2X 3 Note : If X1, X 2 , X 3 − − − − X n are I.I.D. random variables 1 P (one of them is largest) = n 1 P (one of them is smallest) = n Random Process X ( , t ) = {X (1, t), X ( 2 , t )} Collection of sample function of Euemble of sample function Random process or Random signal or stochastic signal X (1 , t1 ) → sample value, values taken by R.V. When R.P. is observed at t = t1 C.T.R.P → It maps the sample points onto continuous time sample function, collection of continuous time sample function. t =t1 X (t ) = Acos(0t + ) ⎯⎯⎯ → X (t1) = Acos(0t1 + ) C.T.R.P R.V., D.R.V. t → Continuous time, A → Constant, 0 → constant  U [−, ] → CRV Any typical R.P can be understood as x(t ) = f (t , ) (Function of time and R.V.) x(n) = f (n, ) Statistical parameter of R.P. Case 1 : X (t ) X (t0 ) − CRV t = t0  E [ X (t0 )] =  x f X (t0 ) ( x) dx −  E [ x2 (t0 )] = x 2X (t0 ) = E [ X 2 (t0 )] − 9E ( X (t0 ))2 ] 2 f X (t0 ) ( x) dx , − X (t ) X (t 0 ) − DRV Case 2 : t = t0 CTRP GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.33  Communication Systems E[ X (t0 )] =  xi PX (t0 ) ( xi ) =  xi P{X (t0 ) = xi ) i i E[ x 2 (t0 )] =  xi2 PX (t0 ) ( xi ) =  xi2 P{X (t0 ) = xi ) i i 2X (t0 ) = E [ X 2 (t0 )] − ( E [ X [t0 ])2 Case 3 : DTRP → CRV E [ X (n0 )], E [ x2 (n0 )], 2X (n0 ) → Some as case 1, replace to by n0 Case 4 : DTRP → DRV E [ X (n0 )], E [ x2 (n0 )], 2X (n0 ) → Same as case-2, Replace to by n0 CTR.P t = t1 X (t1 ) X (t ) t = t2 X ( t2 ) E[ X (t1) X (t2 )] = RX (t1) X (t2 ) = RXX (t1, t2 ) Auto correlation of RP X(t) X (t ) X (t ) X (t + ) Then E[ X (t1 ) X (t + )] = RXX (t , t + ) Cov ( X (t1 ) X (t2 )) = E [ X (t1 ) X (t2 )] − E[ X (t1)]E[ X (t2 )]] XX (t1, t2 ) = RXX (t1, t2 ) −X (t1)X (t2 ) Auto covariance of R.P. X (t ) XX (t, t + ) = RXX (t, t + ) −X (t )X (t +) Cross Correlation X (t ) ⎯⎯⎯ t =t → X (t1), Y (t ) ⎯⎯⎯ t =t →Y (t2 ) R.P 1 R.V R.P 2 R.V E [ X (t1 )Y (t2 )] = RXY (t1, t2 ) (1) If RXY (t1, t2 ) = 0 V t1  TR X (t ) and Y (t ) R.P. will t2  TR Become orthogonal. (2) Cov [ X (t1),Y (t2 )] = RXY (t1, t2 ) − X (t1) X (t2 ) = 0 V t1  TR t2  TR GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.34  Communication Systems RP X (t ) and Y (t ) are uncorrelated. If X (t1 ) and X (t2 ) are independent  E[ X (t1 ) E[ X (t2 )] t1  t2 E[ X (t1 ) X (t2 )]  2  E[ X (t1 )] t1 = t2 Same for DRV, replace t by n. Types of R.P. (1) Strict sense stationary R.P. → R.P. should be independent of time shift X (t1 ) X (t2 ) − − − X (tk ) X (t ) → kRV.. Kth order Joint PDF- ( x1, x2 − − − xk ) …(i) f X (t1 ) X (t2 ) − − − X (tk ) X (t1 + ), X (t2 + ) − − − X (tk + ) X (t ) : k R.V. Kth order Joint PDF- ( x1, x2 − − − xk ) …(ii) f X (t1+) X (t2 +)−−− X (tk +) (i) = (ii) → X (t ) is solid to be SSSRP. f X (t1) ( x) = f X (t2 +) ( x) independent of time 2nd order joint PDF is independent of time shift. f X (t1) X (t2 ) ( x1, x2 ) → f X (0) X (t2 −t1) ( x1, x2 ) Does not depend on individual sampling instances t1 and t2 Depends on time difference between sampling instances t1 and t2 E[ X (t1 ) X (t2 )] = E[ X (0) X (t2 − t1 ) = RXX (t1, t2 ) E[ X (0) X (t2 − t1 )] = RXX (0, t1 − t2 ) = RXX (t1 t2 ) XX (t1 t2 ) = RXX (t1 t2 ) − 2X WSSRP → There are stationary RP which are stationary at least upto 2nd order. GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.35  Communication Systems (1) E[ X (t )] =  X Constant (2) E [ X 2 (t )] = Constant (3) 2X (t ) = Constant E[RP] = E[RV] MSV(RP) = MSV(RV) Var (RP)=Var (RV) (4) E [ X (t1 ) X (t2 )] = RXX (t1 t2 ) E [ X (t + ) X (t )] = RXX (−) E [ X (t ) X (t + )] = RXX () RX X () = RXX (−) 2X (t ) = RXX (0) − 2X Cov [ X (t ) X (t + )] = RX X () − 2X (5) E[ X 2 (t )] = RXX (0) =0  E[ X (t )]E[ X (t + )] =  X 2 (  0) (6) E[ X (t ) X (t + )] = RXX ()ACF =   E[ X (t )] = RXX (0) ( = 0) 2  (7) cov[ X (t ) X (t + )] = RXX (t , t + ) −  X (t ) X (t + )  0 0 C XX () = RXX () − 2X =   RXX (0) −  X =0 2    2 0 RX X () =  X RX X (0)  = 0  Important point : (1) If X(t) is zero mean WSSRP. E [X (t) = 0 GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.36  Communication Systems 2X (t ) = E [ X 2 (t )] Var [ X (t )] = MSV{X (t )} Var { X (t = t1 )} = MSV{ X (t = t1 )} (2) If X (k) is zero mean WSSRP E[ X (k )] = 0 2X (k ) = E[ X 2 (k )] Var [ X (k )] = MSV[X (k )] X (t ) : WSSRP+IIDRP E [ X (t )] =  X , E[ X (t + )] =  X , E[ X 2 (t )] = RXX (0) = Constant 2X (t ) = Constant Let Y (t ) = X (at + b) E [Y (t ) =  X , E[ y2 (t )] = RXX (0) Y2 (t ) = RXX (0) −  X 2 E[Y (t )Y (t + )] = RXX (a) = RYY () Cov [Y (t )Y (t + )] = CYY () = RXX (a) − 2X → Y =  X = Constant For Y (t ) →  ,Y (t ) → WSSRP  RYY () = RXX (a) Time shift, Time reversal, time scaling does not affect stationary nature of R.P. Let Y (t ) = aX (t ) + b, X (t ) is WSSRP E [Y (t )] = a X + b = Constant E[Y 2 (t )] = a2 RXX (0) + b2 + 2ab X = Constant Y2 (t ) = a22X (t ) Cov [Y (t )Y (t + )] = RYY () − Y2 E[Y (t )Y (t + )] = a2 RXX () + 2ab X + b2 = RYY () y(t ) → WSSRP Linear transformation of WSSRP does not change its stationarity. If WSSRP passed through LTI system, output is also a WSSRP. GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.37  Communication Systems E [Y (t )] = (a + b) X E [Y 2 (t )] = a2 RXX (0) + b2 RXX (0) + 2abRXX (0 ) Y2 (t ) = (a2 + b2 )2X (t ) + 2ab[ RXX (T0 ) − 2X ] RYY () = (a2 + b2 )RXX () + abRXX ( − T0 ) + abRXX ( + T0 ) CYY () = a2CXX () + b2CXX () + abRXX ( − T0 ) + abRXX ( + T0 ) − 2ab2X E[ X (n) X (n + k )] = [k ] = RXX (k ) IIDRP E [ X (n) X (n + k )] = E [ X 2 (n))(k = 0) A, 0 → constant,  U (0,2)OR  U [−, ] X (t ) = A cos(0t + ) E [ A cos(0t + )] = 0 E [ A cos(0t + + )] = 0 A2 2 A2 E [ X 2 (t )] = ,  X (t ) = 2 2 A2 E [ X (t ) X (t + )] = cos 0 = RXX () 2 A2 Cov [ X (t ) X (t + )] = cos 0 2 X (t ) : WSSRP + periodic with 0 → RXX () will also be periodic with same T.P. ERGODIC Random Process : Time Avg = Statistical Aug. GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.38  Communication Systems 1 T X (t ) dt = E [ X (t )] Auto Correlation and its properties Similarity between 2 Samples Let X (t ) is WSSRP, X (t ) is observed  duration apart (1) E [ X (t ) X (t + ) = RXX () (2) RXX (−) = RXX () : Even (3) RXX ()  RXX (0) (4) {RXX ()}max = RXX (0) = Maximum similarity   0 (5)  RXX ()d  = 2  RXX ()d  = 2  RXX ()d  − 0 − (6) (7) (8) X (t ) is power signal RXX (0) = E[ X 2 (t )] = 2X ( t ) + 2X ( t ) Total power A.C power D.C power of R.P of R.P of R.P (9) If X (t ) is ergodic and WSSRP, it has no periodic component E [ X (t ) =  X  0 2X = lim RXX () = lim RXX () → ||→ If not ergodic but WSSRP then RXX (0) = E[ X 2 (t )] RXX ()  2X , GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.39  Communication Systems Important point: X (t )  X () X (t )  X ( f )   X () =  x(t )e− jt dt X( f ) =  x(t )e − j 2ft dt − −   1 jt j 2ft x(t ) =  X ()e dt 2 − x(t ) =  X ( f )e dt −   X (0) =  x(t )dt x(0) =  X ( f )df − −   1 x(0) =  X ()d  2 − X (0) =  x(t)dt − 3.2. Parserval Theorem    1 Ex(t ) =  x2 (t )dt =  | X () | d  =  | X ( f ) | df 2 2 − 2 − − Energy spectral density X (t ) → WSSRP, Engery ESD x(t ) ⎯⎯→ F.T X ( )  | X () | 2 = G XX ( ) x( t) ⎯⎯ → X ( f )  | X ( f ) | 2 = G XX ( f ) ESD of x(t) F.T. E[ X (t ) X (t + )] = RXX () ⎯⎯⎯ →GXX () F.T. RXX () ⎯⎯⎯ →GXX ( f ) F.T ACF( X (t )) ⎯⎯ ⎯ ESD[ x(t )]   GXX (0) =  RXX () d  = 2  RXX () d  − 0 Zero freq. value of ESD = Area under ACF GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.40  Communication Systems  1 Area under ESDG XX () RXX (0) =   2 − GXX ()d  = 2 E[ X 2 (t )]   GXX ( f ) df = Area under ESD G XX ( f ) − Energy Calculation :    1 EX (t ) =  | X (t ) |2dt =  | X () | d  =  | X ( f ) | df 2 2 − 2 − −   1 =  GXX () d =  GXX ( f ) df 2 − − GXX () = GXX (−) Power spectral density – (PSD) X (t ) → Power signal, WSSRP PSD 1 X (t ) ⎯⎯ → S XX () = lim | X T () |2 T → T PSD 1 X (t ) ⎯⎯ → S XX ( f ) = lim | X T ( f ) |2 T → T  − j (1) E [ X (t ) X (t + )] = RXX () S XX () =  RXX ()e d − F.T. (2) ACF [ X (t )]⎯⎯ →PSD[ X (t )]  − j 2f   RXX ()e F.T. RXX ⎯⎯ → XX () S XX ( f ) = d −   (3) S XX (0) =  RXX () d  = 2  RXX () d  − 0 Zero freq. value of = Area under ACF PSD   1 (4) RXX () =  S XX ()e jd  =  S XX ( f )e jdf 2 − − 1    S XX ()d   2 − RXX (0) =      S XX ( f )df  − GATE WALLAH ELECTRONICS & COMMUNICATION HANDBOOK 8.41  Communication Systems (5) RXX () = RXX (−), S XX () = S XX (−) (6) Calculation of power 1  Area under PSD   2  S XX ()d  = 2 E[ X 2 (t )] = RXX (0) =  −     S XX ( f )df = Area under PSD  −   1  0 E [ X 2 (t )] = RXX (0) = S XX ()d  = 2  S XX ( f )df 0 Total power A.C. Power = 2 X (t ),D.C.Power = 2 X (t ) Mean or Aug value 0+ 1 E[ X (t )] = 2 − S XX ()d  0  0+