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Random process practice, Cheat Sheet of Digital Signal Processing

Anna UniversityDigital Signal Processing

Practice problems on random process

Typology: Cheat Sheet

2024/2025

Uploaded on 08/19/2025

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Random Signals & Noise Qi Q2 Q3 Q4 If the density function of a continuous RVX is given by ax; O; O1 is otherwise () &) Qs Ole wl Ale wly © @) x) 6 The probability density function of X is defined as, pe os > forx>1 0; elsewhere fxts)-| The value of ‘p’ is (a) 0.7956 (c) 0.5967 (b) 0.9567 () 0.6975 Consider the Cauchy density function K x)=; -w T alRoaIN The mean value E(x) and the variance of respectively are the mean value and the variance of the Cauchey density function respectively are @) 0,0 (b) «0,0 (©) 0,0 (d) 2,20 The joint probability density function of the random variables X and Y is given as xe ryt) ; O01/2. (a) 1 ) 2 © 3 (4 The joint probability density function of two random variables is fyy(x, y) = K[x? + y]; K=3/44,xe(2,3), ye(0, 2). Then the mean value E(x) is 88 225 @) 5 © 35 85 229 © a9 ) 5 The input X(f) to a diode with a transfer characteristic Y = X? is a zero mean stationary Gaussian random process with an autocorrelation function Ryx(t) = exp(-|t)). The mean py(#) is (a) 0 (b) 0.5 @1 (d) 2 The input X(t) to a diode with a transfer characteristic Y = X? is a zero mean stationary Gaussian random process with an autocorrelation function Ryx(t) = exp(-|t}). The autocorrelation function, Rylty t) is (@) 1+exp(-2|r)) —(b)_ 1+2exp(-2|r)) (©) 1+2exp(-|t)) — @)_1+exp(-Ir)) Consider a random variable Z such that Z=pX + qY, where p and qare arbitrary constant. If X and Y are independent random variable, then which of the following is true? @) of =poz +40, 2 22 22 2.22.2 () of =p’ of +9" 0, +2p"q? oxo, © o=pot+q@ oy @) of =p’ oz +9? 0, +2pqoz oy Q.25 Q.26 Q.27 Q.28 Q.29 Which of the following can be suitable autocorrelation functions? 1. A tri(t/t,); where tri(x) is unit triangular function 2. Asin(@,t) 3. Acos(@,t) 4. Arect(t/t) (a) 1 only &) 1,3 © 1,3,4 (d) 1,2,3,4 Let X, and X, be two independent random variables uniformly distributed between 0 and 1. The probability that X, + X,>1is . Two random variable X and Y are to be uncorrelated if @ E(KY=0 () EX) E(Y)=0 (9 E(XY)-E(X) E()=0 (@) E(KY) +E) EX) For the power spectral density shown below, the total power (in W) is . Sxlf)A 45(F) 2x10° 2x10° 2 3 (kHz) The random process X(t) is defined as X(t) = Y cos(2nfyt + 0) where Y and @ are 2 independent random variables, Y distributed uniformly on [-3, 3] and 0 distributed uniformly on [0, 2x]. The autocorrelation function (Ry(é)) and power spectral density of X(i), respectively are (a) Fcos2afos, 18S + fo)+5(F— fo] (0) 3cos2nfys, S8(F+ fo)+3(F— fo) (¢) Seos2afos, 8+ fo)+5(F— fo) (d) 3cos2nfot, 3[8(f + fo) +5(f - fo)] 32 Electronics Engineering ¢ Communication Systems Q.30 Consider the functions given below: Q.31 Q.32 Q.33 P. f(a) =sin2nfyt Q faa? _fi-ls [<1 R reo-{r |e|>1 FO) 1 Ss. 0 t Which of the above functions cannot be the ACF of arandom process? (a) Ponly (b) P,Q, Ronly (©) P,Qonly @ P,QORS A Gaussian pulse, g(t) is given by: 9(t) =V2exp(-n#2) Its autocorrelation function R,(t) is @) Vaew{-Z2) () Been?) (© exp(-n7’) (d) 2exp(-n7) The input to an RC low-pass network is a zero mean stationary Gaussian random process X(t) with R,,(t) = exp(—a|t|). The output of the low pass network is Y(t). The mean value of the output random process Y(f) is @1 (b) 0 ©) 05 2 The input to an RC low pass network is a zero mean stationary Gaussian random process X(t) with Ry,(t) = exp(-a|z|). The output of the low 1 pass network is Y(é). If B = Re’ then power spectral density of the output random process Y(fis Q.34 Q.35 Q.36 Q.37 207 \f_ iB @ a2 + (2nf) ) |B? + (2nf)? a2 2 ) a? + (2nf) )| B? + (2nf)? 2? \(_ © 2s pe) Oa? 2a _ Be @) a? + (2nf) )| B? + (2nf)? The input to an RC low pass network is a zero mean stationary Gaussian random process X(t) with Ryy(t) = exp(-a|z|). The output of the low pass network is Y(t). The variance of the output random process Y(f) is a B @ cap ©) cap o2 p2 © cap @ asp The stationary random process X(t) has a power spectral density denoted by S,(f). What is the power spectral density of Y(t) = X(#) - X(t-T)? (a) 2S4(f) (1 cos(2nfT)) (©) Sq(f) (1 ~cos(2nfM) (© Se(f) (1-cos(4nfN) (€) 254(f) (1-cos(4nfN) The stationary random process X(t) has a power spectral density denoted by S,(f). What is the power spectral density of Z(t) = X’(t) - X(t)? (@) Sx(f) (1+2n37%) (0) 28 e(f) (1+ 4039) © Sf O+eP) @) Sf) +4n2?) The stationary random process X(t) has a power spectral density denoted by S,(f). IEZ(t) = X'(t)-X(, Y() = X(f) -X(t-T) then what is the power spectral density of W(t) = Y(t) + Z(#)? (a) Sy(f) (1 + 4n2f? + 4nfsin(2nfT)) ©) Sy(f) (1 + 2n2f? + 4nfsin(2nfT)) (©) Sf) (1 +4n2/? + 2nfsin(2nfT)) d) Sy(f) (1 + 2n2f? + 2nfsin(2nfT)) Electronics Engineering ¢ Communication Systems Q.45 Q.46 © 2a 1 _ 1 Bro? [asda fp? peed f? @ a 1 _ 1 2 a2 oe +4n? f? 2 +4n? f? Consider the random process: X(t) =A cos(@,t + 6) where A and @, are constants, while 0 is a random variable with an uniform pdf, fo(0)=, -n 0 is a constant. We further assume at t=0, X(0) is equally likely to be 0 or 1. The value of m,(t) is Q.a9 Q.50 Q51 Q.52 A random process X(t) takes values 0 and 1. A transition from 0 to 1 or from 1 to 0 occurs randomly and the probability of having ‘n’ transitions in a time interval of duration t is given by (= 1 (4 PNW) = sec (4k where, k > Ois a constant. We further assume at t=0, X(0) is equally likely to be 0 or 1. R(t +1, f) is (142k) 4 its) @ al aeke ©) Blas aK 1 1+k) a (142s) © 2\1s2ke @ ol ahh Statement-I : A random process may be wide sence stationary. n ) jn=0,1,2.... Statement-II : WSS random process shows constant mean and ACF depends only on tie. (ht) The Rayleigh density function is given by 2 x? /207 Fx(x) =} 07 x>0 0; otherwise The mean value E(X) is given as: t o ft @) off © aV2 s © ve @) ovm The Rayleigh density function is given by Xx? 262 se : x>0 fx (2) =5 07 0; otherwise The variance of X is VAR(X) is givenas, oe o@3)F ofge oe KILLERGENIX | 35 Q.53 Q.54 Q.55 Q.56 Let @ bea random variable uniformly distributed on [0, 7] and let random variables X and Y be defined by X = cos@ and Y = sin@. Then which of the following is true with respect to X and Y? (a) X and Y are uncorrelated as well as independent. (b) X and Y are correlated as well as independent. (¢) X and Y are uncorrelated but they are not independent. (d) X and Y are correlated but they are not independent. Two random variables X and Y are distributed according to Fry (*/¥) = xe > J, xy20 0; xy <0 The value of E(XY)is______. In a binary communication system (figure below), 0 or 1 is transmitted. Because of channel noise, 0 can be received as 1 and vice-versa. Let m0 and m1 denote the events of transmitting 0 and 1, respectively. Let ry and r, denote the events of receiving 0 and 1, respectively. Let P(m,) = 0.5, P(m,) = 0.5, P(r, | mg) = 0.1 and P(r) |m,) = 0.2. P(ro| mo) Pag) my P “nm Pel) P(r.) and P(r,) respectively are (a) 0.45 and 0.55 (b) 0.55 and 0.45 (©) 0.5 and 0.5 (d) 0.6 and 0.4 If‘p’ is the probability of occurrence of the event, then information associated with it (b) -log,)p Hartleys (d) log, p nats (a) log, p bits (©) 4og, p Hartleys Q57 Q.58 Q.59 Q.60 Q.61 The joint probability density of the random variables X and Y is f(x,y) = ke“**) in the range 4. Ifa message consists of 10 bits, 100 bits, 1000 bits, 10,000 bits, then the probability of the message being in error in each of the above cases respectively are___,__,___and (a) 0.0043, 0.043, 0.43, 0.999 (b) 0.0063, 0.063, 0.63, 0.999 (c) 0.00995, 0.0952, 0.952, 0.999 (d) 0.00995, 0.0952, 0.632, 0.999 Consider the probability density fx (x)= pet, where X is a random variable whose allowable values range is from x = -00 to x = +o. The relationship between p and q and the value of probability, P(-1 < X <2) (for q=1) respectively are © p=4,0.4873 (a) p= 4, 0.7483 3 © P= £, 0.7483 @) p= 4, 0.4873 The joint probability density of the random variables X and Yis f(x,y) = ke "1-1 co mn) 1 and o(r=3)=3. If mean, 2 uig(0) = EUX(D} then p, (1) is (a) 0 (b) 0.5 © 1 (d) 2 A random process X(t) is defined by X(t) = 2cos(2nt + Y) where Y is discrete random variable with PY=0)=3 and r(y=5)-3. If auto- correlation function Ryy(t,, t,) = E{X(t,) X(t,)} then R,,(0, 1) is (a) 0 (b) 0.5 @1 (d) 2 Consider an SSB system shown below which transmits the lower sideband modulated signal s(t). 38 | Electronics Engineering ¢ Communication Systems m(t) —>( ; )—>} rn A, cos2rf.t Q.76 m(t) A, cos2nf.t Q7 LPF s). Dynlt LPF © a ”| 3 kHz 200) cos2nf.t Assume that the message signal m(t) has PSD IM(f, iP and its Fourier transform is 0.003 ; |f|<1.5 kHz M(f)=40.001; 15kHz 3 kHz Find the value of ‘A,’ such that the power of signal s(t) is equal to 100 mW. (a) 3.65V () 7.3V (© 1.825 V (@) 3V Consider an SSB sysetm shown below which transmits the lower sideband modulated signal s(t). LPF | s(#) n(t) fe 3 kHz 20) cos2nf.t Assume that the message signal m(t) has PSD IM(f i? and its Fourier transform is 0.003; |f|<1.5 kHz M(f)=40.001; 1.5kHz3 kHz N Ifn(#) is AWGN with PSD Sy/(f) aaa whatis the noise power in the demodulator output when No =10+ (=) : 2 Hz (a) 0.15 mW (b) 0.075 mw (© 0.3mWw (d) 0.5mWw A low frequency signal of bandwidth 4000(Hz) having strength of 0.001 (Watts) is passed through a distorting channel defined as 4000 A) = F000 F and is also corrupted with AWGN with two sided PSD of 10° W/Hz. At the receiver side there is an equalizer which exactly matches the channel within frequency of message signal. The strength of the noise at the output of equalizer is (a) 32x 105 Ww (& 16 x 105W © 2 a0 Ww 4) a0 Ww Q.78 The PSD of a signal x(t) is as shown in figure. sf) 4 -5 kHz 0 5 kHz f Elx(#lis (a) zero 4 (©) 20 x 10% (d) 40 x 10° Q.79 White noise with a two sided spectral density of 10° W/Hz is passed through an ideal differentiator. The output of the differentiator is again passed through a LPF having a cut-off frequency of 10 Hz. The noise power at the output of the LPF is (a) 0.0263 W (b) 0.0283 W (©) 0.0163 W (4) 0.263 W Q.80 Statement-I : Slope overload distortion and granular noise can be considered as one form of quantization error. Statement-II : Error results because of approximation of sampled value of nearest quantization level is known as quantization error. Q.81 A random process has power spectral density given by This noise is passed through an ideal BPF with a bandwidth of 2 MHz and centered at 50 MHz.