# GATE Questions & Answers of Communications Electronics and Communication Engg

#### Communications 109 Question(s) | Weightage 14 (Marks)

The block diagram of a frequency synthesizer consisting of a Phase Locked Loop (PLL) and a divide-by-N counter (comprising ÷2 ,÷4, ÷8, ÷16 outputs) is sketched below. The synthesizer is excited with a 5 kHz signal (Input 1). The free-running frequency of the PLL is set to 20 kHz. Assume that the commutator switch makes contacts repeatedly in the order 1-2-3-4.

The corresponding frequencies synthesized are:

Consider binary data transmission at a rate of 56 kbps using baseband binary pulse amplitude modulation (PAM) that is designed to have a raised-cosine spectrum. The transmission bandwidth (in kHz) required for a roll-off factor of 0.25 is ________

A superheterodyne receiver operates in the frequency range of 58 MHz − 68 MHz. The intermediate frequency ${f}_{IF}$ and local oscillator frequency ${f}_{LO}$ are chosen such that ${f}_{IF}\le {f}_{LO}$. It is required that the image frequencies fall outside the 58 MHz − 68 MHz band. The minimum required ${f}_{IF}$(in MHz) is ________

The amplitude of a sinusoidal carrier is modulated by a single sinusoid to obtain the amplitude modulated signal. The value of the modulation index is __________

Consider a discrete memoryless source with alphabet S={s0,s1,s2,s3,s4,......} and respective probabilities of occurrence $P=\left\{\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{16},\frac{1}{32},....\right\}$. The entropy of the source (in bits) is _______

A digital communication system uses a repetition code for channel encoding/decoding. During transmission, each bit is repeated three times (0 is transmitted as 000, and 1 is transmitted as 111). It is assumed that the source puts out symbols independently and with equal probability. The decoder operates as follows: In a block of three received bits, if the number of zeros exceeds the number of ones, the decoder decides in favor of a 0, and if the number of ones exceeds the number of zeros, the decoder decides in favor of a 1. Assuming a binary symmetric channel with crossover probability p = 0.1, the average probability of error is ________

An analog pulse $s(t)$ is transmitted over an additive white Gaussian noise (AWGN) channel. The received signal is $r(t) = s(t) + n(t)$, where $n(t)$ is additive white Gaussian noise with power spectral density $\frac{{N}_{0}}{2}$. The received signal is passed through a filter with impulse response $h(t).$ Let $E_s$ and $E_h$ denote the energies of the pulse $s(t)$ and the filter $h(t)$ respectively. When the signal-to-noise ratio (SNR) is maximized at the output of the filter (SNRmax), which of the following holds?

An antenna pointing in a certain direction has a noise temperature of 50 K. The ambient temperature is 290 K. The antenna is connected to a pre-amplifier that has a noise figure of 2 dB and an available gain of 40 dB over an effective bandwidth of 12 MHz. The effective input noise temperature Te for the amplifier and the noise power Pao at the output of the preamplifier, respectively, are

A discrete memoryless source has an alphabet {a1, a2, a3, a4} with corresponding probabilities . The minimum required average codeword length in bits to represent this source for error-free reconstruction is ________

A speech signal is sampled at 8 kHz and encoded into PCM format using 8 bits/sample. The PCM data is transmitted through a baseband channel via 4-level PAM. The minimum bandwidth (in kHz) required for transmission is ________

An information source generates a binary sequence $\left\{{\alpha }_{n}\right\}$. ${\alpha }_{n}$ can take one of the two possible values −1 and +1 with equal probability and are statistically independent and identically distributed. This sequence is precoded to obtain another sequence $\left\{{\beta }_{n}\right\}$, as . The sequence $\left\{{\beta }_{n}\right\}$ is used to modulate a pulse $g(t)$ to generate the baseband signal

$X\left(t\right)={\textstyle\sum_{n=-\infty}^\infty}\beta_ng\left(t-nt\right)$ where $g\left(t\right)=\left\{\begin{array}{lc}1,\;\;&0\leq t\leq T\\0,&\mathrm{otherwise}\end{array}\right.$

If there is a null at $f=\frac{1}{3T}$in the power spectral density of $X(t)$ then $k$ is ________

An ideal band-pass channel 500 Hz - 2000 Hz is deployed for communication. A modem is designed to transmit bits at the rate of 4800 bits/s using 16-QAM. The roll-off factor of a pulse with a raised cosine spectrum that utilizes the entire frequency band is ________

Consider a random process X(t)=3V(t)-8, where V(t) is a zero mean stationary random process with autocorrelation ${R}_{v}\left(\tau \right)=4{e}^{-5\left|\tau \right|}$. The power in X(t) is ________

A binary communication system makes use of the symbols “zero” and “one”. There are channel errors. Consider the following events:
x0 : a "zero" is transmitted
x1 : a "one" is transmitted
y0 : a "zero" is received
y1 : a "one" is received
The following probabilities are given: $P\left(x_0\right)=\frac12,\;P\left(y_0\vert x_0\right)=\frac34$ and $P\left(y_0\vert x_1\right)=\frac12.$ The information in bits that you obtain when you learn which symbol has been received (while you know that a “zero” has been transmitted) is ________

For a superheterodyne receiver, the intermediate frequency is 15 MHz and the local oscillator frequency is 3.5 GHz. If the frequency of the received signal is greater than the local oscillator frequency, then the image frequency (in MHz) is __________

An analog baseband signal, bandlimited to 100 Hz, is sampled at the Nyquist rate. The samples are quantized into four message symbols that occur independently with probabilities p1 = p4 = 0.125 and p2 = p3. The information rate (bits/sec) of the message source is __________

A binary baseband digital communication system employs the signal

$p\left(t\right)=\left\{\begin{array}{l}\frac1{\sqrt{{\mathrm T}_\mathrm s}},\;0\leq\mathrm t\leq{\mathrm T}_\mathrm s\;\;\;\\0,\;\;\;\;\;\;\;\;\;\;\;otherwise\;\end{array}\right.$

for transmission of bits. The graphical representation of the matched filter output y(t) for this signal will be

A wide sense stationary random process X(t) passes through the LTI system shown in the figure. If the autocorrelation function of X(t) is R($\tau$), then the autocorrelation function R($\tau$) of the output Y(t) is equal to

A voice-grade AWGN (additive white Gaussian noise) telephone channel has a bandwidth of 4.0 kHz and two-sided noise power spectral density $\frac{\eta }{2}=2.5×{10}^{-5}$ Watt per Hz. If information at the rate of 52 kbps is to be transmitted over this channel with arbitrarily small bit error rate, then the minimum bit-energy Eb (in mJ/bit) necessary is __________

The bit error probability of a memoryless binary symmetric channel is 10−5. If 105 bits are sent over this channel, then the probability that not more than one bit will be in error is __________

A sinusoidal signal of 2 kHz frequency is applied to a delta modulator. The sampling rate and step-size $\triangle$ of the delta modulator are 20,000 samples per second and 0.1 V, respectively. To prevent slope overload, the maximum amplitude of the sinusoidal signal (in Volts) is

Consider the signal $s\left(t\right)=m\left(t\right)\mathrm{cos}\left(2\pi {f}_{\mathrm{c}}\mathrm{t}\right)+\stackrel{^}{m}\left(t\right)\mathrm{sin}\left(2\mathrm{\pi }{f}_{\mathrm{c}}\mathrm{t}\right)$ where $\stackrel{^}{m}\left(t\right)$ denotes the Hilbert transform of m(t) and band width of m(t) is very small compared to fc. The signal s(t) is a.

The input X to the Binary Symmetric Channel (BSC) shown in the figure is ‘1’ with probability 0.8. The cross-over probability is 1/7. If the received bit Y = 0, the conditional probability that ‘1’ was transmitted is _______.

The transmitted signal in a GSM system is of 200 kHz bandwidth and 8 users share a common bandwidth using TDMA. If at a given time 12 users are talking in a cell, the total bandwidth of the signal received by the base station of the cell will be at least (in kHz) _______.

In the system shown in Figure (a), m(t) is a low-pass signal with bandwidth W Hz. The frequency response of the band-pass filter H(f) is shown in Figure (b).If it is desired that the output signal z(t) = 10x(t), the maximum value of W (in Hz) should be strictly less than ________.

A source emits bit 0 with probability $\frac{1}{3}$ and bit 1 with probability $\frac{2}{3}$ . The emitted bits are communicated to the receiver. The receiver decides for either 0 or 1 based on the received value $R$. It is given that the conditional density functions of $R$ are as

The minimum decision error probability is

A sinusoidal signal of amplitude A is quantized by a uniform quantizer. Assume that the signal utilizes all the representation levels of the quantizer. If the signal to quantization noise ratio is 31.8 dB, the number of levels in the quantizer is __________.

Let the random variable X represent the number of times a fair coin needs to be tossed till two consecutive heads appear for the first time. The expectation of X is _____

A zero mean white Gaussian noise having power spectral density $\frac{{N}_{0}}{2}$ is passed through an LTI filter whose impulse response h(t) is shown in the figure. The variance of the filtered noise at t = 4 is

${\left\{{X}_{n}\right\}}_{n=-\infty }^{n=\infty }$ is an independent and distributed (i.i.d) random process with Xn equally likely to be +1 or -1. ${\left\{{Y}_{n}\right\}}_{n=-\infty }^{n=\infty }$ is another random process obtained as Yn = Xn+0.5Xn-1 . The autocorrelation function of ${\left\{{X}_{n}\right\}}_{n=-\infty }^{n=\infty }$ , denoted by RY[K], is

Consider a binary, digital communication system which uses pulses g(t) and – g(t) for transmitting bits over AWGN channel. If the receiver uses a matched filter, which one of the following pulses will give the minimum probability of bit error?

Let $X\in \left\{0,1\right\}$ and $Y\in \left\{0,1\right\}$ be two independent binary random variables. If $P(X=0)=P$ and $P(Y=0)=q$ , then $P(X+Y\geq1)$ is equal to

The modulation scheme commonly used for transmission from GSM mobile terminals is

A message signal $m\left(t\right)={A}_{m}\mathrm{sin}\left(2\pi {f}_{\mathit{m}}t\right)$ is used to modulate the phase of a carrier ${A}_{c}\mathrm{cos}\left(2\pi {f}_{\mathit{c}}t\right)$ to get the modulated signal $y\left(t\right)={A}_{c}\mathrm{cos}\left(2\pi {f}_{\mathit{c}}t+m\left(\mathrm{t}\right)\right)$. The bandwidth of $y\left(t\right)$

A fair die with faces {1, 2, 3, 4, 5, 6} is thrown repeatedly till ‘3’ is observed for the first time. Let X denote the number of times the die is thrown. The expected value of X is _____.

The variance of the random variable X with probability density function $f\left(x\right)=\frac{1}{2}\left|x\right|{e}^{-\left|x\right|}$ is_______.

A random binary wave $y\left(t\right)$ is given by

$y\left(t\right)=\sum_\limits{n=-\infty}^\infty X_np\left(t-nT-\phi\right)$

Where $p\left(t\right)=u\left(t\right)-u\left(t-T\right),u\left(t\right)$ is the unit step function and $\phi$ is an independent random variable with uniform distribution in $\left[0,T\right]$.The sequence $\left\{{X}_{n}\right\}$ consists of independent and identically distributed binary valued random variables with $P\left\{{X}_{n}=+1\right\}=P\left\{{X}_{n}=-1\right\}=0.5$ for each n

The value of the autocorrelation ${R}_{yy}\left(\frac{3T}{4}\right)\triangleq E\left[y\left(t\right)y\left(t-\frac{3T}{4}\right)\right]$ equals_______.

Let x be a real-valued random variable with E[X] and E[X2] denoting the mean values of X and X2, respectively. The relation which always holds true is

Consider a random process $X\left(t\right)=\sqrt{2}\mathrm{sin}\left(2\pi t+\phi \right)$ where the random phase $\mathit{\phi }$ is uniformly distributed in the interval [0,2$\mathit{\pi }$ ]. The auto-correlation E[X(t1)X(t2)] is

Let $Q\left(\sqrt{\gamma }\right)$ be the BER of a BPSK system over an AWGN channel with two-sided noise power spectral density No/2 . The parameter $\gamma$ is a function of bit energy and noise power spectral density.
A system with two independent and identical AWGN channels with noise power spectral density No/2 is shown in the figure. The BPSK demodulator receives the sum of outputs of both the channels.

If the BER of this system is $Q\left(b\sqrt{\gamma }\right)$, then the value of b is _____.

Consider sinusoidal modulation in an AM system. Assuming no overmodulation, the modulation index ($\mu$) when the maximum and minimum values of the envelope, respectively, are 3 V and 1 V, is ________.

The input to a 1-bit quantizer is a random variable X with pdf ${f}_{x}\left(x\right)=2{e}^{-2x}$ for x ≥ 0 and ${f}_{x}\left(x\right)$= 0 for x < 0. For outputs to be of equal probability, the quantizer threshold should be _____.

Coherent orthogonal binary FSK modulation is used to transmit two equiprobable symbol waveforms and , where $\mathrm{\alpha }$=4 mV. Assume an AWGN channel with two-sided noise power spectral density $\frac{{\mathrm{N}}_{\mathrm{o}}}{2}=0.5×{10}^{-12}$ W/Hz. Using an optimal receiver and the relation , the bit error probability for a data rate of 500 kbps is

The power spectral density of a real stationary random process X(t) is given by

${S}_{X}\left(f\right)=\left\{\begin{array}{ll}\frac{1}{w},& \left|f\right|\le W\\ 0,& \left|f\right|>W\end{array}\right\.$

The value of the expectation $E\left[\mathrm{\pi X}\left(\mathrm{t}\right)\mathrm{X}\left(\mathrm{t}-\frac{1}{4\mathrm{W}}\right)\right]$ is_________.

In the figure, $M\left(f\right)$ is the Fourier transform of the message signal $m\left(t\right)$ where A = 100 Hz and B = 40 Hz. Given $v\left(t\right)=\mathrm{cos}\left(2\pi {f}_{\mathit{c}}t\right)$ and $w\left(t\right)=\mathrm{cos}\left(2\mathit{\pi }\left({\mathit{f}}_{\mathit{c}}+\mathit{A}\right)\mathit{t}\right),$, where ${\mathit{f}}_{\mathit{c}}>\mathit{A}$. The cutoff frequencies of both the filters are ${\mathit{f}}_{\mathit{c}}$.

The bandwidth of the signal at the output of the modulator (in Hz) is _____.

An analog voltage in the range 0 to 8 V is divided in 16 equal intervals for conversion to 4-bit digital output. The maximum quantization error (in V) is _________

A real band-limited random process X(t) has two-sided power spectral density

${S}_{X}\left(f\right)\left\{\begin{array}{ll}{10}^{-6}\left(3000-\left|f\right|\right)Watts/Hz& for\left|f\right|\le 3kHz\\ 0& otherwise\end{array}\right\$

where f is the frequency expressed in Hz. The signal X(t) modulates a carrier cos16000$\mathrm{\pi t}$ and the resultant signal is passed through an ideal band-pass filter of unity gain with centre frequency of 8 kHz and band-width of 2 kHz. The output power (in Watts) is _______.

In a PCM system, the signal m(t)={sin(100$\mathrm{\pi t}$ )+cos(100$\mathrm{\pi t}$ )} V is sampled at the Nyquist rate. The samples are processed by a uniform quantizer with step size 0.75 V. The minimum data rate of the PCM system in bits per second is _____.

A binary random variable X takes the value of 1 with probability 13⁄. X is input to a cascade of 2 independent identical binary symmetric channels (BSCs) each with crossover probability 12⁄. The output of BSCs are the random variables Y1 and Y2 as shown in the figure.

The value of H(Y1)+H(Y2) in bits is _____.

If calls arrive at a telephone exchange such that the time of arrival of any call is independent of the time of arrival of earlier or future calls, the probability distribution function of the total number of calls in a fixed time interval will be

In a double side-band (DSB) full carrier AM transmission system, if the modulation index is doubled, then the ratio of total sideband power to the carrier power increases by a factor of ______.

Consider a communication scheme where the binary valued signal X satisfies P{X= +1}=0.75 and P{X= -1}=0.25. The received signal Y = X + Z, where Z is a Gaussian random variable with zero mean and variance ${\sigma }^{2}$. The received signal Y is fed to the threshold detector. The output of the threshold detector $\widehat X$ is:

$\widehat X=\left\{\begin{array}{lc}+1,&Y>\tau\\-1,&Y\leq\tau.\end{array}\right.$

To achieve a minimum probability of error $P\left\{\widehat X\;\neq X\right\}$, the threshold τ should be

Consider the Z-channel given in the figure. The input is 0 or 1 with equal probability.

If the output is 0, the probability that the input is also 0 equals ______________

An M-level PSK modulation scheme is used to transmit independent binary digits over a band-pass channel with bandwidth 100 kHz. The bit rate is 200 kbps and the system characteristic is a raised-cosine spectrum with 100% excess bandwidth. The minimum value of M is ________.

Consider a discrete-time channel Y = X+ Z, where the additive noise Z is signal-dependent.
In particular, given the transmitted symbol X{−a,+a} at any instant, the noise sample Z is chosen independently from a Gaussian distribution with mean $\beta X$ and unit variance. Assume a threshold detector with zero threshold at the receiver.

When $\beta$=0, the BER was found to be Q(a)=1 × 10−8.

When $\beta$=−0.3, the BER is closest to

The bit rate of a digital communication system is R kbits/s. The modulation used is 32-QAM. The minimum bandwidth required for ISI free transmission is

Consider two identically distributed zero-mean random variables U and V . Let the cumulative distribution functions of U and 2V be F(x) and G(x) respectively. Then, for all values of x

The power spectral density of a real process X(t) for positive frequencies is shown below. The values of E[X2(t)] and |E[X(t)]| , respectively, are

In a baseband communications link, frequencies upto 3500 Hz are used for signaling. Using a raised cosine pulse with 75% excess bandwidth and for no inter-symbol interference, the maximum possible signaling rate in symbols per second is

A source alphabet consists of N symbols with the probability of the first two symbols being the same. A source encoder increases the probability of the first symbol by a small amount $\epsilon$ and decreases that of the second by $\epsilon$. After encoding, the entropy of the source

A BPSK scheme operating over an AWGN channel with noise power spectral density of N0/2, uses equiprobable signals ${s}_{1}\left(t\right)=\sqrt{\frac{2E}{T}}\mathrm{sin}\left({\omega }_{c}t\right)$ and ${s}_{2}\left(t\right)=-\sqrt{\frac{2E}{T}}\mathrm{sin}\left({\omega }_{c}t\right)$ over the symbol interval (0, T). If the local oscillator in a coherent receiver is ahead in phase by 45$°$ with respect to the received signal, the probability of error in the resulting system is

A binary symmetric channel (BSC) has a transition probability of 1/8. If the binary transmit symbol X is such that P(X=0) = 9/10, then the probability of error for an optimum receiver will be

The signal m(t) as shown is applied both to a phase modulator (with kp as the phase constant) and a frequency modulator (with kf as the frequency constant) having the same carrier frequency.

The ratio kp/kf (in rad/Hz) for the same maximum phase deviation is

The Column-1 lists the attributes and the Column-2 lists the modulation systems. Match the attribute to the modulation system that best meets it

 Column-1 Column-2 P.  Power efficient transmission of signals I.   Conventional AM Q. Most bandwidth efficient transmission of voice signals II.   FM R. Simplest receiver structure III.  VSB S.  Bandwidth efficient transmission of signals with Significant dc component IV.  SSB-SC

An analog signal is band-limited to 4 kHz, sampled at the Nyquist rate and the samples are quantized into 4 levels. The quantized levels are assumed to be independent and equally probable. If we transmit two quantized samples per second, the information rate is

A message signal $m\left(t\right)=\mathrm{cos}2000\pi t+4\mathrm{cos}4000\pi t$ modulates the carrier where ${f}_{c}=1$ MHz to produce an AM signal. For demodulating the generated AM signal using an envelope detector, the time constant RC of the detector circuit should satisfy

X(t) is a stationary random process with autocorrelation function ${R}_{X}\left(\tau \right)=\mathrm{exp}\left(-\pi {f}^{2}\right)$.This process is passed through the system shown below. The power spectral density of the output process Y(t) is

A four-phase and an eight-phase signal constellation are shown in the figure below.

For the constraint that the minimum distance between pairs of signal points be d for both constellations, the radii r1, and r2 of the circles are

A four-phase and an eight-phase signal constellation are shown in the figure below.

Assuming high SNR and that all signals are equally probable, the additional average transmitted signal energy required by the 8-PSK signal to achieve the same error probability as the 4-PSK signal is

Suppose that the modulating signal is m(t) = 2cos (2$\pi$ fmt) and the carrier signal is xC(t) = AC cos(2$\pi$fct), which one of the following is a conventional AM signal without over-modulation?

Consider an angle modulated signal x(t) = 6cos[2$\pi$x106t+2sin(8000$\pi$t) + 4cos(8000pt)] V. The average power of x(t) is.

Consider the pulse shape s(t) as shown. The impulse response h(t) of the filter matched to this pulse is

The Nyquist sampling rate for the signal $s\left(t\right)=\frac{\mathrm{sin}\left(500\mathrm{\pi t}\right)}{\mathrm{\pi t}}×\frac{\mathrm{sin}\left(700\right)\mathrm{\pi t}}{\mathrm{\pi t}}$ is given by

X(t) is a stationary process with the power spectral density Sx(f)>0 for all f. The process is passed through a system shown below.

Let Sy(f) be the power spectral density of Y(t). Which one of the following statements is correct?

Consider a baseband binary PAM receiver shown below. The additive channel noise n(t) is whit with power spectral density Sk(f)=N0/2=10-20 W/Hz. The low-pass is ideal with unity gain and cutoff frequency  1MHZ Let Yk represent  the random variable y(tk)

Yk=Nk   if transmitted bit bk=0

Yk=a+N if transmitted bit bk=1

Where Nk represents the noise sample value. The noise sample has a probability density function PNk(n)=0.5αe-α|n| (This has mean zero and variance 2/α2) Assume transmitted bits to be equiprobable and thresold Z is set to a/2=10-6v

The value of the parameter α(in  V-1) is