GATE Electronics & Communication

Consider a frequency-modulated (FM) signal $$ f(t)=A_c \cos \left(2 \pi f_c t+3 \sin \left(2 \pi f_1 t\right)+4 \sin \left(6 \pi f_1 t\right)\right) $$ where $A_c$ and $f_c$ are, r...

The generator matrix of a $(6,3)$ binary linear block code is given by $$ G=\left[\begin{array}{llllll} 1 & 0 & 0 & 1 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 & 1 \\ 0 & 0 & 1 & 1 & 1 & 0 \end...

Consider a message signal $m(t)$ which is bandlimited to $[-W, W]$, where $W$ is in Hz . Consider the following two modulation schemes for the message signal: Double sideband-suppr...

The random variable $X$ takes values in $\{-1,0,1\}$ with probabilities $P(X=-1)=P(X=1)$ and $\alpha$ and $P(X=0)=1-2 \alpha$, where $0 Let $g(\alpha)$ denote the entropy of $X$ (i...

$X$ and $Y$ are Bernoulli random variables taking values in $\{0,1\}$. The joint probability mass function of the random variables is given by: $$ \begin{aligned} & P(X=0, Y=0)=0.0...

A source transmits symbol $S$ that takes values uniformly at random from the set $\{-2,0,2\}$. The receiver obtains $Y=S+N$, where $N$ is a zero-mean Gaussian random variable indep...

Consider a real-valued random process $$ f(t)=\sum\limits_{n=1}^N a_n p(t-n T), $$ where $T>0$ and $N$ is a positive integer. Here, $p(t)=1$ for $t \in[0,0.5 T]$ and 0 otherwise. T...

Consider an additive white Gaussian noise (AWGN) channel with bandwidth $W$ and noise power spectral density $\frac{N_o}{2}$. Let $P_{a v}$ denote the average transmit power constr...

A digital communication system transmits through a noiseless bandlimited channel $[-W, W]$. The received signal $z(t)$ at the output of the receiving filter is given by $z(t) = \su...

A white Gaussian noise $w(t)$ with zero mean and power spectral density $\frac{N_0}{2}$, when applied to a first-order RC low pass filter produces an output $n(t)$. At a particular...

An amplitude modulator has output (in Volts) $$s(t) = A \cos(400 \pi t) + B \cos(360 \pi t) + B \cos(440 \pi t)$$. The carrier power normalized to $1\Omega$ resistance is 50 Watts....

A source transmits symbols from an alphabet of size 16. The value of maximum achievable entropy (in bits) is _______ .

A source transmits a symbol $s$, taken from $\\{-4, 0, 4\\}$ with equal probability, over an additive white Gaussian noise channel. The received noisy symbol $r$ is given by $r = s...

The information bit sequence {1 1 1 0 1 0 1 0 1} is to be transmitted by encoding with Cyclic Redundancy Check 4 (CRC-4) code, for which the generator polynomial is $C(x) = x^4 + x...

Let $X(t) = A\cos(2\pi f_0 t+\theta)$ be a random process, where amplitude $A$ and phase $\theta$ are independent of each other, and are uniformly distributed in the intervals $[-2...

For a real signal, which of the following is/are valid power spectral density/densities?

Let a frequency modulated (FM) signal $$x(t) = A\cos ({\omega _c}t + {k_f}\int_{ - \infty }^t {m(\lambda )d\lambda )} $$, where $$m(t)$$ is a message signal of bandwidth W. It is p...

Let X(t) be a white Gaussian noise with power spectral density $$\frac{1}{2}$$W/Hz. If X(t) is input to an LTI system with impulse response $$e^{-t}u(t)$$. The average power of the...

The frequency of occurrence of 8 symbols (a-h) is shown in the table below. A symbol is chosen and it is determined by asking a series of "yes/no" questions which are assumed to be...

Let H(X) denote the entropy of a discrete random variable X taking K possible distinct real values. Which of the following statements is/are necessarily true?

A symbol stream contains alternate QPSK and 16-QAM symbols. If symbols from this stream are transmitted at the rate of 1 mega-symbols per second, the raw (uncoded) data rate is ___...

Consider communication over a memoryless binary symmetric channel using a (7, 4) Hamming code. Each transmitted bit is received correctly with probability (1 $$-$$ $$\in$$), and fl...

Consider a channel over which either symbol x A or symbol x B is transmitted. Let the output of the channel Y be the input to a maximum likelihood (ML) detector at the receiver. Th...

Consider a super heterodyne receiver tuned to 600 kHz . If the local oscillator feeds a 1000 kHz signal to the mixer. The image frequency (in integer) is $\_\_\_\_$ kHz .

A message signal having peak-to-peak value of 2 V , root mean square value of 0.1 V and bandwidth of 5 kHz is sampled and fed to a pulse code modulation (PCM) system that uses a un...

A 4 kHz sinusoidal message signal having amplitude 4 V is fed to a delta modulator (DM) operating at a sampling rate of 32 kHz . The minimum step size required to avoid slope overl...

Consider a carrier signal which is amplitude modulated by a single - tone sinusoidal message signal with a modulation index of $50 \%$. If the carrier and one of the side bands are...

Consider a polar non-return to Zero (NRZ) waveform, using +2 V and -2 V for representing binary ' 1 ' and ' 0 ' respectively, is transmitted in the presence of additive zero-mean w...

A sinusoidal message signal having root mean square value of 4 V and frequency of 1 kHz is fed to a phase modulator with phase deviation constant $2 \mathrm{rad} /$ volt. If the ca...

A binary random variable $X$ takes the value +2 or -2 . The probability $P(X=+2)=\alpha$. The value of $\alpha$ (rounded off to one decimal place), for which the entropy of $X$ is...

A digital communication system transmits a block of $N$ bits. The probability of error in decoding a bit is $\alpha$. The error event of each bit is independent of the error events...

The random variable $$ Y=\int_{-\infty}^{\infty} W(t) \phi(t) d t, \quad \text { where } \phi(t)=\left\{\begin{array}{cc} 1, & 5 \leq t \leq 7 \\ 0, & \text { otherwise } \end{arra...

In a digital communication system, a symbol $S$ randomly chosen from the set $\left\{s_1, s_2, s_3, s_4\right\}$ is transmitted. It is given that $s_1=-3, s_2=-1, s_3=+1$ and $s_4=...

The un-modulated carrier power in an AM transmitter is 5kW. This carrier is modulated by a sinusoidal modulating signal. The maximum percentage of modulation is 50%. If it is reduc...

A modulating signal given By $$x\left(t\right)=5\sin\left(4\mathrm\pi10^3\mathrm t\;-\;10\mathrm{πcos}2\mathrm\pi10^3\mathrm t\right)$$ V is fed to a phase modulator with phase dev...

A sinusoidal message signal is converted to a PCM signal using a uniform quantizer. The required signal to quantization noise ratio (SQNR) at the output of the quantizer is 40dB. T...

Which one of the following graphs shows the Shannon capacity (channel capacity) in bits of a memory less binary symmetric channel with crossover probability P?

In binary frequency shift keying (FSK), the given signal wave forms are $$\,{u_o}(t) = 5\,\cos \,(20000\,\pi \,t);\,0 \le \,\,t\, \le \,T,$$ and $${u_o}(t) = 5\,\cos \,(22000\,\pi...

Consider the random process x(t) = U + Vt. Where U is a zero mean Gaussian random variable and V is a random variable uniformly distributed between 0 and 2. Assume that U and V are...

In a digital communication system, the overall pulse shape p(t) at the receiver before the sampler has the Fourier transform P(f). If the symbols are transmitted at the rate of 200...

Let $$\left( {{X_1},\,{X_2}} \right)$$ be independent random variables, $${X_1}$$ has mean 0 and variance 1, while $${X_2}$$ has mean 1 and variance 4. The mutual information I $$\...

Which one of the following statements about differential pulse code modulation (DPCM) is true?

A binary baseband digital communication system employs the signal $$$p\left( t \right) = \left\{ {\matrix{ {{1 \over {\sqrt {{T_s}} }},} & {0 \le t \le {T_s}} \cr {0,} & {otherwise...

A super heterodyne receiver operates in the frequency range of 58 MHz – 68 MHz. The intermediate frequency f 1F and local oscillator frequency f L0 are chosen such that f 1F $$\leq...

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 loca...

A discrete memoryless source has an alphabet $$({a_1},\,{a_2},\,{a_3},\,{a_4})\,$$ with corresponding probabilities$$\left( {{1 \over 2}\,\,,{1 \over 4},\,{1 \over 8},\,\,{1 \over...

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 (...

A voice-grade AWGN (additive white Gaussian noise) telephone channel has a bandwidth of 4.0 kHz and two-sided noise power spectral density $${\eta \over 2} = 2.5\, \times \,{10^{ -...

The bit error probability of a memoryless binary symmetric channel is $${10^{ - 5}}$$. If $${10^{ - 5}}$$ bits are sent over this channel, then the probability that not more than o...

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 ba...

Consider 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 \r...

An antenna pointing in a certain direction has a noise temperature of 50K. The ambient temperature is 290K. The antenna is connected to a pre-amplifier that has a noise figure of 2...

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 wit...

An analog baseband signal, band limited to 100 Hz, is sampled at the Nyquist rate. The samples are quantized into four message symbols that occur independently with probabilities $...

An ideal band-pass channel 500Hz-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...

A binary communication system makes use of the symbols “zero” and “one”. There are channel errors. Consider the following events: $${x_0}$$ : a " zero " is transmitted $${x_1}$$ :...

Consider a discreet memoryless source with alphabet $$S = \left\{ {{s_0},\,{s_1},\,{s_2},\,{s_3},\,{s_{4......}}} \right\}$$ and respective probabilities of occurrence $$P = \left\...

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 st...

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 transm...

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

A source emits bit 0 with probability $${1 \over 3}$$ and bit 1 with probability $${2 \over 3}$$. The emitted bits are communicated to the receiver. The receiver decides for either...

Consider the signal $$s\left(t\right)\;=\;m\left(t\right)\;\cos\left(2{\mathrm{πf}}_\mathrm c\mathrm t\right)\;+\widehat m\left(t\right)\;\sin\left(2\mathrm\pi\;{\mathrm f}_\mathrm...

$$\mathop {\left\{ {{X_n}} \right\}}\nolimits_{n = - \infty }^{n = \infty } $$ is an independent and identically distributed (i.i.d) random process with $${X_n}$$ equally likely to...

Let $$X \in \left\{ {0,1} \right\}$$ and $$Y \in \left\{ {0,1} \right\}$$ be two independent binary random variables. If $$P\left( {X\,\, = 0} \right)\,\, = p$$ and $$P\left( {Y\,\...

Consider a binary, digital communication system which uses pulses g (t) and − g (t)for transmitting bits over an AWGN channel. If the receiver uses a matched filter, which one of t...

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 quantizatio...

A random binary wave $$y(t)$$ is given by $$$y\left( t \right) = \sum\limits_{n = - \infty }^\infty {{X_n}p\left( {t - nT - \phi } \right)} $$$ where $$p(t) = u(t) - u(t - T)$$, $$...

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 bandwid...

A sinusoidal signal of 2 kHz frequency is applied to a delta modulator. The sampling rate and step-size $$\Delta $$ of the delta modulator are 20,000 samples per second and 0.1 V,...

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 fun...

Consider an FM signal $$f\left(t\right)\;=\;\cos\left[2{\mathrm{πf}}_\mathrm c\mathrm t\;+\;{\mathrm\beta}_1\sin\;2{\mathrm{πf}}_1\mathrm t\;+\;{\mathrm\beta}_2\sin\;2{\mathrm{πf}}...

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 characteri...

Consider a random process $$X\left( t \right) = \sqrt 2 \sin \left( {2\pi t + \varphi } \right),$$ where the random phase $$\varphi $$ is uniformly distributed in the interval $$\l...

Let $$X(t)$$ be a wide sense stationary $$(WSS)$$ random procfess with power spectral density $${S_x}\left( f \right)$$. If $$Y(t)$$ is the process defined as $$Y(t) = X(2t - 1)$$,...

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 var...

The capacity of band-limited additive white Gaussian noise (AWGN) channel is given by $$C = \,W\,\,{\log _2}\left( {1 + {P \over {{\sigma ^2}\,W}}} \right)$$ bits per second (bps),...

The capacity of a Binary Symmetric Channel (BSC) with cross - over probability 0.5 is ______ .

Consider a discrete-time channel Y = X + Z, where the additive noise Z is signal- dependent. In particular, given the trasmitted symbol $$X\, \in \,\{ \, - \,a,\,\, + \,a\} $$ at a...

In a code - division multiple access (CDMA) system with N = 8 chips, the maximum number of users who can be assigned mutually orthogonal signature sequences is _______.

In a PCA system, the signal $$m(t) = \{ \sin (100\,\pi \,t)\, + \cos (100\,\pi \,t\} $$ V is sampled at the Nyquist rate. The samples are processed by a uniform quantizer with step...