The corresponding frequencies synthesized are:
The amplitude of a sinusoidal carrier is modulated by a single $s\left(t\right)=5\mathrm{cos}1600\mathrm{\pi t}+20\mathrm{cos}1800\mathrm{\pi t}+5\mathrm{cos}2000\mathrm{\pi t}$ sinusoid to obtain the amplitude modulated signal. The value of the modulation index 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 (SNR_{max}), which of the following holds?
$ 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 ________
A binary baseband digital communication system employs the signal
$p\left(t\right)=\left\{\begin{array}{l}\frac1{\sqrt{{\mathrm T}_\mathrm s}},\;0\leq\mathrm t\leq{\mathrm T}_\mathrm s\;\;\;\\0,\;\;\;\;\;\;\;\;\;\;\;otherwise\;\end{array}\right.$
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 $s\left(t\right)=m\left(t\right)\mathrm{cos}\left(2\pi {f}_{\mathrm{c}}\mathrm{t}\right)+\hat{m}\left(t\right)\mathrm{sin}\left(2\mathrm{\pi}{f}_{\mathrm{c}}\mathrm{t}\right)$ where $\hat{m}\left(t\right)$ denotes the Hilbert transform of m(t) and band width of m(t) is very small compared to f_{c}. 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
${f}_{R|0}\left(r\right)=\left\{\begin{array}{ll}\frac{1}{4},& -3\le x\le 1,\\ 0& \mathrm{Otherwise}\end{array}\right.\mathrm{and}{f}_{R|1}\left(r\right)=\left\{\begin{array}{ll}\frac{1}{6},& -1\le x\le 5,\\ 0& \mathrm{Otherwise}\end{array}\right.$
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 X_{n} equally likely to be +1 or -1. ${\left\{{Y}_{n}\right\}}_{n=-\infty}^{n=\infty}$ is another random process obtained as Y_{n} = X_{n}+0.5X_{n-}_{1} . The autocorrelation function of ${\left\{{X}_{n}\right\}}_{n=-\infty}^{n=\infty}$ , denoted by R_{Y}[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[X^{2}] denoting the mean values of X and X^{2}, 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(t_{1})X(t_{2})] 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 N_{o}/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 N_{o}/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 ${s}_{1}\left(t\right)=\alpha \mathrm{cos}2{\mathrm{\pi f}}_{1}\mathrm{t}$ and ${s}_{2}\left(t\right)=\alpha \mathrm{cos}2{\mathrm{\pi f}}_{2}\mathrm{t}$, where $\mathrm{\alpha}$=4 mV. Assume an AWGN channel with two-sided noise power spectral density $\frac{{\mathrm{N}}_{\mathrm{o}}}{2}=0.5\times {10}^{-12}$ W/Hz. Using an optimal receiver and the relation $Q\left(v\right)=\frac{1}{\sqrt{2\mathrm{\pi}}}{\int}_{v}^{\infty}{e}^{\raisebox{1ex}{$-{u}^{2}$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}du$, 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 Y_{1} and Y_{2} as shown in the figure.
The value of H(Y_{1})+H(Y_{2}) 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}.
$\left(Q\left(v\right)=\frac{1}{\sqrt{2\mathrm{\pi}}}{\int}_{v}^{\infty}{e}^{-{u}^{2}/2}du,andforv1,useQ\left(v\right)\approx {e}^{-{v}^{2}/2}\right)$
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[X^{2}(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 N_{0}/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$\xb0$ 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 k_{p} as the phase constant) and a frequency modulator (with k_{f} as the frequency constant) having the same carrier frequency.
The ratio k_{p}/k_{f }(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
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 $c\left(t\right)=\mathrm{cos}2\pi t{f}_{c}t$ 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 r_{1}, and r_{2} of the circles are
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 $ f_{m}t) and the carrier signal is x_{C}(t) = A_{C} cos(2$\pi $f_{c}t), which one of the following is a conventional AM signal without over-modulation?
Consider an angle modulated signal x(t) = 6cos[2$\pi $x10^{6}t+2sin(8000$\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 $s\left(t\right)=\frac{\mathrm{sin}\left(500\mathrm{\pi t}\right)}{\mathrm{\pi t}}\times \frac{\mathrm{sin}\left(700\right)\mathrm{\pi t}}{\mathrm{\pi t}}$ 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 S_{y}(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 S_{k(}f)=N_{0}/2=10^{-20} W/Hz. The low-pass is ideal with unity gain and cutoff frequency 1MHZ Let Y_{k} represent the random variable y(t_{k})
Y_{k}=N_{k} if transmitted bit b_{k}=0
Y_{k}=a+N_{k } if transmitted bit b_{k}=1
Where Nk represents the noise sample value. The noise sample has a probability density function P_{Nk}(n)=0.5αe^{-α|n| }(This has mean zero and variance 2/α^{2}) Assume transmitted bits to be equiprobable and thresold Z is set to a/2=10^{-6}v
The value of the parameter α(in V^{-1}) is
The probability of bit error is