Consider the following amplitude modulated signal:
$ s\left(t\right)=\cos\left(2000\;\pi t\right)+4\;\cos\left(2400\;\pi t\right)+\cos\left(2800\pi t\right)\; $.
The ratio (accurate to three decimal places) of the power of the message signal to the power of the carrier signal is __________.
Consider a binary channel code in which each codeword has a fixed length of 5 bits. The Hamming distance between any pair of distinct codewords in this code is at least 2. The maximum number of codewords such a code can contain is _________.
A binary source generates symbols $ X\in\left\{-1,1\right\} $ which are transmitted over a noisy channel. The probability of transmitting $X$ = 1 is 0.5. Input to the threshold detector is $ R=X+N $. The probability density function $ f_N\left(n\right) $ of the noise $ N $ is shown below.
If the detection threshold is zero, then the probability of error (correct to two decimal places) is __________.
Let $ c\left(t\right)=A_c\cos\left(2\pi f_ct\right) $ and $ m\left(t\right)=\cos\left(2\pi f_mt\right) $. It is given that $ f_c\gg5f_m $ . The signal $ c\left(t\right)+m\left(t\right) $ is applied to the input of a non-linear device, whose output $ v_o\left(t\right) $ is related to the input $ v_i\left(t\right) $ as $ v_o\left(t\right)=av_i\left(t\right)+bv_i^2\left(t\right) $ , where $a$ and $b$ are positive constants. The output of the non-linear device is passed through an ideal band-pass filter with center frequency $f_c$ and bandwidth $ 3f_m $ , to produce an amplitude modulated (AM) wave. If it is desired to have the sideband power of the AM wave to be half of the carrier power, then $ a/b $ is
Consider a white Gaussian noise process $N(t)$ with two-sided power spectral density $ S_N\left(f\right)=0.5 $ W/Hz as input to a filter with impulse response $ 0.5e^{-t^2/2} $ (where $t$ is in seconds) resulting in output $Y(t)$. The power in $Y(t)$ in watts is
The input 4sinc$\left(2t\right)$ is fed to a Hilbert transformer to obtain $ y\left(t\right) $, as shown in the figure below:
Here sinc$ \left(x\right)=\frac{\sin\left(\pi x\right)}{\pi x} $ . The value (accurate to two decimal places) of $ \int_{-\infty}^\infty\left|y\left(t\right)\right|^2dt $ is _______.
A random variable $X$ takes values −0.5 and 0.5 with probabilities $\frac14$ and $\frac34$ , respectively. The noisy observation of $X$ is $Y$ = $X$ + $Z$, where $Z$ has uniform probability density over the interval (−1, 1). $X$ and $Z$ are independent. If the MAP rule based detector outputs $\widehat x$ as
$ \widehat X=\left\{\begin{array}{lc}-0.5,&Y<\alpha\\\;\;\;0.5,&Y\geq\alpha,\end{array}\right. $
then the value of $\alpha$ (accurate to two decimal places) is _______.
A band limited low-pass signal $ x\left(t\right) $ of bandwidth 5 kHz is sampled at a sampling rate $ f_s $ . The signal $ x\left(t\right) $ is reconstructed using the reconstruction filter $ H\left(f\right) $ whose magnitude response is shown below:
The minimum sampling rate $ f_s $ (in kHz) for perfect reconstruction of $ x\left(t\right) $ is _______.
Which one of the following statements about differential pulse code modulation (DPCM) is true?
In a digital communication system, the overall pulse shape p(t) at the receiver before the sampler has the Fourier transformation P(f) . If the symbols are transmitted at the rate of 2000 symbols per second, for which of the following cases is the inter symbol interface zero?
In binary frequency shift keying (FSK), the given signal waveforms are ${u}_{0}\left(t\right)=5\mathrm{cos}\left(20000\mathrm{\pi t}\right);0\le \mathrm{t}\le \mathrm{T}$ ,and ${u}_{1}\left(t\right)=5\mathrm{cos}\left(22000\mathrm{\pi t}\right);0\le \mathrm{t}\le \mathrm{T}$ , where T is the bit-duration interval and t is in seconds. Both u_{0} (t) and u_{1} (t) are zero outside the interval 0 ≤ t ≤ T. With a matched filter (correlator) based receiver, the smallest positive value of T (in milliseconds) required to have u_{0} (t) and u_{1} (t) uncorrelated is
Let X (t ) be a wide sense stationary random process with the power spectral density S_{x}(f ) as shown in Figure (a), where f is in Hertz(Hz). The random process X (t ) is input to an ideal lowpass filter with the frequency response $H\left(f\right)=\left\{\begin{array}{ll}1,& \left|f\right|\le \frac{1}{2}Hz\\ 0,& \left|f\right|>\frac{1}{2}Hz\end{array}\right.$ as shown in Figure(b). The output of the lowpass filter is Y (t ).
Let E be the expectation operator and consider the following statement:
I. E (X(t)) = E (X (t))
II. E (X^{2}(t)) = E (Y^{2 }(t))
III. E (Y^{2}(t)) = 2
Select the correct option:
Which one of the following graphs shows the Shannon capacity (channel capacity) in bits of a memoryless binary symmetric channel with crossover probability p?
Consider the random process $X\left(t\right)=U+Vt$ 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 statistically independent. The mean value of the random process at t = 2 is________
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 40 dB. The minimum number of bits per sample needed to achieve the desired SQNR is_________
Passengers try repeatedly to get a seat reservation in any train running between two stations until they are successful. If there is 40% chance of getting reservation in any attempt by a passenger, then the average number of attempts that passengers need to make to get a seat reserved is__________
The unmodulated carrier power in an AM transmitter is 5 kW. This carrier is modulate by a sinusoidal modulating signal. The maximum percentage of modulation is 50%. If it is reduced to 40%, then the maximum unmodulated carrier power (in kW) that can be used without overloading the transmitter is__________
A modulating signal given by $ x\left(t\right)=5\;\sin\left(4\pi10^3t-10\pi\;\cos\;2\pi\;10^3t\right) $V is fed to a phase modulator with phase deviation constant k_{p} = 5 rad/V. If the carrier frequency is 20 kHz, the instantaneous frequency (in kHz) at t = 0.5 ms is___________
Consider a binary memoryless channel characterized by the transition probablility diagram shown in the figure
The channel is
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