GATE EC Communications questions

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 ___________.

In the circuit shown, choose the correct timing diagram of the output (y) from the given waveforms $W1$, $W2$, $W3$ and $W4$.

The input-output relationship of a causal stable LTI system is given as $$y[n] = \alpha \: y[n-1] + \beta \: x[n]$$. If the impulse response $h[n]$ of this system satisfies the condition $\sum_{n=0}^{\infty}h[n]= 2$, the relationship between $\alpha$ and $\beta$ is

Coherent orthogonal binary FSK modulation is used to transmit two equiprobable symbol waveforms $s_{1}( t )= \alpha \cos 2\pi f_1 t$, and $s_2(t) = \alpha \cos 2 \pi f_2 t$ where $\alpha = 4$ mV. Assume an AWGN channel with two-sided noise power spectral density $\frac{N_o}{2}= 0.5 \times 10^{-12}$ W/Hz. Using an optimal receiver and the relation …

The power spectral density of a real stationary random process $X(t)$ is given by $$S_X (f) = \begin{cases} \frac{1}{W}, & \mid f \mid \leq W \\ 0, & \mid f \mid > W \end{cases}$$ The value of the expectation $E \left [ \pi X(t)X\left ( t-\frac{1}{4W} \right ) \right ]$ is __________.

A modulated signal is $y(t)= m(t)\cos(40000\pi t)$, where the baseband signal $m(t)$ has frequency components less than $5$ KHz only. The minimum required rate (in KHz) at which $y(t)$ should be sampled to recover $m(t)$ is _______.

Consider an $FM$ signal $f(t)= \cos\left [ 2\pi f_{c} t+\beta _{1}\sin2\pi f_{1}t+\beta _{2}\sin2\pi f_{2}t\right ].$ The maximum deviation of the instantaneous frequency from the carrier frequency $f_{c}$ is

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

A real band-limited random process $X(t)$ has two-sided power spectral density $$S_{X}(f)= \begin{cases} 10^{-6} (3000-\mid f \mid) \text{Watts/Hz} & \text{for } \mid f \mid \leq 3 kHz \\ 0 & \text{otherwise} \end{cases}$$ where $f$ is the frequency expressed in Hz.The signal $X(t)$ modulates a carrier $\cos16000\pi t$ and the resultant …

In a $PCM$ system, the signal $m(t)=\left \{\sin(100\pi t)+\cos(100\pi t) \right \}V$ 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 ______.

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 __________.

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 __________

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

Let $U$ and $V$ be two independent and identically distributed random variables such that $P(U=+1)=P(U=-1) = \dfrac{1}{2}.$ The entropy $H(U + V)$ in bits is

Bits $1$ and $0$ are transmitted with equal probability. At the receiver, the pdf of the respective received signals for both bits are as shown below. If the detection threshold is $1,$ the $BER$ will be

Bits $1$ and $0$ are transmitted with equal probability. At the receiver, the pdf of the respective received signals for both bits are as shown below. The optimum threshold to achieve minimum bit error rate $(BER)$ is

The power spectral density of a real process $X(t)$ for positive frequencies is shown below. The values of $E[X^2(t)]$ and $\mid E[X(t)] \mid$, 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 the symbols per second is

A BPSK scheme operating over an AWGN channel with noise power spectral density of $\frac{N_o}{2}$, uses equiprobable signals $s_1(t)=\sqrt{\frac{2E}{T}}\sin(\omega_ct)$ and $s_2(t)=-\sqrt{\frac{2E}{T}}\sin(\omega_ct)$ over the symbol interval $(0,T)$. If the local oscillator in a coherent receiver is ahead in phase by $45^\circ$ with respect to the …

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 $\frac{k_p}{k_f}$ (in $rad/Hz$) for the same maximum phase deviation is