Digital Communication Systems Practice Questions

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 _______ mega-bits per second (rounded off to one decimal place).

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. The conditional probability density functions for Y given $x_A$ and $x_B$ are: $f_{Y|x_A}(y)=e^{-(y+1)}u(y+1),$ $f_{Y|x_B}(y)=e^{(y-1)}(1-u(y-1)),$ where $u(\cdot )$ is the standard unit step function. The probability of symbol error for this system is _________ (rounded off to two decimal places).

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 message signal having peak-to-peak value of $2\:V$ , root mean square value of $0.1\:V$ and bandwidth of $\text{5 kHz}$ is sampled and fed to a pulse code modulation $(\text{PCM})$ system that uses a uniform quantizer. The $\text{PCM}$ output is transmitted over a channel that can support a maximum transmission rate of $\text{50 kbps}$ . Assuming that the quantization error is uniformly distributed, the maximum signal to quantization noise ratio that can be obtained by the $\text{PCM}$ system (rounded off to two decimal places) is ______

A single bit, equally likely to be 0 and 1, is to be sent across an additive white Gaussian noise (AWGN) channel with power spectral density $N_0/2$ . Binary signaling, with $0\rightarrow p(t)$ and $1\rightarrow q(t)$ , is used for the transmission, along with an optimal receiver that minimizes the bit-error probability. Let $\varphi _1(t),\varphi _2(t)$ form an orthonormal signal set. If we choose $p(t)=\varphi _1(t)\; and \; q(t)=-\varphi _2(t)$ , we would obtain a certain bit-error probability $P_b$ . If we keep $p(t)=\varphi _1(t)$ , but take $q(t)=\sqrt{E}\varphi _2(t)$ , for what value of E would we obtain the same bit-error probability $P_b$ ?

A random variable X takes values -1 and +1 with probabilities 0.2 and 0.8, respectively. It is transmitted across a channel which adds noise N, so that the random variable at the channel output is Y=X+N. The noise N is independent of X, and is uniformly distributed over the interval [-2,2]. The receiver makes a decision

where the threshold $\theta \in [-1,1]$ is chosen so as to minimize the probability of error $Pr[\hat{X}\neq X]$ . The minimum probability of error, rounded off to 1 decimal place, is ______.

A voice signal m(t) is in the frequency range 5 kHz to 15 kHz. The signal is amplitude-modulated to generate an AM signal $f(t)=A(1+m(t))cos 2\pi f_c t$ , where $f_c$ =600kHz. The AM signal f(t) is to be digitized and archived. This is done by first sampling f(t) at 1.2 times the Nyquist frequency, and then quantizing each sample using a 256-level quantizer. Finally, each quantized sample is binary coded using K bits, where K is the minimum number of bits required for the encoding. The rate, in Megabits per second (rounded off to 2 decimal places), of the resulting stream of coded bits is _______ Mbps.

A random variable X takes values -0.5 and 0.5 with probabilities 1/4 and 3/4, 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 $\hat{X}$ as

then the value of $\alpha$ (accurate to two decimal places) is _______.

A binary source generates symbols $X \in \{-1,1\}$ 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 $f_{N}(n)$ of the noise N is shown below. If the detection threshold is zero, then the probability of error (correct to two decimal places) 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 40dB. The minimum number of bits per sample needed to achieve the desired SQNR is _______

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 2000 symbols per second, for which of the following cases is inter symbol interference zero?

In binary frequency shift keying (FSK), the given signal waveform are $u_{0}(t)=5cos(20000\pi t);o\leq t\leq T$ , and $u_{1}(t)=5cos(22000\pi t);o\leq t\leq T$ , 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 $0\leq t \leq T$ . 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

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

A binary baseband digital communication system employs the signal

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

An ideal band-pass channel 500Hz - 2000Hz 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 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 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 _______

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 one bit will be in 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 ( $SNR_{max}$ ), which of the following holds?