A source transmits symbol  that takes values uniformly at random from the set . The receiver obtains , where  is a zero-mean Gaussian random variable independent of . The receiver uses the maximum likelihood decoder to estimate the transmitted symbol .

Suppose the probability of symbol estimation error  is expressed as follows:

where  denotes the probability that  exceeds 1.

What is the value of  ?

A source transmits a symbol , taken from  with equal probability, over an additive white Gaussian noise channel. The received noisy symbol  is given by , where the noise  is zero mean with variance 4 and is independent of .

Using , the optimum symbol error probability is_______.

A full scale sinusoidal signal is applied to a 10-bit ADC. The fundamental signal component in the ADC output has a normalized power of 1W, and the total noise and distortion normalized power is . The effective number of bits (rounded off to the nearest integer) of the ADC is ________.

A digital communication system transmits through a noiseless bandlimited channel . The received signal  at the output of the receiving filter is given by  where  are the symbols and  is the overall system response to a single symbol. The received signal is sampled at . The Fourier transform of  is . The Nyquist condition that  must satisfy for zero inter symbol interference at the receiver is _______.

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  or symbol  is transmitted. Let the output of the channel  be the input to a maximum likelihood (ML) detector at the receiver. The conditional probability density functions for  given  and  are:

where  is the standard unit step function. The probability of symbol error for this system is __________ (rounded off to two decimal places).

A speech signal, band limited to , is sampled at 1.25 times the Nyquist rate. The speech samples, assumed to be statistically independent and uniformly distributed in the range  to , are subsequently quantized in an 8-bit uniform quantizer and then transmitted over a voice-grade AWGN telephone channel. If the ratio of transmitted signal power to channel noise power is , the minimum channel bandwidth required to ensure reliable transmission of the signal with arbitrarily small probability of transmission error (rounded off to two decimal places) is __________ .

A message signal having peak-to-peak value of , root mean square value of  and bandwidth of  is sampled and fed to a pulse code modulation (PCM) system that uses a uniform quantizer. The PCM output is transmitted over a channel that can support a maximum transmission rate of , Assuming that the quantization error is uniformly distributed, the maximum signal to quantization noise ratio that can be obtained by the PCM system (rounded off to two decimal places) is _________

Consider a polar non-return to zero (NRZ) waveform using  and  for representing binary ' 1 ' and ' 0 ' respectively, is transmitted in the presence of additive zero-mean white Gaussian noise with variance . If the a priori probability of transmission of a binary ' 1 ' is 0.4, the optimum threshold voltage for a maximum a posteriori (MAP) receiver (rounded off to two decimal places) is _______V.

A digital communication system transmits a block of  bits. The probability of error in decoding a bit is . The error event of each bit is independent of the error events of the other bits. The received block is declared erroneous if at least one of the its bits is decoded wrongly. The probability that the received block is erroneous is

 In a digital communication system, a symbol  randomly chosen from the set  is transmitted. It is given that  and . The received symbol is  is a zero-mean unit-variance Gaussian random variable and is independent of . , is the conditional probability of symbol error for the maximum likelihood (ML) decoding when the transmitted symbol . The index  for which the conditional symbol error probability  is the highest is _______.

A voice signal m(t) is in frequency range 5kHz to 15kHz. The signal is amplitude modulated to generate an AM signal, where . 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 quantizer 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 and , 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  as

Then the value of  (accurate to two decimal places) is ________

In 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 the inter symbol interference zero?

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

In binary frequency shift keying (FSK), the given signal waveforms are

where T is the bit-duration interval and t is in seconds. Both  and  are zero outside the interval .With a matched filter (correlator) based receiver, the smallest positive value of T (in milliseconds) required to have  and  uncorrelated 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 ________

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 ________.

An ideal band-pass channel 500 Hz - 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 with a raised cosine spectrum that utilizes the entire frequency band is ________.

An analog baseband signal, band limited to 100Hz, is sampled at the Nyquist rate. The samples are quantized into four message symbols that occur independently with probabilities   and  . The information rate (bits/sec) of the message source is ____________

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

A sinusoidal signal of 2 kHz frequency is applied to a delta modulator. The sampling rate and step-size Δ 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

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) ________

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 _______.

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 the following pulses will give the minimum probability of bit error?

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

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 ___________.

Let   be the BER of a BPSK system over an AWGN channel with two-sided noise power spectral density . The parameter γ 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  is shown in the figure. The BPSK demodulator receives the sum of outputs of both the channels.

If the BER of this system is , then the value of b is _____.

The input to a 1-bit quantizer is a random variable X with PDF  for  and  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  and , where

∝= 4 mV. Assume an AWGN channel with two-sided noise power spectral density . Using an optimal receiver and the relation, the bit error probability for a data rate of 500 kbps is

In a PCM system, the signal  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 _____.

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 , where the additive noise Z is signal-dependent. In particular, given the transmitted symbol  at any instant, the noise sample Z is chosen independently from a Gaussian distribution with mean βX and unit variance. Assume a threshold detector with zero threshold at the receiver.

When β = 0, the BER was found to be .         

, and for v>1, use  

When β = −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

 Let , and h(t) is a filter matched to g(t). If g(t) is applied as input to h(t), then the Fourier transform of the output 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

In a baseband communications link, frequencies up to 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 BPSK scheme operating over an AWGN channel with noise power spectral density of , uses equiprobable signals  and  over the symbol interval (0, 1). If the local oscillator in a coherent receiver is ahead in phase by 45° with respect to the received signal, the probability of error in the resulting system is

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 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  and  of the circles are

A four-phase and an eight-phase signal constellation are shown in the figure below.

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

Consider the pulse shape s(t) as shown. The impulse response h(t) of the filter matched to this pulse is

Consider a baseband binary PAM receiver shown below. The additive channel noise n(t) is white with power spectral density . The low-pass filter is ideal with unity gain and cut-off frequency 1MHz. Let  represent the random variable .

 if transmitted bit

 if transmitted bit

Where represents the noise sample value. The noise sample has a probability density function,  (This has mean zero and variance ). Assume transmitted bits to be equiprobable and threshold Z is set to .

The value of the parameter  is

The probability of bit error is

Four messages band limited to W, W, 2W and 3W respectively are to be multiplexed using Time Division Multiplexing (TDM). The minimum bandwidth required for transmission of this TDM signal is

A speech signal, band limited to 4 kHz and peak voltage varying between +5V and -5V, is sampled at the Nyquist rate. Each sample is quantized and represented by 8 bits.

If the bits 0 and 1 are transmitted using bipolar pulses, the minimum bandwidth required for distortion free transmission is

A speech signal, band limited to 4 kHz and peak voltage varying between +5V and -5V, is sampled at the Nyquist rate. Each sample is quantized and represented by 8 bits.

Assuming the signal to be uniformly distributed between its peak to peak value, the signal to noise ratio at the quantizer output is

A speech signal, band limited to 4 kHz and peak voltage varying between +5V and -5V, is sampled at the Nyquist rate. Each sample is quantized and represented by 8 bits.

The number of quantization levels required to reduced to reduce the quantization noise by a factor of 4 would be

In delta modulation, the slope overload distortion can be reduced by

The raised cosine pulse p(t) is used for zero ISI in digital communications. The expression for p(t) with unity roll-off factor is given by

The value of p(t) at  is

During transmission over a certain binary communication channel, bit errors occur independently with probability p. The probability of AT MOST one bit in error in a block of n bits is given by

In a GSM system, 8 channels can co-exist in 200 kHz bandwidth using TDMA. A GSM based cellular operator is allocated 5 MHz bandwidth. Assuming a frequency reuse factor of i.e. a five-cell repeat pattern, the maximum number of simultaneous channels that can exist in one cell is

In a Direct Sequence CDMA system the chip rate is  chips per second. If the processing gain is desired to be AT LEAST 100, the data rate

The ratio of the average energy of Constellation 1 to the average energy of Constellation 2 is

If these constellations are used for digital communications over an AWGN channel, then which of the following statements is true?

An input to a 6-level quantizer has the probability density function f(x) as shown in the

figure. Decision boundaries of the quantizer are chosen so as to maximize the entropy of the quantizer output, It is given that 3 consecutive decision boundaries are ‘-1’, ‘0’ and ’1’.

The values of a and b are

An input to a 6-level quantizer has the probability density function f(x) as shown in the

figure. Decision boundaries of the quantizer are chosen so as to maximize the entropy of the quantizer output, It is given that 3 consecutive decision boundaries are ‘-1’, ‘0’ and ’1’.

Assuming that the reconstruction levels of the quantizer are the mid-points of the decision boundaries, the ratio of signal power to quantization noise power is

The minimum sampling frequency (in samples/sec) required to reconstruct the following signal from its samples without distortion.

would be:

The minimum step-size required for a Delta-Modulator operating at 32 K samples/sec to track the signal (here u (t) is the unit-step function)

So that slope-overhead is avoided, would be

 In the following figure the minimum value of the constant “C”, which is to be added to such that and are different, is

A signal as shown in figure is applied to a matched filter. Which of the following does represent the output of this matched filter?

A symmetric three-level midtread quantizer is to be designed assuming equiprobable occurrence of all quantization levels.

f the input probability density function is divided into three regions as shown in figure, the value of a in the figure is

A symmetric three-level midtread quantizer is to be designed assuming equiprobable occurrence of all quantization levels.

The quantization noise power for the quantization region between –a and +a in the figure is

In a PCM system, if the code word length is increased from 6 to 8 bits, the signal to quantization noise ratio improves by the factor

In the output of a DM speech encoder, the consecutive pulses are of opposite polarity:  during time interval . This indicates that during this interval

Consider the signal x (t) shown in Fig. Let h (t) denote the impulse response of the filter matched to x (t), with h (t) being non-zero only in the interval 0 to 4 sec. The slop of h (t) in the interval is

Match the following

Three analog signals, having bandwidths 1200 Hz, 600 Hz and 600 Hz, are sampled at their respective Nyquist rates, encoded with 12 bit words, and time division multiplexed. The bit rate for the multiplexed signal is

A source produces binary data at the rate of 10 kbps. The binary symbols are represented as shown in Figure.

The source output is transmitted using two modulation schemes, namely Binary PSK (BPSK) and Quadrature PSK (BPSK) and Quadrature PSK (QPSK). Let  and  be the bandwidth requirements of BPSK and QPSK respectively. Assuming that the bandwidth of the above rectangular pulses is 10 kHz,  and  are

At a given probability of error, binary coherent FSK is inferior to binary coherent PSK by

Let . x(t) is sampled with the rectangular pulse train shown in figure. The only spectral components (in kHz) present in the sampled signal in the frequency range 2.5 KHz to 3.5 KHz are

A sinusoidal signal with peak-to-peak amplitude of 1.536 V is quantized into 128 levels using a mid-rise uniform quantizer. The quantization noise power is

If , the energy per bit of a binary digital signal, is  watt-sec and the one-sided power spectral density of the white noise, W/Hz, then the output SNR of the matched filter is

The input to a linear delta modulator having a step-size  is a sine wave with frequency  and peak amplitude . If the sampling frequency , the combination of the sine-wave frequency and the peak amplitude, where slope overload will take place is

If S represents the carrier synchronization at the receiver and ρ represents the bandwidth efficiency, then the correct statement for the coherent binary PSK is

A signal is sampled at 8 kHz and is quantized using 8-bit uniform quantizer. Assuming  for a sinusoidal signal, the correct statement for PCM signal with a bit rate of R is

For a bit-rate of 8 Kbps, the best possible values of the transmitted frequencies in a coherent binary FSK system are

A signal  is ideally sampled with a sampling period of 50μsec and then passed through an ideal low-pass filter with cut-off frequency of 15 KHz. Which of the following frequencies is/are present at the filter output?

A video transmission system transmits 625 picture frames per second. Each frame consists of a 400 × 400 pixel grid with 64 intensity levels per pixel. The data rate of the system is

The Nyquist sampling interval, for the signal  is

The frequency range for satellite communication is

In a digital communication system employing Frequency Shift Keying (FSK), the 0 and 1 bit are represented by sine waves of 10 KHz and 25 KHz respectively. These waveforms will be orthogonal for a bit interval of        

The Nyquist sampling frequency (in Hz) of a signal given by  is

The peak-to-peak input to an 8-bit PCM coder is 2 volts. The signal power-to-quantization noise power ratio (in dB) for an input of  is

The input to a matched filter is given by

The peak amplitude of the filter output is        

Four independent messages have bandwidths of 100 Hz, 200 Hz, 100 Hz and 400 Hz, respectively. Each is sampled at the Nyquist rate, and the samples are time division multiplexed (TDM) and transmitted. The transmitted sample rate (in Hz) is

Input to a linear delta modulator is a sinusoidal signal whose frequency can vary from 200 Hz to 4000 Hz. The input is sampled at eight times the Nyquist rate. The peak amplitude of the sinusoidal signal is 1 volt.

(a) Determine the minimum value of the step size in order to avoid slope overload when the input signal frequency is 800 Hz.

(b) What is the peak amplitude of the input signal, to just overload the modulator, when the input signal frequency is 200 Hz?

(c) Is the modulator overloaded when the input signal frequency is 4000 Hz?

In a PCM system with uniform quantization, increasing the number of bits from 8 to 9 will reduce the quantization noise power by factor of

Flat top sampling of low pass signals

Quadrature multiplexing is

Compression in PCM refers to relative compression of

For a given data rate, the bandwidth  of a BPSK signal and the bandwidth  of the OOK signal are related as

The pulse rate in a DM system is 56,000 per sec. The input signal is, with t in sec.

Find the minimum value of step size which will avoid slop overload distortion. What will be the disadvantages of choosing a value of larger than the minimum?

The line code that has zero dc components for pulse transmission of random binary data is

A deterministic signal has the power spectrum given in the figure is. The minimum sampling rate needed to completely represent this signal is        

A Communication channel has first order low pass transfer function. The channel is used to transmit pulses at a symbol rate greater than the half-power frequency of the low pass function. Which of the network shown in the figure is can be used to equalise the received pulses?

A rectangular pulse of duration T is applied to a filter matched to this input. The output of the filter is a