Signal processing

1. Consider a six-point decimation-in-time Fast Fourier Transform (FFT) algorithm, for which the signal-flow graph corresponding to X[I] is shown in the figure. Let $${W_6} = \exp \left( { – \frac{{j2\pi }}{6}} \right).$$ In the figure, what should be the values of the coefficients a1, a2, a3 in terms of W6 so that X[I] is obtained correctly?

[amp_mcq option1=”$${a_1} = 1,{a_2} = W_6^2,{a_3} = {W_6}$$” option2=”$${a_1} = – 1,{a_2} = W_6^2,{a_3} = {W_6}$$” option3=”$${a_1} = – 1,{a_2} = {W_6},{a_3} = W_6^2$$” option4=”$${a_1} = 1,{a_2} = {W_6},{a_3} = W_6^2$$” correct=”option1″]

Detailed SolutionConsider a six-point decimation-in-time Fast Fourier Transform (FFT) algorithm, for which the signal-flow graph corresponding to X[I] is shown in the figure. Let $${W_6} = \exp \left( { – \frac{{j2\pi }}{6}} \right).$$ In the figure, what should be the values of the coefficients a1, a2, a3 in terms of W6 so that X[I] is obtained correctly?

3. Given that $$L\left[ {f\left( t \right)} \right] = {{s + 2} \over {{s^2} + 1}},L\left[ {g\left( t \right)} \right] = {{{s^2} + 1} \over {\left( {s + 3} \right)\left( {s + 2} \right)}},$$ $$h\left( t \right) = \int\limits_0^t {f\left( \tau \right)} g\left( {t – \tau } \right)d\tau $$ L[h(t)] is

[amp_mcq option1=”$${{{s^2} + 1} \over {s + 3}}$$” option2=”$${1 \over {s + 3}}$$” option3=”$${{{s^2} + 1} \over {\left( {s + 3} \right)\left( {s + 2} \right)}} + {{s + 2} \over {{s^2} + 1}}$$” option4=”None of the above” correct=”option1″]

Detailed SolutionGiven that $$L\left[ {f\left( t \right)} \right] = {{s + 2} \over {{s^2} + 1}},L\left[ {g\left( t \right)} \right] = {{{s^2} + 1} \over {\left( {s + 3} \right)\left( {s + 2} \right)}},$$ $$h\left( t \right) = \int\limits_0^t {f\left( \tau \right)} g\left( {t – \tau } \right)d\tau $$ L[h(t)] is

4. A 1.0 kHz signal is flat-top sampled at the rate of 1800 samples/sec and the samples are applied to an ideal rectangular LPF with cut-off frequency of 1100 Hz, then the output of the filter contains

[amp_mcq option1=”Only 800 Hz component” option2=”800 Hz and 900 Hz components” option3=”800 Hz and 1000 Hz components” option4=”800 Hz, 900 Hz and 1000 Hz components” correct=”option1″]

Detailed SolutionA 1.0 kHz signal is flat-top sampled at the rate of 1800 samples/sec and the samples are applied to an ideal rectangular LPF with cut-off frequency of 1100 Hz, then the output of the filter contains

5. The input x(t) and the output y(t) of a continuous- time system are related as $$y\left( t \right) = \int\limits_{t – T}^t {x\left( u \right)du} $$ The system is

[amp_mcq option1=”Linear and time-variant” option2=”Linear and time-invariant” option3=”Non-linear and time-variant” option4=”Non-linear and time-invariant” correct=”option1″]

Detailed SolutionThe input x(t) and the output y(t) of a continuous- time system are related as $$y\left( t \right) = \int\limits_{t – T}^t {x\left( u \right)du} $$ The system is

6. The input to a channel is a bandpass signal. It is obtained by linearly modulating a sinusoidal carrier with a single-tone signal. The output of the channel due to this input is given by $$y\left( t \right) = \left( {{1 \over {100}}} \right)\cos \left( {100t – {{10}^{ – 6}}} \right)\cos \left( {{{10}^6}t – 1.56} \right)$$ The group delay (tg) and the phase delay (tp) in seconds, of the channel are

[amp_mcq option1=”tg = 10-6, tp = 1.56″ option2=”tg = 1.56, tp = 10-6″ option3=”tg = 10-8, tp = 1.56 × 10-6″ option4=”tg = 108, tp = 1.56″ correct=”option4″]

Detailed SolutionThe input to a channel is a bandpass signal. It is obtained by linearly modulating a sinusoidal carrier with a single-tone signal. The output of the channel due to this input is given by $$y\left( t \right) = \left( {{1 \over {100}}} \right)\cos \left( {100t – {{10}^{ – 6}}} \right)\cos \left( {{{10}^6}t – 1.56} \right)$$ The group delay (tg) and the phase delay (tp) in seconds, of the channel are

8. The response of an initially relaxed linear constant parameter network to a unit impulse applied at t = 0 is 4e-2tu(t). The response of this network to a unit step function will be

[amp_mcq option1=”2[1 – e-2t]u(t)” option2=”4[e-t – e-2t]u(t)” option3=”sin2t” option4=”(1 – 4e-4t)u(t)” correct=”option1″]

Detailed SolutionThe response of an initially relaxed linear constant parameter network to a unit impulse applied at t = 0 is 4e-2tu(t). The response of this network to a unit step function will be

9. A network consisting of a finite number of linear resistor (R), inducer (L), and capacitor (C) elements, connected all in series or all in parallel, is excited with a source of the form $$\sum\limits_{k = 1}^3 {{a_x}\,\cos \left( {k{\omega _0}t} \right),{\rm{were}}\,{a_k} \ne 0,} \,{\omega _0} \ne 0.$$ The source has nonzero impedance. Which one of the following is a possible form of the output measured across a resistor in the network?

[amp_mcq option1=”$$\sum\limits_{k = 1}^3 {{b_x}\,\cos \left( {k{\omega _0}t + {\phi _k}} \right),{\rm{were}}\,{b_k} \ne {a_k},} \,\forall K$$” option2=”$$\sum\limits_{k = 1}^3 {{b_x}\,\cos \left( {k{\omega _0}t + {\phi _k}} \right),{\rm{were}}\,{b_k} \ne 0,} \,\forall K$$” option3=”$$\sum\limits_{k = 1}^3 {{a_x}\,\cos \left( {k{\omega _0}t + {\phi _k}} \right)} $$” option4=”$$\sum\limits_{k = 1}^2 {{a_x}\,\cos \left( {k{\omega _0}t + {\phi _k}} \right)} $$” correct=”option1″]

Detailed SolutionA network consisting of a finite number of linear resistor (R), inducer (L), and capacitor (C) elements, connected all in series or all in parallel, is excited with a source of the form $$\sum\limits_{k = 1}^3 {{a_x}\,\cos \left( {k{\omega _0}t} \right),{\rm{were}}\,{a_k} \ne 0,} \,{\omega _0} \ne 0.$$ The source has nonzero impedance. Which one of the following is a possible form of the output measured across a resistor in the network?

Page 1Page 2Page 3Page 4Page 5
Test 1Test 2Test 3