# Questions & Answers of Signals and Systems

#### Topics of Signals and Systems 117 Question(s) | Weightage 12 (Marks)

Which one of the following is an eigen function of the class of all continuous-time, linear, timeinvariant systems ($u\left(t\right)$ denotes the unit-step function)?

A continuous-time function x(t) is periodic with period T. The function is sampled uniformly with a sampling period Ts. In which one of the following cases is the sampled signal periodic?

Consider the sequence $x\left[n\right]={a}^{n}u\left[n\right]+{b}^{n}u\left[n\right]$, where $u\left[n\right]$ denotes the unit-step sequence and $0<\left|a\right|<\left|b\right|<1$. The region of convergence (ROC) of the z-transform of $x\left[n\right]$ is

A continuous-time sinusoid of frequency 33 Hz is multiplied with a periodic Dirac impulse train of frequency 46 Hz. The resulting signal is passed through an ideal analog low-pass filter with a cutoff frequency of 23 Hz. The fundamental frequency (in Hz) of the output is _________

The Laplace transform of the causal periodic square wave of period T shown in the figure below is

A network consisting of a finite number of linear resistor (R), inductor (L), and capacitor (C) elements, connected all in series or all in parallel, is excited with a source of the form

$\sum_{k=1}^3a_k\;\cos\left(k\omega_0t\right),\;\mathrm{where}\;a_k\neq0\;,\omega_0\neq0.$

The source has nonzero impedance. Which one of the following is a possible form of the output measured across a resistor in the network?

A first-order low-pass filter of time constant T is excited with different input signals (with zero initial conditions up to t = 0). Match the excitation signals X, Y, Z with the corresponding time responses for t ≥ 0:
 X: Impulse $P:1-{e}^{-t/T}$ Y: Unit step $Q:t-T\left(1-{e}^{-t/T}\right)$ Z: Ramp ${\mathrm{R:e}}^{-t/T}$

Consider the signal

$x\left[n\right]=6\;\delta\left[n+2\right]+3\;\delta\left[n+1\right]+8\;\delta\left[n\right]+7\;\delta\left[n-1\right]+4\;\delta\left[n-2\right]$

If $X\left({e}^{j\omega }\right)$ is the discrete-time Fourier transform of x[n],

then $\frac1n\int_{-\mathrm\pi}^\mathrm\pi X\left(e^{j\omega}\right)\sin^2\left(2\omega\right)d\omega$ is equal to _________

The energy of the signal is ________

A continuous-time filter with transfer function $H\left(s\right)=\frac{2s+6}{{s}^{2}+6s+8}$ is converted to a discretetime filter with transfer function so that the impulse response of the continuous-time filter, sampled at 2 Hz, is identical at the sampling instants to the impulse response of the discrete time filter. The value of k is ________

The Discrete Fourier Transform (DFT) of the 4-point sequence

$X\left[k\right]=\left\{X\left[0\right],X\left[1\right],X\left[2\right],X\left[3\right]\right\}=\left\{12,2j,0-2j\right\}.$

If${X}_{1}\left[k\right]$ is the DTF of 12-point sequence ${x}_{1}\left[n\right]=\left\{3,0,0,2,0,0,3,0,0,4,0,0\right\}$

the value of $\left|\frac{{X}_{1}\left[8\right]}{{X}_{1}\left[11\right]}\right|$ is ________

Consider the signal $x\left(t\right)=\mathrm{cos}\left(6\mathrm{\pi t}\right)+\mathrm{sin}\left(8\mathrm{\pi t}\right)$, where t is in seconds. The Nyquist sampling rate (in samples/second) for the signal $y\left(t\right)=x\left(2t+5\right)$ is

If the signal $x\left(t\right)=\frac{\mathrm{sin}\left(t\right)}{\mathrm{\pi t}}*\frac{\mathrm{sin}\left(t\right)}{\mathrm{\pi t}}$ with * ; denoting the convolution operation, then x(t) is equal to

A discreate-time signal  $x\left[n\right]=\delta \left[n-3\right]+\delta \left[n-5\right]$ has z-transform X(z). If Y(z)=X(-z) is the z-transform of another signal y[n],then

A signal $2\cos\left(\frac{2\mathrm\pi}3t\right)-\;\cos\;\left(\mathrm{πt}\right)$ is the input to an LTI system with the transfer function

$H\left(s\right)=e^s+e^{-s}$

If Ck denotes the kth coefficient in the exponential Fourier series of the output signal, then C3 is equal to

The ROC (region of convergence) of the z-transform of a discrete-time signal is represented by the shaded region in the z-plane. If the signal $x\left[n\right]={\left(2.0\right)}^{\left|n\right|},-\infty , then the ROC of its z-transform is represented by

A continuous-time speech signal xa(t) is sampled at a rate of 8 kHz and the samples are subsequently grouped in blocks, each of size N. The DFT of each block is to be computed in real time using the radix-2 decimation-in-frequency FFT algorithm. If the processor performs all operations sequentially, and takes 20 μs for computing each complex multiplication (including multiplications by 1 and −1) and the time required for addition/subtraction is negligible, then the maximum value of N is __________

The direct form structure of an FIR (finite impulse response) filter is shown in the figure.

The filter can be used to approximate a

The result of the convolution x (–t) * $\delta$(–t – to) is

The waveform of a periodic signal x(t) is shown in the figure.

A signal g(t) is defined by $g\left(t\right)=x\left(\frac{t-1}{2}\right)$ . The average power of g(t) is ______.

Two sequences [a,b,c] and [A,B,C] are related as,

If another sequence [pqr] is derived as,

$\left[\begin{array}{c}p\\ q\\ r\end{array}\right]=\left[\begin{array}{ccc}1& 1& 1\\ 1& {W}_{3}^{1}& {W}_{3}^{2}\\ 1& {W}_{3}^{2}& {W}_{3}^{4}\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& {W}_{3}^{2}& 0\\ 0& 0& {W}_{3}^{4}\end{array}\right]\left[\begin{array}{c}A}{3}\\ B}{3}\\ C}{3}\end{array}\right],$

then the relationship between the sequences [p, q, r] and [a,b,c] is

For the discrete time system shown in the figure, the poles of the system transfer function are Located at

The pole-zero diagram of a causal and stable discrete-time system is shown in the figure. The zero at the origin has multiplicity 4. The impulse response of the system is h[n]. If h[0] = 1, we can conclude.

The bilateral Laplace transform of a function is

The magnitude and phase of the complex Fourier series coefficients ak of a periodic signal x(t) are shown in the figure. Choose the correct statement from the four choices given. Notation C is the set of complex numbers, R is the set of purely real numbers, and P is the set of purely imaginary numbers.

Let the signal f(t) = 0 outside the interval [T1, T2], where T1 and T2 are finite. Furthermore, $\left|f\left(t\right)\right|<\infty$ . The region of convergence (RoC) of the signal’s bilateral Laplace transform F(s) is

Two casual discrete-time signals x[n] and y[n] are related as $y\left[n\right]=\sum _{m=0}^{n}x\left[m\right]$ . If the z-transform of , the value of x[2] is _______.

The signal $\mathrm{cos}\left(10\mathrm{\pi t}+\frac{\mathrm{\pi }}{4}\right)$ is ideally sampled at a sampling frequency of 15 Hz. The sampled signal is passed through a filter with impulse response $\left(\frac{\mathrm{sin}\left(\mathrm{\pi t}\right)}{\mathrm{\pi t}}\right)\mathrm{cos}\left(40\mathrm{\pi t}-\frac{\mathrm{\pi }}{2}\right)$ . The filter output is

Consider the differential equation $\frac{dx}{dt}$ = 10-0.2x with initial condition x(0) = 1. The response x(t) for t>0 is

Input x(t) and output y(t) of an LTI system are related by the differential equation y''(t) - y-(t) - 6y(t) = x(t). If the system is neither causal nor stable, the impulse response h(t) of the system is

Consider two real sequences with time – origin marked by the bold value,

x1[n] ={1,2,3,0} , x2[n] ={1,3,2,1}

Let X1(k) and X2(k) be 4-point DFTs of x1[n] and x2[n] , respectively . Another sequence x3[n] is derived by taking 4-point inverse DFT of X3(k) =X1(k)X2(k) .

The value of x3[2] is_____.

Let x(t) = $\alpha$ s(t) +s(–t) with s(t) = $\beta$e-4t u(t) , where u(t) is unit step function . If the bilateral Laplace transform of x(t) is

Then the value of $\beta$ is______.

Consider the function $g\left(t\right)={e}^{-t}\mathrm{sin}\left(2\pi t\right)u\left(t\right)$ where u(t) is the unit step function. The area under g(t) is _____.

The value of $\sum\limits_{n=0}^\infty n\left(\frac12\right)^n$ is _____.

The impulse response of an LTI system can be obtained by

Consider a four-point moving average filter defined by the equation $y\left[n\right]=\sum _{i=0}^{3}{\alpha }_{i}x\left[n-i\right]$ . The condition on the filter coefficients that results in a null at zero frequency is

Suppose x[n] is an absolutely summable discrete-time signal. Its z-transform is a rational function with two poles and two zeroes. The poles are at $z=±2j$ . Which one of the following statements is TRUE for the signal x[n]?

A realization of a stable discrete time system is shown in the figure. If the system is excited by a unit step sequence input x[n], the response y[n] is

Let $\stackrel{~}{x}\left[n\right]=1+\mathrm{cos}\left(\frac{\mathrm{\pi n}}{8}\right)$ be a periodic signal with period 16. Its DFS coefficients are defined by $a_k=\frac1{16}\sum\limits_{n=0}^{15}\widetilde x\left[n\right]exp\left(-j\frac{\mathrm\pi}8kn\right)$ for all k. The value of the coefficients a31 is_____.

Consider a continuous-time signal defined as
$x\left(t\right)=\left(\frac{\sin\left(\pi t/2\right)}{\left(\pi t/2\right)}\right)\ast\sum_\limits{n=-\infty}^\infty\delta\left(t-10n\right)$
where '*' denotes the convolution operation and t is in seconds. The Nyquist sampling rate (in samples/sec) for x(t) is ___________.

Two sequences ${x}_{1}\left[n\right]$ and ${x}_{2}\left[n\right]$ have the same energy. Suppose , where $\alpha$ is a positive real number and $u\left[n\right]$ is the unit step sequence. Assume

Then the value of $\alpha$ is_______.

The complex envelope of the bandpass signal $x\left(t\right)=-\sqrt{2}\left(\frac{\mathrm{sin}\left(\pi t/5\right)}{\pi t/5}\right)\mathrm{sin}\left(\pi t-\frac{\pi }{4}\right)$,centered about $f=\frac{1}{2}\mathrm{Hz}$, is

C is a closed path in the z-plane given by |z|=3. The value of the integral $\oint_C\left(\frac{z^2-z+4j}{z+2j}\right)dz$

A discrete-time signal $x\left[n\right]=\mathrm{sin}\left({\mathrm{\pi }}^{2}n\right),n$ being an integer is

Consider two real valued signals, x(t) band-limited to [ –500 Hz, 500 Hz] and y(t) bandlimited to [ –1 kHz, 1 kHz]. For z(t) = x(t)•y(t), the Nyquist sampling frequency (in kHz) is ______.

A continuous, linear time-invariant filter has an impulse response h(t) described by

When a constant input of value 5 is applied to this filter, the steady state output is_____.

For a function g(t), it is given that ${\int }_{-\infty }^{+\infty }g\left(t\right){e}^{-jwt}dt=\omega {e}^{-2{\omega }^{2}}$ for any real value $\omega$.
If $y\left(t\right)={\int }_{-\infty }^{t}g\left(t\right)d\tau ,$, then is

Let  $x\left[n\right]={\left(-\frac{1}{9}\right)}^{n}u\left(n\right)-{\left(-\frac{1}{3}\right)}^{n}u\left(-n-1\right)$ The Region of Convergence (ROC) of the z-transform of x[n]

Consider a discrete time periodic signal $x\left[n\right]=sin\left(\frac{\mathit{\pi }\mathit{n}}{\mathit{5}}\right)$. Let ak be the complex Fourier series coefficients of $x\left[n\right]$. The coefficients $\left\{{a}_{k}\right\}$ are non-zero when $k=BM±1$ where M is any integer. The value of B is______.

A system is described by the following differential equation, where u(t) is the input to the system and y(t) is the output of the system

$\stackrel{.}{y}\left(t\right)+5y\left(t\right)=u\left(t\right)$

When y(0) =1 and u(t) is a unit step function, y(t) is

An FIR system is described by the system function

$H\left(z\right)=1+\frac{7}{2}{z}^{-1}+\frac{3}{2}{z}^{-2}$

The system is

Let x[n]=x[-n]. Let X(z) be the z-transform of x[n]. If 0.5+j 0.25 is a zero of ,X(z) which one of the following must also be a zero of X(z)

Consider the periodic square wave in the figure shown

The ratio of the power in the 7th harmonic to the power in the 5th harmonic for this waveform is closest in value to _______.

Consider a discrete-time signal

If y[n] is the convolution of x[n] with itself, the value of y[4] is _________.

The input-output relationship of a causal stable LTI system is given as

If the impulse response h[n] of this system satisfies the condition ${\sum }_{n=0}^{\infty }h\left[n\right]=2$, the relationship between α and β is

The value of the integral ${\int }_{-\infty }^{\infty }\mathrm{sin}{c}^{2}\left(5t\right)dt$ is ________.

Let $x\left(t\right)=cos\left(10\mathrm{\pi t}\right)+\mathrm{cos}\left(30\mathrm{\pi t}\right)$ be sampled at 20 Hz and reconstructed using an ideal low-pass filter with cut-off frequency of 20 Hz. The frequency/frequencies present in the reconstructed signal is/are

For an all-pass system $H\left(z\right)=\frac{\left({z}^{-1}-b\right)}{\left(1-a{z}^{-1}\right)}$, where $\left|H\left({e}^{-j\omega }\right)\right|=1$ for all $\omega$
if $Re\left(a\right)\ne 0,Im\left(a\right)\ne 0,$then b equals

The input $-3{e}^{2t}u\left(t\right)$, where $u\left(t\right)$ is the unit step function, is applied to a system with transfer function $\frac{s-2}{s+3}$. If the initial value of the output is −2, then the value of the output at steady state is _______.

Let ${H}_{1}\left(z\right)={\left(1-p{z}^{-1}\right)}^{-1},{H}_{2}\left(z\right)={\left(1-q{z}^{-1}\right)}^{-1},H\left(z\right)={H}_{1}\left(z\right)+r{H}_{2}\left(z\right)$. The quantities $p,q,r$ are real numbers. Consider $p=\frac{1}{2}$,$q=-\frac{1}{4}$,$\left|r\right|<1$. If the zero of $H\left(z\right)$ lies on the unit circle, then r = ________

Let $h\left(t\right)$ denote the impulse response of a causal system with transfer function $\frac{1}{s+1}$. Consider the following three statements.

S1: The system is stable.

S2: $\frac{h\left(t+1\right)}{h\left(t\right)}$ is independent of t for t >0.

S3: A non-causal system with the same transfer function is stable.

For the above system,

The z-transform of the sequence x[n] is given by $X\left(z\right)=\frac{1}{{\left(1-2{z}^{-1}\right)}^{2}}$  , with the region of convergence $\left|z\right|>2$. Then, $x\left[2\right]$ is ________.
Let $X\left(t\right)$ be a wide sense stationary (WSS) random process with power spectral density ${S}_{x}\left(f\right)$. If $Y\left(t\right)$ is the process defined as $Y\left(t\right)=X\left(2t-1\right)$, the power spectral density ${S}_{y}\left(f\right)$ is