Signal and Systems | Subject Wise

Answer Ranges: –27.01 to –26.99

For the minimization of the energy in the error signal there are different approaches like, Prony’s method, Pade approximation. As g(n) has three samples.

Consider them as g(-1) , g(0) , g(1)
E(h,g) can be minimized by making h[n] = g[n] using rectangular window and parsval’s theorem of DTFT.
Now, 10g[–1] + g[1] = 10 x (–3) + 3 = –27

We are obtaining X(1) correctly

∴ k = 1

We know that

∴ comparing with given graph

a₁ = 1, a₂ = W₆,

For an input-output relation if the present output depends on present and past input values then the given system is “Causal”.

For the given relation,

For n ranging from 0 to 10 present output depends on present input only.
At all other points present output depends on present and past input values.
Thus the system is “Causal”.

Stability
If x[n] is bounded for the given finite range of n i.e. 0 Similarly x [n] - x [n- 1] is also bounded at all other values of n
Thus the system is “stable”.

We have to determine the characteristics of the system given by the equation in question.
First let us check for Linear/Non-Linear characteristics:
For input x₁(t) we have=> y₁(t)=x₁(2t)+3
For input x₂(t) we have=> y₂(t)=x₂(2t)+3
Now, the weighted sum of outputs can be given as: ay₁(t)+by₂(t)=a[x₁(2t)+3]+b[x₂(2t)+3]
The output due to weighted sum of inputs is:
y₃(t)=T[ax₁(2t)+bx₂(2t)]=[ax₁(t)+bx₂(t)]+3
So, we can see that=> y₃(t)ay₁(t)+by₂(t)
So, the system is non-linear
Next let us check for Time variant/invariant characteristics:
Delay the signal by T=> x(t-T)=X(2t-T)+3
y(t-T)=x(2(t-T))+3
So, from both the equations, we can interpret that: x(t-T)y(t-T)
So, the system is time variant.

The Correct Answer Among All the Options is B
According to data given, we can draw the poles in z-domain as follows,

For the system to be stable, ROC should include the unit circle. From the given pole pattern, it is clear that to make the system stable, the ROC should be two-sided and hence the impulse response of the system should be also two-sided.
For LTI system to be stable ROC of Z transform of unit impulse response must include unity circle: Required ROC for above condition: 0.25 < |z| <2
Since ROC is two sided so unit impulse response must also be two sided.

Change in only time period or frequency does not change in the value of Fourier series coefficients
So,

Method 2

Hilbert transform does not alter the amplitude spectrum of the signal and using CTFT to determine the amplitude,

If the input to a system is its eigen signal, the response has the same form as the eigen signal

For periodic signal
T/T_s -> rational number
Here T=1.2 T_s
T/T_s = 12/10
= 6/5 (Which is rational number)

13 Hz. When 33 Hz and 46 Hz are multiplied with each other, we get signals of frequency 13 Hz and 79 Hz so on. Since we are passing the output through low pass filter, it allows only low frequency signals till it’s cut off frequency(23 Hz). So only 13 Hz is passed.

I_rms = I_max because it is half wave rectifier.

x(-t) is an even function
i.e,.

Given
So,

Given,
We can directly find the DFT of given sequence

DFT repeats itself 2 times as 2 zeros are added after each sample

DFT repeats itself 2 times as 2 zeros are added after each sample

Thus, Common ROC does not exist for x[n]

The number of computations required for each multiplication = 
Each computation takes 20μs.

For any LTI system,

h(t) = impulse response
s(t) = step response
Hence, the correct option is (B).

Let us redraw the given system as

From the circuit, we have

Again,

……(2)
Multiplying equations (1) and (2),

For unit step response,


Hence,

Given

Let X_C be complex envelope of above signal. So, we have

Given characteristic equation,

Applying the Routh stability criterion,
s⁵     1       3         - 4
s⁴     2       6         - 8
s ³    0       0           0
s²
s¹
s⁰
It contains complete zero row, so we obtain the auxiliary equation as
Put x = s²,
2x²+ 6x − 8 = 0
x = 1, -4
So, s² = 1 or s = ±1
and
Hence, one root s =− 1 lies on the left side. Taking differential of auxiliary equation,
Now, the Routh array is redrawn as
s⁵     1           3       -4
s⁴     2           6       -8
s³     8           12      0
s²     3          -8       0
s¹    33.33       0
s^o     -8
Since, there is only one sign change in the first column of Routh array, so one pole lie in R.H.S and two poles lie on imaginary axis. Hence, the remaining two poles lies in L.H.S.

only changes the sign of the magnitude |a_k|. Since the magnitude spectrum |a_k| is even
the corresponding time-domain signal is real.

Given x(t ) is band limited to [−500Hz, 500Hz] and
y(t )is band limited to [−1000Hz, 1000Hz]
z(t ) = x (t ).y(t )
Multiplication in time domain results convolution in frequency domain.
The range of convolution in frequency domain is [−1500Hz, 1500Hz]
So maximum frequency present in z(t) is 1500Hz
Nyquist rate = 3000Hz or 3 kHz

The Correct Answer Among All the Options is B

Fourier series co-efficients are also periodic with period N =10


Refer the Topic Wise Question for DFT and Its Applications Signal and Systems

The Correct Answer Among All the Options is C
Minimum phase system has all zeros inside unit circle maximum phase system has all zeros outside unit circle mixed phase system has some zero outside unit circle and some zeros inside unit circle.

H(z)=(2z² + 7z + 3)/z²
      = (2z+1)(z+3)/z<sup>2

So zeroes are -1/2 and -3</sup>
One zero is inside and one zero outside unit circle hence mixed phase system
Refer the Topic Wise Question for DFT and Its Applications Signal and Systems

The Correct Answer Among All the Options is B
Given x[n] = x[−n]

Refer the Topic Wise Question for Z-Transform Signal and Systems

The Correct Answer Among All the Options is A

Refer the Topic Wise Question for DFT and Its Applications Signal and Systems

The Correct Answer Among All the Options is A
Given system equation as


Refer the Topic Wise Question for Responses and Stability Signals and Systems

The Correct Answer Among All the Options is B
m(t)M(f)

After multiplication with


After high pass filter

After multiplication with and low pass filter of cut off f_c

Bandwidth = A – B
= 100 – 40 = 60
Refer the Topic Wise Question for DFT and Its Applications Signal and Systems

The Correct Answer Among All the Options is B


Refer the Topic Wise Question for Z-Transform Signal and Systems

The Correct Answer Among All the Options is D

Refer the Topic Wise Question for DFT and Its Applications Signal and Systems

The Correct Answer Among All the Options is A



Refer the Topic Wise Question for Convolution and Its Properties Signal and Systems

The Correct Answer Among All the Options is D



Refer the Topic Wise Question for Responses and Stability Signals and Systems

The Correct Answer Among All the Options is B

It is given that system is stable thus its ROC includes axis. This implies it cannot be causal, because for causal system ROC is right side of the rightmost pole.
⇒ Poles at s = 2 must be removes so that it can be become causal and stable simultaneously.

Refer the Topic Wise Question for Responses and Stability Signals and Systems

The Correct Answer Among All the Options is B
Given differential equation


Refer the Topic Wise Question for Responses and Stability Signals and Systems

The Correct Answer Among All the Options is B
This can be solve by directly using option and satisfying the condition given in question


DFT of (x) will not result in [1 2 3 4]
Try with DFT of Y 1 2 3 2]


Same as x
Then ‘B’ is right option
Refer the Topic Wise Question for DFT and Its Applications Signal and Systems

The Correct Answer Among All the Options is C
If the two systems with impulse response and are connected in cascaded configuration as shown in figure, then the overall response of the system is the convolution of the individual impulse responses.



Hence correct option is C.
Refer the Topic Wise Question for Responses and Stability Signals and Systems

The Correct Answer Among All the Options is C
Given, the input

Its Laplace transform is

The impulse response of system is given

Its Laplace transform is

Hence, the overall response at the output is

Its inverse Laplace transform is

Hence correct option is C.
Refer the Topic Wise Question for Responses and Stability Signals and Systems

The Correct Answer Among All the Options is A
Given, the signal

So we have

Therefore, the respective time periods are


So, the fundament time period of the signal is

or,
Hence, the fundamental frequency in rad/sec is

Hence correct option is A.
Refer the Topic Wise Question for Responses and Stability Signals and Systems

The Correct Answer Among All the Options is A
Given, the maximum frequency of the band-limited signal

According to the Nyquist sampling theorem, the sampling frequency must be greater than the Nyquist frequency which is given as

So, the sampling frequency f_s must satisfy

Only the option A does’nt satisfy the condition therefore, 5 kHz is not a valid sampling frequency.
Hence correct option is A.
Refer the Topic Wise Question for Sampling Theorem and Applications Signal and Systems

The Correct Answer Among All the Options is D

The matched filter is characterized by a frequency response that is given as

where
Now, consider a filter matched to a known signal g(t). The fourier transform of the resulting matched filter output g_o(t) will be

T is duration of
Assume
So,
Since, the given Gaussian function is

Fourier transform of this signal will be

Therefore, output of the matched filter is

Refer the Topic Wise Question for Fouries Series and Its Application Signal and Systems

The Correct Answer Among All the Options is B
Given, the impulse response of continuous time system

From the convolution property, we know

So, for the input
(Unit step funⁿ)
The output of the system is obtained as

Hence correct option is B.
Refer the Topic Wise Question for Responses and Stability Signals and Systems

The Correct Answer Among All the Options is B
Given, the differential equation

Taking its Laplace transform with zero initial conditions, we have
…(1)
Now, the input signal is

i.e,
Taking its Laplace transform, we obtain

Substituting it in equation (1), we get

Hence correct option is B.
Refer the Topic Wise Question for Convolution and Its Properties Signal and Systems

The Correct Answer Among All the Options is A
Given, the DFT of vector as
D.F.T.
Also, we have
…(i)
For matrix circular convolution, we know

where are three point signals for and similarly for and are three point signals. Comparing this transformation to Eq(1), we get

Now, we know that

So,

Hence correct option is A.
Refer the Topic Wise Question for DFT and Its Applications Signal and Systems

The Correct Answer Among All the Options is D
If then

In this problem,
Given,

We need


Refer the Topic Wise Question for Convolution and Its Properties Signal and Systems

The Correct Answer Among All the Options is C





So overall ROC will be intersection of there ROCs i.e

Refer the Topic Wise Question for Z-Transform Signal and Systems

The Correct Answer Among All the Options is D


for input is


so system is not time invariant
for input x(τ) = cos (3τ) (bounded i/p)
as
So for bounded i/p, o/p is not bounded therefore system is not stable.
Refer the Topic Wise Question for Sampling Theorem and Applications Signal and Systems

The Correct Answer Among All the Options is C
….. (1)

……(2)
Where


So




So h(0) = 1
Refer the Topic Wise Question for Fouries Series and Its Application Signal and Systems

The Correct Answer Among All the Options is A






will be zero for k > 1 and g[k] will be zero for k = 0 as it is casual sequence.

g[1] = 0
Refer the Topic Wise Question for Convolution and Its Properties Signal and Systems

The Correct Answer Among All the Options is D
Parabolic function is defined as

∴ Laplace transform will be

∴ A parabolic function is degree faster than the ramp function.
Refer the Topic Wise Question for Basic of Signals and Systems Signal and Systems

The Correct Answer Among All the Options is A

∴ h(n) = aⁿu(n) + bⁿu(–n–1)

For the system to be stable
| a | < 1 & | b | > 1
∴ Option A
Refer the Topic Wise Question for Z-Transform Signal and Systems

The Correct Answer Among All the Options is A
Given x(n) =
∴ x(n) = 2(n+1) + 4(n) + 3 (n – 2)
∴ Option A
Refer the Topic Wise Question for Basic of Signals and Systems Signal and Systems

The Correct Answer Among All the Options is A
Frequency response for rectangular window is
Main lobe width for rectangular window is
∴ Option A
Refer the Topic Wise Question for DFT and Its Applications Signal and Systems

The Correct Answer Among All the Options is C
In N-point DFT,
no. of multiplication = →256
in N-point FFT,
no. of multiplication= →32
required difference =256-32=224
Refer the Topic Wise Question for DFT and Its Applications Signal and Systems

The Correct Answer Among All the Options is D
Formula for calculating no. of stages (S)=integer
Given
For LPF1 :
Transition band() = |passband frequency-stopband frequency|
= |4-1.8|
=2.2MHz


For LPF2:
input x(n) is fed after decimation by 50. So,
x() x()
therefore ,
now, Transition band()=|2-0.8| =0.2MHz
= 120
+= 545+120 => 665
Refer the Topic Wise Question for DFT and Its Applications Signal and Systems

The Correct Answer Among All the Options is A
For given LTI system,
f(n)=x(n) and
in frequency domain,
F(w)=X(w). and


Similarly for the other LTI system,


f(n)= I.F [X(w).]
g(n)=I.F[X(w).]
f(n) g(n) because ], so statement 2 is not true.
Refer the Topic Wise Question for Basic of Signals and Systems Signal and Systems

The Correct Answer Among All the Options is A
Bandpass sampling require less bandwidth to reconstruct the signal compare to low pass sampling. Bandpass sampling time is more i.e. speed requirement is less because it will store less sample compared to low pass sampling . it will decrease the memory requirement , because it store less sample when compared to low pass sampling
Refer the Topic Wise Question for Sampling Theorem and Applications Signal and Systems

The Correct Answer Among All the Options is D
, m is an even integer.

For minimum sampling frequency ,



Therefore , from the given option spectrum inversion occurs at 1600MHz.
Refer the Topic Wise Question for Sampling Theorem and Applications Signal and Systems

The Correct Answer Among All the Options is B

Given ,

………………..(1)
We have , X(m)=
X(0)=
=
As N=8, therefore , n=0,1,2,3,4,5,6,7
X(0)=x(0)+x(1)+……..x(7)
On putting different values of n in (1) we get values of x(0)….x(7). But as equation (1) is sum of sinusoidal function , sinusoidal has real values lies in[-1,1]. So all values of x(n) will also be real.
Refer the Topic Wise Question for Sampling Theorem and Applications Signal and Systems

The Correct Answer Among All the Options is B
X(w)=
As we know the properties of F.T
(-jt)x(t)

(-jt)
t
=
Refer the Topic Wise Question for Fouries Series and Its Application Signal and Systems

The Correct Answer Among All the Options is A
expression for DTFT is :
X[] =
x[n] =.dw

DTFT of , when |a|<1
Therefore,
X[] =
Using partial fraction,

On solving we get, A=2,B=-1

Now taking inverse DTFT of we get ,
. 2(0.2)ⁿu(n)-(0.1)ⁿu(n)
Refer the Topic Wise Question for Z-Transform Signal and Systems

The Correct Answer Among All the Options is D
x(t)=A sin (wt)

=
Given, Duty cycle =

Therefore,
=
Peak power =
=

10
Refer the Topic Wise Question for Basic of Signals and Systems Signal and Systems

The Correct Answer Among All the Options is C
y(t) = x(t) * y(t) .

y(t)=
= [T() - ] +0
= T
Refer the Topic Wise Question for Convolution and Its Properties Signal and Systems

The Correct Answer Among All the Options is C
In a convolution encoder, convolution of the input signal and impulse response is done. A convolution encoder is a time invariant system, and non-recursive codes are simply non-systematic codes. The given transfer functions clearly represents a non-recursive encoder(first), and z transform is what connects a transfer function with impulse response. Also, the constraint length of the given encoder is 3.
Refer the Topic Wise Question for Convolution and Its Properties Signal and Systems

The Correct Answer Among All the Options is B
We have to find the response of a matched filter at t=T.
Now, matched filter response is given as: h(T-t)
Let us first draw h(t)

Now, h(t+T)='+' left shift by 'T'


Next: h(-t+T)=> h(T-t)
So, now it can be drawn as:

Refer the Topic Wise Question for Responses and Stability Signals and Systems

The Correct Answer Among All the Options is B
Auto-correlation function is also called as serial correlation.
For auto-correlation function, following property exist:
R_xx(т)= F.T(S_xx(f))
And, R_xx(т)= dт and
S_xx(f)=
Refer the Topic Wise Question for Convolution and Its Properties Signal and Systems