ece iv signals & systems [10ec44] assignment
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signal systemTRANSCRIPT
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Signals and Systems 10EC44
Dept of ECE/SJBIT Page 1
Assignment Questions
UNIT 1: INTRODUCTION
1. What are even and Odd signals 2. Find the even and odd components of the following signals
a. tcostsintsintcos)t(x
b. 432 9531)( ttttx
c. t10tcos)t1()t(x 33
3. What are periodic and A periodic signals. Explain for both continuous and discrete cases. 4. Determine whether the following signals are periodic. If they are periodic find the fundamental
period.
a. 2))t2((cos)t(x
b. )n2cos()n(x
c. n2cos)n(x
5. Define energy and power of a signal for both continuous and discrete case. 6. Which of the following are energy signals and power signals and find the power or energy of the
signal identified.
a.
otherwise0
2t1,t2
1t0,t
)t(x
b.
otherwise0
10n5,n10
5n0,n
)n(x
c.
0
5.0t5.0tcos5)t(x
d.
otherwise0
4n4,nsin)n(x
7. The raised cosine pulse is shown in fig.1 and is defined by
0
1w
1)t(cos2
1
)t(x
Determine the total energy of the signal x(t)
8. Explain with examples various operations that can be performed on dependent and independent
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variables of a continuous time function and discrete time function.
9. A sinusoidal signal x(t)=3 cos )6/t200( is passed through a square law device defined by the
input output relation y(t)=x2(t).
10. A rectangular pulse x(t) is defined by
otherwise0
Tt0A)t(x
This pulse is applied to an integrator whose output is defined by
1
0
d)(x)t(y
find the total energy of the output.
11. A Trapezoidal signal as shown in fig. 2 is time scaled producing output y(t)=x(t). Sketch y(t) for a=5, a=0.2.
12. Problem no.1.14 (a,e,f,g) page 6.3, signals and systems by Simon Haykin. 13. A continuous-time signal x ( t ) is shown in Fig. 1-17. Sketch and label each of the following
signals.
( a ) x(t - 2); ( b ) x(2t); ( c ) x(t/2); (dl x ( - t )
14. A discrete-time signal x [ n ] is shown in Fig. 1-19. Sketch and label each of the following
signals.
(a) x[n 2] (b) x[2n] ( c ) x [ - n ] ( d ) x [ - n + 2]
15. Determine the even and odd components of the following signals:
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16. Let x(t) be an arbitrary signal with even and odd parts denoted by xe(t) and xo(t), respectively. Show that
17. Let x[n] be an arbitrary sequence with even and odd parts denoted by xe[n] and xo[n] respectively.
Show that
18. Determine whether or not each of the following signals is periodic. If a signal is periodic, determine its fundamental period.
19. Show that if x[n] is periodic with period N, then
20. Evaluate the following integrals:
21. Consider a continuous-time system with the input-output relation
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Signals and Systems 10EC44
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Determine whether this system is ( a ) linear, ( b ) time-invariant, ( c ) causal.
22. Consider a continuous-time system with the input-output relation
Determine whether this system is ( a ) linear, ( b ) time-invariant.
23. Give an example of a linear time-varying system such that with a periodic input the corresponding output is not periodic.
24. A system is called invertible if we can determine its input signal x uniquely by observing its output signal y. This is illustrated in Fig. 1-43. Determine if each of the following systems is invertible. If
the system is invertible, give the inverse
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UNITS 2&3: TIME-DOMAIN REPRESENTATIONS FOR LTI SYSTEMS 1 & 2
1. Show that if x(n) is input of a linear time invariant system having impulse response h(n), then the output of the system due to x(n) is
k
)kn(h)k(x)n(y
2. Use the definition of convolution sum to prove the following properties 1. x(n) * [h(n)+g(n)]=x(n)*h(n)+x(n)*g(n) (Distributive Property) 2. x(n) * [h(n)*g(n)]=x(n)*h(n) *g(n) (Associative Property) 3. x(n) * h(n) =h(n) * x(n) (Commutative Property)
3. Prove that absolute summability of the impulse response is a necessary condition for stability of a discrete time system.
4. Compute the convolution y(t)= x(t)*h(t) of the following pair of signals:
5. Compute the convolution sum y[n] =x[n]* h[n] of the following pairs of sequences:
6. Show that if y (t) =x(t)* h(t), then
7. Let y[n] = x[n]* h[n]. Then show that
8. Show that
for an arbitrary starting point no.
9. The step response s ( t ) of a continuous-time LTI system is given by
Find the impulse response h(r) of the system.
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10. The system shown in Fig. 2-31 is formed by connection two systems in parallel. The impulse responses of the systems are given by
(a) Find the impulse response h(t) of the overall system.
(b) Is the overall system stable?
11. Consider an integrator whose input x(t) and output y ( t ) are related by Hhgk
(a) Find the impulse response h(t) of the integrator.
( b) Is the integrator stable?
12. Consider a discrete-time LTI system with impulse response h[n] given by
Is this system memory less?
13. The impulse response of a discrete-time LTI system is given by
Let y[n] be the output of the system with the input. Find y[1] and y[4].
14. Consider a discrete-time LTI system with impulse response h[n] given by
( a ) Is the system causal?
( b ) Is the system stable?
15. Let y(t)= x(t)*h(t). Then show that
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16. The input x ( t ) and the impulse response h ( t ) of a continuous time LTI system are given by
(a) Compute the output y (t) by Eq. (2.6).
(b) Compute the output y (t) by Eq. (2.10).
17. Compute the output y(t) for a continuous-time LTI system whose impulse response h (t) and the input x (t) are given by
18. Show that
19. Evaluate y (t) = x (t) * h(t), where x (t) and h (t) are shown in Fig. 2-6 (a) by analytical technique,
and (b) by a graphical method.
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20. Consider a continuous-time LTI system described by
(a) Find and sketch the impulse response h(t) of the system.
(b) Is this system causal?
21. Let y (t) be the output of a continuous-time LTI system with input x(t) . Find the output of the system if the input is x
l(t) , where x
l (t) is the first derivative of x(t) .
22. Verify the BIBO stability condition for continuous-time LTI systems. 23. Consider a stable continuous-time LTI system with impulse response h ( t ) that is real and even. Show
that cos wt and sin wt are eigenfunctions of this system with the same real eigenvalue.
24. The continuous-time system shown in Fig. 2-19 consists of two integrators and two scalar multipliers. Write a differential equation that relates the output y(t) and theinput x( t ).
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UNITS 4 & 5: FOURIER REPRESENTATION FOR SIGNALS 1 & 2
1. A continuous time periodic signal x(t) is real valued and has a fundamental period T=8, the non-
zero Fourier series coefficients for x(t) are j4aa,2aa 3311
Express x(t) in form
0kkkk )1w(cosA)t(x
2. For a continuous time signal (periodic)
)t3/5(sin4)t3/2cos(2)t(X determine the fundamental frequency 0w and Fourier
series coefficients ka such that
k
jkwk 0ea)t(x
3. Consider the following three continuous time periodic signals whose Fourier series representation are as follows:
100
0k
jkk
1 t50
2e.
2
1)1(x
100
100k
t50
2jk
2 e.)k(cos)t(x
100
100k
t50
2jk
3 e.2
ksinj)t(x
Using Fourier Series properties
i. Which of the three signals is / are real valued ii. Which of the three signals is / are even
4. State and prove the properties of continuous time Fourier Series 5. Obtain an expression to express a continuous time periodic signal in Fourier series representation
in (i) Trigonometric form (ii) Exponential form
6. State and prove parsevels relation for continuous time periodic signals 7. Obtain an expression for Fourier series representation of a discrete time periodic signals in
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complex exponential form.
8. What is Gibbs phenomenon? State and prove the properties of discrete time Fourier series. 9. Obtain Fourier transform of
i. 0afor)t(u.e)t(x )t(a
ii. 0afor,e)t(x ta
iii. |||)t()t(x
10. Consider a rectangular pulse
1
1
T|t|,0
T|t|,1)t(x
Obtain Fourier transform of x(t)
11. Obtain the analysis and synthesis equation of Fourier Transform of n periodic signal.
12. Obtain x(t) if x(jw)
w|w|for0
w|w|for1
13. Obtain an expression for Fourier transform of a periodic signal 14. State and prove properties of continuos time Fourier transform. 15. A Discrete time periodic signal x(n) is real value and has fundamental period N=5. The non-zero
discrete time Fourier series coefficients. For x(n) are 3/j
444/j
220 e2aaeaa,1A
Express x(n) in the form
1kkkk0 )nw(sinAA)n(x
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UNIT 6: APPLICATIONS OF FOURIER REPRESENTATIONS
1. The output of a system in response to an input )t(u.e)t(x t2 is )t(u.e)t(y t . Find the
frequency response and impulse response of this system.
2. Find the frequency response and impulse response of a system described by the differential equation
)t(x)t(xdt
d2)t(y2)t(y
dt
d3)t(y
dt
d
2
2
3. Find the frequency response and impulse response of a discrete time system described by the difference equation y(n-2)+5y(n-1)+6y(n)=8x(n-1)+18x(n)
4. Find the Fourier transform of impulse train given by
n
)nTt()t(p
T is fundamental period
5. Obtain an expression that relates discrete time Fourier transform and Discrete time Fourier series.
6. Find both DTFS & DTFT for the signal. )n2/(sin4)3/n8/3(cos2)n(x
7. Show that x(n) * h(n) )e(H)e(x jnjn where x(n) are the input and impulse response of a LTI
systems and )e(x jn and )e(H jn are their respective DFTF
8. If )n(u)2/j()n(h n is impulse response of a LTI system, then obtain the response of this
system to the input )3/n(cos3)n(x
9. Consider x(n)=cos )n16/7( )n16/9(cos using modulation property. Evaluate the effect of
computing the DFTF using only 2M+1 values of x(n) where |n| M
10. Consider x(t) =cos t . Assume that this signal is sampled at internals T=1/4, T=1 & T=1/2. Find the Fourier Transform of the sampled data for all the three cases.
11. Determine Fourier Transform pair associated with the signal whose DTFT is
jn
jn
e1
1)(x
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UNITS 7&8: Z-Transforms 1&2
1. State and prove the properties of unilateral Z transform & ROC 2. Find Z transform of
)n(u)2/1()1n(u)n(X n
3. Determine Z transform, ROC pole and zero location of x(t) for
i. )n(u)3/1()n(u)2/1()n(x nn
ii. )n(ue)n(xn0
j
4. Find Z transform of X(n)={n(-1/2)
n x(n)} * (1/4)
-n u(n)
X(n)=an cos (0n)
5. Find inverse Z transform of
1z21
2
1z2
11
1)z(x
Assume a. signal is casual b. Signal has DTFT
4Z2Z2
4Z4Z10Z)z(x
2
23
with ROC |z| 1
6. Find the inverse Z transform of
)ze9.01()ze9.01(
Z1)z(H
14/j14/j
1
7. A system is described by the difference equation
)2n(x8/1)1n(x4/1)n(x)2n(y4
1)1yn)n(Y
Find the Transfer function of the Inverse system
Does a stable and causal Inverse system exists
8. Sketch the magnitude response for the system having transfer functions.
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9. Find the z-transform of the following x[n]:
10. Given
(a) State all the possible regions of convergence.
(b) For which ROC is X (z) the z-transform of a causal sequence?
11. Show the following properties for the z-transform.
12. Derive the following transform pairs:
13. Find the z-transforms of the following x[n]:
14. Using the relation
find the z-transform of the following x[n]:
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15. Using the z-transform
16. Find the inverse z-transform of X(z)= ea/z , z > 0 17. Using the method of long division, find the inverse z-transform of the following X ( z ) :
18. Consider the system shown in Fig. 4-9. Find the system function H ( z ) and its impulse response
h[n]
19. Consider a discrete-time LTI system whose system function H(z) is given by
(a) Find the step response s[n].
(b) Find the output y[n] to the input x[n] = nu[n].
20. Consider a causal discrete-time system whose output y[n] and input x[n] are related by
(a) Find its system function H(z).
(b) Find its impulse response h[n].