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COIMBATORE-10
DEPARTMENT OF IT
UNIT – I
CLASSIFICATION OF SIGNALS AND SYSTEMS
PART – A
2. What is the total energy of the discrete time sinal x(n) which takes the
value of unity at n= -1,0,1?
3. Draw the waveform x(-t) and x(2-t) of the signal x(t) = t 0≤t≤3
0 t>3
Ans:
1
X(t) X(-t)
0 1 2 3 t -3 -2 -1 0 t
X(2-t)
-1 0 1 2 t
2 π f1 t = 3 π t
f1 = 3 / 2
T1 = 2 / 3
2 π f2 t = 9 t
f1 = 9 / 2 π
T1 = 2 π / 9 which is not a ratio of a integer.
Hence the given signal is not periodic.
The system squares input. The invertible system has to take square root
(i.e.,) π √ x2(t). but [-x(t)]2 = [x(t)]2 = x2(t). This means output of x2(t) is
btained for two inputs -x(t) as well as x(t). Hence this system is not invertible.
2
8. Check whether the system having input-output relation
y(t)= ∞ ∫ -∞ x(t) dt is linear time invariant or not.
Ans:
u(n)
0 1 2 3 4 5 6 7 n
u(n-3)
0 1 2 3 4 5 6 7 8 9 n
X(n)=u(n)-u(n-3)
0 1 2 n
Ans :Signal is a physical quantity that varies with respect to time , space or
any other independent variable. Or It is a mathematical representation of the
system Eg y(t) = t. and x(t)= sin t.
3
12. What are the classifications of the signal?
Ans:
1. Deterministic and Random Signal
2. Energy and Power Signal
3. Periodic and nonperidic Signal
4. Analog and Digital signal
j 4 πn / 7
13. Determine the fundamental period of x(n)=1 + e - e j 2 πn / 5
Ans:
1 = 4 π / 7
f 1 = 2 / 7 = k 1 / N1
2 = 2 π / 5
f 2 = 1 / 5 = k 2 / N2
N1 = 7 and N2 = 5. The fundamental period will be least common multiple
of N1 and N2 (i.e.,) 35.
j3π(n+(1/2)
14. Find the fundamental period of the signal x(n) = { 3 e } / 5.
Ans :
4
y1(n) + y2(n) = nx1(n) + nx2(n)
a1y1(n) + a2 y2(n) = a1 nx1(n) + a2 nx2(n)
= a1y1(n) + a2 y2(n)
Hence system is linear.
–j2πft
18. Check the following signal is power signal e
Ans:
lim T/2
P = T ∞ (1/T) ∫ │ e –j2πft │2 dt
-T/2
lim T/2
P = T ∞ (1/T) ∫ 1 dt
-T/2
=1 ( since │e-jө │= 1 )
Hence this is power signal.
n = 6 π n
2πfn=6πn
f=3
f = k / N = 3 / 1 ( rational ) . Hence this signal is periodic.
N=1
Put n = n-k
Y(n-k +2) = a x(n-k +1) + b x(n-k +3)
Hence the system is time invariant.
Y(t) depends upon x(t+10) (i.e.,) future input. Hence the system is
noncausal.
5
23. Determine whether the system is stable or not.
∞
Y(n) = ∑ x(k)
n = -∞
Ans :
= 1/8
f = / 2 π = { (1/8) x (1 / 2π ) } = 1 / 16π (rational)
Hence this is a periodic signal.
PART – B
2. (i) What is the periodicity of the signak x(t) = sin 100πt + cos 150 πt?
(3 Marks)
(ii) What are the basic continuous time signals? Draw any four Waveforms
and write their equations. (9 Marks)
3. (i) Determine the energy of the discrete time signal. X(n) = (1/2) n , n≥0
3n , n<0
(8 marks)
(ii) Verify the following system is linear. Y(n) = x(n) + n x(n+1) (5 marks)
4. (i) Test whether the system described by the equation y(n) = n x(n) is
linear (4 marks)
(ii) Verify the linearity , causality and time invariance of the system.
6
Y(n+2) = a x(n+1) + b x(n+3). (9 marks)
5. (i) Test whether the system described by the equation y(n) = n x(n) is Shift
invariant. (4 marks)
(ii) Determine whether or not each of the following signals is periodic. If the
signal is periodic, determine the fundamental period.
(i) x(t) = [ cos((2t – (π/3)) ]2
(ii) x(n) = ∑(n) – 4k - (n-1-4k) (8 Marks)
7. (i) Determine whether the system are linear, time invariant , causal and
stable.
a. y(n) = nx(n)
b. y(t) = x(t) + x(t-2) for t≥0 and 0 for t<0 (8 Marks)
(ii) Consider a continuous time signal x(t) = (t+2) - (t-2). Calculate the
t
value of Eα for the signals y(t) = ∫ x(t)dt (4 Marks)
-∞
n
8. (i) Express x[n] = (-1) -2≤n≤2 as a sum of scaled and shifted unit step
function. (8 Marks)
(ii) Given y[n] = x[n] + n x[n+1]. Determine whether the system is causal,
linear,time invariant and memoryless. (8 Marks)
9. Find even and odd parts of the signal x(t) given in fig.(a) (12 Marks)
Fig. (a)
X(t)
-2 -1 0 1 2 3 4 t
10. (i) Given y(t)= x(t+10) + x2(t). Determine whether the system is causal,
linear,time invariant and memoryless. (8 Marks)
2
(ii) Given y(t)= x(t ). Determine whether the system is causal,
linear,time invariant and memoryless. (8 Marks)
UNIT – I I
Analysis of continuous Time Signals
7
PART – A
∫ x(f) dt < ∞
<T >
0
4. What are the differences between fourier series and fourier
transform?
Ans:
S.No Fourier Series Fourier Transform
It is calculated for the periodic It is calculated for the non periodic
1.
signals. and periodic signals.
Expands the signal in time Represents the signal in frequency
2.
domain domain
X(t) cos(2 πfct + ø) < --- > ( ej ø/2 ) X(f-fc) + ( e-j ø/2 ) X(f+fc)
8
2. Convolution property :
FT
x1(t) * x2(t) < ---- > X1(f) . X2(f)
FT FT
x(t) < --- > X(f) (or) x(t) < --- > X()
∞
9
-∞
11. Find the laplace transform of x(t) = t e-at u(t) , where (a>0)
Ans:
LT
e-at u(t) < ---- > ( 1 / (s+a’) ) , ROC: Re(s) > (-a)
LT
-t e-at u(t) < --- > (d(1/(s+a)) / ds )
LT
t e-at u(t) < --- > 1/(s+a)2 ROC: Re(s) > (-a)
12. Find the Fourier transform of x(t) = t e-at u(t) , where (a>0)
Ans:
∞
X(f) = ∫ x(t) e-j2πf t dt
-∞
∞
= ∫ e-at e-j2πf t dt
0
∞
= ∫ e-(a+j2πf) t dt
0
= (1 / (a+j2πf) )
13. State the initial and final value theorem of laplace transform.
Ans:
initial value theorem
lt
f(0+) = s∞ [ s F(s) ]
final value theorem
lt lt
t∞ f(t) = s0 [ s F(s) ]
10
∞
= ∫ e-st dt [ since u(t) =0 for t<0 ]
0
∞
= (-1/s) [e-st] = 1/s
0
It states that the total average power of the periodic signal x(t) , is equal to the
sum of the average powers of its phasor components (i.e.,)
∞
P= ∑ │cn)│2
n = -∞
17. Find the laplace transform of the signal x(t) = -t e -2t u(t)
Ans:
n=2
LT
{ t / (n-1)! } e-at u(t) < ----- > 1 / (s+a) n’ , Re(s) > -a
2-1
a=2
LT
t e-at u(t) < ----- > 1 / (s+2) n’
LT
-t e-at u(t) < ----- > - 1 / (s+2) n’ , Re(s) > -2
18. let x(t)=t, 0≤t≤1 be a periodic signal with fundamental period T=1 and
fourier series coefficients ak , Find the value of a0
Ans:
t+T0
C0 = (1/T0) ∫ x(t) dt
t
1 1
= (1/1) ∫ t dt = [ t2 / 2 ] = 1/2
1 0
Fourier series
∞
11
k = -∞
Fourier Transform :
∞
j2 f t
= ∫e e - j2 fc t dt
-∞
∞
j2 (f- fc) t
= ∫e dt
-∞
= (f - fc)
∞
X(f) = ∫ x(t) e - j2 f t dt
-∞
∞
= ∫ (t) e - j2 f t dt
-∞
12
f(t)= e - j2 f t and t0=0
∞
X(f) = ∫ e - j2 f t (t-0) dt = e - 2 f . 0 = 1
-∞
Let x(t0 and X(f) be a fourier transform pair and ‘a’ is some constant.
Then by time scaling property
∞
= ∫1. e - j2 f t dt
-∞
∞
- j2 f t
= (1/- j2 f ) [e ]
0
=(1/- j2 f )
PART-B
1. (i)Find the fourier series for the periodic signal x(t)=t , 0≤t≤1 and repeats
every 1sec. (10 Marks)
(ii) Determine the fourier series representation for x(t)=2sin(2t-3)+sin(6t).
(6 Marks)
(ii) State and prove parsaval’s theorem for complex exponential fourier
series. (6 Marks)
3. Find the fourier transform of the signal x(t) and plot the amplitude
spectrum. (16 Marks)
13
4. Find the fourier series of the signal x(t) = ∫ sin(2f0 m)t . cos(2f0 n)t dt,
0
where
f0 is the fundamental frequency and m and n are any positive integer.
(16 Marks)
5. (i) Determine the trigonometric fourier series representation of the full wave
rectified output. (10 Marks)
6 (i) Find the laplace transform of x(t) = e-b│t│ for (b<0) and e-b│t│ for (b>0)
(10 Marks)
(ii) Determine the fourier transform of x(t) = 1 for (-1≤t≤1) and zero for other
value of t. (6 Marks)
.
7. (i) State and prove convolution theorem of laplace transform. (8 Marks)
(ii) Prove that the convolution of two signals is equivalent to multiplication of
their respective spectrum in frequency domain. (8 Marks)
8. (i) Find the Laplace transform of tx(t) and x(t-t0) where t0 is a constant term
and
x(t)< --- > X(s). (10 Marks)
(ii) Determine the laplace transform of x(t) = 2t for (0≤t≤1)
0 Otherwise
(6 Marks)
10. (i) Find the laplace transform of x(t) = e-at u(t) (8 Marks)
(ii) Determine the fourier series coefficients of (a) x(t) = sin 0t
(b) x(t) = sin 0t (8 Marks)
14
Unit-III
Linear Time Invariant – Continuous Time System
PART –A
2. What is the overall impulse response h(t) when two systems with
impulse response h1(t) and h2(t) are in parallel and in series?
Ans:
For parallel connection h(t)= h1(t) + h2(t)
For series connection h(t)= h1(t)*h2(t)
15
8. What is the relationship between input and output of an LTI system?
Ans:
9. What is the transfer function of a system whose poles are at -0.3 ± j0.4
and a zero at -0.2 ?
Ans:
= (s+0.2) / (s+0.3-j0.4)(s+0.3+j0.4)
10. Find the impulse response of the system given by H(s) = 1/(s+9)
Ans:
FT { h(t) } = FT { x (t)*h(t) }
16
13. Given y(t) = ∫ x(ι) h(t-ι)dι . If x(t) = u(t-1) and h(t) = u(t+3). The upper
limit in the integral changes to-----------and lower limit changes
to-----------.
Ans:
The convolution integral will be,
= ∫ 1x 1 dι
t+3
= ∫ dι
Thus the upper & lower limits will be t+3 and 1 respectively.
The impulse response is the output produced by the system when unit
impulse is applied at the input. The impulse response is denoted by h(t).
17. The impulse response of the LTI –CT system is given as h(t) =e-t
u(t). Determine transfer function and check whether the system is causal
and stable.
Ans:
H(s) = 1/(s+1)
17
Here pole at s=-1, ie located in left half of s—plane. Hence this system is
causal and stable.
18. Define eigen value and eigen function of LTI –CT systems.
Ans:
Y(t) = H(s) e st
Thus the output is equal to input multiplied by H(s). Hence e st is called eigen
function and H(s) is called eigen value.
H(s) = Y(s)/X(s)
= e-st0
Taking inverse laplace transform of the above equation
h(t) = δ (t-t0).
PART ---B
X(t) =1 0≤t≤2
0 otherwise
h(t) = 1 0≤t≤3
0 otherwise
2. a) How do you represent any arbitrary signal interms of delta function and
its delayed function (8 Marks)
b) Determine the response of the system with impulse response h(t) = u(t)
for the input x(t)= e-2t u(t) (8 Marks)
3. a) Find the output of an LTI system with impulse response h(t) =δ(t-3) for
the input x(t)= cos 4t + cos 7t. (8 Marks)
b) Using laplace transform find the impulse response of an LTI system
described by the differential equation d2y(t)/dt2 – dy(t)/dt -2y(t) =x(t) (8 Marks)
18
b)Find the response of the system x(t) = δ(t)-δ(t-1.5).Here h(t) is the
impulse response of the system. (8 Marks)
19
UNIT - IV
ANALYSIS OF DISCRETE TIME SIGNALS
PART –A
X(0) = Lt z∞X(z)
5. What is the difference between the spectrum of the CT signal and the
spectrum of the corresponding sampled signal.
Ans:
X(f) = fs ∑ X (f-nf s)
n=-∞
20
x(∞) = Lt z1 (1-z-1)X(z)
= 1+2z-1+3z-2+4z-3
X(0) = Lt z∞X(z)
21
j Ωn
Synthesis equation : x(n) = 1/2∏ ∫X(Ω) e dΩ.
FT { δ (n)}=1
PART --B
22
2. a) How will you evaluate fourier transform from pole zero plot of z-
transform. (6 Marks)
b) Find the inverse z- transform of X(z) = 1/(1-1.5z-1+0.5z-2) for ROC :
0.5<|z| <1 (10 Marks)
3. a) Write down any 4 properties of z- transform and explain (6 Marks)
b) Obtain the relation between z-transform and DTFT (10 Marks)
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UNIT – V
LINEAR TIME INVARIANT DISCRETE TIME SYSTEMS
PART – A
1. Is the output sequence of an LTI system finite when the input x(n0 is
finite? Justify the answer.
Ans:
If the impulse response of the system is infinite, then output sequence is
infinite even though inout is finite.
For ex: input , x(n) = (n) finite length
Impulse response h(n) = an u(n) Infinite length
Output Sequence y(n) = hn) * x(n)
= an u(n) * (n)
= an u(n)
24
Y(z) - { (1/2) z-1 y(z) } + { (1/4) z-2 y(z) } = x(z) – z-1x(z)
Y[n] = {2 -1 0 -5 4}
Y[n] = {1 4 7 6}
N M
Y(n) = - ∑ ak y(n-k) = ∑ bk y(n-k)
K=1 k=0
8. Write the difference equation and the transfer function of the system
in fig.
X(n)
Z-1 Z-1 Y(n)
B2
Z-1
A1
Z-1
A2
Ans:
25
X(t)
B2=0
Y(t)
-5
B1=1
∫
-4 B0=0
12. Draw direct form-II representation of H(z) = (1+z-1 + 3z-2 ) / (1+z-2 + z-3 )
Ans:
X(n) 1 y(n)
Z-1
-1 1
Z-1
-1 -3
13. Find the convolution sum for x(n) = {1,1,1,1} and h(n)={1,2,2,1}
Ans:
Y(n) = { 1 3 5 6 5 3 1}
26
14. Draw the radix-2 basic butterfly diagram.
Ans:
a A=a+ WNrb
WNr
b B=a- WNrb
15. Draw the black diagram for H(z) = (1+2z-1+4 z-4) / (1-z-1+4 z-2)
Ans:
X(n) 1 Y(n)
Z-1
2 Z-1
1
-1
Z
Z-1 Z-1
-1
Z-1
4
1jhdfDraw the block diagram for the system specified by the difference
equation y[n]+ay[-2]=b0x[n]+b1x[n-1]
Ans:
27
X(n) B0
Y(n)
z-1
B1
z-1
-a
17. For a state space representation of the system. Find the transfer
function of the system.
A= 0 1 B=0 C=1 2
-3 -2 1
Ans:
X(n) Y(n)
Q1(n+1)
Z-1
-a1 Q1(n) b1
Q2(n+1)
-1
Z
-a2
Q2(n) b2
19. What are the properties of convolution?
Ans:
1. Commutative property of convolution
2. Associative property of convolution
3. Distributive property of convolution
20. What are Impulse response and properties of LTI systems?
Ans:
1. Causality
2. Stability
28
PART – B
2. Find the impulse response of the stable system whose input-output relation
is given by the equation y(n) - 4 y(n-1) + 3 y(n-2) = x(n) + 2 x(n-1) (16 Marks)
4. Find the output sequence y(n) of the system described by the equation
Y(n) = 0.7 y(n-1) – 0.1 y(n-2) + 2 x(n) – x(n-2).
For the input sequence x(n) = u(n). (16 Marks)
5. (i) What is the impulse response x(n) of the system if the poles and zeros
are
radially moved k times their original location? (3 Marks)
(ii) Find the overall impulse response of the causal system in fig.
h1(n) = (1/3)n u(n) , h2(n) = (1/2)n u(n) and h3(n) = (1/5)n u(n) (12 Marks)
6. Realize direct form-I , direct form-II , cascade and parallel realization of the
discrete time system having system function
H(z) = 2(z+2) / {z(z-0.1) (z+0.5)0 (z+0.4)} (16
Marks)
8. (i) Find the impulse of the discrete time system described by the difference
equation. Y(n-2) – 3y(n-1) + 2 y(n) = x(n-1) (6 Marks)
(ii) Describe radix-2 DIT FFT algorithm (10 Marks)
9. (i) Explain the state variable description of discrete time system. (8 Marks)
(ii) Compute the linear convolution of x(n) = { 1,1,0,1,1} and
h(n) ={1,-2,-3,4} (8 Marks)
10. Given H(z) = (0.3 +z-1 – 0.47 z--2) / (1-0.5 z-1 + z-2 + 6 z-3) . Draw the block
diagram representation using DFI and DF II realization. (16 Marks)
29