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ANAND INSTITUTE OF HIGHER TECHNOLOGYKAZHIPATTUR, CHENNAI –603 103
DEPARTMENT OF ECEDate : 17.05.2011
Academic year 2011-12(Odd semester)PART-A UNIVERSITY QUESTIONS WITH ANSWERS
Subject : Signals and systems Sub Code : 147303Staff Name : S. Nagarajan Class : III sem ECESection : A & B
PART-AUNIT-I CLASSIFICATION OF SIGNALS AND SYSTEMS
1.Waveform [Nov-2010] [Nov-2007](i)Prove that δ(n)=u(n)-u(n-1) [Nov-2010]Ans: (i) u(n) = 1 for n ≥ 0
= 0 for n <0u(n-1) = 1 for n ≥ 1 δ (n) = u(n) – u(n-3)= 1 ,n=0
= 0 for n < 1 = 0, n ≠0
0 1 2 3 4 5 6 7 n
(ii)Draw the waveforms δ(t-2) and u(t+2) [Nov-2007]Ans: u(t+2) = 1, t ≥ -2 δ(t-2) = 1 ,t=2
= 0 ,elsewhere = 0 , t ≠2u(t+2) δ(t-2)
1 1
-2 -1 0 1 2 3 4 t -1 0 1 2 3 t
2. Periodicity [Nov-2010,Nov-2009,May-2009, Apr-2008,Nov-2007,Nov-2006](i)Check the periodicity of cos(0.01n) [Nov-2010]Ans(i) Here x(n) = cos(0.01πn)
N=(2π/ ω)mω=0.01π
N=(2π/0.01π )m =200m Let m=1Therefore N=200.It is rational number .So it is periodic.
(ii)Show that the complex exponential signal x(t)= ej0
t is periodic and that thefundamental period is (2π/ 0 ) [Nov-2009,May-2009,Nov-2006]
1
δ (n) = u(n) – u(n-3)
Ans: x(t)= ejω0t
x(t+T)= ejω0
(t+T)
x(t+T)= ejω0
t ejω0
T
Since ejω0
T = ejω0
(2π/ ω0
)= ej2π =1
x(t +T)= ejω0
t =x(t)x(t +T)= x(t).So,it is periodic
(iii).Determine whether the signal x(n)=cos(0.1n) is periodic or not [Apr-2008]Ans: Here x(n) = cos(0.1πn)
N=(2π/ ω)mω=0.1π
N=(2π/0.1π )m =20m Let m=1Therefore N=20.It is rational number .So it is periodic.
(iv)What is the periodicity of x(t)=e j100πt+30? [Nov-2007]Ans:Time period T=(2π/ω0)
T=2π / 100 π =1/50=0.02.It is rational number. So it is periodic
(v) Find the fundamental period of the signal x(n) ={3ej3π[n+(1/2)] } / 5 [Nov-2006]Ans:Fundamental period N/m =(2π/ ω0) =(2π/ 3π) = (2/3)
N/m = (2/3) let m=3Therefore fundamental period N=2.
3. Systems [Apr-2010,Nov-2009,Nov-2007,May-2007,May-2006](i)When is a system said to be memory less? Give a example? [Apr-2010]
Ans: (i) A system is said to be static (or) memory less, if the output of thesystem at any time t depends only on a present value of the inputsignals. Ex : y(t)=x2(t)
(ii)What is the classification of system? [Nov-2009]Ans: (1) Linear and non-linear system (2) Time invariant and Time variant
system (3) Causal and Non-causal system (4) Static and Dynamicsystem (5) Stable and Unstable system
(iii) Determine whether the system described by the following input-outputrelationship is linear and causal y(t)=x(-t) [Nov-2007]Ans: LINEARITY:
H[α1x1(t)+ α2x2(t)] = α1y1(t)+ α2y2(t)L.H.S : H[α1x1(t)+ α2x2(t)] = α1 x1 (-t)+ α2x2(-t)
RHS: α1y1(t)+ α2y2(t) = α1 x1 (-t)+ α2x2(-t)LHS =RHS
So the system is linearCAUSALITY:H[x(t-τ)] = y(t-τ)L.H.S H[x(t-τ)] = x[-(t-τ)] =x[-t+τ]R.H.S y(t-τ) = x[-t-τ)]
L.H.S≠R.H.S.so this system is time variant
(iv)Check whether the system classified by y(t) =ex(t) is time invariant or not[May-2007]
Ans :H[x(t- τ)] = ex(t- τ)
y(t- τ) = ex(t- τ)
H[x(t- τ)] = y(t- τ)] .so it is time invariant
(v)State the condition for causal system(or)Define causality and stability of asystem with an examples for each [May-2006]Ans: Causality: A system said to be causal if the output of the system at any time ’t
depends only on the present and past values of the inputs are calledcausal . A system said to be non causal if the output of the system at anytime ’t ‘ depends only on future values of the inputs are called non-causal .For examples : y(t) =x(t)+x(t-1) is a causal system but y(t) =x(t+1) is not a
causal system.Stability: A system is stable ,if and only if every bounded input produces abounded Output. Let the input signal x(t) is bounded (finite) i.e|x(t)|<Mx<∞. Where Mx is a positive real number. If |y(t)|<My<∞. i.e. y(t) is
also bounded, then the system is BIBO stable. Otherwise ,it is unstable
4.Type of signal [Apr-2010,May-2009,Nov-2007,May-2007,May- 2006](i)Define step function and delta function(or) Define unit impulse and unit step
signals [Apr-2010,May-2009,Nov-2007,May-2007]Ans:CT unit step function , u(t) = 1 for t≥ 0
= 0 for t<0DT unit step function, u(n) = 1 for n ≥ 0
= 0 for n<0CT delta(Unit Impulse) function , δ(t) =1 for t=0
= 0 for t ≠0DT delta (Unit Impulse) function , δ(n) =1 for n=0
= 0 for n ≠0
(ii)Represent a ramp signal in continuous time and discrete time, mathematically[May- 2006]
Ans:CT ramp r(t) = t ,t ≥0 DT ramp r(n) = n ,n≥0= 0 ,t<0 = 0 ,n<0
5.Shifting waveform [Nov-2009](i)For the signal shown in Fig.find x(2t+3) [Nov-2009]
x(t)
2
1
-1 0 1 2 tAns:lower limit 2t+3= -1 Upper limit 2t+3= 2
2t=-4 2t=-1t=-2 t= -0.5
x(2t+3)
2
1
-2 -1.5 -1 -0.5 0 t
6.Even and Odd signal [Nov-2008](i)Find the even and odd components of the signal x(t)=ejt [Nov-2008]Ans: x(t)=ejt
x(-t)= e-jt
xe(t)=(1/2)[x(t)+x(-t)]=(1/2)[ ejt + e-jt ] =costx0(t)= (1/2)[x(t)-x(-t)]=(1/2)[ ejt - e-jt ] =jsintx(t)= xe(t)+x0(t)= cost+jsint
7. Define linear time invariant system [Nov-2007]Ans: A system is said to be linear as well as Time invariant called linear time
invariant system.
8. Energy and power signal [May-2007,Nov-2006](i) Determine the power and RMS value of the signal x(t) =ejat cos0t [May-2007]
Ans: + Tp = lim (1/2T) ∫ |x(t)|2 dt
T→∞ -T+ T + T
p= lim (1/2T) ∫ [ejat cosω0t]2 dt = lim (1/4T) ∫ (1+cos2ω0t) dt
T -T T -T+T
P = lim (1/4T) ∫ 1.dt +0 = (1/4T)[T+T] =(1/4T)(2T)T -T
Power P = (1/2) watt.R.M.S value = = [1 /√p ]= [1 /√2 ]=1 / 1.414=0.707
(ii)Verify whether x(t)=A e-2t u(t) , α > 0is an energy signal or not [Nov-2006]Ans: +T
E = lim ∫ |x(t)|2 dtT→∞ -T
+TE = A2 lim ∫ | e-2t u(t)|2 dt
T→∞ -T
∞E = A2 lim ∫ e-4t |u(t)|2 dt = A2 / 4
T→∞ 0+ T
p = lim (1/2T) ∫ |x(t)|2 dtT→∞ -T
+TP = A2 lim (1/2T) ∫ | e-2t u(t)|2 dt
T→∞ -TT
P = A2 lim (1/2T) ∫ e-4t dtT→∞ 0
TP = A2 lim (1/2T) [ e-4t /-4] = e-∞ =0
T→∞ 0P=0.
E= A2 / 4 and P=0.So it is Energy signal.
UNIT-II (ANALYSIS OF CT SIGNALS )
1.Fourier Transform [Nov-2010,May-2009,Nov-2007](i)Find the Fourier Transform of x(t)= ej2πf
ct [Nov-2010]
Ans: +X(jω) = ∫ x(t) e-j2πft dt
-∞+
X(jω) = ∫ ej2πfct e-j2πft dt
-∞+
X(jω) = ∫ e-j2π(f-fc)t dt=[e--e]
-∞X(jω) = -
(ii) Obtain the Fourier transform of x(t)=eat u(-t),a>0 ] [May-2009]Ans: ∞ 0 ∞
x(jω) = ∫ x(t) e-jωt dt = ∫ e-at e-jωt.dt = - ∫ e-( jω +a)t dt-∞ -∞ 0
∞=-[ e- (jω +a)t /-(jω+a)] =1/(jω+a)[e-∞-e0]=-1/( jω+a)[0-1]=-1/ (jω+a)
0
(iii) Define Fourier transform [Nov-2007]Ans: Let x(t) be a signal such that -∞<t<+∞ and x(t) is absolutely
integrable then the Fourier Transform of x(t) is defined as+
X(jω) = ∫ x(t) e-jωt dt-∞
2.Relation ship between Fourier transform and Laplace transform(i)What is relationship between Fourier transform and Laplace transform?
(or)Define Laplace transform .In what way it is different from Fourier Transform?[Nov-2010,Nov-2009,Nov-2008,Nov-2007,May-2007]
Ans: ∞(i)Laplace transform is denoted by x(s) = ∫ x(t) e-st dt
-∞+
Continuous Time Fourier transform : X(jω) = ∫ x(t) e-jωt dt-∞
Fourier transform and Laplace transform relationship is s=jω
3. Laplace Transform [Apr-2010, Nov-2009,May-2009,Nov-2008,Nov-2007][Nov-2006]
(i) Find the Laplace transform of the signal x(t)=e-atu(t) [Apr-2010]Ans: ∞ ∞ ∞ ∞
x(s) = ∫ x(t) e-st dt = ∫ e-at e-st.dt = ∫ e-(s+a)t dt =[ e-(s+a)t /-(s+a)]-∞ 0 0 0
=-1/(s+a)[e-∞-e0]=-1/(s+a)[0-1]=1/s+a.
(ii)What is the Laplace transform of the function x(t)=u(t)-u(t-2)? [ Nov-2009]Ans:
L.T[u(t)] ∞ ∞= ∫ 1. e-st dt = [e-st /-s] = (-1/s)[e-∞-e0]=(-1/s)[0-1]=1/s.0 0
L.T[u(t-2)]∞ ∞
= ∫ 1. e-st dt = [e-st /-s] = (-1/s)[ e-∞-e-s2]= (-1/s)[ e-∞-e-s2]= e-s2/s.2 2
Therefore L.T[x(t)=u(t)-u(t-2]= 1/s(1- e-s2 ).
(iii)Determine laplace transform of x(t) =e-at sin(t) u(t) [May-2009]Ans: ∞ ∞
x(s) = ∫ x(t) e-st dt = ∫ e-at 1/2j (ejωt-e-jωt) e-st.dt-∞ 0
∞ ∞= (1/2j )[ ∫ e-at ejωt e-st.dt-∫ e-at e-jωt e-st.dt]
0 0∞ ∞
= (1/2j ) [ ∫ e-(s+a-jω)t.dt-∫ e-(s+a+jω)t dt]0 0
∞ ∞= (1/2j ) [ {e-(s+a-jω)t. /-( s+a-jω)}-{e-(s+a+jω)t. /-( s+a+jω)}
0 0= (1/2j ) [ {-1/( s+a-jω)}[e-∞-e0] ]+{1/( s+a+jω)}[e-∞-e0] ]= (1/2j ) [ {-1/( s+a-jω)}[0-1] ]+{1/( s+a+jω)}[0-1] ]=(1/2j ) [ {1/( s+a-jω)}-{1/( s+a+jω)}]=(1/2j ) [ {( s+a+jω-s-s+jω)/( s+a-jω)( s+a+jω)}]=(1/2j ) {(2jω)/ [(s+a)2 + ω2] }= ω / [(s+a)2 + ω2]
(iv)Find the Laplace transform of unit ramp function [Nov-2008]Ans: x(t) =r(t)=t u(t)
L.T[u(t)] ∞ ∞= ∫ 1. e-st dt = [e-st /-s] = (-1/s)[e-∞-e0]=(-1/s)[0-1]=1/s.0 0
L[t u(t)] = -(d/ds)(1/s) = [s.0-1.1]/s2 =1/ s2
(v) Find the laplace transform of u(t-2) [Nov-2007]Ans: ∞ ∞ ∞
x(s) = ∫ x(t) e-st dt = ∫ e-st.dt = [e-st/-s] =(-1/s)[ e-∞-e-s2]-∞ 2 2=(-1/s) [0-e-s2]= e-s2/s.
(vi). Find the laplace transform for the signal x(t)=-t e-2t u(t) [Nov-2006]Ans: ∞ ∞ ∞ ∞
x(s) = ∫ x(t) e-st dt = ∫ e-at e-st.dt = ∫ e-(s+2)t dt =[ e-(s+2)t /-(s+2)]-∞ 0 0 0=-1/(s+2)[e-∞-e0]=-1/(s+2)[0-1]=1/s+2.
L[-te-2t u(t)] = (d/ds)x(s) =(d/ds)[1/s+2]=[(s+2).0-1(1)]/ (s+2)2 =-1/(s+2)2
4.Properties of Fourier Transform [Apr-2010,Nov-2009,Nov-2006](i)State modulation property and convolution(time) property of Fourier
transform(or)State any two properties of Continuous – Time Fourier Transform.[Apr-2010,Nov-2009]
Ans (i) 1.Modulation property :-
x(t) cos(2πfct + ) X( f-fc) + X( f+fc)2.Convoltion property:-This property states that convolution in time doman corresponds tomultiplication in frequency domain that isy(t)=x(t) * h(t) y(jω)= X(jω). H(jω)
(ii) Define parseval’s relation for continuous time periodic signals [Nov-2006]Ans: If x(t) and x(jω) are Fourier transform pair, then
∞ ∞∫ |x(t)|2 dt = (1/2π) ∫ |x(jω)|2 dω
-∞ -∞
(iii) Write the differentiation and integration property of Fourier transform[Nov-2006]
Ans: Differeniation : (d/dt) x(t)-- jω X(jω)
Integration: t∫ |x(τ)|2 dτ = (1/jω) x(jω) + π x(0)δ(ω)
-∞5.Fourier series [Nov2009,Apr-2008, Nov-2007,May-2006](i)State Dirchlet’s condition for Fourier series
(or)Write the condition to be satisfied for the existence of Fourier transform ofaperiodic signal [Nov2009,Apr-2008,May-2006]
Ans: 1.The function x(t) should be within the interval T0
2. The function x(t) should have finite number of maxima and minima in theinterval T0
3. The function x(t) should have at the most finite number of discontinuitiesin the interval.
4. The function should be absolutely integral
(ii)Define Fourier series [Nov-2007]Ans:Fourier series:
Let us consider a periodic signal x(t) with fundamental period T.If thereexists a convergent series
+ ∞x(t) = Σ ak ejkω
0t, ω0 =(2π /T)
k = - ∞then the series is called Fourier series.
6.Fourier series coefficient [Nov-2008,Apr-2008](i)Determine the complex Fourier series representation of the signal x(t)=sin0t
[Nov-2008]Ans: x(t)=sinω0t
X(t)=(1/2j)[ejω0
t - e-jω0
t] )=(1/2j)ejω0t - (1/2j)e-jω
0t]
+ ∞Fourier series x(t) = Σ ak ejkω
0t
k = - ∞If k=1,then a1=(1/2j) If k=-1,then a-1=(-1/2j)
Fourier series coefficient a1=(1/2j) a-1=(-1/2j)
(ii)Find the Fourier series representation for the signal x(t)=3 cos[(π/2)t+( π/4)][Apr-2008]
Ans: x(t)= (3/2)[ej(π/2)t+( π/4)] +e-j[π/2)t+( π/4)]]Take ω
0 = π/2= (3/2)[ejω
0t+( π/4)] +e-j[ω
0t+( π/4)]]
+ ∞Fourier series x(t) = Σ ak ejkω
0t
k = - ∞If k=1,then a1=(3/2) ej π/4 If k=-1,then a-1=(3/2) e-j π/4
Fourier series coefficient a1=(3/2) ej π/4 and a-1=(3/2) e-j π/4
7.Inverse Fourier Transform [Nov-2007,May-2006](i)Find the inverse Fourier Transform of x(ω)=2πδ(ω) [Nov-2007]
Ans: +∞x(t) =(1/2π ) ∫ X(jω) ejωt dω
-∞
+∞x(t) =(1/2π ) ∫ δ(ω) ejωt dω
-∞δ(ω) = 1, ω=0
= 0, ω≠0X(t)=(1/2π).1= ( 1/2π )
F-1[2πδ(ω)]= 2π(1/2π) =1
(ii)Determine the inverse Fourier transform of x(j) = δ() [May-2006].Ans: F-1[δ(ω)]= (1/2π)(1) =1/2π.
8.Using Properties of Laplace transform [May-2007](i)Find the initial and final value for x(s) =[s+5] /[s2+3s+2] [May-2007]Ans: The initial value x(0) = lim s x(s)
s→∞= lim s [[s+5] /[s2+3s+2]
s→∞= lim [ s2+5s] /[ s2+3s+2] = lim [ (1)+(5/s)] /[ (1)+(3/s)+(2/s2)]
s→∞ s→∞=1
The final value theorem is lim x(t) = lim s x(s)t→∞ s→0
= lim [ s2+5s] /[ s2+3s+2] = 0s→0
9.Properties of Laplace transform [May-2006](i)List out any four properties of Laplace transform used in signal analysis[May-06]
Ans: 1. Time shifting 2.Differentiation in Time 3.Convolution 4. Linearity
UNIT-III (LTI-CT SYSTEMS)
1. Differential equation. [Nov-2010](i)Write the Nth order differential equation. [Nov-2010]Ans: : The general form of a constant coefficient differential equation
N MΣ ak [dk/dtk] y(t)= Σ bk [dk/dtk] x(t)k=0 k=0
Here N is the order of the differential equation
2.Convolution Integral [Nov-2010,Apr-2010, Nov-2009,May-2009, Nov-2008,Nov-2007,May-2007,Nov-2006]
(i)What is convolution integral?(or) State the convolution integral for continuoustime LTI systems. (or)Write down the convolutional integral to find the output of thecontinuous time systems. [Nov-2010,Apr-2010,May-2007,Nov-2006]
Ans Convolution integral is given by
+∞y(t) = ∫ x(τ)h(t-τ) dτ
-∞
3.Properties of Convolution Integral[Apr-2010,Nov-2009,May-2009, Nov-2008, Nov-2007]
(i)What is the impulse response of two LTI systems connected inparallel?(or)What is the overall impulse response h(t) when two systems withimpulse response h1(t) and h2(t) are in parallel and in series?
[Apr-2010,Nov-2008]Ans: For parallel connection x(t)*[h1(t)+h2(t)] =[x(t)*h1(t) ]+[x(t)*h2 (t)]
Where h(t) =h1(t)+h2(t)For series connection: x(t)*[h1(t)*h2(t)] =[x(t)*h1(t) ]*h2 (t)
Where h(t)=h1(t)*h2(t)
(ii)Give four step to compute convolution integral [Nov-2009,May-2009,Nov-2007]Ans: 1.Folding: One of the signal is first folded at t=0
2. Shifting : The folded signal is shifted right or left depending upon time at
which output is to be calculated.3.Multiplication:The shifted signal is multiplied other signals4.Integration: The multiplied signals are integrated to get the convolution
Output
4.Transfer function [Nov-2009,Apr-2008,Nov-2007,May-2006](i)What are the transfer function of the following ? [Nov-2009]
(a)An ideal integrator (b)An ideal delay of T secondsAns: (a)An ideal integrator
ty(t) = ∫ x(τ) dτ
-∞Applying Laplace Transform Y(s)=X(s)/s
H(s)=[Y(s)/X(s)]=1/sb)An ideal delay of T seconds
y(t) = x(t-T)Applying Laplace Transform
Y(s) = e-sT X(s)H(s)=[Y(s)/X(s)] = e-ST
(ii).Find the transfer function of LTI system described by the differential equation[d2y(t)/dt2] + 3 [dy(t)/dt]+2y(t) =2 [dx(t)/dt]-3 x(t) [Apr-2008]
Ans: s2y(s)+3sy(s)+2y(s)=2sx(s)-3x(s)Y(s)[s2+3s+2]=x(s)[2s-3]
[Y(s)/x(s)]=[2s-3] / [s2+3s+2 ]H(s)= [2s-3] / [s2+3s+2 ]
(iii) Define transfer function(or)system function in continuous time systems[Nov-2007,May-2006]
Ans:Transfer function is defined as the ratio of L.T of output to the L.T of inputH(s) =[Y(s)/X(s)]
5. Impulse response [Apr-2008,Nov-2007](i)Define impulse response of a LTI system [Apr-2008,Nov-2007]
Ans: Impulse response of LTI system is denoted by h(t) .It is the response of thesystem to unit impulse input.
UNIT-IV (ANALYSIS OF DT SIGNALS)1.Discrete Time Fourier Transform(DTFT)
[Nov-2010,Apr-2010,Apr-2008,Nov-2007,May-2007,Nov-2006](i)Find the DTFT of u(n). [Nov-2010]
Ans: (i) ∞X(ejω )= Σ x(n) e-jωn
n =-∞∞
X(ejω )= Σ u(n) e-jωn
n =-∞
∞X(ejω )= Σ (e-jω)n =1/[1- e-jω]
n =0
(ii) State the sufficient condition for the existence of DTFT for an aperiodicsequence x(n) [Apr-2010]Ans: Sufficient condition for the existence of DTFT for an aperiodic sequence
is + ∞Σ │ x(n)│ < ∞
n = - ∞
(iii)Compute discrete time Fourier transform of the signal x(n)=u(n-2)-u(n-6)[Apr-2008]
Ans: u(n-2) =1,n≥2=0,n<2
u(n-6) =1,n≥6=0,n<6
x(n)=u(n-2)-u(n-6) =1,n=2,3,4,5= 0 ,all other ‘n’
∞X(ejω )= Σ x(n) e-jωn
n =-∞5
X(ejω )= Σ x(n) e-jωn
n =2= x(2) e-jω2 +x(3) e-jω3 +x(4) e-jω4 +x(5) e-jω5
= e-jω2 + e-jω3 + e-jω4 + e-jω5
(iv)Define discrete time Fourier transform? Define DTFT pair (or)Write theanalysis and synthesis equation of DTFT [Nov-2007May-2007,Nov-2006]Ans: DTFT : ∞
X(ejω )= Σ x(n) e-jωn
n =-∞+π
IDTFT : x(n) = (1/2π) ∫ X(ejω )ejωn dω-π
(v) Find the Fourier transform of h(n)=(n-no) [ Nov-2006]Ans: ∞
X(ejω )= Σ x(n) e-jωn
n =-∞∞
X(ejω )= Σ (n-no) e-jωn
n =-∞
δ(n-no) =1,n= no
=0,n≠ no
Therefore X(ejω ) = e-jωn0
2.Alaising [Nov-2010,Apr-2010,Nov-2009](i)Define aliasing effect(or)Write a note on aliasing [Nov-2010,Apr-2010]
Ans: Aliasing :When the high frequency interfaces with low frequency andappears as low frequency, then the phenomenon is calledaliasing.
Effect of aliasing:(1)Since high and low frequencies interface with each other,distortion is
generated.(2)The data is lost and it cannot be recovered.
To avoid aliasing: 1. Sampling rate fs≥2W2. Strictly band limit the signal to ‘W’
(ii)What is an anti-aliasing filter? [Nov-2009]Ans: A filter that is used to reject high frequency signals before it is sampled to reduce
the aliasing is called anti aliasing filter.
3.Sampling theorem [Apr-2010,Nov-2009,May-2007](i) State the Sampling theorem. [Apr-2010,Nov-2009,May-2007]Ans: Statement of sampling theorem
(1)A band limited signal of finite energy, which has no frequency componentshigher than W hertz, is completely described by specifying the values of thesignal at instants of time separated by 1/2W seconds and
(2)A band limited signal of finite energy, which has no frequency componentshigher than W hertz, may be completely recovered from the knowledge of its
samples taken at the rate of 2W samples per second.
4. Z-transform [Apr-2010, Nov-2009,Nov-2008,Nov-2007,May-2007,May 2006](i)Define Z-transform(or) Define one sided Z-transform and Two-sided
Z-transform. [Apr-2010,Nov-2008,Nov-2007]Ans: (i) Two –sided (or) bilateral z-transform is given by
+ ∞x(z) = Σ x(n) z-n
n = - ∞One-sided(or) unilateral z-transform is given by
∞x(z) = Σ x(n) z-n
n = 0
(ii)What is the z-transform of an u(n) and –an u(-n-1).In what way the z-transforms of
these two functions are different? [Nov-2009,Nov-2008]Ans: Z[an u(n)]=
+ ∞x(z) = Σ x(n) z-n
n = - ∞+ ∞ + ∞
x(z) = Σ an. z-n = Σ (az-1)n = 1/1-az-1=z/(z-a) ROC :z>an =0 n =0
Z[–an u(-n-1)]=+ ∞
x(z) = Σ x(n) z-n
n = - ∞-1 + ∞
x(z) = Σ -an. z-n = - Σ a-nzn
n = - ∞ n =1+ ∞
x(z) = - Σ (a-1z)n = -{[a/(a-z)]-1} =z/(z-a) ROC:z<an = 1
These two values will be same. But ROC is different.
(iii)What is the z-transform and the associated ROC of the signal x(n)=u(n-n0)?[Nov-2008]
Ans: + ∞x(z) = Σ x(n) z-n
n = - ∞+ ∞
x(z) = Σ u(n-n0)z-n
n = - ∞=z-no z[u(n)]=(z-no)z/(z-1).ROC is z>1
(iv)What is the z-transform of u(n) and δ(n)? [Nov-2007]Ans: (1) u(n):
+ ∞x(z) = Σ x(n) z-n
n = - ∞u(n) = 1 for n ≥ 0
= 0 for n<0+ ∞ + ∞
x(z) = Σ 1 z-n = Σ (z-1)n = z/(z-1)n =0 n =0
(2) δ(n) :+ ∞
x(z) = Σ x(n) z-n
n = - ∞ δ(n) =1 for n=0= 0 for n ≠0
+ ∞x(z) = Σ δ(n) z-n = δ(0) z0=1 since δ(0) =1 & z0 =1
n =0
(v)Find the z-transform of x(n)=u(n)-u(n-3)? [Nov-2007]Ans: u(n) =1,n≥0
=0,n<0u(n-3) =1,n≥3
=0,n<3x(n)=u(n)-u(n-3) =1,n=0,1,2
= 0 ,all other ‘n’+ ∞
x(z) = Σ x(n) z-n
n = - ∞
2= Σ 1. z-n = 1 +z-1 +z-2
n = 0
(vi)Find the z-transform for x(n) =an-1 u(n-1) [May-2007]Ans: + ∞x(z) = Σ x(n) z-n
n = - ∞+ ∞ + ∞
x(z) = a-1 Σ an. z-n = a-1 Σ (az-1)n =a-1 [ z/(z-a)-1]n =1 n =1
(vii)Find the z-transform of the given data sequence, x(n)=1, 0<n<10= 0,otherwise ? [May 2006]
Ans: + ∞x(z) = Σ x(n) z-n
n = - ∞10 10
x(z) = Σ 1 z-n = Σ z-n = 1+z-1+z-2+z-3+z-4+z-5+z-6+z-7+z-8+z-9+z-10
n =0 n =05.Properties of DTFT [Nov-2009,Nov-2007,Nov-2006,May-2006]
(i)State Parseval’s relation for discrete-time aperiodic signals. [Nov-2009]Ans: + ∞ π
E = Σ │ x(n)│2 = (1/2π) ∫ │X(ejω ) │2 dωn = - ∞ -π
(ii)State and prove time shifting property of DTFT [Nov-2007]Ans:Statement:Time shifting: DTFT[x(n-n0] = e-jωn0 x(ejω)
Proof: F[ x(n-n0 ]= Σ x(n- n0) e
-jn
n =-∞
F[ x(n-n0) ]= Σ x(P)e-j(P+ n
0)
P =-∞
=e-jωn0 Σ x(P)e-jP
P =-∞=e-jωn
0 x(ejω)
(iii)State the linearity and periodicity properties of Discrete –Time FourierTransform [Nov-2006]Ans: Linearity : DTFT[ax(n)+b y(n) ]=a x(ejω)+b y(ejω)
Periodicity : x(ejω) is periodic with period 2π i.e x[ej(ω+2π)] = x(ejω)
(iv)State the time shifting and frequency shifting properties of DTFT. [May-2006]Ans:Time shifting: DTFT[x(n-n0] = e-jωn
0 x(ejω)Frequency shifting : DTFT [ejω
0n x(n) ] = x[ej(ω-ω
0)]
6.Properties of Z-transform[Nov-2009,May-2009,Nov-2008,May-2007,May 2006](i)State Parseval relation in Z-transform [Nov-2009,May-2009]
Ans: Let us consider two complex valued sequences x1(n) and x2(n).Parseval’s relationstated that
+ ∞Σ x1(n)x2*(n) = (1/2πj) ∫ X1(V)x2(1/v*)v-1 dv
n = - ∞ c
(ii)State shifting property and scaling property of the z-transform(or) List any two properties of z-transform [Nov-2008,May-2007]
Ans: Time shifting: z[x(n-k)] = z-k x(z)scaling(i) z[anu(n)]=x(a-1z) (ii)z[a-nu(n)]=x(az)
(iii) What is the mathematical expression for the convolution property ofZ -transform? [May 2006]Ans: Z[x(n)* h(n)] = X(z)H(z)
7.Properties of ROC [May-2009,Apr-2008,Nov-2007,Nov-2006](i) Write the properties of the ROC for the z-transform
[May-2009,Apr-2008,Nov-2007,Nov-2006]Ans(i) (1)The ROC is concentric ring in the z-plane centered at the origin.
(2)The ROC can not contain any poles.
8.Region of convergence w.r.t ,z-transform [Nov-2008,Apr-2008](i)Define Region of convergence w.r.t ,z-transform. [Nov-2008,Apr-2008]
Ans(i) The range of value of z for which the z-transform converge is calledRegion of convergence.
9.What are the different methods of evaluating inverse z-transform? [Nov-2007]
Ans: 1. Contour Integral 2.Power series 3.Partial fraction method
10. What is the relation between z-transform and Fourier transform of discrete timesignal? [Nov-2008]Ans: z-transform is given by
+ ∞x(z) = Σ x(n) z-n
n = - ∞
DTFT is given by∞
X(ejω )= Σ x(n) e-jωn
n =-∞z-transform and Fourier transform of discrete time signal relationship is z= ejω
UNIT-V LTI-DT SYSTEMS
1.Prove that x(n)* δ(n)=x(n) [Nov-2010]Ans: The convolution of x(n) and h(n) is given as
+ ∞X(n)*h(n)= Σ x(n)h(n-k) ----1
k=- ∞since convolution is commutative
h(k)= δ(k) =1 at k=0= 0 at k not equal to zero
Hence the summation of eqn 1 is evaluated only at k=0 i.e.,X(n)*h(n)=h(k)x(n-k) at k=0
=1.x(n-0)=x(n)
Thus x(n)*h(n)=x(n)* δ(n)=x(n) if h(n)= δ(n).
2.Causal and Stable[Nov-2010,Nov-2008,Apr-2008, Nov-2007,Nov-2006,May-2006](i)Determine the range of ‘a’ for which the LTI system with impulse response
h(n)=an u(n) is stable. [Nov-2010]Ans: ∞
Σ h(n)< ∞n = -∞
∞Σ an =1+a+a2+a3+……….
n = -∞This is geometric series and it converges to 1 / [1-a] if a<1 .If the series does notconverge,then the system becomes unstable.Thus the given system is stable if a<1
(ii)State the conditions for Discrete time LTI system to be causal and stable[Nov-2008,Apr-2008,Nov-2006,May-2006]
Ans:Causal: h(n)=0 for n<0,this condition are satisfied means causal. Otherwise non-
CausalStable: ∞
Σ h(n)< ∞ ,this condition are satisfied means causal. Otherwise unstablen = -∞
(iii)Prove that for the causal LSI system the impulse response h[n]=0 ,for n<0[Nov-2007]
Ans: y(n) = Σ h(k) x(n-k)
k= -∞-1
y(n) = Σ h(k) x(n-k) + Σ h(k) x(n-k)k= -∞ k=0
= ……...h(-2)x(n+2)+h(-1)x(n+1)+h(0)x(n)+h(1)x(n-1)+…………..
y(n) = Σ h(k) x(n-k) so it is causal h[n]=0 ,for n<0k=0
3. Properties of convolution(or) Convolution SUM(i)State the properties of convolution(or)Mention the properties of convolution sum(or)State the prop[erties needed for interconnecting LTI systems
[Apr-2010,May-2009,May-2007,May-2006]Ans(i):1.Commutative property :x(n)*h(n)=h(n)*x(n)
2.Associative property : [x(n)*h1(n)]*h2(n)=x(n)*[h1(n)*h2(n)]3. Distributive property: [x(n)*h1(n)]+[x(n)*h2(n)]=x(n)*[h1(n)+h2(n)]
4.Distinguish between IIR and FIR systems [Nov-2009,May-2009,May-2006]Ans:
Sl.no FIR (Finite Impulse Response)systems
IIR(Infinite Impulse Response)systems )
1 Length of impulse response islimited
Length of impulse response is infinite
2 There is no feedback of output Feedback of output is taken
3 These systems are non-recursive These systems are recursive
5.Transfer function(or)System function [Nov-2009, Nov-2008,Nov-2007,May-2006](i)Define system function [Nov-2009,Nov-2007,May-2006]Ans: (i)System function is defined as the ratio of the z-transform of output to the z-
transform of inputH(Z)=Z[y(n)]/Z[x(n)]=[Y(Z)/X(Z)]
(ii)Find the transfer function of the discrete time systemy(n)-(3/4)y(n-1)+(1/8)y(n-2)=x(n) [Nov-2008]
Ans: y(n)-(3/4)y(n-1)+(1/8)y(n-2)=x(n)Taking z-transform on both sidesY(z)-(3/4)z-1y(z)+(1/8)z-2y(z)=x(z)Y(z)[1-(3/4)z-1+(1/8) z-2 ]=x(z)Transfer function H(Z)=1/[1-(3/4)z-1+(1/8) z-2]
(iii)Find the transfer function of the system governed by the difference equationy(n)-2y(n-1)-6y(n-2)=x(n)-3x(n-1) [Nov-2008]
Ans: y(n)-2y(n-1)-6y(n-2)=x(n)-3x(n-1)Taking z-transform on both sideY(z)-2z-1y(z)-6z-2y(z)=x(z)[1-3z-1]
Y(z)[1-2z-1-6z-2] =x(z)[1-3z-1]H(Z)= [1-3z-1] / [1-2z-1-6z-2]
6.Block Diagram Representation( Direct Form-I) [May-2009,Nov-2006](i)Realize the following system y(n) = 2 y(n-1) - x(n) + 2 x(n-1) in Direct Form-I
[May-2009,Nov-2006]Ans: : (i) y(n) = 2 y(n-1) - x(n) + 2 x(n-1)
Taking z-tranform on both sideY(z)=2z-1y(z)-x(z)+2z-1x(z)
7.Block Diagram Representation [Nov-2008,Apr-2008,Nov-2006](i)List of various forms of realization of IIR system [Nov-2008]Ans: (i) 1.Direct form-I 2.Direct form-II 3. Cascade form 4.Parallel form
(ii)What are the basic building blocks to realize any structure? [Apr-2008]Ans:1.Adder 2.Multiplier 3.Constant multiplier 4.Time delay 5.Time advance
(iii)Draw the block diagram for H(Z) = [1+2Z-1+4Z-4] / [1+-Z-1+Z-2] [Nov-2006]Ans: y(z)/x(z) =[1+2z-1+4z-4] / [1+-z-1+z-2]
y(z)[ 1+ z-1+z-2] = x(z)[ 1+2z-1+4z-4]y(z)+z-1y(z)+z-2y(z)=x(z)+2z-1x(z)+4z-4x(z)y(z)= x(z)+2z-1x(z)+4z-4x(z) -z-1y(z)-z-2y(z)
8.Differentiate between recursive and Non-recursive difference equations[Nov-2008,Apr-2008,May-2007]
Ans:Sl.no Recursive system Non-recursive system1 When the output y(n) of the system
depends upon the present and pastinputs as well as past outputs, it iscalled recursive system
When the output y(n) of the systemdepends upon the present and pastinputs ,then it is called Non-recursivesystem
2 Ex:y(n) = 0.5 y(n-1) +0.75x(n)+2x(n-1)
Ex:y(n) = 2x(n)+1.5 x(n-1) +0.5x(n-2)
9.Block Diagram Representation (Direct Form-II) [Nov-2007,May-2007 ](i)Realize the following system using direct form-II method
y[n]-(1/2)y[n-1]=x[n]+(1/2)x[n-1] [Nov-2007]Ans: y[n]-(1/2)y[n-1]=x[n]+(1/2)x[n-1]
Taking z-transform on both sidey(z)-(1/2)z-1 y(z)=x(z)+(1/2)z-1 x(z)y(z)[1-(1/2) z-1 ] =x(z)[1+(1/2)z-1]y(z)/x(z) = [1+(1/2)z-1] / [1-(1/2) z-1 ][w(z)/x(z)][y(z)/w(z) = [1+(1/2)z-1] / [1-(1/2) z-1 ]w(z)/x(z)=1/ [1-(1/2) z-1 ]Y(z)/w(z)= [1+(1/2)z-1]
x(n) y(n)
1/2 1/2
+ +
z-1
(ii) What is meant by canonic structure? [May-2007 ]Ans: Direct form –II structure requires the minimum number of delays for the
realization of the system, it is called a canonic structure
10. Draw the state variable model for discrete system [Nov-2007]
11.Block Diagram Representation (Cascade and parallel form) [Nov-2006](i)Draw the general block diagrams of the parallel and cascade form structures
[Nov-2006]
12.How many number of additions, multiplications and memory locations arerequired to realize a IIR system with transfer function ,H(Z) having M zeros andN poles in direct form –I and Direct form -II [May-2006]
Ans:Direct form-I Direct form-IIAddition:M+NMultiplication:N+M+1Delays:M+N
M+NN+M+1Max{N,M}
PART-B UNIVERSITY QUESTIONSUNIT-I CLASSIFICATION OF SIGNALS AND SYSTEMS
1.CT and DT systems[Nov-2010,Nov-2009,Apr-2008,Nov-2007,May-2007,Nov-2006,May-2006]
(i)Check the linearity,time invariance,causality and stabilityY(n)=x(n2) [Nov-2010]
(ii)Determine whether the following systems are linear or not.1. (dy/dt)+3ty(t)=t2x(t)
2. y(n)=2x(n)+(1/(x(n-1)) [Nov-2009](iii)Determine whether the following systems are time invariant or not
(1)y(t)=t x(t) (2)y(n)=x(2n) [Nov-2009](iv)Check whether or not the given systems are linear, time invariant. causal,
memory less and stable (1)y(t) =x(t-2)+x(2-t) (2) y(t)=[dx(t)/dt] [Apr-2008](v)Verify whether the systems given are causal,instantaneous,linear and
shiftinvariant(1) y(t)=x(t) cost (2) y(n)=log10 x(n) (3) y(n)=x(n) u(n) [Nov-2007]
(vi) Verify whether the system y(n)=x2(n) is linear and time invariant system[Nov-2007]
(vii)Test whether y(t)(d2y(t)/dt2)+3t(dy(t)/dt)+y(t) =x(t) is linear ,causal and timeinvariant or not [May-2007]
(viii) Determine whether the given systems y(t) = 0 ,t >0= x(t) + x(t-2) , t ≥ 0
(1)Memoryless (2)Time invariant (3) Linear (4) Causal and (5) Stable[Nov-2006]
(ix) Verify whether the system given by y(t) = x(t2),is causal, instantaneous, linearand shift invariant [May-2006]
2.CT and DT signals [Nov-2010,Apr-2010,Nov-2009](i)Derive the relationship between unit step and delta function. [Nov-2010](ii) Write about elementary continuous-time signals in detail(or)
Distinguish between the following:(1)Continuous Time Signal and Discrete Time Signal(2)Unit step and unit ramp functions(3)Periodic and aperiodic signal(4)Deterministic and random signals [Apr-2010,Nov-2009]
(iii)Explain the properties of unit impulse function [Apr-2010]
3.Shifting [Nov-2010,Nov-2008,Apr-2008](i)If x(n)={0,2,-1,0,2,1,2,0,-1}.What is x(n-3) and x(1-n)? [Nov-2010](ii)A discrete time signal x(n) is shown in figure i. sketch and label each of the
following signals. (1)x(n)u(1-n) (2)x(n)δ(n-1) (3)x(n)[u(n+2)-u(n)
x(n)3
2
1
-4 -3 -2 -1 0 1 2 3 4 5 n
[Nov-2008]
(iii)A continuous time signal x(t) is shown in fig.Sketch and label carefully each of
the following signals: (1)x(t-1) (2) x(2-t) (3) x(t)[(t+3/2)- (t-3/2)] (4)x(2t+1]
x(t)
2
1
-2 -1 0 1 2 t[Apr-2008]
4.Periodic or aperiodic [Apr-2010,May-2009,Nov-2007](i)Find whether the signal x(t)=2cos(10t+1)-sin(4t-1) is periodic or not.(ii)Find the fundamental period T of the continuous time signal
x(t)=20cos[ . [Apr-2010]
(iii)Determine if the following signals are periodic;if periodic give the period(1)x(t)=cos(4t)+2sin(8tr) (2)x(t)=3 cos(4t)+sin(πt) [May-2009]
(iv)Determine whether or not the given signals are periodic(1) x(t)=sin2t (2) x(n)=cos((1/4)n) [Nov-2007]
(v) Find the periodicity of the signal x(n)=sin(2πn/3)+cos(πn/2) [Nov-2007](vi) Test whether the following signals are periodic or not and if the signal is
periodic,calculate the fundamental period(1) x(t)=2 u(t)+2 sin2t(2)x(t)=20 cos[10πt+(π/6)(3)x(n)=cos(π/2)n+sin(π/8)n+3 cos[(π/4)n +(π/3)]
(4)x(n)=ej(2π/3)n+ej(3π/4)n [Nov-2007]
5.Energy and power signal[Nov-2009,Nov-2008,Apr-2008,Nov-2007,May-2007](i)Compare energy and power signal [Nov-2009]
(ii)Determine ,whether x(t)=rect(t/10)cos0t is energy signal or power signal.[Nov-2009]
(iii).Determine the power and RMS value of the following signals.x1(t)=5cos(50t+π/3)
x2(t)=10cos5tcos10t. [Nov-2009](iv)Determine the energy and power of the following signals:
1.x(t)=t u(t) 2.x(n)=2ej3n [Nov-2008](v)Determine the value of power and energy for each of the signals
(1)x1(n)=e j[(n/2)+(/8)] (2)x2(n)=(1/2)n u(n) [Apr-2008](vi)Determine the power and RMS value of the signals
x1(t)= 5 cos[50t+(π/4)+ 16 sin[100t+(π/3)] andx2(t)=10 cos5t cos10t [Nov-2007]
(vii)A trapezoidal pulse x(t) is defined byx(t)= 5-t ,4≤t≤5
= 1,-4≤t≤4=t+5,-5≤t≤-4(1)Determine total energy of x(t) (2) Sketch x(2t-3)(3)If y(t)=[dx(t)/dt],determine the total energy of y(t) [Nov-2007]
(viii) Determine the energy of the signals x1(t)=e-3t u(t) and x2(t)=(1/3)n u(n)[Nov-2007]
(ix).Check whether x(t)=ej[2t+(π/4)] is an energy or power signals [May-2007]
6.Exponential signal [Nov-2007](i).Explain real exponential and complex exponential signal [Nov-2007]
7. Even and odd signal [Nov-2007,May-2007](i)Find the even and odd component of the signal x(n)={-2,1,2,-1,3} [Nov-2007](ii)Find the even and odd components of the signal x(t)=cost +sint+costsint[May-07]
8.Evaluate the signal [Nov-2006]∞
(i) Find the summation ∑ e2nδ(n-2)n=-8 [Nov-2006]4
(ii)Evaluate the integral (t+t2) δ(t-3)dt [Nov-2006]-2
(iii) Consider a continuous time signal x(t) =δ(t+2) - δ(t-2) .Calculate the value of+
Ea for the signals, y(t) = x(τ) dτ [Nov-2006]-
UNIT-II (ANALYSIS OF CT SIGNALS )1.Fourier Transform
[Nov-2010,Apr-2010,Nov-2009,May-2009,Nov-2008,Nov-2007 ,Nov-2006,May-2006](i)Find the Fourier Transform of the signal shown in fig
(or)Obtain the Fourier transform of rectangular pulse of duration T andamplitude A.
[Nov-2010,Nov-2009,Nov-2008,Nov-2007](ii)Explain the Fourier spectrum of a periodic signal x(t). [Apr-2010](iii)Find the Fourier transform of
x(t)=e-|t|
for -1 ≤ t ≤ 1= 0 otherwise. [Apr-2010]
(iv)Find the Fourier transform to the following signal and plot magnitude
x(t)
1
1 t-1
-1[May-2009]
(v)Find the Fourier transform of the signam function shown in figure .
[Nov-2006](vi) Find the Fourier transform of the signal x(t) = e-at u(t),a>0 and plot the
magnitude and phase spectrum [Nov-2006]+ ∞
(vii) For the sampling signal δs(t)= ∑ δ(t-nTs),use the Fourier transform to findn=- ∞
its spectrum. [May-2006]
2.Fourier series[Nov-2010,Apr-2010,Nov-2009,May-2009,Apr-2008,Nov-2007,May-2007,May-2006]
(i)Find out the exponential Fourier series of a Impulse train.Plot its magnitude andphase spectrum [Nov-2010]
(ii)What are the two types of Fourier series representations? Give the relevantmathematical representations. [Nov-2010]
(iii) Find the trigonometric Fourier series for the periodic signal x(t) shown in thefigure given below.
x(t)+1
t-3 -2 -1 0 1 2 3
-1T [Apr-2010,Nov-2009]
(iv) For the signal; x(t)=t2,find the trigonometric Fourier series over theinterval(-1,1). [May-2009,May-2007]
(v).Let x(t)=t ,0t1=2-t,1t2 be a periodic signal with fundamental period T=2 and
Fourier coefficients ak (i) Determine the value of a0 (ii)Determine the Fourierseries representation of [dx(t)/dt] (iii)Use the result of part(2) and thedifferentiation property of continuous time Fourier series to help determinethe Fourier series coefficients of x(t) [Apr-2008]
(vi)Find the exponential fourier series for the signal shown
[Nov-2007](vii)Obtain the Fourier series expansion of a half wave sine wave [Nov-2007](viii)Find the Fourier series representation and sketch of the amplitude and phase
spectrum for the signal x(n)=5=sin(nπ/2)+cos(nπ/4) [May-2007](ix) Find the complex exponential Fourier series expansion of x(t)=e-0.1t over the
internal -5<t<15 sec. [May-2006]
3.Inverse Lapalce Transform [Nov-2010, May-2009, Nov-2007, May-2006](i)Find out inverse Lapalce transform of F(S)= [S-2] /[S(S+1)3 ] [Nov-2010](ii)Find the inverse Laplace transform of [3s2+8s+23] / [(s+3)(s2+2s+10)] [May-2009](iii)Find the inverse Laplace transform of
x(s)=1/[(s+1)(s+2)] [Nov-2007](iv )Find the inverse laplace transform of x(s)=[3s2+8s+6]/[(s+2)(s2+2s+1)]
[Nov-2007](v) Find the inverse transform of x(s) =[2s3+8s2+11s+3] / [(s+2)(s+1)3] [May-2006](vi)Find the inverse Laplace transform of x(s)={4/(s+2)(s+4)} if the ROC is
Case(I) -2 > Re(s) > -4 Case (II) Re(s) < -4 [May-2006]
4.Laplace transform [Apr-2010, Nov-2009, Nov-2007, Nov-2006](i)Find the Laplace transform of the signal
x(t)=e-at
u(t)+e-bt
u(-t) [ Apr-2010]
(ii)Determine the Laplace transform of the following signals.x1(t)=u(t-2) x2(t)=t2e-2tu(t) [ Nov-2009]
(iii).Determine the Laplace transform of the signal.X(t)=sinπt; 0<t<1
=0 otherwise. [Nov-2009](iv)Find the Laplace transform of x(t)=te-at u(t) [Nov-2007](v) x1(t) =e-2t u(t) ,x2(t) = e-3t u(t) determine Y(s) ,where
y(t) =x1(t-2)* x2(-t+3) [Nov-2006](vi) Find the Laplace transform for x(t) = δ(t) +u(t) [Nov-2006]
5.Properties of Laplace transform [May-2009, Nov-2008, Nov-2007](i)Find the Laplace transform of tx(t) and x(t-t0) where t0 is a constant term
x(t)-x(s) [May-2009](ii)State and prove the time scaling property of Laplace transform [Nov-2008](iii)State and prove convolution in time and convolution in frequency
properties of Laplace transform [Nov-2007](iv) State and prove any four properties of Laplace transform [Nov-2007]
6.Properties of Fourier series [May-2009, Nov-2007,May-2007](i)Explain the properties of Fourier series (1)Time shift (2)Time Differentiation.
[May-2009,May-2007](ii)State and prove any three properties of continuous time Fourier series[Nov-2007]
7.Properties of Fourier transform [Nov-2008,Nov-2007,Nov-2006,May-2006](i)State and prove the differentiation property of Fourier transform[Nov-2008](ii)State and prove any two properties of continuous time Fourier transform(iii)State and prove convolution property of Fourier transform [Nov-2007]
(iv) State and prove the parseval’s relation for continuous time signals using
Fourier Transform [Nov-2006,May-2006]
8.Using Properties of Fourier transform [Nov-2007,Nov-2006](i)Using the properties of continuous time Fourier transform, determine the
time domain signal x(t),if the frequency domain signalx(jω)=j(d/dω){ej2ω/[1+(jω/3)] [Nov-2007]
(ii)Using Fourier transform properties find the Fourier transform of the signalin fig
X(t)
2A
A
0 T/4 T/2 3T/4 T t[Nov-2007]
(iii)Prove that the time scaling property of Fourier transform and hence findthe Fourier of f(t) = e-0.5t u(t) [Nov-2006]
9.Inverse Fourier transform [May-2007](i).Find the inverse fourier transform of δ(-0) [May-2007]
10. Using Properties of Laplace transform [May-2006](i)Use the initial value theorem to find the initial value of the signal
corresponding to the laplace transform Y(s)=[s+1] / [s(s+2)].Verify usingInverse transform. [May-2006]
UNIT-III (LTI-CT SYSTEMS)
1.Output response using differential equation[Nov-2010,Nov-2007,May-2007,Nov-2006]
(i)Solve the differential equation[d2y(t)/dt2]+4[dy(t)/dt]+5y(t)=5x(t)With y(0)=1 and [dy(t)/dt]=2 and x(t)=u(t) [Nov-2010]
(ii)Solve [d2y(t)/dt2] + 4[dy(t)/dt]+4 y(t)=[dx(t)/dt]+x(t) if the initial conditions arey(0)=(9/4),[dy(0)/dt]=5 if the input is e-3t u(t) [Nov-2007]
(iii) A systems is described by the differential equation [d2y(t)/dt2] +7[dy(t)/dt]+12y(t)=x(t).Use L.T to determine the response of the system to a unit step input appliedto at t=0.Assume the initial conditions as y(o)=-2 and y’(0)=0 [May-2007]
(iv) Consider an LTI system whose input x(t) and output y(t) are related by thedifferential equation [dy(t)/dt]+4y(t)=x(t).The system also satisfies the conditionat initial rest, if x(t)=e-(1+3j)t u(t) find y(t) [Nov-2006]
(v)Find y(t) for the differential equation [dy(t)/dt]+5y(t)\=x(t),where x(t)=3e-2t u(t)
and y(0)=-2 [Nov-2006]
2.Impulse response [Nov-2010,Apr-2010, Nov-2009, Nov-2007](i)The system is described by the input output relation
[d2y(t)/dt2]+[dy(t)/dt]-3y(t)=[dx(t)/dt]+2x(t).Find the system transfer function,frequency response and impulse response. [Nov-2010]
(ii)Using Laplace transform, find the impulse response of an LTI system describedby the differential equation (d2y(t)/dt2)-(dy(t)/dt)-2y(t)=x(t). [Apr-2010]
(iii)Find the impulse response of the system shown in Fig.R
.
Input c outputvi(t) v0(t)
[Nov-2009](iv)Consider an LTI system whose response to the input x(t)=(e-t +e-3t)u(t) is
y(t)=(2e-t-2e-4t)u(t).Find the systems impulse response [Nov-2007](v) A system has the transfer function H(S)=[3S-1]/[(S+3)(S-2)].Find the
impulse response assuming the system is stable, and the system iscausal. [Nov-2007]
(vi)Find the impulse response of a system characterized by the differential equationτ0[d
2y(t)/dt2]+y(t) =x(t).Where x(t) is the input and y(t) is the output andy(0)=y’(0)=0
3.Block diagram representation (Direct form I and II) [Nov-2010, May-2007](i)Draw the direct form I and II implementations of the system described by
[dy(t)/dt]+5y(t)=3x(t). [Nov-2010](ii)Realize the following differential differential equation as a direct form-II structure
[d3y(t)/dt3]+4[d2y(t)/dt2]+7[dy(t)/dt]+8y(t)=5[d2x(t)/dt2]+4[dx(t)/dt]+7x(t)
4.State variable equation and matrix [Nov-2010,Nov-2009,May-2006](i)The realization of the second order LTI-CT system is shown below. Obtain the
state variable descriptions .Hence obtain the transfer function of the systemx(t) y(t)
2
2 3
-1[Nov-2010]
(ii)For the circuit shown in Fig., obtain state variable equation. The input voltagesource is x(t) and the output y(t) is taken across capacitor C2.
+ +
∫
∫
+ +
R R+ .
x(t) c1 c2 y(t)
- [Nov-2009](iii) Discuss on representation of state variable [May-2006](iv)Find the state transition matrix for the system parameter matrix A= -3 0
0 -2[May-2006]
5.Convolution Integral [Apr-2010,Nov-2009, May-2009, Nov-2006](i)Explain the steps to compute the convolution integral. [Apr-2010](ii)Find the convolution of the following signals:
x(t)=e-2tu(t) h(t)=u(t+2). [Apr-2010](iii)Explain the properties of convolution integral [Apr-2010](iv)Find the convolution of the following signals.
x1(t)=e-atu(t); x2(t)=e-btu(t). [Nov-2009](v)Find the convolution of x(t) and h(t) given
X(t)= sint u(t) h(t)= u(t) andX(t)=e-at u(t) h(t)=e-bt u(t) [ May-2009]
(vi)Convolve x(t) and h(t) as shown in figX(t) h(t)
+12
0 1 2 t
-1 0 1 2 3 t
[Nov-2006]
6.Block diagram representation (cascade form) [May-2009,Nov-2007](i)Realize[s(s+2)]/[(s+1)(s+3)(s+4)] in cascade form [May-2009,Nov-2007]
7.System function [Nov-2008](i)The output response y(t) of a continuous time LTI system is 2e-3t u(t) when the
input x(t) is u(t).Find the system function [Nov-2008](ii)Consider a continuous time LTI system [d2y(t)/dt2]+[dy(t)/dt]-2y(t)=x(t)
(1)Find the system function H(S)(2)Determine if the impulse response h(t) for the system is causal, the system is
stable and the system is neither causal nor stable. [Nov-2008](iii)The output response y(t) of a continuous time LTI system is 2e-2t u(t) when the
input x(t) is u(t).Find the system function. [Nov-2008]
8.Response(or) output [Apr-2008, Nov-2007, May-2007]
(i)Consider an LTI system with input x(t)=e-t u(t) and impulse response h(t)=e-2t u(t)(1)Determine the Laplace transform of x(t) and h(t)(2)Using the convolution property ,determine the Laplace transform y(s) of output
y(t)(3)From the Laplace transform of y(t) as obtained in part(2),determine y(t)(4)Verify your result in part(2) by explicitly convolving x(t) and h(t) [Apr-2008]
(ii)Find the unit step response of the circuit shown in fig
[Nov-2007](iii)Find the step response of the system whose impulse response is t u(t)[Nov-2007](iv)Find the step response of the system whose impulse response is given as
h(t)=u(t+1)-u(t-1) [May-2007](v)Find the response of the system with impulse response h(t)=e-3t u(t) for the
input x(t)=u(t-3)-u(t-5) [May-2007]
9.Input Response [Apr-2008](i) Consider a causal LTI system with frequency response H(j)=1/[j+3].For a
particular input x(t) this system is to produce the outputy(t)=e-3t u(t)-e-4t u(t).Determine x(t). [Apr-2008]
10.Causal and stable [Nov-2006,May-2006](i) Check whether the following systems are stable and causal:
(1) h(t) = e-2t u(t-1) (2) h(t) = e-4t u(t+10) (3) h(t) = te-t u(t) [Nov-2006](ii)Find whether the system with the impulse response h(t)=(1/RC)e-t/RC u(t) is BIBO
Stable [May-2006]
UNIT-IV (ANALYSIS OF DT SIGNALS)1. Properties of Z-transform [Nov-2010,Nov-2009,Apr-2008,Nov-2007, May-2006]
(i) State and prove time shifting property and time-convolution property of z-transform [Nov-2010,Nov-2009,Apr-2008,Nov-2007]
(ii) State and prove the time shifting and differentiation in frequency properties ofz – transform [May-2006]
2.Z-transform[Nov-2010,Apr-2010,Nov-2009,Apr-2008,Nov-2007,[May-2007,Nov-2006]
(i)Determine the Z transform of(1)x(n)=an/n and(2) nu(n) For n ≥ 0 [Nov-2010]
(ii)Find the Z-transform of the given signal x(n) and find ROC.x(n)=[sin(w0n)]u(n). [Apr-2010,Nov-2009]
(iii)Determine the z-transform for the sequence(n)=4n cos[(2n/6)+(/4)] u(-n-1).
Sketch the pole-zero plot and indicate the ROC [Apr-2008](iv)Find the z-transform of
x(n)=[(1/2)n –(1/4)n]u(n) and plot the pole-zero pattern [Nov-2007](v) Find the z-transform of x(n)=(1/2)n u(-n) [Nov-2007](vi) Determine z-transform for the signal x(n) =(2/3)n u(n)+(-1/2)n u(n) and plot the
ROC and pole-zero locations of x(z) [May-2007](vii) Determine the z-transform of the signal x(n)=nan u(n) and hence determine
z- transform of the unit ramp signal n u(n) . [Nov-2006]
3.Sampling theorem and Aliasing [Nov-2010,Apr-2010,May-2007](i)State and prove sampling theorem.(or)Describe the sampling operation and
explain how aliasing error can be prevented. (or)What is meant by aliasing andhow it is avoided? [Nov-2010,Apr-2010,May-2007]
4.Inverse z-transform using partial fraction method[Nov-2010, Nov-2008,Apr-2008, Nov-2007]
(i)Find the inverse Z transform of the following:(1) X(Z) = [1-(1/2)Z-1] / [1+(3/4)Z-1+(1/8)Z-2] Z>(1/2)
(2) X(Z) = [1-(1/2)Z-1] / [ 1-(1/4)Z-2] Z > (1/2) [Nov-2010](ii) Find the inverse z-transform of
x(z)=[z3-z2+z-(1/16)] / [z3-(5/4)z2+(1/2)z-(1/16)],z>(1/2) [Nov-2008](iii)Find the inverse z-transform of x(z)=(1/1024)[(1024-z-10)/[1-(1/2)z-1]],z>0
[Apr-2008](iv)Suppose that the algebraic expression for the z-transform of x(n) is
x(z)=[1-(1/4)z-1]/{[1+(1/4)z-2] [1+(5/4)z-1+(3/8)z-2].How many different regionsof convergence could correspond to x(z)? [Apr-2008]
(v)Find the inverse z-transform of x(z)=[z2]/[(1-az)(z-a)] [Nov-2007]
5.DTFT(Discrete Time Fourier Transform) [Apr-2010,May-2009, Nov-2007](i) Find the Fourier Transform of
X(n)=A |n| ≤ N.=0 |n| › N. [Apr-2010]
(ii)Find the DTFT of x(n)=an u(n) and x(n)=a-n u(-n-1) [May-2009](iii) Find the DTFT of the given x(n)=(0.5)n u(n) + 2-n u(-n-1) [Nov-2007]
6.Properties of DTFT [Apr-2010](i)Explain any four properties of DTFT. [Apr-2010
7.Nyquist rate and Nyquist interval [Nov-2009, Nov-2008, May-2007](i)Determine the Nyquist sampling interval for the signal x(t)=sin c2(200πt).
[Nov-2009](ii)Find the Nyquist frequency and Nyquist rate for each for each of the following
Signals (1)x(t)=25 cos(100πt) (2)x(t)=15sect(t/2) [Nov-2008](iii)Consider a signal x(t)= cos2000πt + 10 sin10000πt+20 cos5000πt
Determine the(1)Nyquist rate for the signal(2) If the sampling rate is 5000 samples/sec,then what is the DT signal
obtained after sampling? [May-2007]
8.Inverse z-transform using residual method [May-2009,May-2007](i)Find the Inverse z-transform of x(z)=2/{z(z-(1/2)}.For ROC Z>(1/2) using
Cauchy residue method. [May-2009](ii)Use residue method to find the inverse z-transform of
x(z)={1-(1/4)z-1/{1-(1/9)z-2} [May-2009](iii) Using residue method, find the inverse z-transform for
x(z)=[1+3z-1] /(1+3z-1+2z-2)],z>2 [May-2007]
9.Using Properties of Z-transform [Nov-2008,Nov-2006,May-2006](i)Determine the initial and final vaules for the signal x(n) with transfer function
x(z)=[z3-(3/4)z2+2z-(5/4)] / [(z-1)(z-(1/3))(z2-(1/2)z+1)] [Nov-2008](ii)State and prove Initial value theorem [Nov-2006](iii)Using Final value theorem of z-transform, find the final value of the signal
for which Y(Z)=[2Z-1] /[1-1.8Z-1+0.8Z-2] [May-2006]
10.Difference Equation [Apr-2008](i)Consider a system consisting of cascade of two LTI systems with
frequency responseH1(e
j)=[2-e j]/[1-(1/2)e -j] and H2(ej)=1/[[1-(1/2)e -j+(1/4)e -j2]
Find the difference equation describing the overall system [Apr-2008]
11.Inverse z-transform using power series method [Nov-2007](i) Determine the inverse z-transform of x(z)=log(1-2Z), Z<(1/2) by using
the power series ∞log(1-x) = - Σ (xi/i ,x<1 and by first differentiating x(z) and the
i=1using this to recover x[n] [Nov-2007]
12.Relation between z-transform and Fourier Transform [Nov-2007](i)Briefly explain the relationship between z-transform and Fourier
Transform [Nov-2007]
13.Using Properties of DTFT [May-2007,Nov-2006, May-2006](i)Use Fourier transform to find the output of the system whose impulse
response, h(n)=(1/3)n u(n) and the input to the system is x(n)=(1/2)n u(n)[May-2007]
(ii) Consider a discrete time LTI system with impulse response h(n) =(1/2)n u(n)Using Fourier Transform determine the response for the inputx(n) = (n+1)(1/4)n u(n) . [Nov-2006]
(ii)Determine h2(n) for the system shown in fig
X(n)H(ejω)
Where H(ejω)= [-12+5e-jω ] / [12-7e-jω+e-2jω] [May-2006]
14.Properties of ROC with respect to z-transform [May-2007](i)What is meant by ROC of z-transform? Explain its significance [May-2007]
15.Impulse Reponse and frequency response [May-2006](i) The system is described by y(n) +(5/6)y(n-1) +(1/6)y(n-2) =x(n)
Determine (1) Frequency response of the system(1) Impulse response of the system [May-2006]
16.Inverse z-transform using differentiation property [May-2006](i) Using differentiation property ,determine the inverse transform for
x(z) ={az-1/(1-az-1)2}; z > a [May-2006]
UNIT-V LTI-DT SYSTEMS
1.Impulse Response [Nov-2010,Apr-2010, Nov-2009, Nov-2008,Nov-2007, May-2007](i)A discrete time causal system has a transfer function
H(Z) = [1-Z-1] / [1-0.2Z-1-0.15Z-2](i)Determine the difference equation of the system(ii)Show pole zero diagram(iii)Find the impulse response. [Nov-2010]
(ii)The system function of the LTI system is given asH(Z) = [3-4Z-1] / [1-3.5Z-1+1.5Z-2]Specify the ROC of H(Z) and determine h(n) for the following condition.(1)Stable system (2) Causal system [Nov-2010]
(iii)Find the impulse response of the discrete time system described by thedifference equation y(n-2)-3y(n-1)+2y(n)=x(n-1). [Apr-2010,Nov-2007]
(iv)Find the impulse and step response of the following system:y(n)-(3/4)y(n-1)+(1/8)y(n-2)=x(n). [Nov-2009]
(v)When an input x(n)=3δ(n-2) is applied to a causal LTI system, whose output isfound to be y(n)=2(-1/2)n +8(1/4)n. Find the impulse response h(n) of the system
[Nov-2008](vi)A causal LTI system described by the difference equation
y(n)=y(n-1)+y(n-2)+x(n-1)(1)Find the system function for the system(2)Find the unit impulse impulse response of the system [Nov-2008]
h1(n)=(1/3)n u(n)
h2(n)=?
+
(vii)Determine the impulse and frequency response of the system described by thedifference equation y(n)-(1/6)y(n-1)-(1/6)y(n-2)=x(n-1) [May-2007]
2.Block diagram representation(Direct Form-I & II)[Nov-2010,May-2009, Apr-2008, Nov-2007, May-2007]
(i) Obtain the direct form I & II structure forY(n)-(3/4)y(n-1)+(1/8)y(n-2)=x(n)+(1/2)x(n-1) [Nov-2010,May-2009]
(ii) Develop a direct form-I realization of the difference equationy(n)=b0x(n)+b1x(n-1)+b2x(n-2)+b3x(n-3)-a1y(n-1)-a2y(n-2)-a3y(n-3) [Apr-2008]
(iii)y(n)-(5/6)y(n-2)+(1/6)y(n-2)=x(n)+2x(n-1) in direct form-I [Nov-2007](iv)Differentiate between Direct form-I and Direct for-II structures. [May-2007]
3. Block diagram representation [Apr-2010,Nov-2006](i)Discuss the block diagram representation for LTI discrete time systems. Give the
summary blocks used to represent discrete time systems. [Apr-2010,Nov-2006]
4.State variable equation and matrix [Apr-2010, Apr-2008,Nov-2007](i)Describe the state variable model for discrete time systems. [Apr-2010,Nov-2007](ii)Find the state variable matrices A, B, C, D for the equation
y(n)-3y(n-1)-2y(n-2)=x(n)+5x(n-1)+6x(n-2) [Apr-2010,Nov-2007](iii)Consider a causal LTI system,whose input x(n) and output y(n) are related
through the block diagram representation shown in fig(1)Determine a difference equation relating y(n) and x(n)(2)Is the system stable?
[Apr-2008]5.Block diagram representation(Cascade form and Parallel Form)
[Nov-2009,May-2009,Nov-2008, Apr-2008, May-2007](i) Obtain the cascade and parallel form realization of the following system.
y(n)-(1/4)y(n-1)-(1/8)y(n-2)=x(n)+3x(n-1)+2x(n-2) [Nov-2009,May-2009](ii) Realize the given discrete time system in cascade and parallel forms
H(z)=[1+(1/2)z-1] / [(1-z-1+(1/4)z-2][1-z-1+(1/2)z-2] [Nov-2008](iii) Develop Cascade and Parallel realization structures for
H(z)=[(z/6)+(5/24)+(5/24)z-1+(1/24)z-2 ] / [1-(1/2)z-1+(1/4)z-2] [Apr-2008](iv)Obtain the parallel structure realization of the system described by the
difference equation y(n)-(13/12)y(n-1)+(9/24)y(n-2)-(1/24)y(n-3)=x(n)+2x(n-1)
[May-2007]6.Output Response [May-2009,Nov-2007,May-2006]
(i) Find the output sequence y(n) of the system described by the equationy(n) =0.7y(n-1) –0.1y(n-2) +2x(n) –x(n-2) for the sequence x(n) = u(n) [May-2009]
(ii) Find the output of the system whose input-output is related by the differenceequation y(n)-(5/6)y(n-1)+(1/6)y(n-2)=x(n)-(1/2)x(n-1) for the step input[Nov-2007]
(iii) Consider the following linear constant coefficient difference equationy[n]-(3/4)y[n-1]+(1/8)y[n-2] =2x[n-1].Determine y[n] when x[n]=δ[n]and y[n]=0,n<0. [Nov-2007]
(iv)Using z-transform ,compute the response of the systemy(n) = 0.7 y(n-1) -0.12y(n-2) +x(n-1)+x(n-2) to the input x(n) = n u(n) .
Is the system stable? [May-2006]
7.Using Properties of convolution SUM [Nov-2008, Nov-2006](i)Find the overall response of the system shown in figure with
h1(n)= δ(n),h2(n)=(n-1)u(n) and h3(n)= δ(n)+nu(n-1)+δ(n-2)
x(n) y(n)
[Nov-2008](ii). Two discrete time LTI system are connected in cascade as shown in the figure.
Determine the unit sample response of this cascade connection .unit sampleresponse of overall system =h(n)
x(n) y(n)
[Nov-2006]
8.Convoltion SUM(summation) [Nov-2007 ,May-2007 ,Nov-2006](i)Find the linear convolution of
(1) X(n)={1,2,3,4,5} with h(n)={1,2,3,3,2,1}(2)X(n)={1,-1,2,3} with h(n)={0,1,2,3} [Nov-2007]
(ii)Find the convolution of the signals x(n) =cos(πn) u(n) and h(n) =(1/2)n u(n)[May-2007]
(iii) Find the convolution sum for the x(n) =(1/3)-n u(-n-1) and h(n)=u(n-1)[Nov-2006]
(iv) Consolve the following two sequences linearly x(n) and h(n) to get y(n).x(n)= {1,1,1,1} and h(n) ={2,2}.Also give the illustration[Nov-2006]
(v)Compute the convolution of the 2 sequence given and plot the output
h1(n)
h2(n)
h3(n)+
h1(n)=(1/2)n u(n) h2(n)=(1/4)n u(n)
[Nov-2006]9. Properties of convolution SUM [Nov-2007,Nov-2006,May-2006](i) Explain the properties of convolution (or)Prove that order of convolution is
unimportant, that is x1[n]*x2[n] =x2[n]*x1[n][Nov-2007,Nov-2006,May-2006]
10.System function and Frequency response [May-2007, Nov-2007](i)Acausal system is represented by the difference equation
y(n)+(1/4)y(n-1)=x(n)+(1/2)x(n-1).Use z-transform to determine the (i) systemfunction (ii) unit sample response of the system(iii) frequency response of thesystem [May-2007]
(ii)The input to a causal linear time invariant system isx[n]=u[-n-1]+(1/2)n u[n],the z-transform of the output of the system isY(Z)=[-(1/2)Z-1] /{[1-(1/2)Z-1][(1+Z-1)]}.Determine H(Z),theZ-transform of the impulse response and also determine the output y[n].
[Nov-2007]
